Version bump for release 0.1.9 RC 3

Fix integer detection
Clingo treats operations that were assumed to be “invalid” not as processing errors but as operations returning an empty set. This changes how formulas have to be evaluated. This commit implements an explicit function for retrieving the return type of an expression, that is, both the domain of the result as well as whether it’s an empty, unit, or general set with multiple values.
2018-04-22 17:31:43 +02:00 · 2018-04-22 17:04:15 +02:00 · 2018-04-22 14:57:16 +02:00 · 2018-04-22 14:56:58 +02:00 · 2018-04-21 18:48:57 +02:00 · 2018-04-21 18:12:43 +02:00
37 changed files with 1627 additions and 563 deletions
--- a/.ci/Dockerfile-ubuntu-18.04
+++ b/.ci/Dockerfile-ubuntu-18.04
@@ -3,7 +3,7 @@ FROM ubuntu:18.04
 ARG toolchain

 RUN apt-get update
-RUN apt-get install -y cmake git ninja-build re2c
+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

--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1,10 +1,21 @@
 # Change Log

-## 0.1.8 RC 1 (2018-04-10)
+## 0.1.9 RC 3 (2018-04-22)

 ### Features

-* more, advanced simplification rules
+* optional detection of integer variables and integer predicate parameters
+* command-line option `--detect-integers` to enable integer variable detection
+* support for declaring functions integer with the `#external` directive
+* new simplification rule applying to integer variables
+
+## 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)

--- a/README.md
+++ b/README.md
@@ -9,10 +9,11 @@
 ## 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

--- a/app/main.cpp
+++ b/app/main.cpp
@@ -18,6 +18,7 @@ int main(int argc, char **argv)
 		("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"));
@@ -50,6 +51,7 @@ int main(int argc, char **argv)

 		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>();
@@ -70,7 +72,7 @@ int main(int argc, char **argv)

 	if (version)
 	{
-		std::cout << "anthem version 0.1.8-rc.1" << std::endl;
+		std::cout << "anthem version 0.1.9-rc.3" << std::endl;
 		return EXIT_SUCCESS;
 	}

--- a/examples/graph-coloring.lp
+++ b/examples/graph-coloring.lp
@@ -3,7 +3,8 @@
 #external vertex(1).
 #show color/2.

-{color(V,C)} :- vertex(V), color(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(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
@@ -1,7 +1,9 @@
+#show in/2.
+%#external integer(n(0)).
+%#external integer(r(0)).
+
 {in(1..n, 1..r)}.
 covered(I) :- in(I, S).

 :- I = 1..n, not covered(I).
 :- in(I, S), in(J, S), in(I + J, S).
-
-#show in/2.
--- a/include/anthem/AST.h
+++ b/include/anthem/AST.h
@@ -2,6 +2,7 @@
 #define __ANTHEM__AST_H

 #include <anthem/ASTForward.h>
+#include <anthem/Utils.h>

 namespace anthem
 {
@@ -32,7 +33,8 @@ struct BinaryOperation
 		Minus,
 		Multiplication,
 		Division,
-		Modulo
+		Modulo,
+		Power
 	};

 	explicit BinaryOperation(Operator operator_, Term &&left, Term &&right)
@@ -102,32 +104,15 @@ 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 +122,37 @@ 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::Noninteger};
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
 // TODO: refactor (limit element type to primitive terms)
 struct In
 {
@@ -203,13 +213,13 @@ 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 +229,47 @@ 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 +314,30 @@ 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 +381,7 @@ 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
@@ -36,12 +36,6 @@ class VariableStack
 		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, Context &context);

 ////////////////////////////////////////////////////////////////////////////////////////////////////
 // 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::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++)
@@ -93,7 +96,8 @@ struct BodyTermTranslateVisitor
 	}

 	template<class T>
-	std::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>());

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

 struct BodyLiteralTranslateVisitor
 {
-	std::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::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::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)
@@ -132,15 +136,15 @@ 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::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::nullopt;
@@ -151,16 +155,16 @@ struct BodyLiteralTranslateVisitor

 struct BodyBodyLiteralTranslateVisitor
 {
-	std::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::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::nullopt;
--- a/include/anthem/Context.h
+++ b/include/anthem/Context.h
@@ -16,9 +16,9 @@ namespace anthem
 //
 ////////////////////////////////////////////////////////////////////////////////////////////////////

