Version bump for release 0.1.9 RC 1

Add integer variable detection to change log
Add suffix to variables with known domain
2018-04-20 16:39:54 +02:00 · 2018-04-20 16:38:57 +02:00 · 2018-04-20 16:37:49 +02:00 · 2018-04-20 16:37:49 +02:00 · 2018-04-20 16:37:49 +02:00 · 2018-04-20 16:37:49 +02:00
47 changed files with 2651 additions and 578 deletions
--- a/.ci/Dockerfile-arch-latest
+++ b/.ci/Dockerfile-arch-latest
@@ -0,0 +1,9 @@
+FROM archimg/base-devel:latest
+
+ARG toolchain
+
+RUN pacman -Sy
+RUN pacman -S --noconfirm cmake git ninja re2c
+RUN if [ "${toolchain}" = "clang" ]; then pacman -S --noconfirm clang; fi
+
+VOLUME /app
--- a/.ci/Dockerfile-ubuntu-18.04
+++ b/.ci/Dockerfile-ubuntu-18.04
@@ -0,0 +1,10 @@
+FROM ubuntu:18.04
+
+ARG toolchain
+
+RUN apt-get update
+RUN apt-get install -y bison cmake git ninja-build re2c
+RUN if [ "${toolchain}" = "gcc" ]; then apt-get install -y g++; fi
+RUN if [ "${toolchain}" = "clang" ]; then apt-get install -y clang; fi
+
+VOLUME /app
--- a/.ci/ci.sh
+++ b/.ci/ci.sh
@@ -0,0 +1,17 @@
+#!/bin/bash
+
+if [ "$1" = "gcc" ]
+then
+	cxx=g++
+	cc=gcc
+elif [ "$1" = "clang" ]
+then
+	cxx=clang++
+	cc=clang
+fi
+
+git submodule update --init --recursive
+mkdir -p build/debug
+cd build/debug
+cmake ../.. -GNinja -DANTHEM_BUILD_TESTS=ON -DCMAKE_CXX_COMPILER=${cxx} -DCMAKE_C_COMPILER=${cc}
+ninja anthem-app && ninja run-tests
--- a/.gitmodules
+++ b/.gitmodules
@@ -3,4 +3,7 @@
 	url = https://github.com/potassco/clingo
 [submodule "lib/catch"]
 	path = lib/catch
-	url = https://github.com/philsquared/Catch
+	url = https://github.com/catchorg/Catch2
+[submodule "lib/cxxopts"]
+	path = lib/cxxopts
+	url = https://github.com/jarro2783/cxxopts
--- a/.travis.yml
+++ b/.travis.yml
@@ -1,47 +1,25 @@
-# Use container-based distribution
-sudo: false
-language: c++
-addons:
-  apt:
-    sources: &default_sources
-      - ubuntu-toolchain-r-test
-      - boost-latest
-    packages: &default_packages
-      - libboost-program-options1.55-dev
-      - libboost-iostreams1.55-dev
-      - libboost-system1.55-dev
-      - libboost-filesystem1.55-dev
-      - re2c
+sudo: required
+
+services:
+  - docker
+
 matrix:
  include:
-    - env: COMPILER_NAME=g++ _CXX=g++-5 _CC=gcc-5
+    - env: distribution=arch-latest toolchain=gcc
      os: linux
      language: cpp
-      addons:
-        apt:
-          sources:
-            - *default_sources
-          packages:
-            - *default_packages
-            - g++-5
-    - env: COMPILER_NAME=g++ _CXX=g++-6 _CC=gcc-6
+    - env: distribution=arch-latest toolchain=clang
      os: linux
      language: cpp
-      addons:
-        apt:
-          sources:
-            - *default_sources
-          packages:
-            - *default_packages
-            - g++-6
+    - env: distribution=ubuntu-18.04 toolchain=gcc
+      os: linux
+      language: cpp
+    - env: distribution=ubuntu-18.04 toolchain=clang
+      os: linux
+      language: cpp
+
+before_install:
+  - docker build --build-arg toolchain=${toolchain} -t ${distribution} - < .ci/Dockerfile-${distribution}
+
 script:
-  - if [[ "${TRAVIS_OS_NAME}" == "linux" ]]; then
-      CMAKE_URL="http://www.cmake.org/files/v3.7/cmake-3.7.0-Linux-x86_64.tar.gz";
-      mkdir cmake-bin && wget --quiet --no-check-certificate -O - ${CMAKE_URL} | tar --strip-components=1 -xz -C cmake-bin;
-      export PATH=${PWD}/cmake-bin/bin:${PATH};
-    fi
-  - git submodule update --init --recursive
-  - mkdir -p build/debug
-  - cd build/debug
-  - cmake ../.. -DCMAKE_BUILD_TYPE=Debug -DCMAKE_CXX_COMPILER=$_CXX -DCMAKE_C_COMPILER=$_CC -DANTHEM_BUILD_TESTS=ON
-  - make -j3 anthem-app && make -j3 run-tests
+  - docker run --mount source=$(pwd),target=/app,type=bind -w /app ${distribution} /bin/bash -c ".ci/ci.sh ${toolchain}"
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1,10 +1,33 @@
 # Change Log

-## (unreleased)
+## 0.1.9 RC 1 (2018-04-20)
+
+### Features
+
+* optional detection of integer variables and integer predicate parameters
+* command-line option `--detect-integers` to enable integer variable detection
+
+## 0.1.8 (2018-04-20)
+
+### Features
+
+* more and advanced simplification rules
+* adds support for exponentiation (power) and modulus (absolute value)
+* new examples: prime numbers, permutation generator, and graph coloring (extended)
+
+## 0.1.7 (2018-04-08)
+
+### Features
+
+* support for declaring placeholders with the `#external` directive
+
+### Internal
+
+* drops Boost dependency in favor of the header-only command-line option library [cxxopts](https://github.com/jarro2783/cxxopts)

 ## 0.1.6 (2017-06-12)

-Features:
+### Features

 * unique IDs for all variables (user-defined variables are renamed)
 * support for hiding predicates from completed output by using `#show` statements
@@ -12,7 +35,7 @@ Features:
 * command-line option `--parentheses` to fully parenthesize the output
 * adds multiple example instances for experimenting

-Bug Fixes:
+### Bug Fixes

 * adds missing error message when attempting to read inaccessible file
 * removes unnecessary parentheses after simplification
@@ -20,52 +43,52 @@ Bug Fixes:

 ## 0.1.5 (2017-05-04)

-Bug Fixes:
+### Bug Fixes

 * fixes lost signs with negated 0-ary predicates

 ## 0.1.4 (2017-04-12)

-Features:
+### Features

 * completion of input programs (optional)
 * command-line option `--complete` to turn on completion

 ## 0.1.3 (2017-03-30)

-Features:
+### Features

 * support for anonymous variables

-Bug Fixes:
+### Bug Fixes

 * fixes incorrectly simplified rules with comparisons
 * fixes misleading error message concerning negated, unsupported body literals

 ## 0.1.2 (2017-03-23)

-Features:
+### Features

 * simplification of output formulas (optional)
 * command-line option `--simplify` to turn on simplification

-Bug Fixes:
+### Bug Fixes

 * fixes incorrectly translated choice rules with multiple elements in the head aggregate

-Internal:
+### Internal

 * explicit syntax tree representation for first-order formulas

 ## 0.1.1 (2017-03-06)

-Features:
+### Features

 * support for choice rules (without guards)

 ## 0.1.0 (2016-11-24)

-Features:
+### Features

 * initial support for translating rules in *Essential Gringo* (excluding aggregates) to first-order logic formulas
 * command-line option `--color` to autodetect, enable, or disable color output
--- a/CMakeLists.txt
+++ b/CMakeLists.txt
@@ -7,7 +7,7 @@ option(ANTHEM_BUILD_STATIC "Build static binaries" OFF)
 set(CMAKE_CXX_FLAGS "-Wall -Wextra -Wpedantic ${CMAKE_CXX_FLAGS}")
 set(CMAKE_CXX_FLAGS_DEBUG "-g ${CMAKE_CXX_FLAGS_DEBUG}")

-set(CMAKE_CXX_STANDARD 14)
+set(CMAKE_CXX_STANDARD 17)
 set(CMAKE_CXX_STANDARD_REQUIRED ON)
 set(CMAKE_CXX_EXTENSIONS OFF)

--- a/LICENSE.md
+++ b/LICENSE.md
@@ -1,6 +1,6 @@
 # The MIT License (MIT)

-Copyright © 2016–2017 Patrick Lühne
+Copyright © 2016–2018 Patrick Lühne

 Permission is hereby granted, free of charge, to any person obtaining a copy
 of this software and associated documentation files (the "Software"), to deal
--- a/README.md
+++ b/README.md
@@ -9,15 +9,16 @@
 ## Usage

 ```bash
-$ anthem [--simplify] file...
+$ anthem [--complete] [--simplify] file...
 ```

-With the option `--simplify`, output formulas are simplified by applying several basic transformation rules.
+`--complete` instructs `anthem` to perform Clark’s completion on the translated formulas.
+With the option `--simplify`, the output formulas are simplified by applying several basic transformation rules.

 ## Building

-`anthem` requires [CMake](https://cmake.org/) and [Boost](http://www.boost.org/) for building.
-After installing the dependencies, `anthem` is built with a C++14 compiler (GCC ≥ 6.1 or clang ≥ 3.8).
+`anthem` requires [CMake](https://cmake.org/) for building.
+After installing the dependencies, `anthem` is built with a C++17 compiler (GCC ≥ 7.3 or clang ≥ 5.0).

 ```bash
 $ git clone https://github.com/potassco/anthem.git
--- a/app/CMakeLists.txt
+++ b/app/CMakeLists.txt
@@ -1,7 +1,5 @@
 set(target anthem-app)

-find_package(Boost 1.55.0 COMPONENTS program_options system filesystem REQUIRED)
-
 file(GLOB core_sources "*.cpp")
 file(GLOB core_headers "*.h")

@@ -11,11 +9,10 @@ set(sources
 )

 set(includes
-	${Boost_INCLUDE_DIRS}
+	${PROJECT_SOURCE_DIR}/lib/cxxopts/include
 )

 set(libraries
-	${Boost_LIBRARIES}
 	anthem
 )

--- a/app/main.cpp
+++ b/app/main.cpp
@@ -1,6 +1,6 @@
 #include <iostream>

-#include <boost/program_options.hpp>
+#include <cxxopts.hpp>

 #include <anthem/AST.h>
 #include <anthem/Context.h>
@@ -10,63 +10,72 @@ int main(int argc, char **argv)
 {
 	anthem::Context context;

-	namespace po = boost::program_options;
+	cxxopts::Options options("anthem", "Translate ASP programs to the language of first-order theorem provers.");

-	po::options_description description("Allowed options");
-	description.add_options()
-		("help,h", "Display this help message")
-		("version,v", "Display version information")
-		("input,i", po::value<std::vector<std::string>>(), "Input files")
-		("simplify,s", po::bool_switch(&context.performSimplification), "Simplify the output")
-		("complete,c", po::bool_switch(&context.performCompletion), "Perform completion")
-		("color", po::value<std::string>()->default_value("auto"), "Colorize output (always, never, auto)")
-		("parentheses", po::value<std::string>()->default_value("normal"), "Parenthesis style (normal, full)")
-		("log-priority,p", po::value<std::string>()->default_value("warning"), "Log messages starting from this priority (debug, info, warning, error)");
+	options.add_options()
+		("h,help", "Display this help message")
+		("v,version", "Display version information")
+		("i,input", "Input files", cxxopts::value<std::vector<std::string>>())
+		("s,simplify", "Simplify the output")
+		("c,complete", "Perform completion")
+		("d,detect-integers", "Detect integer variables")
+		("color", "Colorize output (always, never, auto)", cxxopts::value<std::string>()->default_value("auto"))
+		("parentheses", "Parenthesis style (normal, full)", cxxopts::value<std::string>()->default_value("normal"))
+		("p,log-priority", "Log messages starting from this priority (debug, info, warning, error)", cxxopts::value<std::string>()->default_value("info"));

-	po::positional_options_description positionalOptionsDescription;
-	positionalOptionsDescription.add("input", -1);
-
-	po::variables_map variablesMap;
+	options.parse_positional("input");
+	options.positional_help("[<input file...>]");

 	const auto printHelp =
 		[&]()
 		{
-			std::cout
-				<< "Usage: anthem [options] file..." << std::endl
-				<< "Translate ASP programs to the language of first-order theorem provers." << std::endl << std::endl
-				<< description;
+			std::cout << options.help();
 		};

+	bool help;
+	bool version;
+	std::vector<std::string> inputFiles;
+	std::string colorPolicyString;
+	std::string parenthesisStyleString;
+	std::string logPriorityString;
+
 	try
 	{
-		po::store(po::command_line_parser(argc, argv)
-			.options(description)
-			.positional(positionalOptionsDescription)
-			.run(),
-			variablesMap);
-		po::notify(variablesMap);
+		const auto parseResult = options.parse(argc, argv);
+
+		help = (parseResult.count("help") > 0);
+		version = (parseResult.count("version") > 0);
+
+		if (parseResult.count("input") > 0)
+			inputFiles = parseResult["input"].as<std::vector<std::string>>();
+
+		context.performSimplification = (parseResult.count("simplify") > 0);
+		context.performCompletion = (parseResult.count("complete") > 0);
+		context.performIntegerDetection = (parseResult.count("detect-integers") > 0);
+		colorPolicyString = parseResult["color"].as<std::string>();
+		parenthesisStyleString = parseResult["parentheses"].as<std::string>();
+		logPriorityString = parseResult["log-priority"].as<std::string>();
 	}
-	catch (const po::error &e)
+	catch (const std::exception &exception)
 	{
-		context.logger.log(anthem::output::Priority::Error) << e.what();
+		context.logger.log(anthem::output::Priority::Error) << exception.what();
+		context.logger.errorStream() << std::endl;
 		printHelp();
 		return EXIT_FAILURE;
 	}

-	if (variablesMap.count("help"))
+	if (help)
 	{
 		printHelp();
 		return EXIT_SUCCESS;
 	}

-	if (variablesMap.count("version"))
+	if (version)
 	{
-		std::cout << "anthem version 0.1.7-git" << std::endl;
+		std::cout << "anthem version 0.1.9-rc.1" << std::endl;
 		return EXIT_SUCCESS;
 	}

-	const auto colorPolicyString = variablesMap["color"].as<std::string>();
-
 	if (colorPolicyString == "auto")
 		context.logger.setColorPolicy(anthem::output::ColorStream::ColorPolicy::Auto);
 	else if (colorPolicyString == "never")
@@ -81,22 +90,18 @@ int main(int argc, char **argv)
 		return EXIT_FAILURE;
 	}

-	const auto parenthesisStyle = variablesMap["parentheses"].as<std::string>();
-
-	if (parenthesisStyle == "normal")
+	if (parenthesisStyleString == "normal")
 		context.parenthesisStyle = anthem::ast::ParenthesisStyle::Normal;
-	else if (parenthesisStyle == "full")
+	else if (parenthesisStyleString == "full")
 		context.parenthesisStyle = anthem::ast::ParenthesisStyle::Full;
 	else
 	{
-		context.logger.log(anthem::output::Priority::Error) << "unknown parenthesis style “" << parenthesisStyle << "”";
+		context.logger.log(anthem::output::Priority::Error) << "unknown parenthesis style “" << parenthesisStyleString << "”";
 		context.logger.errorStream() << std::endl;
 		printHelp();
 		return EXIT_FAILURE;
 	}

-	const auto logPriorityString = variablesMap["log-priority"].as<std::string>();
-
 	try
 	{
 		const auto logPriority = anthem::output::priorityFromName(logPriorityString.c_str());
@@ -112,11 +117,8 @@ int main(int argc, char **argv)

 	try
 	{
-		if (variablesMap.count("input"))
-		{
-			const auto &inputFiles = variablesMap["input"].as<std::vector<std::string>>();
+		if (!inputFiles.empty())
 			anthem::translate(inputFiles, context);
-		}
 		else
 			anthem::translate("std::cin", std::cin, context);
 	}
--- a/examples/graph-coloring.lp
+++ b/examples/graph-coloring.lp
@@ -1,12 +1,10 @@
-colored(V, red) :- vertex(V), not colored(V, green), not colored(V, blue).
-colored(V, green) :- vertex(V), not colored(V, red), not colored(V, blue).
-colored(V, blue) :- vertex(V), not colored(V, red), not colored(V, green).
+#external color(1).
+#external edge(2).
+#external vertex(1).
+#show color/2.

-:- edge(V1, V2), colored(V1, C), colored(V2, C).
-
-vertex(a).
-vertex(b).
-vertex(c).
-
-edge(a, b).
-edge(a, c).
+{color(V, C)} :- vertex(V), color(C).
+covered(V) :- color(V, _).
+:- vertex(V), not covered(V).
+:- color(V1, C), color(V2, C), edge(V1, V2).
+:- color(V, C1), color(V, C2), C1 != C2.
--- a/examples/permutations.lp
+++ b/examples/permutations.lp
@@ -0,0 +1,13 @@
+#show p/2.
+#external integer(n(0)).
+
+{p(1..n, 1..n)}.
+
+:- p(X, Y1), p(X, Y2), Y1 != Y2.
+:- p(X1, Y), p(X2, Y), X1 != X2.
+
+q1(X) :- p(X, _).
+q2(Y) :- p(_, Y).
+
+:- not q1(X), X = 1..n.
+:- not q2(Y), Y = 1..n.
--- a/examples/prime.lp
+++ b/examples/prime.lp
@@ -0,0 +1,5 @@
+#show prime/1.
+#external integer(n(0)).
+
+composite(I * J) :- I = 2..n, J = 2..n.
+prime(N) :- N = 2..n, not composite(N).
--- a/examples/schur-numbers.lp
+++ b/examples/schur-numbers.lp
@@ -1,3 +1,7 @@
+#show in/2.
+#external integer(n(0)).
+#external integer(r(0)).
+
 {in(1..n, 1..r)}.
 covered(I) :- in(I, S).

--- a/include/anthem/AST.h
+++ b/include/anthem/AST.h
@@ -2,6 +2,8 @@
 #define __ANTHEM__AST_H

 #include <anthem/ASTForward.h>
+#include <anthem/Domain.h>
+#include <anthem/Tristate.h>

 namespace anthem
 {
@@ -32,7 +34,8 @@ struct BinaryOperation
 		Minus,
 		Multiplication,
 		Division,
-		Modulo
+		Modulo,
+		Power
 	};

 	explicit BinaryOperation(Operator operator_, Term &&left, Term &&right)
@@ -102,32 +105,15 @@ struct Comparison

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-struct Constant
-{
-	explicit Constant(std::string &&name)
-	:	name{std::move(name)}
-	{
-	}
-
-	Constant(const Constant &other) = delete;
-	Constant &operator=(const Constant &other) = delete;
-	Constant(Constant &&other) = default;
-	Constant &operator=(Constant &&other) = default;
-
-	std::string name;
-};
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
 struct Function
 {
-	explicit Function(std::string &&name)
-	:	name{std::move(name)}
+	explicit Function(FunctionDeclaration *declaration)
+	:	declaration{declaration}
 	{
 	}

-	explicit Function(std::string &&name, std::vector<Term> &&arguments)
-	:	name{std::move(name)},
+	explicit Function(FunctionDeclaration *declaration, std::vector<Term> &&arguments)
+	:	declaration{declaration},
 		arguments{std::move(arguments)}
 	{
 	}
@@ -137,12 +123,37 @@ struct Function
 	Function(Function &&other) noexcept = default;
 	Function &operator=(Function &&other) noexcept = default;

-	std::string name;
+	FunctionDeclaration *declaration;
 	std::vector<Term> arguments;
 };

 ////////////////////////////////////////////////////////////////////////////////////////////////////

+struct FunctionDeclaration
+{
+	explicit FunctionDeclaration(std::string &&name)
+	:	name{std::move(name)}
+	{
+	}
+
+	explicit FunctionDeclaration(std::string &&name, size_t arity)
+	:	name{std::move(name)},
+		arity{arity}
+	{
+	}
+
+	FunctionDeclaration(const FunctionDeclaration &other) = delete;
+	FunctionDeclaration &operator=(const FunctionDeclaration &other) = delete;
+	FunctionDeclaration(FunctionDeclaration &&other) noexcept = default;
+	FunctionDeclaration &operator=(FunctionDeclaration &&other) noexcept = default;
+
+	std::string name;
+	size_t arity;
+	Domain domain{Domain::General};
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
 // TODO: refactor (limit element type to primitive terms)
 struct In
 {
@@ -203,13 +214,13 @@ struct Interval

 struct Predicate
 {
-	explicit Predicate(std::string &&name)
-	:	name{std::move(name)}
+	explicit Predicate(PredicateDeclaration *declaration)
+	:	declaration{declaration}
 	{
 	}

-	explicit Predicate(std::string &&name, std::vector<Term> &&arguments)
-	:	name{std::move(name)},
+	explicit Predicate(PredicateDeclaration *declaration, std::vector<Term> &&arguments)
+	:	declaration{declaration},
 		arguments{std::move(arguments)}
 	{
 	}
@@ -219,35 +230,47 @@ struct Predicate
 	Predicate(Predicate &&other) noexcept = default;
 	Predicate &operator=(Predicate &&other) noexcept = default;

-	std::size_t arity() const
-	{
-		return arguments.size();
-	}
-
-	std::string name;
+	PredicateDeclaration *declaration{nullptr};
 	std::vector<Term> arguments;
 };

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-// TODO: make more use of this class
-struct PredicateSignature
+struct PredicateDeclaration
 {
-	explicit PredicateSignature(std::string &&name, size_t arity)
+	enum class Visibility
+	{
+		Default,
+		Visible,
+		Hidden
+	};
+
+	struct Parameter
+	{
+		Domain domain{Domain::Unknown};
+	};
+
+	explicit PredicateDeclaration(std::string &&name, size_t arity)
 	:	name{std::move(name)},
-		arity{arity}
+		parameters{std::vector<Parameter>(arity)}
 	{
 	}

-	PredicateSignature(const PredicateSignature &other) = delete;
-	PredicateSignature &operator=(const PredicateSignature &other) = delete;
-	// TODO: make noexcept again
-	// GCC versions before 7 don’t declare moving std::string noexcept and would complain here
-	PredicateSignature(PredicateSignature &&other) = default;
-	PredicateSignature &operator=(PredicateSignature &&other) = default;
+	PredicateDeclaration(const PredicateDeclaration &other) = delete;
+	PredicateDeclaration &operator=(const PredicateDeclaration &other) = delete;
+	PredicateDeclaration(PredicateDeclaration &&other) noexcept = default;
+	PredicateDeclaration &operator=(PredicateDeclaration &&other) noexcept = default;
+
+	size_t arity() const noexcept
+	{
+		return parameters.size();
+	}

 	std::string name;
-	size_t arity;
+	std::vector<Parameter> parameters;
+	bool isUsed{false};
+	bool isExternal{false};
+	Visibility visibility{Visibility::Default};
 };

 ////////////////////////////////////////////////////////////////////////////////////////////////////
@@ -292,6 +315,30 @@ struct String

 ////////////////////////////////////////////////////////////////////////////////////////////////////

+struct UnaryOperation
+{
+	enum class Operator
+	{
+		Absolute
+	};
+
+	explicit UnaryOperation(Operator operator_, Term &&argument)
+	:	operator_{operator_},
+		argument{std::move(argument)}
+	{
+	}
+
+	UnaryOperation(const UnaryOperation &other) = delete;
+	UnaryOperation &operator=(const UnaryOperation &other) = delete;
+	UnaryOperation(UnaryOperation &&other) noexcept = default;
+	UnaryOperation &operator=(UnaryOperation &&other) noexcept = default;
+
+	Operator operator_;
+	Term argument;
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
 struct Variable
 {
 	explicit Variable(VariableDeclaration *declaration)
@@ -335,6 +382,7 @@ struct VariableDeclaration
 	VariableDeclaration &operator=(VariableDeclaration &&other) = default;

 	Type type;
+	Domain domain{Domain::Unknown};
 	std::string name;
 };

--- a/include/anthem/ASTCopy.h
+++ b/include/anthem/ASTCopy.h
@@ -22,7 +22,6 @@ namespace ast
 BinaryOperation prepareCopy(const BinaryOperation &other);
 Boolean prepareCopy(const Boolean &other);
 Comparison prepareCopy(const Comparison &other);
-Constant prepareCopy(const Constant &other);
 Function prepareCopy(const Function &other);
 In prepareCopy(const In &other);
 Integer prepareCopy(const Integer &other);
--- a/include/anthem/ASTForward.h
+++ b/include/anthem/ASTForward.h
@@ -24,10 +24,10 @@ struct BinaryOperation;
 struct Biconditional;
 struct Boolean;
 struct Comparison;
-struct Constant;
 struct Exists;
 struct ForAll;
 struct Function;
+struct FunctionDeclaration;
 struct Implies;
 struct In;
 struct Integer;
@@ -35,8 +35,10 @@ struct Interval;
 struct Not;
 struct Or;
 struct Predicate;
+struct PredicateDeclaration;
 struct SpecialInteger;
 struct String;
+struct UnaryOperation;
 struct Variable;
 struct VariableDeclaration;
 using VariableDeclarationPointer = std::unique_ptr<VariableDeclaration>;
@@ -62,12 +64,12 @@ using Formula = Clingo::Variant<
 using Term = Clingo::Variant<
 	BinaryOperation,
 	Boolean,
-	Constant,
 	Function,
 	Integer,
 	Interval,
 	SpecialInteger,
 	String,
+	UnaryOperation,
 	Variable>;

 ////////////////////////////////////////////////////////////////////////////////////////////////////
--- a/include/anthem/ASTUtils.h
+++ b/include/anthem/ASTUtils.h
@@ -1,10 +1,11 @@
 #ifndef __ANTHEM__AST_UTILS_H
 #define __ANTHEM__AST_UTILS_H

-#include <experimental/optional>
+#include <optional>

 #include <anthem/AST.h>
 #include <anthem/ASTVisitors.h>
+#include <anthem/Context.h>

 namespace anthem
 {
@@ -28,19 +29,13 @@ class VariableStack
 		void push(Layer layer);
 		void pop();

-		std::experimental::optional<VariableDeclaration *> findUserVariableDeclaration(const char *variableName) const;
+		std::optional<VariableDeclaration *> findUserVariableDeclaration(const char *variableName) const;
 		bool contains(const VariableDeclaration &variableDeclaration) const;

 	private:
 		std::vector<Layer> m_layers;
 };

-////////////////////////////////////////////////////////////////////////////////////////////////////
-
-bool matches(const Predicate &lhs, const Predicate &rhs);
-bool matches(const Predicate &predicate, const PredicateSignature &signature);
-bool matches(const PredicateSignature &lhs, const PredicateSignature &rhs);
-void collectPredicateSignatures(const Formula &formula, std::vector<PredicateSignature> &predicateSignatures);

 ////////////////////////////////////////////////////////////////////////////////////////////////////
 // Replacing Variables
--- a/include/anthem/ASTVisitors.h
+++ b/include/anthem/ASTVisitors.h
@@ -123,12 +123,6 @@ struct RecursiveTermVisitor
 		return T::accept(boolean, term, std::forward<Arguments>(arguments)...);
 	}

-	template <class... Arguments>
-	ReturnType visit(Constant &constant, Term &term, Arguments &&... arguments)
-	{
-		return T::accept(constant, term, std::forward<Arguments>(arguments)...);
-	}
-
 	template <class... Arguments>
 	ReturnType visit(Function &function, Term &term, Arguments &&... arguments)
 	{
@@ -165,6 +159,14 @@ struct RecursiveTermVisitor
 		return T::accept(string, term, std::forward<Arguments>(arguments)...);
 	}

+	template <class... Arguments>
+	ReturnType visit(UnaryOperation &unaryOperation, Term &term, Arguments &&... arguments)
+	{
+		unaryOperation.argument.accept(*this, unaryOperation.argument, std::forward<Arguments>(arguments)...);
+
+		return T::accept(unaryOperation, term, std::forward<Arguments>(arguments)...);
+	}
+
 	template <class... Arguments>
 	ReturnType visit(Variable &variable, Term &term, Arguments &&... arguments)
 	{
--- a/include/anthem/Body.h
+++ b/include/anthem/Body.h
@@ -43,19 +43,22 @@ ast::Comparison::Operator translate(Clingo::AST::ComparisonOperator comparisonOp
 struct BodyTermTranslateVisitor
 {
 	// TODO: refactor
-	std::experimental::optional<ast::Formula> visit(const Clingo::AST::Function &function, const Clingo::AST::Literal &literal, const Clingo::AST::Term &, RuleContext &ruleContext, ast::VariableStack &variableStack)
+	std::optional<ast::Formula> visit(const Clingo::AST::Function &function,
+		const Clingo::AST::Literal &literal, const Clingo::AST::Term &, RuleContext &ruleContext,
+		Context &context, ast::VariableStack &variableStack)
 	{
 		if (literal.sign == Clingo::AST::Sign::DoubleNegation)
 			throw TranslationException(literal.location, "double-negated literals currently unsupported");

+		auto predicateDeclaration = context.findOrCreatePredicateDeclaration(function.name, function.arguments.size());
+		predicateDeclaration->isUsed = true;
+
 		if (function.arguments.empty())
 		{
-			auto predicate = ast::Formula::make<ast::Predicate>(std::string(function.name));
-
 			if (literal.sign == Clingo::AST::Sign::None)
-				return std::move(predicate);
+				return ast::Predicate(predicateDeclaration);
 			else if (literal.sign == Clingo::AST::Sign::Negation)
-				return ast::Formula::make<ast::Not>(std::move(predicate));
+				return ast::Formula::make<ast::Not>(ast::Predicate(predicateDeclaration));
 		}

 		// Create new body variable declarations
@@ -73,12 +76,12 @@ struct BodyTermTranslateVisitor
 		for (size_t i = 0; i < function.arguments.size(); i++)
 		{
 			auto &argument = function.arguments[i];
-			conjunction.arguments.emplace_back(ast::Formula::make<ast::In>(ast::Variable(parameters[i].get()), translate(argument, ruleContext, variableStack)));
+			conjunction.arguments.emplace_back(ast::Formula::make<ast::In>(ast::Variable(parameters[i].get()), translate(argument, ruleContext, context, variableStack)));
 		}

 		variableStack.pop();

-		ast::Predicate predicate(std::string(function.name));
+		ast::Predicate predicate(predicateDeclaration);
 		predicate.arguments.reserve(function.arguments.size());

 		for (size_t i = 0; i < function.arguments.size(); i++)
@@ -93,12 +96,13 @@ struct BodyTermTranslateVisitor
 	}

 	template<class T>
-	std::experimental::optional<ast::Formula> visit(const T &, const Clingo::AST::Literal &, const Clingo::AST::Term &term, RuleContext &, ast::VariableStack &)
+	std::optional<ast::Formula> visit(const T &, const Clingo::AST::Literal &,
+		const Clingo::AST::Term &term, RuleContext &, Context &, ast::VariableStack &)
 	{
 		assert(!term.data.is<Clingo::AST::Function>());

 		throw TranslationException(term.location, "term currently unsupported in this context, expected function");
-		return std::experimental::nullopt;
+		return std::nullopt;
 	}
 };

@@ -106,18 +110,18 @@ struct BodyTermTranslateVisitor

 struct BodyLiteralTranslateVisitor
 {
-	std::experimental::optional<ast::Formula> visit(const Clingo::AST::Boolean &boolean, const Clingo::AST::Literal &, RuleContext &, ast::VariableStack &)
+	std::optional<ast::Formula> visit(const Clingo::AST::Boolean &boolean, const Clingo::AST::Literal &, RuleContext &, Context &, ast::VariableStack &)
 	{
 		return ast::Formula::make<ast::Boolean>(boolean.value);
 	}

-	std::experimental::optional<ast::Formula> visit(const Clingo::AST::Term &term, const Clingo::AST::Literal &literal, RuleContext &ruleContext, ast::VariableStack &variableStack)
+	std::optional<ast::Formula> visit(const Clingo::AST::Term &term, const Clingo::AST::Literal &literal, RuleContext &ruleContext, Context &context, ast::VariableStack &variableStack)
 	{
-		return term.data.accept(BodyTermTranslateVisitor(), literal, term, ruleContext, variableStack);
+		return term.data.accept(BodyTermTranslateVisitor(), literal, term, ruleContext, context, variableStack);
 	}

 	// TODO: refactor
-	std::experimental::optional<ast::Formula> visit(const Clingo::AST::Comparison &comparison, const Clingo::AST::Literal &literal, RuleContext &ruleContext, ast::VariableStack &variableStack)
+	std::optional<ast::Formula> visit(const Clingo::AST::Comparison &comparison, const Clingo::AST::Literal &literal, RuleContext &ruleContext, Context &context, ast::VariableStack &variableStack)
 	{
 		// Comparisons should never have a sign, because these are converted to positive comparisons by clingo
 		if (literal.sign != Clingo::AST::Sign::None)
@@ -132,18 +136,18 @@ struct BodyLiteralTranslateVisitor

 		ast::And conjunction;
 		conjunction.arguments.reserve(3);
-		conjunction.arguments.emplace_back(ast::Formula::make<ast::In>(ast::Variable(parameters[0].get()), translate(comparison.left, ruleContext, variableStack)));
-		conjunction.arguments.emplace_back(ast::Formula::make<ast::In>(ast::Variable(parameters[1].get()), translate(comparison.right, ruleContext, variableStack)));
+		conjunction.arguments.emplace_back(ast::Formula::make<ast::In>(ast::Variable(parameters[0].get()), translate(comparison.left, ruleContext, context, variableStack)));
+		conjunction.arguments.emplace_back(ast::Formula::make<ast::In>(ast::Variable(parameters[1].get()), translate(comparison.right, ruleContext, context, variableStack)));
 		conjunction.arguments.emplace_back(ast::Formula::make<ast::Comparison>(operator_, ast::Variable(parameters[0].get()), ast::Variable(parameters[1].get())));

 		return ast::Formula::make<ast::Exists>(std::move(parameters), std::move(conjunction));
 	}

 	template<class T>
-	std::experimental::optional<ast::Formula> visit(const T &, const Clingo::AST::Literal &literal, RuleContext &, ast::VariableStack &)
+	std::optional<ast::Formula> visit(const T &, const Clingo::AST::Literal &literal, RuleContext &, Context &, ast::VariableStack &)
 	{
 		throw TranslationException(literal.location, "literal currently unsupported in this context, expected function or term");
-		return std::experimental::nullopt;
+		return std::nullopt;
 	}
 };

@@ -151,19 +155,19 @@ struct BodyLiteralTranslateVisitor

 struct BodyBodyLiteralTranslateVisitor
 {
-	std::experimental::optional<ast::Formula> visit(const Clingo::AST::Literal &literal, const Clingo::AST::BodyLiteral &bodyLiteral, RuleContext &ruleContext, ast::VariableStack &variableStack)
+	std::optional<ast::Formula> visit(const Clingo::AST::Literal &literal, const Clingo::AST::BodyLiteral &bodyLiteral, RuleContext &ruleContext, Context &context, ast::VariableStack &variableStack)
 	{
 		if (bodyLiteral.sign != Clingo::AST::Sign::None)
 			throw TranslationException(bodyLiteral.location, "only positive body literals supported currently");