-struct PredicateSignatureMeta
+struct PredicateDeclarationMeta
 {
-	ast::PredicateSignature predicateSignature;
+	ast::PredicateDeclaration *declaration;
 	bool used{false};
 };

@@ -33,13 +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::optional<std::vector<PredicateSignatureMeta>> visiblePredicateSignatures;
-	std::optional<std::vector<PredicateSignatureMeta>> externalPredicateSignatures;
+	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/Equality.h
+++ b/include/anthem/Equality.h
@@ -3,6 +3,7 @@

 #include <anthem/AST.h>
 #include <anthem/ASTUtils.h>
+#include <anthem/Utils.h>

 namespace anthem
 {
@@ -15,16 +16,6 @@ namespace ast
 //
 ////////////////////////////////////////////////////////////////////////////////////////////////////

-// TODO: move to separate class
-enum class Tristate
-{
-	True,
-	False,
-	Unknown,
-};
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
 Tristate equal(const Formula &lhs, const Formula &rhs);
 Tristate equal(const Term &lhs, const Term &rhs);

@@ -237,7 +228,7 @@ struct FormulaEqualityVisitor

 		const auto &otherPredicate = otherFormula.get<Predicate>();

-		if (!matches(predicate, otherPredicate))
+		if (predicate.declaration != otherPredicate.declaration)
 			return Tristate::False;

 		assert(predicate.arguments.size() == otherPredicate.arguments.size());
@@ -298,18 +289,6 @@ struct TermEqualityVisitor
 			: Tristate::False;
 	}

-	Tristate visit(const Constant &constant, const Term &otherTerm)
-	{
-		if (!otherTerm.is<Constant>())
-			return Tristate::Unknown;
-
-		const auto &otherConstant = otherTerm.get<Constant>();
-
-		return (constant.name == otherConstant.name)
-			? Tristate::True
-			: Tristate::False;
-	}
-
 	Tristate visit(const Function &function, const Term &otherTerm)
 	{
 		if (!otherTerm.is<Function>())
@@ -317,7 +296,7 @@ struct TermEqualityVisitor

 		const auto &otherFunction = otherTerm.get<Function>();

-		if (function.name != otherFunction.name)
+		if (function.declaration != otherFunction.declaration)
 			return Tristate::False;

 		if (function.arguments.size() != otherFunction.arguments.size())
@@ -382,6 +361,19 @@ struct TermEqualityVisitor
 			: 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>())
--- a/include/anthem/Head.h
+++ b/include/anthem/Head.h
@@ -118,8 +118,7 @@ struct HeadLiteralCollectFunctionTermsVisitor

 struct FunctionTermTranslateVisitor
 {
-	// TODO: check correctness
-	std::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");
@@ -130,11 +129,14 @@ 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::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::nullopt;
@@ -145,18 +147,18 @@ struct FunctionTermTranslateVisitor

 struct LiteralTranslateVisitor
 {
-	std::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::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::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::nullopt;
@@ -167,12 +169,12 @@ struct LiteralTranslateVisitor

 struct HeadLiteralTranslateToConsequentVisitor
 {
-	std::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;
@@ -183,7 +185,7 @@ struct HeadLiteralTranslateToConsequentVisitor
 		return ast::Formula::make<ast::Not>(std::move(translatedLiteral.value()));
 	}

-	std::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::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)
@@ -238,7 +240,7 @@ struct HeadLiteralTranslateToConsequentVisitor
 	}

 	template<class T>
-	std::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::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
@@ -12,14 +12,6 @@ namespace anthem
 //
 ////////////////////////////////////////////////////////////////////////////////////////////////////

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

 ////////////////////////////////////////////////////////////////////////////////////////////////////
--- a/include/anthem/SimplificationVisitors.h
+++ b/include/anthem/SimplificationVisitors.h
@@ -3,6 +3,7 @@