-		return literal.data.accept(BodyLiteralTranslateVisitor(), literal, ruleContext, variableStack);
+		return literal.data.accept(BodyLiteralTranslateVisitor(), literal, ruleContext, context, variableStack);
 	}

 	template<class T>
-	std::experimental::optional<ast::Formula> visit(const T &, const Clingo::AST::BodyLiteral &bodyLiteral, RuleContext &, ast::VariableStack &)
+	std::optional<ast::Formula> visit(const T &, const Clingo::AST::BodyLiteral &bodyLiteral, RuleContext &, Context &, ast::VariableStack &)
 	{
 		throw TranslationException(bodyLiteral.location, "body literal currently unsupported in this context, expected literal");
-		return std::experimental::nullopt;
+		return std::nullopt;
 	}
 };

--- a/include/anthem/Context.h
+++ b/include/anthem/Context.h
@@ -1,7 +1,7 @@
 #ifndef __ANTHEM__CONTEXT_H
 #define __ANTHEM__CONTEXT_H

-#include <experimental/optional>
+#include <optional>

 #include <anthem/AST.h>
 #include <anthem/output/Logger.h>
@@ -16,6 +16,14 @@ namespace anthem
 //
 ////////////////////////////////////////////////////////////////////////////////////////////////////

+struct PredicateDeclarationMeta
+{
+	ast::PredicateDeclaration *declaration;
+	bool used{false};
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
 struct Context
 {
 	Context() = default;
@@ -25,12 +33,59 @@ struct Context
 	{
 	}

+	ast::PredicateDeclaration *findOrCreatePredicateDeclaration(const char *name, size_t arity)
+	{
+		const auto matchesExistingPredicateDeclaration =
+			[&](const auto &predicateDeclaration)
+			{
+				return (predicateDeclaration->arity() == arity
+					&& strcmp(predicateDeclaration->name.c_str(), name) == 0);
+			};
+
+		auto matchingPredicateDeclaration = std::find_if(predicateDeclarations.begin(),
+			predicateDeclarations.end(), matchesExistingPredicateDeclaration);
+
+		if (matchingPredicateDeclaration != predicateDeclarations.end())
+			return matchingPredicateDeclaration->get();
+
+		predicateDeclarations.emplace_back(std::make_unique<ast::PredicateDeclaration>(name, arity));
+
+		return predicateDeclarations.back().get();
+	}
+
+	ast::FunctionDeclaration *findOrCreateFunctionDeclaration(const char *name, size_t arity)
+	{
+		const auto matchesExistingFunctionDeclaration =
+			[&](const auto &functionDeclarations)
+			{
+				return (functionDeclarations->arity == arity
+					&& strcmp(functionDeclarations->name.c_str(), name) == 0);
+			};
+
+		auto matchingFunctionDeclaration = std::find_if(functionDeclarations.begin(),
+			functionDeclarations.end(), matchesExistingFunctionDeclaration);
+
+		if (matchingFunctionDeclaration != functionDeclarations.end())
+			return matchingFunctionDeclaration->get();
+
+		functionDeclarations.emplace_back(std::make_unique<ast::FunctionDeclaration>(name, arity));
+
+		return functionDeclarations.back().get();
+	}
+
 	output::Logger logger;

-	bool performSimplification = false;
-	bool performCompletion = false;
+	bool performSimplification{false};
+	bool performCompletion{false};
+	bool performIntegerDetection{false};

-	std::experimental::optional<std::vector<ast::PredicateSignature>> visiblePredicateSignatures;
+	std::vector<std::unique_ptr<ast::PredicateDeclaration>> predicateDeclarations;
+	ast::PredicateDeclaration::Visibility defaultPredicateVisibility{ast::PredicateDeclaration::Visibility::Visible};
+
+	std::vector<std::unique_ptr<ast::FunctionDeclaration>> functionDeclarations;
+
+	bool externalStatementsUsed{false};
+	bool showStatementsUsed{false};

 	ast::ParenthesisStyle parenthesisStyle = ast::ParenthesisStyle::Normal;
 };
--- a/include/anthem/Domain.h
+++ b/include/anthem/Domain.h
@@ -0,0 +1,27 @@
+#ifndef __ANTHEM__DOMAIN_H
+#define __ANTHEM__DOMAIN_H
+
+namespace anthem
+{
+namespace ast
+{
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+//
+// Domain
+//
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+enum class Domain
+{
+	General,
+	Integer,
+	Unknown,
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+}
+}
+
+#endif
--- a/include/anthem/Equality.h
+++ b/include/anthem/Equality.h
@@ -0,0 +1,409 @@
+#ifndef __ANTHEM__EQUALITY_H
+#define __ANTHEM__EQUALITY_H
+
+#include <anthem/AST.h>
+#include <anthem/ASTUtils.h>
+#include <anthem/Tristate.h>
+
+namespace anthem
+{
+namespace ast
+{
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+//
+// Equality
+//
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+Tristate equal(const Formula &lhs, const Formula &rhs);
+Tristate equal(const Term &lhs, const Term &rhs);
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct FormulaEqualityVisitor
+{
+	Tristate visit(const And &and_, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<And>())
+			return Tristate::Unknown;
+
+		const auto &otherAnd = otherFormula.get<And>();
+
+		for (const auto &argument : and_.arguments)
+		{
+			const auto match = std::find_if(
+				otherAnd.arguments.cbegin(), otherAnd.arguments.cend(),
+				[&](const auto &otherArgument)
+				{
+					return equal(argument, otherArgument) == Tristate::True;
+				});
+
+			if (match == otherAnd.arguments.cend())
+				return Tristate::Unknown;
+		}
+
+		for (const auto &otherArgument : otherAnd.arguments)
+		{
+			const auto match = std::find_if(
+				and_.arguments.cbegin(), and_.arguments.cend(),
+				[&](const auto &argument)
+				{
+					return equal(otherArgument, argument) == Tristate::True;
+				});
+
+			if (match == and_.arguments.cend())
+				return Tristate::Unknown;
+		}
+
+		return Tristate::True;
+	}
+
+	Tristate visit(const Biconditional &biconditional, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<Biconditional>())
+			return Tristate::Unknown;
+
+		const auto &otherBiconditional = otherFormula.get<Biconditional>();
+
+		if (equal(biconditional.left, otherBiconditional.left) == Tristate::True
+		    && equal(biconditional.right, otherBiconditional.right) == Tristate::True)
+		{
+			return Tristate::True;
+		}
+
+		if (equal(biconditional.left, otherBiconditional.right) == Tristate::True
+		    && equal(biconditional.right, otherBiconditional.left) == Tristate::True)
+		{
+			return Tristate::True;
+		}
+
+		return Tristate::Unknown;
+	}
+
+	Tristate visit(const Boolean &boolean, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<Boolean>())
+			return Tristate::Unknown;
+
+		const auto &otherBoolean = otherFormula.get<Boolean>();
+
+		return (boolean.value == otherBoolean.value)
+			? Tristate::True
+			: Tristate::False;
+	}
+
+	Tristate visit(const Comparison &comparison, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<Comparison>())
+			return Tristate::Unknown;
+
+		const auto &otherComparison = otherFormula.get<Comparison>();
+
+		if (comparison.operator_ != otherComparison.operator_)
+			return Tristate::Unknown;
+
+		if (equal(comparison.left, otherComparison.left) == Tristate::True
+		    && equal(comparison.right, otherComparison.right) == Tristate::True)
+		{
+			return Tristate::True;
+		}
+
+		// Only = and != are commutative operators, all others don’t need to be checked with exchanged arguments
+		if (comparison.operator_ != Comparison::Operator::Equal
+		    && comparison.operator_ != Comparison::Operator::NotEqual)
+		{
+			return Tristate::Unknown;
+		}
+
+		if (equal(comparison.left, otherComparison.right) == Tristate::True
+		    && equal(comparison.right, otherComparison.left) == Tristate::True)
+		{
+			return Tristate::True;
+		}
+
+		return Tristate::Unknown;
+	}
+
+	Tristate visit(const Exists &, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<Exists>())
+			return Tristate::Unknown;
+
+		// TODO: implement stronger check
+		return Tristate::Unknown;
+	}
+
+	Tristate visit(const ForAll &, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<ForAll>())
+			return Tristate::Unknown;
+
+		// TODO: implement stronger check
+		return Tristate::Unknown;
+	}
+
+	Tristate visit(const Implies &implies, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<Implies>())
+			return Tristate::Unknown;
+
+		const auto &otherImplies = otherFormula.get<Implies>();
+
+		if (equal(implies.antecedent, otherImplies.antecedent) == Tristate::True
+		    && equal(implies.consequent, otherImplies.consequent) == Tristate::True)
+		{
+			return Tristate::True;
+		}
+
+		return Tristate::Unknown;
+	}
+
+	Tristate visit(const In &in, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<In>())
+			return Tristate::Unknown;
+
+		const auto &otherIn = otherFormula.get<In>();
+
+		if (equal(in.element, otherIn.element) == Tristate::True
+		    && equal(in.set, otherIn.set) == Tristate::True)
+		{
+			return Tristate::True;
+		}
+
+		return Tristate::Unknown;
+	}
+
+	Tristate visit(const Not &not_, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<Not>())
+			return Tristate::Unknown;
+
+		const auto &otherNot = otherFormula.get<Not>();
+
+		return equal(not_.argument, otherNot.argument);
+	}
+
+	Tristate visit(const Or &or_, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<Or>())
+			return Tristate::Unknown;
+
+		const auto &otherOr = otherFormula.get<Or>();
+
+		for (const auto &argument : or_.arguments)
+		{
+			const auto match = std::find_if(
+				otherOr.arguments.cbegin(), otherOr.arguments.cend(),
+				[&](const auto &otherArgument)
+				{
+					return equal(argument, otherArgument) == Tristate::True;
+				});
+
+			if (match == otherOr.arguments.cend())
+				return Tristate::Unknown;
+		}
+
+		for (const auto &otherArgument : otherOr.arguments)
+		{
+			const auto match = std::find_if(
+				or_.arguments.cbegin(), or_.arguments.cend(),
+				[&](const auto &argument)
+				{
+					return equal(otherArgument, argument) == Tristate::True;
+				});
+
+			if (match == or_.arguments.cend())
+				return Tristate::Unknown;
+		}
+
+		return Tristate::True;
+	}
+
+	Tristate visit(const Predicate &predicate, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<Predicate>())
+			return Tristate::Unknown;
+
+		const auto &otherPredicate = otherFormula.get<Predicate>();
+
+		if (predicate.declaration != otherPredicate.declaration)
+			return Tristate::False;
+
+		assert(predicate.arguments.size() == otherPredicate.arguments.size());
+
+		for (size_t i = 0; i < predicate.arguments.size(); i++)
+			if (equal(predicate.arguments[i], otherPredicate.arguments[i]) != Tristate::True)
+				return Tristate::Unknown;
+
+		return Tristate::True;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct TermEqualityVisitor
+{
+	Tristate visit(const BinaryOperation &binaryOperation, const Term &otherTerm)
+	{
+		if (!otherTerm.is<BinaryOperation>())
+			return Tristate::Unknown;
+
+		const auto &otherBinaryOperation = otherTerm.get<BinaryOperation>();
+
+		if (binaryOperation.operator_ != otherBinaryOperation.operator_)
+			return Tristate::Unknown;
+
+		if (equal(binaryOperation.left, otherBinaryOperation.left) == Tristate::True
+		    && equal(binaryOperation.right, otherBinaryOperation.right) == Tristate::True)
+		{
+			return Tristate::True;
+		}
+
+		// Only + and * are commutative operators, all others don’t need to be checked with exchanged arguments
+		if (binaryOperation.operator_ != BinaryOperation::Operator::Plus
+		    && binaryOperation.operator_ != BinaryOperation::Operator::Multiplication)
+		{
+			return Tristate::Unknown;
+		}
+
+		if (equal(binaryOperation.left, binaryOperation.right) == Tristate::True
+		    && equal(binaryOperation.right, binaryOperation.left) == Tristate::True)
+		{
+			return Tristate::True;
+		}
+
+		return Tristate::Unknown;
+	}
+
+	Tristate visit(const Boolean &boolean, const Term &otherTerm)
+	{
+		if (!otherTerm.is<Boolean>())
+			return Tristate::Unknown;
+
+		const auto &otherBoolean = otherTerm.get<Boolean>();
+
+		return (boolean.value == otherBoolean.value)
+			? Tristate::True
+			: Tristate::False;
+	}
+
+	Tristate visit(const Function &function, const Term &otherTerm)
+	{
+		if (!otherTerm.is<Function>())
+			return Tristate::Unknown;
+
+		const auto &otherFunction = otherTerm.get<Function>();
+
+		if (function.declaration != otherFunction.declaration)
+			return Tristate::False;
+
+		if (function.arguments.size() != otherFunction.arguments.size())
+			return Tristate::False;
+
+		for (size_t i = 0; i < function.arguments.size(); i++)
+			if (equal(function.arguments[i], otherFunction.arguments[i]) != Tristate::True)
+				return Tristate::Unknown;
+
+		return Tristate::True;
+	}
+
+	Tristate visit(const Integer &integer, const Term &otherTerm)
+	{
+		if (!otherTerm.is<Integer>())
+			return Tristate::Unknown;
+
+		const auto &otherInteger = otherTerm.get<Integer>();
+
+		return (integer.value == otherInteger.value)
+			? Tristate::True
+			: Tristate::False;
+	}
+
+	Tristate visit(const Interval &interval, const Term &otherTerm)
+	{
+		if (!otherTerm.is<Interval>())
+			return Tristate::Unknown;
+
+		const auto &otherInterval = otherTerm.get<Interval>();
+
+		if (equal(interval.from, otherInterval.from) != Tristate::True)
+			return Tristate::Unknown;
+
+		if (equal(interval.to, otherInterval.to) != Tristate::True)
+			return Tristate::Unknown;
+
+		return Tristate::True;
+	}
+
+	Tristate visit(const SpecialInteger &specialInteger, const Term &otherTerm)
+	{
+		if (!otherTerm.is<SpecialInteger>())
+			return Tristate::Unknown;
+
+		const auto &otherSpecialInteger = otherTerm.get<SpecialInteger>();
+
+		return (specialInteger.type == otherSpecialInteger.type)
+			? Tristate::True
+			: Tristate::False;
+	}
+
+	Tristate visit(const String &string, const Term &otherTerm)
+	{
+		if (!otherTerm.is<String>())
+			return Tristate::Unknown;
+
+		const auto &otherString = otherTerm.get<String>();
+
+		return (string.text == otherString.text)
+			? Tristate::True
+			: Tristate::False;
+	}
+
+	Tristate visit(const UnaryOperation &unaryOperation, const Term &otherTerm)
+	{
+		if (!otherTerm.is<UnaryOperation>())
+			return Tristate::Unknown;
+
+		const auto &otherUnaryOperation = otherTerm.get<UnaryOperation>();
+
+		if (unaryOperation.operator_ != otherUnaryOperation.operator_)
+			return Tristate::Unknown;
+
+		return equal(unaryOperation.argument, otherUnaryOperation.argument);
+	}
+
+	Tristate visit(const Variable &variable, const Term &otherTerm)
+	{
+		if (!otherTerm.is<Variable>())
+			return Tristate::Unknown;
+
+		const auto &otherVariable = otherTerm.get<Variable>();
+
+		return (variable.declaration == otherVariable.declaration)
+			? Tristate::True
+			: Tristate::False;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+Tristate equal(const Formula &lhs, const Formula &rhs)
+{
+	return lhs.accept(FormulaEqualityVisitor(), rhs);
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+Tristate equal(const Term &lhs, const Term &rhs)
+{
+	return lhs.accept(TermEqualityVisitor(), rhs);
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+}
+}
+
+#endif
--- a/include/anthem/Head.h
+++ b/include/anthem/Head.h
@@ -2,7 +2,7 @@
 #define __ANTHEM__HEAD_H

 #include <algorithm>
-#include <experimental/optional>
+#include <optional>

 #include <anthem/AST.h>
 #include <anthem/Exception.h>
@@ -118,8 +118,7 @@ struct HeadLiteralCollectFunctionTermsVisitor

 struct FunctionTermTranslateVisitor
 {
-	// TODO: check correctness
-	std::experimental::optional<ast::Formula> visit(const Clingo::AST::Function &function, const Clingo::AST::Term &term, RuleContext &ruleContext, size_t &headVariableIndex)
+	std::optional<ast::Formula> visit(const Clingo::AST::Function &function, const Clingo::AST::Term &term, RuleContext &ruleContext, Context &context, size_t &headVariableIndex)
 	{
 		if (function.external)
 			throw TranslationException(term.location, "external functions currently unsupported");
@@ -130,14 +129,17 @@ struct FunctionTermTranslateVisitor
 		for (size_t i = 0; i < function.arguments.size(); i++)
 			arguments.emplace_back(ast::Variable(ruleContext.freeVariables[headVariableIndex++].get()));

-		return ast::Formula::make<ast::Predicate>(function.name, std::move(arguments));
+		auto predicateDeclaration = context.findOrCreatePredicateDeclaration(function.name, function.arguments.size());
+		predicateDeclaration->isUsed = true;
+
+		return ast::Predicate(predicateDeclaration, std::move(arguments));
 	}

 	template<class T>
-	std::experimental::optional<ast::Formula> visit(const T &, const Clingo::AST::Term &term, RuleContext &, size_t &)
+	std::optional<ast::Formula> visit(const T &, const Clingo::AST::Term &term, RuleContext &, Context &, size_t &)
 	{
 		throw TranslationException(term.location, "term currently unsupported in this context, function expected");
-		return std::experimental::nullopt;
+		return std::nullopt;
 	}
 };

@@ -145,21 +147,21 @@ struct FunctionTermTranslateVisitor

 struct LiteralTranslateVisitor
 {
-	std::experimental::optional<ast::Formula> visit(const Clingo::AST::Boolean &boolean, const Clingo::AST::Literal &, RuleContext &, size_t &)
+	std::optional<ast::Formula> visit(const Clingo::AST::Boolean &boolean, const Clingo::AST::Literal &, RuleContext &, Context &, size_t &)
 	{
 		return ast::Formula::make<ast::Boolean>(boolean.value);
 	}

-	std::experimental::optional<ast::Formula> visit(const Clingo::AST::Term &term, const Clingo::AST::Literal &, RuleContext &ruleContext, size_t &headVariableIndex)
+	std::optional<ast::Formula> visit(const Clingo::AST::Term &term, const Clingo::AST::Literal &, RuleContext &ruleContext, Context &context, size_t &headVariableIndex)
 	{
-		return term.data.accept(FunctionTermTranslateVisitor(), term, ruleContext, headVariableIndex);
+		return term.data.accept(FunctionTermTranslateVisitor(), term, ruleContext, context, headVariableIndex);
 	}

 	template<class T>
-	std::experimental::optional<ast::Formula> visit(const T &, const Clingo::AST::Literal &literal, RuleContext &, size_t &)
+	std::optional<ast::Formula> visit(const T &, const Clingo::AST::Literal &literal, RuleContext &, Context &, size_t &)
 	{
 		throw TranslationException(literal.location, "only disjunctions of literals allowed as head literals");
-		return std::experimental::nullopt;
+		return std::nullopt;
 	}
 };

@@ -167,23 +169,23 @@ struct LiteralTranslateVisitor

 struct HeadLiteralTranslateToConsequentVisitor
 {
-	std::experimental::optional<ast::Formula> visit(const Clingo::AST::Literal &literal, const Clingo::AST::HeadLiteral &, RuleContext &ruleContext, size_t &headVariableIndex)
+	std::optional<ast::Formula> visit(const Clingo::AST::Literal &literal, const Clingo::AST::HeadLiteral &, RuleContext &ruleContext, Context &context, size_t &headVariableIndex)
 	{
 		if (literal.sign == Clingo::AST::Sign::DoubleNegation)
 			throw TranslationException(literal.location, "double-negated head literals currently unsupported");

-		auto translatedLiteral = literal.data.accept(LiteralTranslateVisitor(), literal, ruleContext, headVariableIndex);
+		auto translatedLiteral = literal.data.accept(LiteralTranslateVisitor(), literal, ruleContext, context, headVariableIndex);

 		if (literal.sign == Clingo::AST::Sign::None)
 			return translatedLiteral;

 		if (!translatedLiteral)
-			return std::experimental::nullopt;
+			return std::nullopt;

 		return ast::Formula::make<ast::Not>(std::move(translatedLiteral.value()));
 	}

-	std::experimental::optional<ast::Formula> visit(const Clingo::AST::Disjunction &disjunction, const Clingo::AST::HeadLiteral &headLiteral, RuleContext &ruleContext, size_t &headVariableIndex)
+	std::optional<ast::Formula> visit(const Clingo::AST::Disjunction &disjunction, const Clingo::AST::HeadLiteral &headLiteral, RuleContext &ruleContext, Context &context, size_t &headVariableIndex)
 	{
 		std::vector<ast::Formula> arguments;
 		arguments.reserve(disjunction.elements.size());
@@ -193,7 +195,7 @@ struct HeadLiteralTranslateToConsequentVisitor
 			if (!conditionalLiteral.condition.empty())
 				throw TranslationException(headLiteral.location, "conditional head literals currently unsupported");

-			auto argument = visit(conditionalLiteral.literal, headLiteral, ruleContext, headVariableIndex);
+			auto argument = visit(conditionalLiteral.literal, headLiteral, ruleContext, context, headVariableIndex);

 			if (!argument)
 				throw TranslationException(headLiteral.location, "could not parse argument");
@@ -204,7 +206,7 @@ struct HeadLiteralTranslateToConsequentVisitor
 		return ast::Formula::make<ast::Or>(std::move(arguments));
 	}

-	std::experimental::optional<ast::Formula> visit(const Clingo::AST::Aggregate &aggregate, const Clingo::AST::HeadLiteral &headLiteral, RuleContext &ruleContext, size_t &headVariableIndex)
+	std::optional<ast::Formula> visit(const Clingo::AST::Aggregate &aggregate, const Clingo::AST::HeadLiteral &headLiteral, RuleContext &ruleContext, Context &context, size_t &headVariableIndex)
 	{
 		if (aggregate.left_guard || aggregate.right_guard)
 			throw TranslationException(headLiteral.location, "aggregates with left or right guards currently unsupported");
@@ -215,7 +217,7 @@ struct HeadLiteralTranslateToConsequentVisitor
 				if (!conditionalLiteral.condition.empty())
 					throw TranslationException(headLiteral.location, "conditional head literals currently unsupported");

-				return this->visit(conditionalLiteral.literal, headLiteral, ruleContext, headVariableIndex);
+				return this->visit(conditionalLiteral.literal, headLiteral, ruleContext, context, headVariableIndex);
 			};

 		if (aggregate.elements.size() == 1)
@@ -238,10 +240,10 @@ struct HeadLiteralTranslateToConsequentVisitor
 	}

 	template<class T>
-	std::experimental::optional<ast::Formula> visit(const T &, const Clingo::AST::HeadLiteral &headLiteral, RuleContext &, size_t &)
+	std::optional<ast::Formula> visit(const T &, const Clingo::AST::HeadLiteral &headLiteral, RuleContext &, Context &, size_t &)
 	{
 		throw TranslationException(headLiteral.location, "head literal currently unsupported in this context, expected literal, disjunction, or aggregate");
-		return std::experimental::nullopt;
+		return std::nullopt;
 	}
 };

--- a/include/anthem/HiddenPredicateElimination.h
+++ b/include/anthem/HiddenPredicateElimination.h
@@ -13,7 +13,7 @@ namespace anthem
 //
 ////////////////////////////////////////////////////////////////////////////////////////////////////

-void eliminateHiddenPredicates(const std::vector<ast::PredicateSignature> &predicateSignatures, std::vector<ast::Formula> &completedFormulas, Context &context);
+void eliminateHiddenPredicates(std::vector<ast::Formula> &completedFormulas, Context &context);

 ////////////////////////////////////////////////////////////////////////////////////////////////////

--- a/include/anthem/IntegerVariableDetection.h
+++ b/include/anthem/IntegerVariableDetection.h
@@ -0,0 +1,22 @@
+#ifndef __ANTHEM__INTEGER_VARIABLE_DETECTION_H
+#define __ANTHEM__INTEGER_VARIABLE_DETECTION_H
+
+#include <anthem/AST.h>
+#include <anthem/Context.h>
+
+namespace anthem
+{
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+//
+// IntegerVariableDetection
+//
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+void detectIntegerVariables(std::vector<ast::Formula> &completedFormulas);
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+}
+
+#endif
--- a/include/anthem/Simplification.h
+++ b/include/anthem/Simplification.h
@@ -12,6 +12,14 @@ namespace anthem
 //
 ////////////////////////////////////////////////////////////////////////////////////////////////////

+enum class SimplificationResult
+{
+	Simplified,
+	Unchanged,
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
 void simplify(ast::Formula &formula);

 ////////////////////////////////////////////////////////////////////////////////////////////////////
--- a/include/anthem/SimplificationVisitors.h
+++ b/include/anthem/SimplificationVisitors.h
@@ -0,0 +1,192 @@
+#ifndef __ANTHEM__SIMPLIFICATION_VISITORS_H
+#define __ANTHEM__SIMPLIFICATION_VISITORS_H
+
+#include <anthem/AST.h>
+#include <anthem/Simplification.h>
+
+namespace anthem
+{
+namespace ast
+{
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+//
+// Simplification Visitor
+//
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+template<class T>
+struct FormulaSimplificationVisitor
+{
+	template <class... Arguments>
+	SimplificationResult visit(And &and_, Formula &formula, Arguments &&... arguments)
+	{
+		for (auto &argument : and_.arguments)
+			if (argument.accept(*this, argument, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+				return SimplificationResult::Simplified;
+
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Biconditional &biconditional, Formula &formula, Arguments &&... arguments)
+	{
+		if (biconditional.left.accept(*this, biconditional.left, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		if (biconditional.right.accept(*this, biconditional.right, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Boolean &, Formula &formula, Arguments &&... arguments)
+	{
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Comparison &, Formula &formula, Arguments &&... arguments)
+	{
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Exists &exists, Formula &formula, Arguments &&... arguments)
+	{
+		if (exists.argument.accept(*this, exists.argument, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(ForAll &forAll, Formula &formula, Arguments &&... arguments)
+	{
+		if (forAll.argument.accept(*this, forAll.argument, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Implies &implies, Formula &formula, Arguments &&... arguments)
+	{
+		if (implies.antecedent.accept(*this, implies.antecedent, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		if (implies.consequent.accept(*this, implies.consequent, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(In &, Formula &formula, Arguments &&... arguments)
+	{
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Not &not_, Formula &formula, Arguments &&... arguments)
+	{
+		if (not_.argument.accept(*this, not_.argument, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Or &or_, Formula &formula, Arguments &&... arguments)
+	{
+		for (auto &argument : or_.arguments)
+			if (argument.accept(*this, argument, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+				return SimplificationResult::Simplified;
+
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Predicate &, Formula &formula, Arguments &&... arguments)
+	{
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+template<class T, class SimplificationResult = void>
+struct TermSimplificationVisitor
+{
+	template <class... Arguments>
+	SimplificationResult visit(BinaryOperation &binaryOperation, Term &term, Arguments &&... arguments)
+	{
+		if (binaryOperation.left.accept(*this, binaryOperation.left, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		if (binaryOperation.right.accept(*this, binaryOperation.right, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		return T::accept(term, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Boolean &, Term &term, Arguments &&... arguments)
+	{
+		return T::accept(term, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Function &function, Term &term, Arguments &&... arguments)
+	{
+		for (auto &argument : function.arguments)
+			if (argument.accept(*this, argument, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+				return SimplificationResult::Simplified;
+
+		return T::accept(term, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Integer &, Term &term, Arguments &&... arguments)
+	{
+		return T::accept(term, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Interval &interval, Term &term, Arguments &&... arguments)
+	{
+		if (interval.from.accept(*this, interval.from, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		if (interval.to.accept(*this, interval.to, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		return T::accept(term, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(SpecialInteger &, Term &term, Arguments &&... arguments)
+	{
+		return T::accept(term, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(String &, Term &term, Arguments &&... arguments)
+	{
+		return T::accept(term, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Variable &, Term &term, Arguments &&... arguments)
+	{
+		return T::accept(term, std::forward<Arguments>(arguments)...);
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+}
+}
+
+#endif
--- a/include/anthem/StatementVisitor.h
+++ b/include/anthem/StatementVisitor.h
@@ -73,7 +73,7 @@ struct StatementVisitor

 		// Compute consequent
 		auto headVariableIndex = ruleContext.headVariablesStartIndex;
-		auto consequent = rule.head.data.accept(HeadLiteralTranslateToConsequentVisitor(), rule.head, ruleContext, headVariableIndex);
+		auto consequent = rule.head.data.accept(HeadLiteralTranslateToConsequentVisitor(), rule.head, ruleContext, context, headVariableIndex);

 		assert(ruleContext.headTerms.size() == headVariableIndex - ruleContext.headVariablesStartIndex);

@@ -87,7 +87,7 @@ struct StatementVisitor

 			const auto auxiliaryHeadVariableID = ruleContext.headVariablesStartIndex + i - ruleContext.headTerms.cbegin();
 			auto element = ast::Variable(ruleContext.freeVariables[auxiliaryHeadVariableID].get());
-			auto set = translate(headTerm, ruleContext, variableStack);
+			auto set = translate(headTerm, ruleContext, context, variableStack);
 			auto in = ast::In(std::move(element), std::move(set));

 			antecedent.arguments.emplace_back(std::move(in));
@@ -98,7 +98,7 @@ struct StatementVisitor
 		{
 			const auto &bodyLiteral = *i;

-			auto argument = bodyLiteral.data.accept(BodyBodyLiteralTranslateVisitor(), bodyLiteral, ruleContext, variableStack);
+			auto argument = bodyLiteral.data.accept(BodyBodyLiteralTranslateVisitor(), bodyLiteral, ruleContext, context, variableStack);

 			if (!argument)
 				throw TranslationException(bodyLiteral.location, "could not translate body literal");
@@ -165,8 +165,8 @@ struct StatementVisitor
 		if (signature.negative())
 			throw LogicException(statement.location, "negative #show atom signatures are currently unsupported");

-		if (!context.visiblePredicateSignatures)
-			context.visiblePredicateSignatures.emplace();
+		context.showStatementsUsed = true;
+		context.defaultPredicateVisibility = ast::PredicateDeclaration::Visibility::Hidden;

 		if (std::strlen(signature.name()) == 0)
 		{
@@ -176,7 +176,8 @@ struct StatementVisitor

 		context.logger.log(output::Priority::Debug, statement.location) << "showing “" << signature.name() << "/" << signature.arity() << "”";

-		context.visiblePredicateSignatures.value().emplace_back(std::string(signature.name()), signature.arity());
+		auto predicateDeclaration = context.findOrCreatePredicateDeclaration(signature.name(), signature.arity());
+		predicateDeclaration->visibility = ast::PredicateDeclaration::Visibility::Visible;
 	}

 	void visit(const Clingo::AST::ShowTerm &, const Clingo::AST::Statement &statement, std::vector<ast::ScopedFormula> &, Context &)
@@ -184,6 +185,84 @@ struct StatementVisitor
 		throw LogicException(statement.location, "only #show statements for atoms (not terms) are supported currently");
 	}

+	void visit(const Clingo::AST::External &external, const Clingo::AST::Statement &statement, std::vector<ast::ScopedFormula> &, Context &context)
+	{
+		const auto fail =
+			[&]()
+			{
+				throw LogicException(statement.location, "only #external declarations of the form “#external <predicate name>(<arity>).” or “#external integer(<function name>(<arity>)).” supported");
+			};
+
+		if (!external.body.empty())
+			fail();
+
+		if (!external.atom.data.is<Clingo::AST::Function>())
+			fail();
+
+		const auto &predicate = external.atom.data.get<Clingo::AST::Function>();
+
+		if (predicate.arguments.size() != 1)
+			fail();
+
+		const auto handleIntegerDeclaration =
+			[&]()
+			{
+				// Integer function declarations are treated separately if applicable
+				if (strcmp(predicate.name, "integer") != 0)
+					return false;
+
+				if (predicate.arguments.size() != 1)
+					return false;
+
+				const auto &functionArgument = predicate.arguments.front();
+
+				if (!functionArgument.data.is<Clingo::AST::Function>())
+					return false;
+
+				const auto &function = functionArgument.data.get<Clingo::AST::Function>();
+
+				if (function.arguments.size() != 1)
+					return false;
+
+				const auto &arityArgument = function.arguments.front();
+
+				if (!arityArgument.data.is<Clingo::Symbol>())
+					return false;
+
+				const auto &aritySymbol = arityArgument.data.get<Clingo::Symbol>();
+
+				if (aritySymbol.type() != Clingo::SymbolType::Number)
+					return false;
+
+				const size_t arity = aritySymbol.number();
+
+				auto functionDeclaration = context.findOrCreateFunctionDeclaration(function.name, arity);
+				functionDeclaration->domain = ast::Domain::Integer;
+
+				return true;
+			};
+
+		if (handleIntegerDeclaration())
+			return;
+
+		const auto &arityArgument = predicate.arguments.front();
+
+		if (!arityArgument.data.is<Clingo::Symbol>())
+			fail();
+
+		const auto &aritySymbol = arityArgument.data.get<Clingo::Symbol>();
+
+		if (aritySymbol.type() != Clingo::SymbolType::Number)
+			fail();
+
+		context.externalStatementsUsed = true;
+
+		const size_t arity = aritySymbol.number();
+
+		auto predicateDeclaration = context.findOrCreatePredicateDeclaration(predicate.name, arity);
+		predicateDeclaration->isExternal = true;
+	}
+
 	template<class T>
 	void visit(const T &, const Clingo::AST::Statement &statement, std::vector<ast::ScopedFormula> &, Context &)
 	{
--- a/include/anthem/Term.h
+++ b/include/anthem/Term.h
@@ -23,6 +23,12 @@ ast::BinaryOperation::Operator translate(Clingo::AST::BinaryOperator binaryOpera
 {
 	switch (binaryOperator)
 	{
+		case Clingo::AST::BinaryOperator::XOr:
+			throw TranslationException(term.location, "binary operation “xor” currently unsupported");
+		case Clingo::AST::BinaryOperator::Or:
+			throw TranslationException(term.location, "binary operation “or” currently unsupported");
+		case Clingo::AST::BinaryOperator::And:
+			throw TranslationException(term.location, "binary operation “and” currently unsupported");
 		case Clingo::AST::BinaryOperator::Plus:
 			return ast::BinaryOperation::Operator::Plus;
 		case Clingo::AST::BinaryOperator::Minus:
@@ -33,22 +39,39 @@ ast::BinaryOperation::Operator translate(Clingo::AST::BinaryOperator binaryOpera
 			return ast::BinaryOperation::Operator::Division;
 		case Clingo::AST::BinaryOperator::Modulo:
 			return ast::BinaryOperation::Operator::Modulo;
-		default:
-			throw TranslationException(term.location, "“binary operation” terms currently unsupported");
+		case Clingo::AST::BinaryOperator::Power:
+			return ast::BinaryOperation::Operator::Power;
 	}

-	return ast::BinaryOperation::Operator::Plus;
+	throw TranslationException(term.location, "unknown binary operation");
 }

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-ast::Term translate(const Clingo::AST::Term &term, RuleContext &ruleContext, const ast::VariableStack &variableStack);
+ast::UnaryOperation::Operator translate(Clingo::AST::UnaryOperator unaryOperator, const Clingo::AST::Term &term)
+{
+	switch (unaryOperator)
+	{
+		case Clingo::AST::UnaryOperator::Absolute:
+			return ast::UnaryOperation::Operator::Absolute;
+		case Clingo::AST::UnaryOperator::Minus:
+			throw TranslationException(term.location, "binary operation “minus” currently unsupported");
+		case Clingo::AST::UnaryOperator::Negation:
+			throw TranslationException(term.location, "binary operation “negation” currently unsupported");
+	}
+
+	throw TranslationException(term.location, "unknown unary operation");
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+ast::Term translate(const Clingo::AST::Term &term, RuleContext &ruleContext, Context &context, const ast::VariableStack &variableStack);

 ////////////////////////////////////////////////////////////////////////////////////////////////////

 struct TermTranslateVisitor
 {
-	std::experimental::optional<ast::Term> visit(const Clingo::Symbol &symbol, const Clingo::AST::Term &term, RuleContext &ruleContext, const ast::VariableStack &variableStack)
+	std::optional<ast::Term> visit(const Clingo::Symbol &symbol, const Clingo::AST::Term &term, RuleContext &ruleContext, Context &context, const ast::VariableStack &variableStack)
 	{
 		switch (symbol.type())
 		{
@@ -62,29 +85,29 @@ struct TermTranslateVisitor
 				return ast::Term::make<ast::String>(std::string(symbol.string()));
 			case Clingo::SymbolType::Function:
 			{
-				auto function = ast::Term::make<ast::Function>(symbol.name());
-				// TODO: remove workaround
-				auto &functionRaw = function.get<ast::Function>();
-				functionRaw.arguments.reserve(symbol.arguments().size());
+				auto functionDeclaration = context.findOrCreateFunctionDeclaration(symbol.name(), symbol.arguments().size());
+
+				auto function = ast::Function(functionDeclaration);
+				function.arguments.reserve(symbol.arguments().size());

 				for (const auto &argument : symbol.arguments())
 				{
-					auto translatedArgument = visit(argument, term, ruleContext, variableStack);
+					auto translatedArgument = visit(argument, term, ruleContext, context, variableStack);

 					if (!translatedArgument)
 						throw TranslationException(term.location, "could not translate argument");

-					functionRaw.arguments.emplace_back(std::move(translatedArgument.value()));
+					function.arguments.emplace_back(std::move(translatedArgument.value()));
 				}

 				return std::move(function);
 			}
 		}

-		return std::experimental::nullopt;
+		return std::nullopt;
 	}

-	std::experimental::optional<ast::Term> visit(const Clingo::AST::Variable &variable, const Clingo::AST::Term &, RuleContext &ruleContext, const ast::VariableStack &variableStack)
+	std::optional<ast::Term> visit(const Clingo::AST::Variable &variable, const Clingo::AST::Term &, RuleContext &ruleContext, Context &, const ast::VariableStack &variableStack)
 	{
 		const auto matchingVariableDeclaration = variableStack.findUserVariableDeclaration(variable.name);
 		const auto isAnonymousVariable = (strcmp(variable.name, "_") == 0);
@@ -97,33 +120,36 @@ struct TermTranslateVisitor
 		auto variableDeclaration = std::make_unique<ast::VariableDeclaration>(ast::VariableDeclaration::Type::UserDefined, std::string(variable.name));
 		ruleContext.freeVariables.emplace_back(std::move(variableDeclaration));

+		// TODO: ast::Term::make is unnecessary and can be removed
 		return ast::Term::make<ast::Variable>(ruleContext.freeVariables.back().get());
 	}

-	std::experimental::optional<ast::Term> visit(const Clingo::AST::UnaryOperation &, const Clingo::AST::Term &term, RuleContext &, const ast::VariableStack &)
-	{
-		throw TranslationException(term.location, "“unary operation” terms currently unsupported");
-		return std::experimental::nullopt;
-	}
-
-	std::experimental::optional<ast::Term> visit(const Clingo::AST::BinaryOperation &binaryOperation, const Clingo::AST::Term &term, RuleContext &ruleContext, const ast::VariableStack &variableStack)
+	std::optional<ast::Term> visit(const Clingo::AST::BinaryOperation &binaryOperation, const Clingo::AST::Term &term, RuleContext &ruleContext, Context &context, const ast::VariableStack &variableStack)
 	{
 		const auto operator_ = translate(binaryOperation.binary_operator, term);
-		auto left = translate(binaryOperation.left, ruleContext, variableStack);
-		auto right = translate(binaryOperation.right, ruleContext, variableStack);
+		auto left = translate(binaryOperation.left, ruleContext, context, variableStack);
+		auto right = translate(binaryOperation.right, ruleContext, context, variableStack);

 		return ast::Term::make<ast::BinaryOperation>(operator_, std::move(left), std::move(right));
 	}

-	std::experimental::optional<ast::Term> visit(const Clingo::AST::Interval &interval, const Clingo::AST::Term &, RuleContext &ruleContext, const ast::VariableStack &variableStack)
+	std::optional<ast::Term> visit(const Clingo::AST::UnaryOperation &unaryOperation, const Clingo::AST::Term &term, RuleContext &ruleContext, Context &context, const ast::VariableStack &variableStack)
 	{
-		auto left = translate(interval.left, ruleContext, variableStack);
-		auto right = translate(interval.right, ruleContext, variableStack);
+		const auto operator_ = translate(unaryOperation.unary_operator, term);
+		auto argument = translate(unaryOperation.argument, ruleContext, context, variableStack);
+
+		return ast::Term::make<ast::UnaryOperation>(operator_, std::move(argument));
+	}
+
+	std::optional<ast::Term> visit(const Clingo::AST::Interval &interval, const Clingo::AST::Term &, RuleContext &ruleContext, Context &context, const ast::VariableStack &variableStack)
+	{
+		auto left = translate(interval.left, ruleContext, context, variableStack);
+		auto right = translate(interval.right, ruleContext, context, variableStack);

 		return ast::Term::make<ast::Interval>(std::move(left), std::move(right));
 	}

-	std::experimental::optional<ast::Term> visit(const Clingo::AST::Function &function, const Clingo::AST::Term &term, RuleContext &ruleContext, const ast::VariableStack &variableStack)
+	std::optional<ast::Term> visit(const Clingo::AST::Function &function, const Clingo::AST::Term &term, RuleContext &ruleContext, Context &context, const ast::VariableStack &variableStack)
 	{
 		if (function.external)
 			throw TranslationException(term.location, "external functions currently unsupported");
@@ -132,23 +158,25 @@ struct TermTranslateVisitor
 		arguments.reserve(function.arguments.size());

 		for (const auto &argument : function.arguments)
-			arguments.emplace_back(translate(argument, ruleContext, variableStack));
+			arguments.emplace_back(translate(argument, ruleContext, context, variableStack));

-		return ast::Term::make<ast::Function>(function.name, std::move(arguments));
+		auto functionDeclaration = context.findOrCreateFunctionDeclaration(function.name, function.arguments.size());
+
+		return ast::Term::make<ast::Function>(functionDeclaration, std::move(arguments));
 	}

-	std::experimental::optional<ast::Term> visit(const Clingo::AST::Pool &, const Clingo::AST::Term &term, RuleContext &, const ast::VariableStack &)
+	std::optional<ast::Term> visit(const Clingo::AST::Pool &, const Clingo::AST::Term &term, RuleContext &, Context &, const ast::VariableStack &)
 	{
 		throw TranslationException(term.location, "“pool” terms currently unsupported");
-		return std::experimental::nullopt;
+		return std::nullopt;
 	}
 };

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-ast::Term translate(const Clingo::AST::Term &term, RuleContext &ruleContext, const ast::VariableStack &variableStack)
+ast::Term translate(const Clingo::AST::Term &term, RuleContext &ruleContext, Context &context, const ast::VariableStack &variableStack)
 {
-	auto translatedTerm = term.data.accept(TermTranslateVisitor(), term, ruleContext, variableStack);
+	auto translatedTerm = term.data.accept(TermTranslateVisitor(), term, ruleContext, context, variableStack);

 	if (!translatedTerm)
 		throw TranslationException(term.location, "could not translate term");
--- a/include/anthem/Tristate.h
+++ b/include/anthem/Tristate.h
@@ -0,0 +1,24 @@
+#ifndef __ANTHEM__TRISTATE_H
+#define __ANTHEM__TRISTATE_H
+
+namespace anthem
+{
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+//
+// Tristate
+//
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+enum class Tristate
+{
+	True,
+	False,
+	Unknown,
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+}
+
+#endif
--- a/include/anthem/output/AST.h
+++ b/include/anthem/output/AST.h
@@ -41,37 +41,39 @@ struct PrintContext

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-output::ColorStream &print(output::ColorStream &stream, const BinaryOperation::Operator operator_, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const BinaryOperation &binaryOperation, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const Boolean &boolean, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const Comparison &comparison, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const Constant &constant, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const Function &function, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const In &in, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const Integer &integer, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const Interval &interval, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const Predicate &predicate, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const SpecialInteger &specialInteger, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const String &string, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const Variable &variable, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const VariableDeclaration &variableDeclaration, PrintContext &printContext);
+output::ColorStream &print(output::ColorStream &stream, const BinaryOperation::Operator operator_, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const BinaryOperation &binaryOperation, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const Boolean &boolean, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const Comparison &comparison, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, Comparison::Operator operator_, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const Function &function, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const In &in, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const Integer &integer, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const Interval &interval, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const Predicate &predicate, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const PredicateDeclaration &predicateDeclaration, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const SpecialInteger &specialInteger, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const String &string, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const UnaryOperation &unaryOperation, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const Variable &variable, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const VariableDeclaration &variableDeclaration, PrintContext &printContext, bool omitParentheses = false);

-output::ColorStream &print(output::ColorStream &stream, const And &and_, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const Biconditional &biconditional, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const Exists &exists, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const ForAll &forAll, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const Implies &implies, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const Not &not_, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const Or &or_, PrintContext &printContext);
+output::ColorStream &print(output::ColorStream &stream, const And &and_, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const Biconditional &biconditional, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const Exists &exists, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const ForAll &forAll, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const Implies &implies, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const Not &not_, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const Or &or_, PrintContext &printContext, bool omitParentheses = false);

-output::ColorStream &print(output::ColorStream &stream, const Formula &formula, PrintContext &printContext);
-output::ColorStream &print(output::ColorStream &stream, const Term &term, PrintContext &printContext);
+output::ColorStream &print(output::ColorStream &stream, const Formula &formula, PrintContext &printContext, bool omitParentheses = false);
+output::ColorStream &print(output::ColorStream &stream, const Term &term, PrintContext &printContext, bool omitParentheses = false);

 ////////////////////////////////////////////////////////////////////////////////////////////////////
 // Primitives
 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, BinaryOperation::Operator operator_, PrintContext &)
+inline output::ColorStream &print(output::ColorStream &stream, BinaryOperation::Operator operator_, PrintContext &, bool)
 {
 	switch (operator_)
 	{
@@ -85,6 +87,8 @@ inline output::ColorStream &print(output::ColorStream &stream, BinaryOperation::
 			return (stream << output::Operator("/"));
 		case BinaryOperation::Operator::Modulo:
 			return (stream << output::Operator("%"));
+		case BinaryOperation::Operator::Power:
+			return (stream << output::Operator("**"));
 	}

 	return stream;
@@ -92,22 +96,26 @@ inline output::ColorStream &print(output::ColorStream &stream, BinaryOperation::

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const BinaryOperation &binaryOperation, PrintContext &printContext)
+inline output::ColorStream &print(output::ColorStream &stream, const BinaryOperation &binaryOperation, PrintContext &printContext, bool omitParentheses)
 {
-	stream << "(";
+	if (!omitParentheses || printContext.context.parenthesisStyle == ParenthesisStyle::Full)
+		stream << "(";
+
 	print(stream, binaryOperation.left, printContext);
 	stream << " ";
 	print(stream, binaryOperation.operator_, printContext);
 	stream << " ";
 	print(stream, binaryOperation.right, printContext);
-	stream << ")";
+
+	if (!omitParentheses || printContext.context.parenthesisStyle == ParenthesisStyle::Full)
+		stream << ")";

 	return stream;
 }

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const Boolean &boolean, PrintContext &)
+inline output::ColorStream &print(output::ColorStream &stream, const Boolean &boolean, PrintContext &, bool)
 {
 	if (boolean.value)
 		return (stream << output::Boolean("#true"));
@@ -117,7 +125,7 @@ inline output::ColorStream &print(output::ColorStream &stream, const Boolean &bo

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, Comparison::Operator operator_, PrintContext &)
+inline output::ColorStream &print(output::ColorStream &stream, Comparison::Operator operator_, PrintContext &, bool)
 {
 	switch (operator_)
 	{
@@ -140,7 +148,7 @@ inline output::ColorStream &print(output::ColorStream &stream, Comparison::Opera

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const Comparison &comparison, PrintContext &printContext)
+inline output::ColorStream &print(output::ColorStream &stream, const Comparison &comparison, PrintContext &printContext, bool)
 {
 	if (printContext.context.parenthesisStyle == ParenthesisStyle::Full)
 		stream << "(";
@@ -159,16 +167,9 @@ inline output::ColorStream &print(output::ColorStream &stream, const Comparison

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const Constant &constant, PrintContext &)
+inline output::ColorStream &print(output::ColorStream &stream, const Function &function, PrintContext &printContext, bool)
 {
-	return (stream << constant.name);
-}
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
-inline output::ColorStream &print(output::ColorStream &stream, const Function &function, PrintContext &printContext)
-{
-	stream << function.name;
+	stream << function.declaration->name;

 	if (function.arguments.empty())
 		return stream;
@@ -183,7 +184,7 @@ inline output::ColorStream &print(output::ColorStream &stream, const Function &f
 		print(stream, *i, printContext);
 	}

-	if (function.name.empty() && function.arguments.size() == 1)
+	if (function.declaration->name.empty() && function.arguments.size() == 1)
 		stream << ",";

 	stream << ")";
@@ -193,7 +194,7 @@ inline output::ColorStream &print(output::ColorStream &stream, const Function &f

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const In &in, PrintContext &printContext)
+inline output::ColorStream &print(output::ColorStream &stream, const In &in, PrintContext &printContext, bool)
 {
 	if (printContext.context.parenthesisStyle == ParenthesisStyle::Full)
 		stream << "(";
@@ -210,14 +211,14 @@ inline output::ColorStream &print(output::ColorStream &stream, const In &in, Pri

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const Integer &integer, PrintContext &)
+inline output::ColorStream &print(output::ColorStream &stream, const Integer &integer, PrintContext &, bool)
 {
 	return (stream << output::Number<int>(integer.value));
 }

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const Interval &interval, PrintContext &printContext)
+inline output::ColorStream &print(output::ColorStream &stream, const Interval &interval, PrintContext &printContext, bool)
 {
 	if (printContext.context.parenthesisStyle == ParenthesisStyle::Full)
 		stream << "(";
@@ -234,9 +235,9 @@ inline output::ColorStream &print(output::ColorStream &stream, const Interval &i

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const Predicate &predicate, PrintContext &printContext)
+inline output::ColorStream &print(output::ColorStream &stream, const Predicate &predicate, PrintContext &printContext, bool)
 {
-	stream << predicate.name;
+	stream << predicate.declaration->name;

 	if (predicate.arguments.empty())
 		return stream;
@@ -258,7 +259,14 @@ inline output::ColorStream &print(output::ColorStream &stream, const Predicate &

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const SpecialInteger &specialInteger, PrintContext &)
+inline output::ColorStream &print(output::ColorStream &stream, const PredicateDeclaration &predicateDeclaration, PrintContext &, bool)
+{
+	return (stream << predicateDeclaration.name << "/" << predicateDeclaration.arity());
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+inline output::ColorStream &print(output::ColorStream &stream, const SpecialInteger &specialInteger, PrintContext &, bool)
 {
 	switch (specialInteger.type)
 	{
@@ -273,14 +281,37 @@ inline output::ColorStream &print(output::ColorStream &stream, const SpecialInte

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const String &string, PrintContext &)
+inline output::ColorStream &print(output::ColorStream &stream, const String &string, PrintContext &, bool)
 {
 	return (stream << output::String(string.text.c_str()));
 }

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const Variable &variable, PrintContext &printContext)
+inline output::ColorStream &print(output::ColorStream &stream, const UnaryOperation &unaryOperation, PrintContext &printContext, bool)
+{
+	switch (unaryOperation.operator_)
+	{
+		case UnaryOperation::Operator::Absolute:
+			stream << "|";
+			break;
+	}
+
+	print(stream, unaryOperation.argument, printContext, true);
+
+	switch (unaryOperation.operator_)
+	{
+		case UnaryOperation::Operator::Absolute:
+			stream << "|";
+			break;
+	}
+
+	return stream;
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+inline output::ColorStream &print(output::ColorStream &stream, const Variable &variable, PrintContext &printContext, bool)
 {
 	assert(variable.declaration != nullptr);

@@ -289,10 +320,26 @@ inline output::ColorStream &print(output::ColorStream &stream, const Variable &v

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const VariableDeclaration &variableDeclaration, PrintContext &printContext)
+inline output::ColorStream &print(output::ColorStream &stream, const VariableDeclaration &variableDeclaration, PrintContext &printContext, bool)
 {
+	const auto domainSuffix =
+		[&variableDeclaration]()
+		{
+			switch (variableDeclaration.domain)
+			{
+				case Domain::Unknown:
+					return "";
+				case Domain::General:
+					return "g";
+				case Domain::Integer:
+					return "i";
+			}
+
+			return "";
+		};
+
 	const auto printVariableDeclaration =
-		[&stream, &variableDeclaration](const auto *prefix, auto &variableIDs) -> output::ColorStream &
+		[&](const auto *prefix, auto &variableIDs) -> output::ColorStream &
 		{
 			auto matchingVariableID = variableIDs.find(&variableDeclaration);

@@ -303,7 +350,7 @@ inline output::ColorStream &print(output::ColorStream &stream, const VariableDec
 				matchingVariableID = emplaceResult.first;
 			}

-			const auto variableName = std::string(prefix) + std::to_string(matchingVariableID->second);
+			const auto variableName = std::string(prefix) + std::to_string(matchingVariableID->second) + domainSuffix();

 			return (stream << output::Variable(variableName.c_str()));
 		};
@@ -325,9 +372,10 @@ inline output::ColorStream &print(output::ColorStream &stream, const VariableDec
 // Expressions
 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const And &and_, PrintContext &printContext)
+inline output::ColorStream &print(output::ColorStream &stream, const And &and_, PrintContext &printContext, bool omitParentheses)
 {
-	stream << "(";
+	if (!omitParentheses || printContext.context.parenthesisStyle == ParenthesisStyle::Full)
+		stream << "(";

 	for (auto i = and_.arguments.cbegin(); i != and_.arguments.cend(); i++)
 	{
@@ -337,27 +385,32 @@ inline output::ColorStream &print(output::ColorStream &stream, const And &and_,
 		print(stream, *i, printContext);
 	}

-	stream << ")";
+	if (!omitParentheses || printContext.context.parenthesisStyle == ParenthesisStyle::Full)
+		stream << ")";

 	return stream;
 }

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const Biconditional &biconditional, PrintContext &printContext)
+inline output::ColorStream &print(output::ColorStream &stream, const Biconditional &biconditional, PrintContext &printContext, bool omitParentheses)
 {
-	stream << "(";
+	if (!omitParentheses || printContext.context.parenthesisStyle == ParenthesisStyle::Full)
+		stream << "(";
+
 	print(stream, biconditional.left, printContext);
 	stream << " <-> ";
 	print(stream, biconditional.right, printContext);
-	stream << ")";
+
+	if (!omitParentheses || printContext.context.parenthesisStyle == ParenthesisStyle::Full)
+		stream << ")";

 	return stream;
 }

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const Exists &exists, PrintContext &printContext)
+inline output::ColorStream &print(output::ColorStream &stream, const Exists &exists, PrintContext &printContext, bool)
 {
 	if (printContext.context.parenthesisStyle == ParenthesisStyle::Full)
 		stream << "(";
@@ -383,7 +436,7 @@ inline output::ColorStream &print(output::ColorStream &stream, const Exists &exi

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const ForAll &forAll, PrintContext &printContext)
+inline output::ColorStream &print(output::ColorStream &stream, const ForAll &forAll, PrintContext &printContext, bool)
 {
 	if (printContext.context.parenthesisStyle == ParenthesisStyle::Full)
 		stream << "(";
@@ -409,20 +462,24 @@ inline output::ColorStream &print(output::ColorStream &stream, const ForAll &for

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const Implies &implies, PrintContext &printContext)
+inline output::ColorStream &print(output::ColorStream &stream, const Implies &implies, PrintContext &printContext, bool omitParentheses)
 {
-	stream << "(";
+	if (!omitParentheses || printContext.context.parenthesisStyle == ParenthesisStyle::Full)
+		stream << "(";
+
 	print(stream, implies.antecedent, printContext);
 	stream << " -> ";
 	print(stream, implies.consequent, printContext);
-	stream << ")";
+
+	if (!omitParentheses || printContext.context.parenthesisStyle == ParenthesisStyle::Full)
+		stream << ")";

 	return stream;
 }

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const Not &not_, PrintContext &printContext)
+inline output::ColorStream &print(output::ColorStream &stream, const Not &not_, PrintContext &printContext, bool)
 {
 	if (printContext.context.parenthesisStyle == ParenthesisStyle::Full)
 		stream << "(";
@@ -438,9 +495,10 @@ inline output::ColorStream &print(output::ColorStream &stream, const Not &not_,

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const Or &or_, PrintContext &printContext)
+inline output::ColorStream &print(output::ColorStream &stream, const Or &or_, PrintContext &printContext, bool omitParentheses)
 {
-	stream << "(";
+	if (!omitParentheses || printContext.context.parenthesisStyle == ParenthesisStyle::Full)
+		stream << "(";

 	for (auto i = or_.arguments.cbegin(); i != or_.arguments.cend(); i++)
 	{
@@ -450,7 +508,8 @@ inline output::ColorStream &print(output::ColorStream &stream, const Or &or_, Pr
 		print(stream, *i, printContext);
 	}

-	stream << ")";
+	if (!omitParentheses || printContext.context.parenthesisStyle == ParenthesisStyle::Full)
+		stream << ")";

 	return stream;
 }
@@ -463,24 +522,24 @@ template<class Variant>
 struct VariantPrintVisitor
 {
 	template<class T>
-	output::ColorStream &visit(const T &x, output::ColorStream &stream, PrintContext &printContext)
+	output::ColorStream &visit(const T &x, output::ColorStream &stream, PrintContext &printContext, bool omitParentheses)
 	{
-		return print(stream, x, printContext);
+		return print(stream, x, printContext, omitParentheses);
 	}
 };

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const Formula &formula, PrintContext &printContext)
+inline output::ColorStream &print(output::ColorStream &stream, const Formula &formula, PrintContext &printContext, bool omitParentheses)
 {
-	return formula.accept(VariantPrintVisitor<Formula>(), stream, printContext);
+	return formula.accept(VariantPrintVisitor<Formula>(), stream, printContext, omitParentheses);
 }

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-inline output::ColorStream &print(output::ColorStream &stream, const Term &term, PrintContext &printContext)
+inline output::ColorStream &print(output::ColorStream &stream, const Term &term, PrintContext &printContext, bool omitParentheses)
 {
-	return term.accept(VariantPrintVisitor<Term>(), stream, printContext);
+	return term.accept(VariantPrintVisitor<Term>(), stream, printContext, omitParentheses);
 }

 ////////////////////////////////////////////////////////////////////////////////////////////////////
--- a/lib/catch
+++ b/lib/catch
--- a/lib/clingo
+++ b/lib/clingo
--- a/lib/cxxopts
+++ b/lib/cxxopts
--- a/src/anthem/ASTCopy.cpp
+++ b/src/anthem/ASTCopy.cpp
@@ -103,16 +103,9 @@ Comparison prepareCopy(const Comparison &other)

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-Constant prepareCopy(const Constant &other)
-{
-	return Constant(std::string(other.name));
-}
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
 Function prepareCopy(const Function &other)
 {
-	return Function(std::string(other.name), prepareCopy(other.arguments));
+	return Function(other.declaration, prepareCopy(other.arguments));
 }

 ////////////////////////////////////////////////////////////////////////////////////////////////////
@@ -140,7 +133,7 @@ Interval prepareCopy(const Interval &other)

 Predicate prepareCopy(const Predicate &other)
 {
-	return Predicate(std::string(other.name), prepareCopy(other.arguments));
+	return Predicate(other.declaration, prepareCopy(other.arguments));
 }

 ////////////////////////////////////////////////////////////////////////////////////////////////////
@@ -159,6 +152,13 @@ String prepareCopy(const String &other)

 ////////////////////////////////////////////////////////////////////////////////////////////////////

+UnaryOperation prepareCopy(const UnaryOperation &other)
+{
+	return UnaryOperation(other.operator_, prepareCopy(other.argument));
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
 Variable prepareCopy(const Variable &other)
 {
 	return Variable(other.declaration);
@@ -286,11 +286,6 @@ struct FixDanglingVariablesInTermVisitor
 	{
 	}

-	template <class... Arguments>
-	void visit(Constant &, Arguments &&...)
-	{
-	}
-
 	template <class... Arguments>
 	void visit(Function &function, Arguments &&... arguments)
 	{
@@ -320,6 +315,12 @@ struct FixDanglingVariablesInTermVisitor
 	{
 	}

+	template <class... Arguments>
+	void visit(UnaryOperation &unaryOperation, Arguments &&... arguments)
+	{
+		unaryOperation.argument.accept(*this, std::forward<Arguments>(arguments)...);
+	}
+
 	void visit(Variable &variable, ScopedFormula &scopedFormula, VariableStack &variableStack,
 		std::map<VariableDeclaration *, VariableDeclaration *> &replacements)
 	{
--- a/src/anthem/ASTUtils.cpp
+++ b/src/anthem/ASTUtils.cpp
@@ -27,7 +27,7 @@ void VariableStack::pop()

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-std::experimental::optional<VariableDeclaration *> VariableStack::findUserVariableDeclaration(const char *variableName) const
+std::optional<VariableDeclaration *> VariableStack::findUserVariableDeclaration(const char *variableName) const
 {
 	const auto variableDeclarationMatches =
 		[&variableName](const auto &variableDeclaration)
@@ -45,7 +45,7 @@ std::experimental::optional<VariableDeclaration *> VariableStack::findUserVariab
 			return matchingVariableDeclaration->get();
 	}

-	return std::experimental::nullopt;
+	return std::nullopt;
 }

 ////////////////////////////////////////////////////////////////////////////////////////////////////
@@ -59,7 +59,7 @@ bool VariableStack::contains(const VariableDeclaration &variableDeclaration) con
 		};

 	const auto layerContainsVariableDeclaration =
-		[&variableDeclaration, &variableDeclarationMatches](const auto &layer)
+		[&variableDeclarationMatches](const auto &layer)
 		{
 			return (std::find_if(layer->cbegin(), layer->cend(), variableDeclarationMatches) != layer->cend());
 		};
@@ -150,10 +150,6 @@ struct CollectFreeVariablesVisitor
 		binaryOperation.right.accept(*this, variableStack, freeVariables);
 	}

-	void visit(Constant &, VariableStack &, std::vector<VariableDeclaration *> &)
-	{
-	}
-
 	void visit(Function &function, VariableStack &variableStack, std::vector<VariableDeclaration *> &freeVariables)
 	{
 		for (auto &argument : function.arguments)
@@ -178,6 +174,11 @@ struct CollectFreeVariablesVisitor
 	{
 	}

+	void visit(UnaryOperation &unaryOperation, VariableStack &variableStack, std::vector<VariableDeclaration *> &freeVariables)
+	{
+		unaryOperation.argument.accept(*this, variableStack, freeVariables);
+	}
+
 	void visit(Variable &variable, VariableStack &variableStack, std::vector<VariableDeclaration *> &freeVariables)
 	{
 		if (variableStack.contains(*variable.declaration))
@@ -192,61 +193,5 @@ struct CollectFreeVariablesVisitor

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-struct CollectPredicateSignaturesVisitor : public RecursiveFormulaVisitor<CollectPredicateSignaturesVisitor>
-{
-	static void accept(const Predicate &predicate, const Formula &, std::vector<PredicateSignature> &predicateSignatures)
-	{
-		const auto predicateSignatureMatches =
-			[&predicate](const auto &predicateSignature)
-			{
-				return matches(predicate, predicateSignature);
-			};
-
-		if (std::find_if(predicateSignatures.cbegin(), predicateSignatures.cend(), predicateSignatureMatches) != predicateSignatures.cend())
-			return;
-
-		// TODO: avoid copies
-		predicateSignatures.emplace_back(std::string(predicate.name), predicate.arity());
-	}
-
-	// Ignore all other types of expressions
-	template<class T>
-	static void accept(const T &, const Formula &, std::vector<PredicateSignature> &)
-	{
-	}
-};
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
-bool matches(const Predicate &lhs, const Predicate &rhs)
-{
-	return (lhs.name == rhs.name && lhs.arity() == rhs.arity());
-}
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
-bool matches(const Predicate &predicate, const PredicateSignature &signature)
-{
-	return (predicate.name == signature.name && predicate.arity() == signature.arity);
-}
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
-bool matches(const PredicateSignature &lhs, const PredicateSignature &rhs)
-{
-	return (lhs.name == rhs.name && lhs.arity == rhs.arity);
-}
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
-// TODO: remove const_cast
-void collectPredicateSignatures(const Formula &formula, std::vector<PredicateSignature> &predicateSignatures)
-{
-	auto &formulaMutable = const_cast<Formula &>(formula);
-	formulaMutable.accept(CollectPredicateSignaturesVisitor(), formulaMutable, predicateSignatures);
-}
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
 }
 }
--- a/src/anthem/Completion.cpp
+++ b/src/anthem/Completion.cpp
@@ -35,7 +35,7 @@ ast::Formula buildCompletedFormulaDisjunction(const ast::Predicate &predicate, c

 		auto &otherPredicate = implies.consequent.get<ast::Predicate>();

-		if (!ast::matches(predicate, otherPredicate))
+		if (predicate.declaration != otherPredicate.declaration)
 			continue;

 		assert(otherPredicate.arguments.size() == parameters.size());
@@ -100,22 +100,22 @@ ast::Formula buildCompletedFormulaQuantified(ast::Predicate &&predicate, ast::Fo

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-ast::Formula completePredicate(const ast::PredicateSignature &predicateSignature, std::vector<ast::ScopedFormula> &scopedFormulas)
+ast::Formula completePredicate(ast::PredicateDeclaration &predicateDeclaration, std::vector<ast::ScopedFormula> &scopedFormulas)
 {
 	// Create new set of parameters for the completed definition for the predicate
 	ast::VariableDeclarationPointers parameters;
-	parameters.reserve(predicateSignature.arity);
+	parameters.reserve(predicateDeclaration.arity());

 	std::vector<ast::Term> arguments;
-	arguments.reserve(predicateSignature.arity);
+	arguments.reserve(predicateDeclaration.arity());

-	for (size_t i = 0; i < predicateSignature.arity; i++)
+	for (size_t i = 0; i < predicateDeclaration.arity(); i++)
 	{
 		parameters.emplace_back(std::make_unique<ast::VariableDeclaration>(ast::VariableDeclaration::Type::Head));
 		arguments.emplace_back(ast::Term::make<ast::Variable>(parameters.back().get()));
 	}

-	ast::Predicate predicateCopy(std::string(predicateSignature.name), std::move(arguments));
+	ast::Predicate predicateCopy(&predicateDeclaration, std::move(arguments));

 	auto completedFormulaDisjunction = buildCompletedFormulaDisjunction(predicateCopy, parameters, scopedFormulas);
 	auto completedFormulaQuantified = buildCompletedFormulaQuantified(std::move(predicateCopy), std::move(completedFormulaDisjunction));
@@ -161,28 +161,27 @@ std::vector<ast::Formula> complete(std::vector<ast::ScopedFormula> &&scopedFormu
 			throw CompletionException("cannot perform completion, only single predicates and Booleans supported as formula consequent currently");
 	}

-	std::vector<ast::PredicateSignature> predicateSignatures;
-
-	// Get a list of all predicates
-	for (const auto &scopedFormula : scopedFormulas)
-		ast::collectPredicateSignatures(scopedFormula.formula, predicateSignatures);
-
-	std::sort(predicateSignatures.begin(), predicateSignatures.end(),
+	std::sort(context.predicateDeclarations.begin(), context.predicateDeclarations.end(),
 		[](const auto &lhs, const auto &rhs)
 		{
-			const auto order = std::strcmp(lhs.name.c_str(), rhs.name.c_str());
+			const auto order = std::strcmp(lhs->name.c_str(), rhs->name.c_str());

 			if (order != 0)
 				return (order < 0);

-			return lhs.arity < rhs.arity;
+			return lhs->arity() < rhs->arity();
 		});

 	std::vector<ast::Formula> completedFormulas;

 	// Complete predicates
-	for (const auto &predicateSignature : predicateSignatures)
-		completedFormulas.emplace_back(completePredicate(predicateSignature, scopedFormulas));
+	for (auto &predicateDeclaration : context.predicateDeclarations)
+	{
+		if (!predicateDeclaration->isUsed || predicateDeclaration->isExternal)
+			continue;
+
+		completedFormulas.emplace_back(completePredicate(*predicateDeclaration, scopedFormulas));
+	}

 	// Complete integrity constraints
 	for (auto &scopedFormula : scopedFormulas)
@@ -202,7 +201,7 @@ std::vector<ast::Formula> complete(std::vector<ast::ScopedFormula> &&scopedFormu
 	}

 	// Eliminate all predicates that should not be visible in the output
-	eliminateHiddenPredicates(predicateSignatures, completedFormulas, context);
+	eliminateHiddenPredicates(completedFormulas, context);

 	return completedFormulas;
 }
--- a/src/anthem/HiddenPredicateElimination.cpp
+++ b/src/anthem/HiddenPredicateElimination.cpp
@@ -78,7 +78,7 @@ struct ReplacePredicateInFormulaVisitor : public ast::RecursiveFormulaVisitor<Re
 {
 	static void accept(ast::Predicate &predicate, ast::Formula &formula, const PredicateReplacement &predicateReplacement)
 	{
-		if (!ast::matches(predicate, predicateReplacement.predicate))
+		if (predicate.declaration != predicateReplacement.predicate.declaration)
 			return;

 		auto formulaReplacement = ast::prepareCopy(predicateReplacement.replacement);
@@ -109,15 +109,15 @@ struct ReplacePredicateInFormulaVisitor : public ast::RecursiveFormulaVisitor<Re
 // Detect whether a formula contains a circular dependency on a given predicate
 struct DetectCircularDependcyVisitor : public ast::RecursiveFormulaVisitor<DetectCircularDependcyVisitor>
 {
-	static void accept(ast::Predicate &predicate, ast::Formula &, const ast::PredicateSignature &predicateSignature, bool &hasCircularDependency)
+	static void accept(ast::Predicate &predicate, ast::Formula &, const ast::PredicateDeclaration &predicateDeclaration, bool &hasCircularDependency)
 	{
-		if (ast::matches(predicate, predicateSignature))
+		if (predicate.declaration == &predicateDeclaration)
 			hasCircularDependency = true;
 	}

 	// Ignore all other types of expressions
 	template<class T>
-	static void accept(T &, ast::Formula &, const ast::PredicateSignature &, bool &)
+	static void accept(T &, ast::Formula &, const ast::PredicateDeclaration &, bool &)
 	{
 	}
 };
@@ -125,12 +125,12 @@ struct DetectCircularDependcyVisitor : public ast::RecursiveFormulaVisitor<Detec
 ////////////////////////////////////////////////////////////////////////////////////////////////////

 // Finds the replacement for predicates of the form “p(X1, ..., Xn) <-> q(X1, ..., Xn)”
-PredicateReplacement findReplacement(const ast::PredicateSignature &predicateSignature, const ast::Predicate &predicate)
+PredicateReplacement findReplacement(const ast::PredicateDeclaration &predicateDeclaration, const ast::Predicate &predicate)
 {
 	// Declare variable used, only used in debug mode
-	(void)(predicateSignature);
+	(void)(predicateDeclaration);

-	assert(ast::matches(predicate, predicateSignature));
+	assert(predicate.declaration == &predicateDeclaration);

 	// Replace with “#true”
 	return {predicate, ast::Formula::make<ast::Boolean>(true)};
@@ -139,13 +139,13 @@ PredicateReplacement findReplacement(const ast::PredicateSignature &predicateSig
 ////////////////////////////////////////////////////////////////////////////////////////////////////

 // Finds the replacement for predicates of the form “p(X1, ..., Xn) <-> not q(X1, ..., Xn)”
-PredicateReplacement findReplacement(const ast::PredicateSignature &predicateSignature, const ast::Not &not_)
+PredicateReplacement findReplacement(const ast::PredicateDeclaration &predicateDeclaration, const ast::Not &not_)
 {
 	// Declare variable used, only used in debug mode
-	(void)(predicateSignature);
+	(void)(predicateDeclaration);

 	assert(not_.argument.is<ast::Predicate>());
-	assert(ast::matches(not_.argument.get<ast::Predicate>(), predicateSignature));
+	assert(not_.argument.get<ast::Predicate>().declaration == &predicateDeclaration);

 	// Replace with “#false”
 	return {not_.argument.get<ast::Predicate>(), ast::Formula::make<ast::Boolean>(false)};
@@ -154,13 +154,13 @@ PredicateReplacement findReplacement(const ast::PredicateSignature &predicateSig
 ////////////////////////////////////////////////////////////////////////////////////////////////////

 // Finds the replacement for predicates of the form “forall X1, ..., Xn (p(X1, ..., Xn) <-> ...)”
-PredicateReplacement findReplacement(const ast::PredicateSignature &predicateSignature, const ast::Biconditional &biconditional)
+PredicateReplacement findReplacement(const ast::PredicateDeclaration &predicateDeclaration, const ast::Biconditional &biconditional)
 {
 	// Declare variable used, only used in debug mode
-	(void)(predicateSignature);
+	(void)(predicateDeclaration);

 	assert(biconditional.left.is<ast::Predicate>());
-	assert(ast::matches(biconditional.left.get<ast::Predicate>(), predicateSignature));
+	assert(biconditional.left.get<ast::Predicate>().declaration == &predicateDeclaration);

 	// TODO: avoid copy
 	return {biconditional.left.get<ast::Predicate>(), ast::prepareCopy(biconditional.right)};
@@ -169,78 +169,65 @@ PredicateReplacement findReplacement(const ast::PredicateSignature &predicateSig
 ////////////////////////////////////////////////////////////////////////////////////////////////////

 // Finds a replacement for a predicate that should be hidden
-PredicateReplacement findReplacement(const ast::PredicateSignature &predicateSignature, const ast::Formula &completedPredicateDefinition)
+PredicateReplacement findReplacement(const ast::PredicateDeclaration &predicateDeclaration, const ast::Formula &completedPredicateDefinition)
 {
 	// TODO: refactor
 	if (completedPredicateDefinition.is<ast::ForAll>())
-		return findReplacement(predicateSignature, completedPredicateDefinition.get<ast::ForAll>().argument);
+		return findReplacement(predicateDeclaration, completedPredicateDefinition.get<ast::ForAll>().argument);
 	else if (completedPredicateDefinition.is<ast::Biconditional>())
-		return findReplacement(predicateSignature, completedPredicateDefinition.get<ast::Biconditional>());
+		return findReplacement(predicateDeclaration, completedPredicateDefinition.get<ast::Biconditional>());
 	else if (completedPredicateDefinition.is<ast::Predicate>())
-		return findReplacement(predicateSignature, completedPredicateDefinition.get<ast::Predicate>());
+		return findReplacement(predicateDeclaration, completedPredicateDefinition.get<ast::Predicate>());
 	else if (completedPredicateDefinition.is<ast::Not>())
-		return findReplacement(predicateSignature, completedPredicateDefinition.get<ast::Not>());
+		return findReplacement(predicateDeclaration, completedPredicateDefinition.get<ast::Not>());

-	throw CompletionException("unsupported completed definition for predicate “" + predicateSignature.name + "/" +  std::to_string(predicateSignature.arity) + "” for hiding predicates");
+	throw CompletionException("unsupported completed definition for predicate “" + predicateDeclaration.name + "/" +  std::to_string(predicateDeclaration.arity()) + "” for hiding predicates");
 }

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-void eliminateHiddenPredicates(const std::vector<ast::PredicateSignature> &predicateSignatures, std::vector<ast::Formula> &completedFormulas, Context &context)
+void eliminateHiddenPredicates(std::vector<ast::Formula> &completedFormulas, Context &context)
 {
-	if (!context.visiblePredicateSignatures)
+	if (context.defaultPredicateVisibility == ast::PredicateDeclaration::Visibility::Visible)
 	{
 		context.logger.log(output::Priority::Debug) << "no predicates to be eliminated";
 		return;
 	}

-	const auto &visiblePredicateSignatures = context.visiblePredicateSignatures.value();
+	assert(context.defaultPredicateVisibility == ast::PredicateDeclaration::Visibility::Hidden);

-	// Check for undeclared predicates that are requested to be shown
-	for (const auto &visiblePredicateSignature : visiblePredicateSignatures)
-	{
-		const auto matchesPredicateSignature =
-			[&](const auto &predicateSignature)
-			{
-				return ast::matches(predicateSignature, visiblePredicateSignature);
-			};
-
-		const auto matchingPredicateSignature =
-			std::find_if(predicateSignatures.cbegin(), predicateSignatures.cend(), matchesPredicateSignature);
-
-		if (matchingPredicateSignature == predicateSignatures.cend())
-			context.logger.log(output::Priority::Warning) << "cannot show undeclared predicate “" << visiblePredicateSignature.name << "/" << visiblePredicateSignature.arity <<"”";
-	}
+	// TODO: get rid of index-wise matching of completed formulas and predicate declarations
+	size_t i = -1;

 	// Replace all occurrences of hidden predicates
-	for (size_t i = 0; i < predicateSignatures.size(); i++)
+	for (auto &predicateDeclaration : context.predicateDeclarations)
 	{
-		auto &predicateSignature = predicateSignatures[i];
-
-		const auto matchesVisiblePredicateSignature =
-			[&](const auto &visiblePredicateSignature)
-			{
-				return ast::matches(predicateSignature, visiblePredicateSignature);
-			};
-
-		const auto matchingPredicateSignature =
-			std::find_if(visiblePredicateSignatures.cbegin(), visiblePredicateSignatures.cend(), matchesVisiblePredicateSignature);
-
-		// If the predicate ought to be visible, don’t eliminate it
-		if (matchingPredicateSignature != visiblePredicateSignatures.cend())
+		// Check that the predicate is used and not declared #external
+		if (!predicateDeclaration->isUsed || predicateDeclaration->isExternal)
 			continue;

-		context.logger.log(output::Priority::Debug) << "eliminating “" << predicateSignature.name << "/" << predicateSignature.arity << "”";
+		i++;
+
+		const auto isPredicateVisible =
+			(predicateDeclaration->visibility == ast::PredicateDeclaration::Visibility::Visible)
+			|| (predicateDeclaration->visibility == ast::PredicateDeclaration::Visibility::Default
+				&& context.defaultPredicateVisibility == ast::PredicateDeclaration::Visibility::Visible);
+
+		// If the predicate ought to be visible, don’t eliminate it
+		if (isPredicateVisible)
+			continue;
+
+		context.logger.log(output::Priority::Debug) << "eliminating “" << predicateDeclaration->name << "/" << predicateDeclaration->arity() << "”";

 		const auto &completedPredicateDefinition = completedFormulas[i];
-		auto replacement = findReplacement(predicateSignature, completedPredicateDefinition);
+		auto replacement = findReplacement(*predicateDeclaration, completedPredicateDefinition);

 		bool hasCircularDependency = false;
-		replacement.replacement.accept(DetectCircularDependcyVisitor(), replacement.replacement, predicateSignature, hasCircularDependency);
+		replacement.replacement.accept(DetectCircularDependcyVisitor(), replacement.replacement, *predicateDeclaration, hasCircularDependency);

 		if (hasCircularDependency)
 		{
-			context.logger.log(output::Priority::Warning) << "cannot hide predicate “" << predicateSignature.name << "/" << predicateSignature.arity << "” due to circular dependency";
+			context.logger.log(output::Priority::Warning) << "cannot hide predicate “" << predicateDeclaration->name << "/" << predicateDeclaration->arity() << "” due to circular dependency";
 			continue;
 		}

--- a/src/anthem/IntegerVariableDetection.cpp
+++ b/src/anthem/IntegerVariableDetection.cpp
@@ -0,0 +1,655 @@
+#include <anthem/IntegerVariableDetection.h>
+
+#include <anthem/ASTCopy.h>
+#include <anthem/ASTUtils.h>
+#include <anthem/ASTVisitors.h>
+#include <anthem/Exception.h>
+#include <anthem/Simplification.h>
+#include <anthem/output/AST.h>
+
+namespace anthem
+{
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+//
+// IntegerVariableDetection
+//
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+using VariableDomainMap = std::map<const ast::VariableDeclaration *, ast::Domain>;
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+ast::Domain domain(const ast::Variable &variable, VariableDomainMap &variableDomainMap)
+{
+	if (variable.declaration->domain != ast::Domain::Unknown)
+		return variable.declaration->domain;
+
+	const auto match = variableDomainMap.find(variable.declaration);
+
+	if (match == variableDomainMap.end())
+		return ast::Domain::Unknown;
+
+	return match->second;
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+void clearVariableDomainMap(VariableDomainMap &variableDomainMap)
+{
+	for (auto &variableDeclaration : variableDomainMap)
+		variableDeclaration.second = ast::Domain::Unknown;
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+enum class OperationResult
+{
+	Unchanged,
+	Changed,
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+enum class EvaluationResult
+{
+	True,
+	False,
+	Unknown,
+	Error,
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+EvaluationResult isArithmetic(const ast::Term &term, VariableDomainMap &variableDomainMap);
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct IsTermArithmeticVisitor
+{
+	static EvaluationResult visit(const ast::BinaryOperation &binaryOperation, VariableDomainMap &variableDomainMap)
+	{
+		const auto isLeftArithemtic = isArithmetic(binaryOperation.left, variableDomainMap);
+		const auto isRightArithmetic = isArithmetic(binaryOperation.right, variableDomainMap);
+
+		if (isLeftArithemtic == EvaluationResult::Error || isRightArithmetic == EvaluationResult::Error)
+			return EvaluationResult::Error;
+
+		if (isLeftArithemtic == EvaluationResult::False || isRightArithmetic == EvaluationResult::False)
+			return EvaluationResult::Error;
+
+		if (isLeftArithemtic == EvaluationResult::Unknown || isRightArithmetic == EvaluationResult::Unknown)
+			return EvaluationResult::Unknown;
+
+		return EvaluationResult::True;
+	}
+
+	static EvaluationResult visit(const ast::Boolean &, VariableDomainMap &)
+	{
+		return EvaluationResult::False;
+	}
+
+	static EvaluationResult visit(const ast::Function &function, VariableDomainMap &)
+	{
+		switch (function.declaration->domain)
+		{
+			case ast::Domain::General:
+				return EvaluationResult::False;
+			case ast::Domain::Integer:
+				return EvaluationResult::True;
+			case ast::Domain::Unknown:
+				return EvaluationResult::Unknown;
+		}
+
+		return EvaluationResult::Unknown;
+	}
+
+	static EvaluationResult visit(const ast::Integer &, VariableDomainMap &)
+	{
+		return EvaluationResult::True;
+	}
+
+	static EvaluationResult visit(const ast::Interval &interval, VariableDomainMap &variableDomainMap)
+	{
+		const auto isFromArithmetic = isArithmetic(interval.from, variableDomainMap);
+		const auto isToArithmetic = isArithmetic(interval.to, variableDomainMap);
+
+		if (isFromArithmetic == EvaluationResult::Error || isToArithmetic == EvaluationResult::Error)
+			return EvaluationResult::Error;
+
+		if (isFromArithmetic == EvaluationResult::False || isToArithmetic == EvaluationResult::False)
+			return EvaluationResult::Error;
+
+		if (isFromArithmetic == EvaluationResult::Unknown || isToArithmetic == EvaluationResult::Unknown)
+			return EvaluationResult::Unknown;
+
+		return EvaluationResult::True;
+	}
+
+	static EvaluationResult visit(const ast::SpecialInteger &, VariableDomainMap &)
+	{
+		return EvaluationResult::False;
+	}
+
+	static EvaluationResult visit(const ast::String &, VariableDomainMap &)
+	{
+		return EvaluationResult::False;
+	}
+
+	static EvaluationResult visit(const ast::UnaryOperation &unaryOperation, VariableDomainMap &variableDomainMap)
+	{
+		const auto isArgumentArithmetic = isArithmetic(unaryOperation.argument, variableDomainMap);
+
+		switch (unaryOperation.operator_)
+		{
+			case ast::UnaryOperation::Operator::Absolute:
+				return (isArgumentArithmetic == EvaluationResult::False ? EvaluationResult::Error : isArgumentArithmetic);
+		}
+
+		return EvaluationResult::Unknown;
+	}
+
+	static EvaluationResult visit(const ast::Variable &variable, VariableDomainMap &variableDomainMap)
+	{
+		switch (domain(variable, variableDomainMap))
+		{
+			case ast::Domain::General:
+				return EvaluationResult::False;
+			case ast::Domain::Integer:
+				return EvaluationResult::True;
+			case ast::Domain::Unknown:
+				return EvaluationResult::Unknown;
+		}
+
+		return EvaluationResult::Unknown;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+EvaluationResult isArithmetic(const ast::Term &term, VariableDomainMap &variableDomainMap)
+{
+	return term.accept(IsTermArithmeticVisitor(), variableDomainMap);
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+EvaluationResult evaluate(const ast::Formula &formula, VariableDomainMap &variableDomainMap);
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct EvaluateFormulaVisitor
+{
+	static EvaluationResult visit(const ast::And &and_, VariableDomainMap &variableDomainMap)
+	{
+		bool someFalse = false;
+		bool someUnknown = false;
+
+		for (const auto &argument : and_.arguments)
+		{
+			const auto result = evaluate(argument, variableDomainMap);
+
+			switch (result)
+			{
+				case EvaluationResult::Error:
+					return EvaluationResult::Error;
+				case EvaluationResult::True:
+					break;
+				case EvaluationResult::False:
+					someFalse = true;
+					break;
+				case EvaluationResult::Unknown:
+					someUnknown = true;
+					break;
+			}
+		}
+
+		if (someFalse)
+			return EvaluationResult::False;
+
+		if (someUnknown)
+			return EvaluationResult::Unknown;
+
+		return EvaluationResult::True;
+	}
+
+	static EvaluationResult visit(const ast::Biconditional &biconditional, VariableDomainMap &variableDomainMap)
+	{
+		const auto leftResult = evaluate(biconditional.left, variableDomainMap);
+		const auto rightResult = evaluate(biconditional.right, variableDomainMap);
+
+		if (leftResult == EvaluationResult::Error || rightResult == EvaluationResult::Error)
+			return EvaluationResult::Error;
+
+		if (leftResult == EvaluationResult::Unknown || rightResult == EvaluationResult::Unknown)
+			return EvaluationResult::Unknown;
+
+		return (leftResult == rightResult ? EvaluationResult::True : EvaluationResult::False);
+	}
+
+	static EvaluationResult visit(const ast::Boolean &boolean, VariableDomainMap &)
+	{
+		return (boolean.value == true ? EvaluationResult::True : EvaluationResult::False);
+	}
+
+	static EvaluationResult visit(const ast::Comparison &comparison, VariableDomainMap &variableDomainMap)
+	{
+		const auto isLeftArithmetic = isArithmetic(comparison.left, variableDomainMap);
+		const auto isRightArithmetic = isArithmetic(comparison.right, variableDomainMap);
+
+		if (isLeftArithmetic == EvaluationResult::Error || isRightArithmetic == EvaluationResult::Error)
+			return EvaluationResult::Error;
+
+		if (isLeftArithmetic == EvaluationResult::Unknown || isRightArithmetic == EvaluationResult::Unknown)
+			return EvaluationResult::Unknown;
+
+		if (isLeftArithmetic == isRightArithmetic)
+			return EvaluationResult::Unknown;
+
+		// Handle the case where one side is arithmetic but the other one isn’t
+		switch (comparison.operator_)
+		{
+			case ast::Comparison::Operator::Equal:
+				return EvaluationResult::False;
+			case ast::Comparison::Operator::NotEqual:
+				return EvaluationResult::True;
+			default:
+				// TODO: implement more cases
+				return EvaluationResult::Unknown;
+		}
+	}
+
+	static EvaluationResult visit(const ast::Exists &exists, VariableDomainMap &variableDomainMap)
+	{
+		return evaluate(exists.argument, variableDomainMap);
+	}
+
+	static EvaluationResult visit(const ast::ForAll &forAll, VariableDomainMap &variableDomainMap)
+	{
+		return evaluate(forAll.argument, variableDomainMap);
+	}
+
+	static EvaluationResult visit(const ast::Implies &implies, VariableDomainMap &variableDomainMap)
+	{
+		const auto antecedentResult = evaluate(implies.antecedent, variableDomainMap);
+		const auto consequentResult = evaluate(implies.consequent, variableDomainMap);
+
+		if (antecedentResult == EvaluationResult::Error || consequentResult == EvaluationResult::Error)
+			return EvaluationResult::Error;
+
+		if (antecedentResult == EvaluationResult::False)
+			return EvaluationResult::True;
+
+		if (consequentResult == EvaluationResult::True)
+			return EvaluationResult::True;
+
+		if (antecedentResult == EvaluationResult::True && consequentResult == EvaluationResult::False)
+			return EvaluationResult::False;
+
+		return EvaluationResult::Unknown;
+	}
+
+	static EvaluationResult visit(const ast::In &in, VariableDomainMap &variableDomainMap)
+	{
+		const auto isElementArithmetic = isArithmetic(in.element, variableDomainMap);
+		const auto isSetArithmetic = isArithmetic(in.set, variableDomainMap);
+
+		if (isElementArithmetic == EvaluationResult::Error || isSetArithmetic == EvaluationResult::Error)
+			return EvaluationResult::Error;
+
+		if (isElementArithmetic == EvaluationResult::Unknown || isSetArithmetic == EvaluationResult::Unknown)
+			return EvaluationResult::Unknown;
+
+		if (isElementArithmetic == isSetArithmetic)
+			return EvaluationResult::Unknown;
+
+		// If one side is arithmetic, but the other one isn’t, set inclusion is never satisfied
+		return EvaluationResult::False;
+	}
+
+	static EvaluationResult visit(const ast::Not &not_, VariableDomainMap &variableDomainMap)
+	{
+		const auto result = evaluate(not_.argument, variableDomainMap);
+
+		if (result == EvaluationResult::Error || result == EvaluationResult::Unknown)
+			return result;
+
+		return (result == EvaluationResult::True ? EvaluationResult::False : EvaluationResult::True);
+	}
+
+	static EvaluationResult visit(const ast::Or &or_, VariableDomainMap &variableDomainMap)
+	{
+		bool someTrue = false;
+		bool someUnknown = false;
+
+		for (const auto &argument : or_.arguments)
+		{
+			const auto result = evaluate(argument, variableDomainMap);
+
+			switch (result)
+			{
+				case EvaluationResult::Error:
+					return EvaluationResult::Error;
+				case EvaluationResult::True:
+					someTrue = true;
+					break;
+				case EvaluationResult::False:
+					break;
+				case EvaluationResult::Unknown:
+					someUnknown = true;
+					break;
+			}
+		}
+
+		if (someTrue)
+			return EvaluationResult::True;
+
+		if (someUnknown)
+			return EvaluationResult::Unknown;
+
+		return EvaluationResult::False;
+	}
+
+	static EvaluationResult visit(const ast::Predicate &predicate, VariableDomainMap &variableDomainMap)
+	{
+		assert(predicate.arguments.size() == predicate.declaration->arity());
+
+		for (size_t i = 0; i < predicate.arguments.size(); i++)
+		{
+			const auto &argument = predicate.arguments[i];
+			const auto &parameter = predicate.declaration->parameters[i];
+
+			if (parameter.domain != ast::Domain::Integer)
+				continue;
+
+			const auto isArgumentArithmetic = isArithmetic(argument, variableDomainMap);
+
+			if (isArgumentArithmetic == EvaluationResult::Error || isArgumentArithmetic == EvaluationResult::False)
+				return isArgumentArithmetic;
+		}
+
+		return EvaluationResult::Unknown;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+EvaluationResult evaluate(const ast::Formula &formula, VariableDomainMap &variableDomainMap)
+{
+	return formula.accept(EvaluateFormulaVisitor(), variableDomainMap);
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+OperationResult detectIntegerVariables(ast::Formula &formula, ast::Formula &definition, VariableDomainMap &variableDomainMap);
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct DetectIntegerVariablesVisitor
+{
+	static OperationResult visit(ast::And &and_, ast::Formula &definition, VariableDomainMap &variableDomainMap)
+	{
+		auto operationResult = OperationResult::Unchanged;
+
+		for (auto &argument : and_.arguments)
+			if (detectIntegerVariables(argument, definition, variableDomainMap) == OperationResult::Changed)
+				operationResult = OperationResult::Changed;
+
+		return operationResult;
+	}
+
+	static OperationResult visit(ast::Biconditional &biconditional, ast::Formula &definition, VariableDomainMap &variableDomainMap)
+	{
+		auto operationResult = OperationResult::Unchanged;
+
+		if (detectIntegerVariables(biconditional.left, definition, variableDomainMap) == OperationResult::Changed)
+			operationResult = OperationResult::Changed;
+
+		if (detectIntegerVariables(biconditional.right, definition, variableDomainMap) == OperationResult::Changed)
+			operationResult = OperationResult::Changed;
+
+		return operationResult;
+	}
+
+	static OperationResult visit(ast::Boolean &, ast::Formula &, VariableDomainMap &)
+	{
+		return OperationResult::Unchanged;
+	}
+
+	static OperationResult visit(ast::Comparison &, ast::Formula &, VariableDomainMap &)
+	{
+		return OperationResult::Unchanged;
+	}
+
+	static OperationResult visit(ast::Exists &exists, ast::Formula &definition, VariableDomainMap &variableDomainMap)
+	{
+		auto operationResult = OperationResult::Unchanged;
+
+		if (detectIntegerVariables(exists.argument, definition, variableDomainMap) == OperationResult::Changed)
+			operationResult = OperationResult::Changed;
+
+		for (auto &variableDeclaration : exists.variables)
+		{
+			if (variableDeclaration->domain != ast::Domain::Unknown)
+				continue;
+
+			clearVariableDomainMap(variableDomainMap);
+
+			auto argumentResult = evaluate(exists.argument, variableDomainMap);
+			auto definitionResult = evaluate(definition, variableDomainMap);
+
+			if (argumentResult == EvaluationResult::Error || argumentResult == EvaluationResult::False
+			    || definitionResult == EvaluationResult::Error || definitionResult == EvaluationResult::False)
+			{
+				continue;
+			}
+
+			// As a hypothesis, make the parameter’s domain noninteger
+			variableDomainMap[variableDeclaration.get()] = ast::Domain::General;
+
+			argumentResult = evaluate(exists.argument, variableDomainMap);
+			definitionResult = evaluate(definition, variableDomainMap);
+
+			if (argumentResult == EvaluationResult::Error || argumentResult == EvaluationResult::False
+			    || definitionResult == EvaluationResult::Error || definitionResult == EvaluationResult::False)
+			{
+				// If making the variable noninteger leads to a false or erroneous result, it’s proven to be integer
+				operationResult = OperationResult::Changed;
+				variableDeclaration->domain = ast::Domain::Integer;
+			}
+		}
+
+		return operationResult;
+	}
+
+	static OperationResult visit(ast::ForAll &forAll, ast::Formula &definition, VariableDomainMap &variableDomainMap)
+	{
+		auto operationResult = OperationResult::Unchanged;
+
+		if (detectIntegerVariables(forAll.argument, definition, variableDomainMap) == OperationResult::Changed)
+			operationResult = OperationResult::Changed;
+
+		for (auto &variableDeclaration : forAll.variables)
+		{
+			if (variableDeclaration->domain != ast::Domain::Unknown)
+				continue;
+
+			clearVariableDomainMap(variableDomainMap);
+
+			auto argumentResult = evaluate(forAll.argument, variableDomainMap);
+			auto definitionResult = evaluate(definition, variableDomainMap);
+
+			if (argumentResult == EvaluationResult::Error || argumentResult == EvaluationResult::False
+			    || definitionResult == EvaluationResult::Error || definitionResult == EvaluationResult::False)
+			{
+				continue;
+			}
+
+			// As a hypothesis, make the parameter’s domain noninteger
+			variableDomainMap[variableDeclaration.get()] = ast::Domain::General;
+
+			argumentResult = evaluate(forAll.argument, variableDomainMap);
+			definitionResult = evaluate(definition, variableDomainMap);
+
+			if (argumentResult == EvaluationResult::Error || argumentResult == EvaluationResult::False
+			    || definitionResult == EvaluationResult::Error || definitionResult == EvaluationResult::False)
+			{
+				// If making the variable noninteger leads to a false or erroneous result, it’s proven to be integer
+				operationResult = OperationResult::Changed;
+				variableDeclaration->domain = ast::Domain::Integer;
+			}
+		}
+
+		return operationResult;
+	}
+
+	static OperationResult visit(ast::Implies &implies, ast::Formula &definition, VariableDomainMap &variableDomainMap)
+	{
+		auto operationResult = OperationResult::Unchanged;
+
+		if (detectIntegerVariables(implies.antecedent, definition, variableDomainMap) == OperationResult::Changed)
+			operationResult = OperationResult::Changed;
+
+		if (detectIntegerVariables(implies.consequent, definition, variableDomainMap) == OperationResult::Changed)
+			operationResult = OperationResult::Changed;
+
+		return operationResult;
+	}
+
+	static OperationResult visit(ast::In &, ast::Formula &, VariableDomainMap &)
+	{
+		return OperationResult::Unchanged;
+	}
+
+	static OperationResult visit(ast::Not &not_, ast::Formula &definition, VariableDomainMap &variableDomainMap)
+	{
+		return detectIntegerVariables(not_.argument, definition, variableDomainMap);
+	}
+
+	static OperationResult visit(ast::Or &or_, ast::Formula &definition, VariableDomainMap &variableDomainMap)
+	{
+		auto operationResult = OperationResult::Unchanged;
+
+		for (auto &argument : or_.arguments)
+			if (detectIntegerVariables(argument, definition, variableDomainMap) == OperationResult::Changed)
+				operationResult = OperationResult::Changed;
+
+		return operationResult;
+	}
+
+	static OperationResult visit(ast::Predicate &predicate, ast::Formula &, VariableDomainMap &)
+	{
+		auto operationResult = OperationResult::Unchanged;
+
+		assert(predicate.arguments.size() == predicate.declaration->arity());
+
+		// Propagate integer domains from predicates to variables
+		for (size_t i = 0; i < predicate.arguments.size(); i++)
+		{
+			auto &variableArgument = predicate.arguments[i];
+			auto &parameter = predicate.declaration->parameters[i];
+
+			if (parameter.domain != ast::Domain::Integer)
+				continue;
+
+			if (!variableArgument.is<ast::Variable>())
+				continue;
+
+			auto &variable = variableArgument.get<ast::Variable>();
+
+			if (variable.declaration->domain == ast::Domain::Integer)
+				continue;
+
+			operationResult = OperationResult::Changed;
+			variable.declaration->domain = ast::Domain::Integer;
+		}
+
+		return operationResult;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+OperationResult detectIntegerVariables(ast::Formula &formula, ast::Formula &definition, VariableDomainMap &variableDomainMap)
+{
+	return formula.accept(DetectIntegerVariablesVisitor(), definition, variableDomainMap);
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+// Assumes the completed formulas to be in translated but not simplified form.
+// That is, completed formulas are either variable-free or universally quantified
+void detectIntegerVariables(std::vector<ast::Formula> &completedFormulas)
+{
+	VariableDomainMap variableDomainMap;
+	auto operationResult = OperationResult::Changed;
+
+	while (operationResult == OperationResult::Changed)
+	{
+		operationResult = OperationResult::Unchanged;
+
+		for (auto &completedFormula : completedFormulas)
+		{
+			if (detectIntegerVariables(completedFormula, completedFormula, variableDomainMap) == OperationResult::Changed)
+				operationResult = OperationResult::Changed;
+
+			if (!completedFormula.is<ast::ForAll>())
+				continue;
+
+			auto &forAll = completedFormula.get<ast::ForAll>();
+
+			if (!forAll.argument.is<ast::Biconditional>())
+				continue;
+
+			auto &biconditional = forAll.argument.get<ast::Biconditional>();
+
+			if (!biconditional.left.is<ast::Predicate>())
+				continue;
+
+			auto &predicate = biconditional.left.get<ast::Predicate>();
+			auto &definition = biconditional.right;
+
+			assert(predicate.arguments.size() == predicate.declaration->arity());
+
+			if (detectIntegerVariables(definition, definition, variableDomainMap) == OperationResult::Changed)
+				operationResult = OperationResult::Changed;
+
+			for (size_t i = 0; i < predicate.arguments.size(); i++)
+			{
+				auto &variableArgument = predicate.arguments[i];
+				auto &parameter = predicate.declaration->parameters[i];
+
+				assert(variableArgument.is<ast::Variable>());
+
+				auto &variable = variableArgument.get<ast::Variable>();
+
+				if (parameter.domain != ast::Domain::Unknown)
+					continue;
+
+				clearVariableDomainMap(variableDomainMap);
+
+				auto result = evaluate(definition, variableDomainMap);
+
+				if (result == EvaluationResult::Error || result == EvaluationResult::False)
+					continue;
+
+				// As a hypothesis, make the parameter’s domain noninteger
+				variableDomainMap[variable.declaration] = ast::Domain::General;
+
+				result = evaluate(definition, variableDomainMap);
+
+				if (result == EvaluationResult::Error || result == EvaluationResult::False)
+				{
+					// If making the variable noninteger leads to a false or erroneous result, it’s proven to be integer
+					operationResult = OperationResult::Changed;
+					variable.declaration->domain = ast::Domain::Integer;
+					parameter.domain = ast::Domain::Integer;
+				}
+			}
+		}
+	}
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+}
--- a/src/anthem/Simplification.cpp
+++ b/src/anthem/Simplification.cpp
@@ -1,9 +1,11 @@
 #include <anthem/Simplification.h>

-#include <experimental/optional>
+#include <optional>

 #include <anthem/ASTCopy.h>
-#include <anthem/ASTVisitors.h>
+#include <anthem/Equality.h>
+#include <anthem/output/AST.h>
+#include <anthem/SimplificationVisitors.h>

 namespace anthem
 {
@@ -27,15 +29,15 @@ bool matchesVariableDeclaration(const ast::Term &term, const ast::VariableDeclar

 // Extracts the term t if the given formula is of the form “X = t” and X matches the given variable
 // The input formula is no longer usable after this call if a term is returned
-std::experimental::optional<ast::Term> extractAssignedTerm(ast::Formula &formula, const ast::VariableDeclaration &variableDeclaration)
+std::optional<ast::Term> extractAssignedTerm(ast::Formula &formula, const ast::VariableDeclaration &variableDeclaration)
 {
 	if (!formula.is<ast::Comparison>())
-		return std::experimental::nullopt;
+		return std::nullopt;

 	auto &comparison = formula.get<ast::Comparison>();

 	if (comparison.operator_ != ast::Comparison::Operator::Equal)
-		return std::experimental::nullopt;
+		return std::nullopt;

 	if (matchesVariableDeclaration(comparison.left, variableDeclaration))
 		return std::move(comparison.right);
@@ -43,7 +45,7 @@ std::experimental::optional<ast::Term> extractAssignedTerm(ast::Formula &formula
 	if (matchesVariableDeclaration(comparison.right, variableDeclaration))
 		return std::move(comparison.left);

-	return std::experimental::nullopt;
+	return std::nullopt;
 }

 ////////////////////////////////////////////////////////////////////////////////////////////////////
@@ -97,18 +99,65 @@ struct ReplaceVariableInFormulaVisitor : public ast::RecursiveFormulaVisitor<Rep

 ////////////////////////////////////////////////////////////////////////////////////////////////////

-// Simplifies exists statements by using the equivalence “exists X (X = t and F(X))” == “F(t)”
-// The exists statement has to be of the form “exists <variables> <conjunction>”
-void simplify(ast::Exists &exists, ast::Formula &formula)
+template<class SimplificationRule>
+SimplificationResult simplify(ast::Formula &formula)
 {
-	// Simplify formulas like “exists X (X = Y)” to “#true”
-	// TODO: check that this covers all cases
-	if (exists.argument.is<ast::Comparison>())
+	return SimplificationRule::apply(formula);
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+template<class FirstSimplificationRule, class SecondSimplificationRule, class... OtherSimplificationRules>
+SimplificationResult simplify(ast::Formula &formula)
+{
+	if (simplify<FirstSimplificationRule>(formula) == SimplificationResult::Simplified)
+		return SimplificationResult::Simplified;
+
+	return simplify<SecondSimplificationRule, OtherSimplificationRules...>(formula);
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleExistsWithoutQuantifiedVariables
+{
+	static constexpr const auto Description = "exists () (F) === F";
+
+	static SimplificationResult apply(ast::Formula &formula)
 	{
+		if (!formula.is<ast::Exists>())
+			return SimplificationResult::Unchanged;
+
+		auto &exists = formula.get<ast::Exists>();
+
+		if (!exists.variables.empty())
+			return SimplificationResult::Unchanged;
+
+		formula = std::move(exists.argument);
+
+		return SimplificationResult::Simplified;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleTrivialAssignmentInExists
+{
+	static constexpr const auto Description = "exists X (X = Y) === #true";
+
+	static SimplificationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::Exists>())
+			return SimplificationResult::Unchanged;
+
+		const auto &exists = formula.get<ast::Exists>();
+
+		if (!exists.argument.is<ast::Comparison>())
+			return SimplificationResult::Unchanged;
+
 		const auto &comparison = exists.argument.get<ast::Comparison>();

 		if (comparison.operator_ != ast::Comparison::Operator::Equal)
-			return;
+			return SimplificationResult::Unchanged;

 		const auto matchingAssignment = std::find_if(exists.variables.cbegin(), exists.variables.cend(),
 			[&](const auto &variableDeclaration)
@@ -117,107 +166,378 @@ void simplify(ast::Exists &exists, ast::Formula &formula)
 					|| matchesVariableDeclaration(comparison.right, *variableDeclaration);
 			});

-		if (matchingAssignment != exists.variables.cend())
-			formula = ast::Formula::make<ast::Boolean>(true);
+		if (matchingAssignment == exists.variables.cend())
+			return SimplificationResult::Unchanged;

-		return;
+		formula = ast::Formula::make<ast::Boolean>(true);
+
+		return SimplificationResult::Simplified;
 	}
+};

-	if (!exists.argument.is<ast::And>())
-		return;
+////////////////////////////////////////////////////////////////////////////////////////////////////

-	auto &conjunction = exists.argument.get<ast::And>();
-	auto &arguments = conjunction.arguments;
+struct SimplificationRuleAssignmentInExists
+{
+	static constexpr const auto Description = "exists X (X = t and F(X)) === exists () (F(t))";

-	// Simplify formulas of type “exists X (X = t and F(X))” to “F(t)”
-	for (auto i = exists.variables.begin(); i != exists.variables.end();)
+	static SimplificationResult apply(ast::Formula &formula)
 	{
-		const auto &variableDeclaration = **i;
+		if (!formula.is<ast::Exists>())
+			return SimplificationResult::Unchanged;

-		bool wasVariableReplaced = false;
+		auto &exists = formula.get<ast::Exists>();

-		// TODO: refactor
-		for (auto j = arguments.begin(); j != arguments.end(); j++)
+		if (!exists.argument.is<ast::And>())
+			return SimplificationResult::Unchanged;
+
+		auto &and_ = exists.argument.get<ast::And>();
+		auto &arguments = and_.arguments;
+
+		auto simplificationResult = SimplificationResult::Unchanged;
+
+		for (auto i = exists.variables.begin(); i != exists.variables.end();)
 		{
-			auto &argument = *j;
-			// Find term that is equivalent to the given variable
-			auto assignedTerm = extractAssignedTerm(argument, variableDeclaration);
+			const auto &variableDeclaration = **i;

-			if (!assignedTerm)
-				continue;
+			bool wasVariableReplaced = false;

-			// Replace all occurrences of the variable with the equivalent term
-			for (auto k = arguments.begin(); k != arguments.end(); k++)
+			// TODO: refactor
+			for (auto j = arguments.begin(); j != arguments.end(); j++)
 			{
-				if (k == j)
+				auto &argument = *j;
+				// Find term that is equivalent to the given variable
+				auto assignedTerm = extractAssignedTerm(argument, variableDeclaration);
+
+				if (!assignedTerm)
 					continue;

-				auto &otherArgument = *k;
-				otherArgument.accept(ReplaceVariableInFormulaVisitor(), otherArgument, variableDeclaration, assignedTerm.value());
+				// Replace all occurrences of the variable with the equivalent term
+				for (auto k = arguments.begin(); k != arguments.end(); k++)
+				{
+					if (k == j)
+						continue;
+
+					auto &otherArgument = *k;
+					otherArgument.accept(ReplaceVariableInFormulaVisitor(), otherArgument, variableDeclaration, assignedTerm.value());
+				}
+
+				arguments.erase(j);
+
+				wasVariableReplaced = true;
+				simplificationResult = SimplificationResult::Simplified;
+
+				break;
 			}

-			arguments.erase(j);
-			wasVariableReplaced = true;
-			break;
+			if (wasVariableReplaced)
+			{
+				i = exists.variables.erase(i);
+				continue;
+			}
+
+			i++;
 		}

-		if (wasVariableReplaced)
-		{
-			i = exists.variables.erase(i);
-			continue;
-		}
-
-		i++;
+		return simplificationResult;
 	}
+};

-	// If there are no arguments left, we had a formula of the form “exists X1, ..., Xn (X1 = Y1 and ... and Xn = Yn)”
-	// Such exists statements are useless and can be safely replaced with “#true”
-	if (arguments.empty())
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleEmptyConjunction
+{
+	static constexpr const auto Description = "[empty conjunction] === #true";
+
+	static SimplificationResult apply(ast::Formula &formula)
 	{
+		if (!formula.is<ast::And>())
+			return SimplificationResult::Unchanged;
+
+		auto &and_ = formula.get<ast::And>();
+
+		if (!and_.arguments.empty())
+			return SimplificationResult::Unchanged;
+
 		formula = ast::Formula::make<ast::Boolean>(true);
-		return;
+
+		return SimplificationResult::Simplified;
 	}
+};

-	// If the argument now is a conjunction with just one element, directly replace the input formula with the argument
-	if (arguments.size() == 1)
-		exists.argument = std::move(arguments.front());
+////////////////////////////////////////////////////////////////////////////////////////////////////

-	// If there are still remaining variables, simplification is over
-	if (!exists.variables.empty())
-		return;
+struct SimplificationRuleOneElementConjunction
+{
+	static constexpr const auto Description = "[conjunction of only F] === F";

-	assert(!arguments.empty());
+	static SimplificationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::And>())
+			return SimplificationResult::Unchanged;

-	// If there is more than one element in the conjunction, replace the input formula with the conjunction
-	formula = std::move(exists.argument);
-}
+		auto &and_ = formula.get<ast::And>();
+
+		if (and_.arguments.size() != 1)
+			return SimplificationResult::Unchanged;
+
+		formula = std::move(and_.arguments.front());
+
+		return SimplificationResult::Simplified;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleTrivialExists
+{
+	static constexpr const auto Description = "exists ... ([#true/#false]) === [#true/#false]";
+
+	static SimplificationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::Exists>())
+			return SimplificationResult::Unchanged;
+
+		auto &exists = formula.get<ast::Exists>();
+
+		if (!exists.argument.is<ast::Boolean>())
+			return SimplificationResult::Unchanged;
+
+		formula = std::move(exists.argument);
+
+		return SimplificationResult::Simplified;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleInWithPrimitiveArguments
+{
+	static constexpr const auto Description = "[primitive A] in [primitive B] === A = B";
+
+	static SimplificationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::In>())
+			return SimplificationResult::Unchanged;
+
+		auto &in = formula.get<ast::In>();
+
+		assert(ast::isPrimitive(in.element));
+
+		if (!ast::isPrimitive(in.element) || !ast::isPrimitive(in.set))
+			return SimplificationResult::Unchanged;
+
+		formula = ast::Comparison(ast::Comparison::Operator::Equal, std::move(in.element), std::move(in.set));
+
+		return SimplificationResult::Simplified;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleSubsumptionInBiconditionals
+{
+	static constexpr const auto Description = "(F <-> (F and G)) === (F -> G)";
+
+	static SimplificationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::Biconditional>())
+			return SimplificationResult::Unchanged;
+
+		auto &biconditional = formula.get<ast::Biconditional>();
+
+		const auto leftIsPredicate = biconditional.left.is<ast::Predicate>();
+		const auto rightIsPredicate = biconditional.right.is<ast::Predicate>();
+
+		const auto leftIsAnd = biconditional.left.is<ast::And>();
+		const auto rightIsAnd = biconditional.right.is<ast::And>();
+
+		if (!(leftIsPredicate && rightIsAnd) && !(rightIsPredicate && leftIsAnd))
+			return SimplificationResult::Unchanged;
+
+		auto &predicateSide = (leftIsPredicate ? biconditional.left : biconditional.right);
+		auto &andSide = (leftIsPredicate ? biconditional.right : biconditional.left);
+		auto &and_ = andSide.get<ast::And>();
+
+		const auto matchingPredicate =
+			std::find_if(and_.arguments.cbegin(), and_.arguments.cend(),
+			[&](const auto &argument)
+			{
+				return (ast::equal(predicateSide, argument) == Tristate::True);
+			});
+
+		if (matchingPredicate == and_.arguments.cend())
+			return SimplificationResult::Unchanged;
+
+		and_.arguments.erase(matchingPredicate);
+
+		formula = ast::Formula::make<ast::Implies>(std::move(predicateSide), std::move(andSide));
+
+		return SimplificationResult::Simplified;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleDoubleNegation
+{
+	static constexpr const auto Description = "not not F === F";
+
+	static SimplificationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::Not>())
+			return SimplificationResult::Unchanged;
+
+		auto &not_ = formula.get<ast::Not>();
+
+		if (!not_.argument.is<ast::Not>())
+			return SimplificationResult::Unchanged;
+
+		auto &notNot = not_.argument.get<ast::Not>();
+
+		formula = std::move(notNot.argument);
+
+		return SimplificationResult::Simplified;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleDeMorganForConjunctions
+{
+	static constexpr const auto Description = "(not (F and G)) === (not F or not G)";
+
+	static SimplificationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::Not>())
+			return SimplificationResult::Unchanged;
+
+		auto &not_ = formula.get<ast::Not>();
+
+		if (!not_.argument.is<ast::And>())
+			return SimplificationResult::Unchanged;
+
+		auto &and_ = not_.argument.get<ast::And>();
+
+		for (auto &argument : and_.arguments)
+			argument = ast::Formula::make<ast::Not>(std::move(argument));
+
+		formula = ast::Formula::make<ast::Or>(std::move(and_.arguments));
+
+		return SimplificationResult::Simplified;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleImplicationFromDisjunction
+{
+	static constexpr const auto Description = "(not F or G) === (F -> G)";
+
+	static SimplificationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::Or>())
+			return SimplificationResult::Unchanged;
+
+		auto &or_ = formula.get<ast::Or>();
+
+		if (or_.arguments.size() != 2)
+			return SimplificationResult::Unchanged;
+
+		const auto leftIsNot = or_.arguments[0].is<ast::Not>();
+		const auto rightIsNot = or_.arguments[1].is<ast::Not>();
+
+		if (leftIsNot == rightIsNot)
+			return SimplificationResult::Unchanged;
+
+		auto &negativeSide = leftIsNot ? or_.arguments[0] : or_.arguments[1];
+		auto &positiveSide = leftIsNot ? or_.arguments[1] : or_.arguments[0];
+
+		assert(negativeSide.is<ast::Not>());
+		assert(!positiveSide.is<ast::Not>());
+
+		auto &negativeSideArgument = negativeSide.get<ast::Not>().argument;
+
+		formula = ast::Formula::make<ast::Implies>(std::move(negativeSideArgument), std::move(positiveSide));
+
+		return SimplificationResult::Simplified;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleNegatedComparison
+{
+	static constexpr const auto Description = "(not F [comparison] G) === (F [negated comparison] G)";
+
+	static SimplificationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::Not>())
+			return SimplificationResult::Unchanged;
+
+		auto &not_ = formula.get<ast::Not>();
+
+		if (!not_.argument.is<ast::Comparison>())
+			return SimplificationResult::Unchanged;
+
+		auto &comparison = not_.argument.get<ast::Comparison>();
+
+		switch (comparison.operator_)
+		{
+			case ast::Comparison::Operator::GreaterThan:
+				comparison.operator_ = ast::Comparison::Operator::LessEqual;
+				break;
+			case ast::Comparison::Operator::LessThan:
+				comparison.operator_ = ast::Comparison::Operator::GreaterEqual;
+				break;
+			case ast::Comparison::Operator::LessEqual:
+				comparison.operator_ = ast::Comparison::Operator::GreaterThan;
+				break;
+			case ast::Comparison::Operator::GreaterEqual:
+				comparison.operator_ = ast::Comparison::Operator::LessThan;
+				break;
+			case ast::Comparison::Operator::NotEqual:
+				comparison.operator_ = ast::Comparison::Operator::Equal;
+				break;
+			case ast::Comparison::Operator::Equal:
+				comparison.operator_ = ast::Comparison::Operator::NotEqual;
+				break;
+		}
+
+		formula = std::move(comparison);
+
+		return SimplificationResult::Simplified;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+const auto simplifyWithDefaultRules =
+	simplify
+	<
+		SimplificationRuleDoubleNegation,
+		SimplificationRuleTrivialAssignmentInExists,
+		SimplificationRuleAssignmentInExists,
+		SimplificationRuleEmptyConjunction,
+		SimplificationRuleTrivialExists,
+		SimplificationRuleOneElementConjunction,
+		SimplificationRuleExistsWithoutQuantifiedVariables,
+		SimplificationRuleInWithPrimitiveArguments,
+		SimplificationRuleSubsumptionInBiconditionals,
+		SimplificationRuleDeMorganForConjunctions,
+		SimplificationRuleImplicationFromDisjunction,
+		SimplificationRuleNegatedComparison
+	>;

 ////////////////////////////////////////////////////////////////////////////////////////////////////

 // Performs the different simplification techniques
-struct SimplifyFormulaVisitor : public ast::RecursiveFormulaVisitor<SimplifyFormulaVisitor>
+struct SimplifyFormulaVisitor : public ast::FormulaSimplificationVisitor<SimplifyFormulaVisitor>
 {
-	// Forward exists statements to the dedicated simplification function
-	static void accept(ast::Exists &exists, ast::Formula &formula)
-	{
-		simplify(exists, formula);
-	}
-
-	// Simplify formulas of type “A in B” to “A = B” if A and B are primitive
-	static void accept(ast::In &in, ast::Formula &formula)
-	{
-		assert(ast::isPrimitive(in.element));
-
-		if (!ast::isPrimitive(in.element) || !ast::isPrimitive(in.set))
-			return;
-
-		formula = ast::Comparison(ast::Comparison::Operator::Equal, std::move(in.element), std::move(in.set));
-	}
-
 	// Do nothing for all other types of expressions
-	template<class T>
-	static void accept(T &, ast::Formula &)
+	static SimplificationResult accept(ast::Formula &formula)
 	{
+		return simplifyWithDefaultRules(formula);
 	}
 };

@@ -225,7 +545,7 @@ struct SimplifyFormulaVisitor : public ast::RecursiveFormulaVisitor<SimplifyForm

 void simplify(ast::Formula &formula)
 {
-	formula.accept(SimplifyFormulaVisitor(), formula);
+	while (formula.accept(SimplifyFormulaVisitor(), formula) == SimplificationResult::Simplified);
 }

 ////////////////////////////////////////////////////////////////////////////////////////////////////
--- a/src/anthem/Translation.cpp
+++ b/src/anthem/Translation.cpp
@@ -8,6 +8,7 @@

 #include <anthem/Completion.h>
 #include <anthem/Context.h>
+#include <anthem/IntegerVariableDetection.h>
 #include <anthem/Simplification.h>
 #include <anthem/StatementVisitor.h>
 #include <anthem/output/AST.h>
@@ -67,9 +68,12 @@ void translate(const char *fileName, std::istream &stream, Context &context)
 			for (auto &scopedFormula : scopedFormulas)
 				simplify(scopedFormula.formula);

-		if (context.visiblePredicateSignatures)
+		if (context.showStatementsUsed)
 			context.logger.log(output::Priority::Warning) << "#show statements are ignored because completion is not enabled";

+		if (context.externalStatementsUsed)
+			context.logger.log(output::Priority::Warning) << "#external statements are ignored because completion is not enabled";
+
 		for (const auto &scopedFormula : scopedFormulas)
 		{
 			ast::print(context.logger.outputStream(), scopedFormula.formula, printContext);
@@ -82,6 +86,34 @@ void translate(const char *fileName, std::istream &stream, Context &context)
 	// Perform completion
 	auto completedFormulas = complete(std::move(scopedFormulas), context);

+	for (const auto &predicateDeclaration : context.predicateDeclarations)
+	{
+		if (predicateDeclaration->isUsed)
+			continue;
+
+		// Check for #show statements with undeclared predicates
+		if (predicateDeclaration->visibility != ast::PredicateDeclaration::Visibility::Default)
+			context.logger.log(output::Priority::Warning)
+				<< "#show declaration of “"
+				<< predicateDeclaration->name
+				<< "/"
+				<< predicateDeclaration->arity()
+				<< "” does not match any declared predicate";
+
+		// Check for #external statements with undeclared predicates
+		if (predicateDeclaration->isExternal && !predicateDeclaration->isUsed)
+			context.logger.log(output::Priority::Warning)
+				<< "#external declaration of “"
+				<< predicateDeclaration->name
+				<< "/"
+				<< predicateDeclaration->arity()
+				<< "” does not match any declared predicate";
+	}
+
+	// Detect integer variables
+	if (context.performIntegerDetection)
+		detectIntegerVariables(completedFormulas);
+
 	// Simplify output if specified
 	if (context.performSimplification)
 		for (auto &completedFormula : completedFormulas)
@@ -94,6 +126,38 @@ void translate(const char *fileName, std::istream &stream, Context &context)
 		ast::print(context.logger.outputStream(), completedFormula, printContext);
 		context.logger.outputStream() << std::endl;
 	}
+
+	// Print specifiers for integer predicate parameters
+	for (auto &predicateDeclaration : context.predicateDeclarations)
+	{
+		// Check that the predicate is used and not declared #external
+		if (!predicateDeclaration->isUsed || predicateDeclaration->isExternal)
+			continue;
+
+		const auto isPredicateVisible =
+			(predicateDeclaration->visibility == ast::PredicateDeclaration::Visibility::Visible)
+			|| (predicateDeclaration->visibility == ast::PredicateDeclaration::Visibility::Default
+				&& context.defaultPredicateVisibility == ast::PredicateDeclaration::Visibility::Visible);
+
+		// If the predicate ought to be visible, don’t eliminate it
+		if (!isPredicateVisible)
+			continue;
+
+		for (size_t i = 0; i < predicateDeclaration->parameters.size(); i++)
+		{
+			auto &parameter = predicateDeclaration->parameters[i];
+
+			if (parameter.domain != ast::Domain::Integer)
+				continue;
+
+			context.logger.outputStream()
+				<< output::Keyword("int")
+				<< "(" << predicateDeclaration->name
+				<< "/" << output::Number(predicateDeclaration->arity())
+				<< "@" << output::Number(i)
+				<< ")." << std::endl;
+		}
+	}
 }

 ////////////////////////////////////////////////////////////////////////////////////////////////////
--- a/tests/TestCompletion.cpp
+++ b/tests/TestCompletion.cpp
@@ -152,9 +152,9 @@ TEST_CASE("[completion] Rules are completed", "[completion]")

 		CHECK(output.str() ==
 			"forall V1 (covered(V1) <-> exists U1 in(V1, U1))\n"
-			"forall V2, V3 (in(V2, V3) <-> (V2 in 1..n and V3 in 1..r and in(V2, V3)))\n"
-			"forall U2 not (U2 in 1..n and not covered(U2))\n"
-			"forall U3, U4, U5 not (in(U3, U4) and in(U5, U4) and exists X1 (X1 in (U3 + U5) and in(X1, U4)))\n");
+			"forall V2, V3 (in(V2, V3) -> (V2 in 1..n and V3 in 1..r))\n"
+			"forall U2 (U2 in 1..n -> covered(U2))\n"
+			"forall U3, U4, U5 (not in(U3, U4) or not in(U5, U4) or not exists X1 (X1 in (U3 + U5) and in(X1, U4)))\n");
 	}

 	SECTION("binary operations with multiple variables")
@@ -176,4 +176,20 @@ TEST_CASE("[completion] Rules are completed", "[completion]")
 		CHECK(output.str() ==
 			"forall V1, V2, V3 (p(V1, V2, V3) <-> #true)\n");
 	}
+
+	SECTION("negated comparisons")
+	{
+		input << ":- color(V, C1), color(V, C2), C1 != C2.";
+		anthem::translate("input", input, context);
+
+		CHECK(output.str() == "forall V1, V2 not color(V1, V2)\nforall U1, U2, U3 (not color(U1, U2) or not color(U1, U3) or U2 = U3)\n");
+	}
+
+	SECTION("absolute value operation")
+	{
+		input << "adj(X, Y) :- X = 1..n, Y = 1..n, |X - Y| = 1.";
+		anthem::translate("input", input, context);
+
+		CHECK(output.str() == "forall V1, V2 (adj(V1, V2) <-> (V1 in 1..n and V2 in 1..n and |V1 - V2| = 1))\n");
+	}
 }
--- a/tests/TestHiddenPredicateElimination.cpp
+++ b/tests/TestHiddenPredicateElimination.cpp
@@ -103,9 +103,10 @@ TEST_CASE("[hidden predicate elimination] Hidden predicates are correctly elimin
 			"#show a/1.";
 		anthem::translate("input", input, context);

+		// TODO: simplify further
 		CHECK(output.str() ==
 			"forall V1 (a(V1) <-> not d(V1))\n"
-			"forall V2 (d(V2) <-> not not d(V2))\n"
+			"forall V2 (d(V2) <-> d(V2))\n"
 			"forall V3 (e(V3) <-> e(V3))\n");
 	}

@@ -164,12 +165,11 @@ TEST_CASE("[hidden predicate elimination] Hidden predicates are correctly elimin
 			"#show t/0.";
 		anthem::translate("input", input, context);

-		// TODO: simplify further
 		CHECK(output.str() ==
-			"(s <-> (not #false and s))\n"
-			"(t <-> (not #false and t))\n"
-			"not (s and not t)\n"
-			"not (not #false and not #false and #false)\n");
+			"(s -> not #false)\n"
+			"(t -> not #false)\n"
+			"(s -> t)\n"
+			"(#false or #false or not #false)\n");
 	}

 	SECTION("predicate with more than one argument is hidden correctly")
--- a/tests/TestPlaceholders.cpp
+++ b/tests/TestPlaceholders.cpp
@@ -0,0 +1,62 @@
+#include <catch.hpp>
+
+#include <sstream>
+
+#include <anthem/AST.h>
+#include <anthem/Context.h>
+#include <anthem/Translation.h>
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+TEST_CASE("[placeholders] Programs with placeholders are correctly completed", "[placeholders]")
+{
+	std::stringstream input;
+	std::stringstream output;
+	std::stringstream errors;
+
+	anthem::output::Logger logger(output, errors);
+	anthem::Context context(std::move(logger));
+	context.performSimplification = true;
+	context.performCompletion = true;
+
+	SECTION("no placeholders")
+	{
+		input <<
+			"colored(V, red) :- vertex(V), not colored(V, green), not colored(V, blue).";
+		anthem::translate("input", input, context);
+
+		CHECK(output.str() ==
+			"forall V1, V2 (colored(V1, V2) <-> (V2 = red and vertex(V1) and not colored(V1, green) and not colored(V1, blue)))\n"
+			"forall V3 not vertex(V3)\n");
+	}
+
+	SECTION("single placeholder")
+	{
+		input <<
+			"#external vertex(1).\n"
+			"colored(V, red) :- vertex(V), not colored(V, green), not colored(V, blue).";
+		anthem::translate("input", input, context);
+
+		CHECK(output.str() ==
+			"forall V1, V2 (colored(V1, V2) <-> (V2 = red and vertex(V1) and not colored(V1, green) and not colored(V1, blue)))\n");
+	}
+
+	SECTION("complex example: graph coloring")
+	{
+		input <<
+			"#external color(1).\n"
+			"#external edge(2).\n"
+			"#external vertex(1).\n"
+			"#show color/2.\n"
+			"{color(V, C)} :- vertex(V), color(C).\n"
+			"covered(V) :- color(V, _).\n"
+			":- vertex(V), not covered(V).\n"
+			":- color(V1, C), color(V2, C), edge(V1, V2).";
+		anthem::translate("input", input, context);
+
+		CHECK(output.str() ==
+			"forall V1, V2 (color(V1, V2) -> (vertex(V1) and color(V2)))\n"
+			"forall U1 (vertex(U1) -> exists U2 color(U1, U2))\n"
+			"forall U3, U4, U5 (not color(U3, U4) or not color(U5, U4) or not edge(U3, U5))\n");
+	}
+}
--- a/tests/TestTranslation.cpp
+++ b/tests/TestTranslation.cpp
@@ -296,4 +296,12 @@ TEST_CASE("[translation] Rules are translated correctly", "[translation]")

 		CHECK(output.str() == "((V1 in U1 and V2 in U2 and exists X1, X2 (X1 in U3 and X2 in U4 and q(X1, X2))) -> p(V1, V2))\n");
 	}
+
+	SECTION("exponentiation operator")
+	{
+		input << "p(N, N ** N) :- N = 1..n.";
+		anthem::translate("input", input, context);
+
+		CHECK(output.str() == "((V1 in U1 and V2 in (U1 ** U1) and exists X1, X2 (X1 in U1 and X2 in 1..n and X1 = X2)) -> p(V1, V2))\n");
+	}
 }
Author	SHA1	Message	Date
Patrick Lühne	53c96f0fe7	Version bump for release 0.1.9 RC 1	2018-04-20 16:39:54 +02:00
Patrick Lühne	a9b00dfb7c	Add integer variable detection to change log	2018-04-20 16:38:57 +02:00
Patrick Lühne	d1ee6eca19	Add suffix to variables with known domain This adds suffixes to integer and general variables, where the domain is explicitly known.	2018-04-20 16:37:49 +02:00
Patrick Lühne	aedb7e9bbd	Add option to turn on integer variable detection	2018-04-20 16:37:49 +02:00
Patrick Lühne	8b8dd1b57e	Minor refactoring	2018-04-20 16:37:49 +02:00
Patrick Lühne	5bda14342b	Print rules for integer parameters For every integer parameter of the predicates visible in the output, this prints a short fact to the output to make this explicit.	2018-04-20 16:37:49 +02:00
Patrick Lühne	2f54b7d60e	Minor formatting	2018-04-20 16:37:49 +02:00
Patrick Lühne	2245e139b2	Reimplement integer variable detection This is a reimplementation of the integer variable detection procedure. The idea is to iteratively assume variables to be noninteger, and to prove that this would lead to a false or erroneous result. If the proof is successful, the variable is integer as a consequence.	2018-04-20 16:37:48 +02:00
Patrick Lühne	7ba48044ee	Propagate predicate parameter domains With this change, domains detected for predicate parameters are properly propagated to all occurences of the respective variables, enabling more integer simplifications.	2018-04-20 16:37:48 +02:00
Patrick Lühne	862d03881a	Fix typo	2018-04-20 16:37:48 +02:00
Patrick Lühne	7bde7c498f	Represent predicate parameters explicitly This adds a vector of Parameter structs to PredicateDeclaration. In this way, the domain of each parameter can be tracked individually.	2018-04-20 16:37:48 +02:00
Patrick Lühne	159717f51c	Support declaring functions as integer This adds a new syntax for declaring functions integer: #external integer(<function name>(<arity)). If a function is declared integer, it may enable some variables to be detected as integer as well.	2018-04-20 16:37:48 +02:00
Patrick Lühne	89aec98942	Minor refactoring	2018-04-20 16:37:48 +02:00
Patrick Lühne	55f7fd4ff3	Minor refactoring	2018-04-20 16:37:48 +02:00
Patrick Lühne	ae918a0846	Moved Domain enum to separate header For clarity, this moves the Domain enum class to a separate header, because it’s not just variable-specific but also applicable to functions, for example.	2018-04-20 16:37:48 +02:00
Patrick Lühne	f48802842e	Split functions from their declaration This splits occurrences of functions from their declaration. This is necessary to flag integer functions consistently and not just single occurrences.	2018-04-20 16:37:48 +02:00
Patrick Lühne	b5b05b766c	Remove Constant class Constants are not a construct present in Clingo’s AST and were unintentionally made part of anthem’s AST. This removes the unused classes for clarity.	2018-04-20 16:37:48 +02:00
Patrick Lühne	165f6ac059	Implement basic integer variable detection This adds initial support for detecting integer variables in formulas. The scope is somewhat limited in that variables matching predicate parameters with known integer type aren’t propagated to be integer as well yet. Also, the use of constants and functions prevents variables from being detected as integer, because they have to be assumed to be externally defined in such a way that they evaluate to general values and not necessarily integers.	2018-04-20 16:37:48 +02:00
Patrick Lühne	504f7da4c3	Add unknown variable domain Before being able to tell whether a variable’s domain is general or integer, it is necessary to flag it as “unknown.”	2018-04-20 16:37:48 +02:00
Patrick Lühne	d2b48f9679	Move Tristate class to separate header The Tristate class (representing truth values that are either true, false, or unknown) is used at multiple ends. This moves it to a separate header for reusing it properly.	2018-04-20 16:37:48 +02:00
Patrick Lühne	2372eb24c4	Refactor predicate representation This refactoring separates predicates from their declarations. The purpose of this is to avoid duplicating properties specific to the predicate declaration and not its occurrences in the program.	2018-04-20 16:37:47 +02:00
Patrick Lühne	87dcd6ee93	Add domain specifier to variable declarations With this change, the domain of variable declarations can be specified. While the default is the general domain of all precomputed value, this adds support for integer variables, for which advanced simplification rules apply.	2018-04-20 16:37:47 +02:00
Patrick Lühne	9a59ac17f5	Version bump after release 0.1.8	2018-04-20 16:37:03 +02:00
Patrick Lühne	250942643c	Version bump for release 0.1.8	2018-04-20 16:35:35 +02:00
Patrick Lühne	815bcda367	Update cxxopts to 2.1.0+1+gcc4914f cxxopts 2.1.0 has a bug preventing it from being used with standard main signatures. This updates cxxopts to the commit after release 2.1.0, where this issue was addressed.	2018-04-13 14:03:30 +02:00
Patrick Lühne	04094eee23	Remove unnecessary parentheses The unary modulus operation does not require extra parentheses to be printed in cases like “\|X + Y\|”. This adds a new option to the printing routine to omit parentheses in cases where the parent expression already defines a parenthesis-like scope (currently only with unary operations).	2018-04-12 00:59:03 +02:00
Patrick Lühne	8c250f5c59	Support modulus operation (absolute value) This adds support for computing the absolute value of a term along with an according unit test.	2018-04-12 00:38:48 +02:00
Patrick Lühne	0608748349	Describe --complete option in readme The readme was missing information on the --complete option. This adds a brief mention of Clark’s completion to the readme.	2018-04-11 23:21:56 +02:00
Patrick Lühne	31d4a20491	Update change log with recent additions This updates the change log with the advanced simplification rules, support for the exponentation operator, and the newly added examples.	2018-04-11 21:50:15 +02:00
Patrick Lühne	a01e78a467	Add example program for prime number detection	2018-04-11 21:42:08 +02:00
Patrick Lühne	797660d6de	Add new simplification rule This adds the rule “(not F [comparison] G) === (F [negated comparison] G)” to the simplification rule tableau.	2018-04-11 21:39:27 +02:00
Patrick Lühne	b63ef21849	Add example program generating permutations	2018-04-11 21:35:29 +02:00
Patrick Lühne	cc3c9b642c	Minor formatting in graph coloring example	2018-04-11 21:35:04 +02:00
Patrick Lühne	40ddee8444	Add new simplification rule This adds the rule “(not F or G) === (F -> G)” to the simplification rule tableau.	2018-04-10 22:34:47 +02:00
Patrick Lühne	6f7b021712	Add new simplification rule This adds the rule “(not (F and G)) === (not F or not G)” to the simplification rule tableau.	2018-04-10 22:34:47 +02:00
Patrick Lühne	23624007ec	Add new simplification rule This adds the rule “not not F === F” to the simplification rule tableau.	2018-04-10 22:34:47 +02:00
Patrick Lühne	6d7b91c391	Add new simplification rule This adds the rule “(F <-> (F and G)) === (F -> G)” to the simplification rule tableau.	2018-04-10 22:34:47 +02:00
Patrick Lühne	b88393655a	Iteratively apply simplification tableau rules With this change, the tableau rules for simplifying formula are applied iteratively until a fixpoint is reached.	2018-04-10 22:34:47 +02:00
Patrick Lühne	c4c3156e77	Move simplification rule to tableau This moves the rule “[primitive A] in [primitive B] === A = B” to the simplification rule tableau.	2018-04-10 22:34:47 +02:00
Patrick Lühne	107dae7287	Move simplification rule to tableau This moves the rule “exists () (F) === F” to the simplification rule tableau.	2018-04-10 22:34:47 +02:00
Patrick Lühne	827d6e40fe	Move simplification rule to tableau This moves the rule “[conjunction of only F] === F” to the simplification rule tableau.	2018-04-10 22:34:47 +02:00
Patrick Lühne	4a85fc4b23	Move simplification rule to tableau This moves the rule “exists ... ([#true/#false]) === [#true/#false]” to the simplification rule tableau along with “[empty conjunction] ===	2018-04-10 22:34:46 +02:00
Patrick Lühne	7e3fc007c8	Move simplification rule to tableau This moves the rule “exists X (X = t and F(X)) === exists () (F(t))” to the simplification rule tableau.	2018-04-10 22:34:46 +02:00
Patrick Lühne	5c5411c0ff	Implement simplification rule tableau This implements a tableau containing simplification rules that can be iteratively applied to input formulas until they remain unchanged. First, this moves the rule “exists X (X = Y) === #true” to the tableau as a reference implementation.	2018-04-10 22:34:46 +02:00
Patrick Lühne	eaabeb0c55	Support exponentiation operator Because of a bug in the Clingo API, the exponentation operator was not properly exposed to anthem. This updates Clingo to a version with a fixed API and adds proper support for exponentation within anthem along with a matching unit test.	2018-04-10 22:29:55 +02:00
Patrick Lühne	7b6729acaa	Add missing dependency to Ubuntu image For some reason, Bison is not implicitly installed along with the other dependencies in the Ubuntu 18.04 image used for continuous integration. This adds Bison explicitly.	2018-04-10 22:29:55 +02:00
Patrick Lühne	92fddd6665	Version bump after release 0.1.7	2018-04-08 21:03:20 +02:00
Patrick Lühne	582b6ade6d	Version bump for release 0.1.7	2018-04-08 20:44:43 +02:00
Patrick Lühne	e64b2e70de	Remove unused captured lambda reference	2018-04-08 20:44:43 +02:00
Patrick Lühne	d7e4af98d7	Update copyright year in license file	2018-04-08 20:35:03 +02:00
Patrick Lühne	a406cb43bd	Update graph coloring example with placeholders This replaces the former graph coloring example with a new formulation that makes use of the newly supported placeholders.	2018-04-08 20:28:57 +02:00
Patrick Lühne	c294a29cb2	Support placeholders with #external declarations This adds support for declaring predicates as placeholders through the “#external” directive in the input language of clingo. Placeholders are not subject to completion. This prevents predicates that represent instance-specific facts from being assumed as universally false by default negation when translating an encoding. This stretches clingo’s usual syntax a bit to make the implementation lightweight. In order to declare a predicate with a specific arity as a placeholder, the following statement needs to be added to the program: #external <predicate name>(<arity>). Multiple unit tests cover cases where placeholders are used or not as well as a more complex graph coloring example.	2018-04-08 20:28:57 +02:00
Patrick Lühne	c91cbaf58b	Update Catch to 2.2.2	2018-04-07 00:22:01 +02:00
Patrick Lühne	2a2fec0eac	Update change log with dependency change This adds the dependency change from Boost (for program options) to cxxopts to the change log.	2018-04-06 23:08:57 +02:00
Patrick Lühne	09e56c3bce	Format change log sections with proper headings This makes the change log sections have proper headings, which were just normal text before.	2018-04-06 22:53:59 +02:00
Patrick Lühne	e2c0d6b705	Update cxxopts to 2.0.0+3+gabe9ebd With cxxopts 2.0.0, positional arguments weren’t recognized when other command-line options were passed before. This has been fixed in the meantime, but there is no release with the bug fix yet. This updates cxxopts to a newer commit to ship anthem with this fix.	2018-04-06 22:44:14 +02:00
Patrick Lühne	e01506f9ff	Drop Boost dependency Boost was only used for program option parsing. To avoid this huge dependency, this commit replaces boost::program_options with cxxopts, a header-only library with the same functionality. cxxopts is added as a submodule, and Boost is removed from the dependencies in the code and Travis configuration.	2018-03-25 17:24:06 +02:00
Patrick Lühne	50ebf3c6de	Install g++ package explicitly on Ubuntu Apparently, g++ is only installed because of the Boost dependency. Make the g++ dependency explicit to avoid future package errors.	2018-03-25 17:23:33 +02:00
Patrick Lühne	fde2af5841	Add clang to Travis configurations This adds the clang compiler to the tested Travis configurations.	2018-03-24 18:53:51 +01:00
Patrick Lühne	22238bb398	Switch to C++17 With C++17, optionals, an experimental language feature, were moved to the “std” namespace. This makes C++17 mandatory and drops the now obsolete “experimental” namespace.	2018-03-24 16:09:52 +01:00
Patrick Lühne	c7d1026a31	Switch Travis to Docker As Travis only provides outdated packages (compilers in particular), this changes the Travis configuration to use Docker images to build and test the code. This also has the benefit that multiple distributions can be tested and not just Ubuntu. For the time being, Arch Linux and Ubuntu 18.04 are added as supported platforms.	2018-03-24 15:51:20 +01:00
Patrick Lühne	6b1cf6735e	Update clingo to 5.2.2	2018-03-21 16:41:08 +01:00
Patrick Lühne	addc65e3c5	Update Catch to 2.2.1	2018-03-21 16:35:54 +01:00