 #include <anthem/AST.h>
 #include <anthem/Simplification.h>
+#include <anthem/Utils.h>

 namespace anthem
 {
@@ -19,96 +20,96 @@ template<class T>
 struct FormulaSimplificationVisitor
 {
 	template <class... Arguments>
-	SimplificationResult visit(And &and_, Formula &formula, Arguments &&... arguments)
+	OperationResult 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;
+			if (argument.accept(*this, argument, std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+				return OperationResult::Changed;

 		return T::accept(formula, std::forward<Arguments>(arguments)...);
 	}

 	template <class... Arguments>
-	SimplificationResult visit(Biconditional &biconditional, Formula &formula, Arguments &&... arguments)
+	OperationResult 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.left.accept(*this, biconditional.left, std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+			return OperationResult::Changed;

-		if (biconditional.right.accept(*this, biconditional.right, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
-			return SimplificationResult::Simplified;
+		if (biconditional.right.accept(*this, biconditional.right, std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+			return OperationResult::Changed;

 		return T::accept(formula, std::forward<Arguments>(arguments)...);
 	}

 	template <class... Arguments>
-	SimplificationResult visit(Boolean &, Formula &formula, Arguments &&... arguments)
+	OperationResult visit(Boolean &, Formula &formula, Arguments &&... arguments)
 	{
 		return T::accept(formula, std::forward<Arguments>(arguments)...);
 	}

 	template <class... Arguments>
-	SimplificationResult visit(Comparison &, Formula &formula, Arguments &&... arguments)
+	OperationResult 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)
+	OperationResult visit(Exists &exists, Formula &formula, Arguments &&... arguments)
 	{
-		if (exists.argument.accept(*this, exists.argument, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
-			return SimplificationResult::Simplified;
+		if (exists.argument.accept(*this, exists.argument, std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+			return OperationResult::Changed;

 		return T::accept(formula, std::forward<Arguments>(arguments)...);
 	}

 	template <class... Arguments>
-	SimplificationResult visit(ForAll &forAll, Formula &formula, Arguments &&... arguments)
+	OperationResult visit(ForAll &forAll, Formula &formula, Arguments &&... arguments)
 	{
-		if (forAll.argument.accept(*this, forAll.argument, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
-			return SimplificationResult::Simplified;
+		if (forAll.argument.accept(*this, forAll.argument, std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+			return OperationResult::Changed;

 		return T::accept(formula, std::forward<Arguments>(arguments)...);
 	}

 	template <class... Arguments>
-	SimplificationResult visit(Implies &implies, Formula &formula, Arguments &&... arguments)
+	OperationResult 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.antecedent.accept(*this, implies.antecedent, std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+			return OperationResult::Changed;

-		if (implies.consequent.accept(*this, implies.consequent, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
-			return SimplificationResult::Simplified;
+		if (implies.consequent.accept(*this, implies.consequent, std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+			return OperationResult::Changed;

 		return T::accept(formula, std::forward<Arguments>(arguments)...);
 	}

 	template <class... Arguments>
-	SimplificationResult visit(In &, Formula &formula, Arguments &&... arguments)
+	OperationResult 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)
+	OperationResult visit(Not &not_, Formula &formula, Arguments &&... arguments)
 	{
-		if (not_.argument.accept(*this, not_.argument, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
-			return SimplificationResult::Simplified;
+		if (not_.argument.accept(*this, not_.argument, std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+			return OperationResult::Changed;

 		return T::accept(formula, std::forward<Arguments>(arguments)...);
 	}

 	template <class... Arguments>
-	SimplificationResult visit(Or &or_, Formula &formula, Arguments &&... arguments)
+	OperationResult 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;
+			if (argument.accept(*this, argument, std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+				return OperationResult::Changed;

 		return T::accept(formula, std::forward<Arguments>(arguments)...);
 	}

 	template <class... Arguments>
-	SimplificationResult visit(Predicate &, Formula &formula, Arguments &&... arguments)
+	OperationResult visit(Predicate &, Formula &formula, Arguments &&... arguments)
 	{
 		return T::accept(formula, std::forward<Arguments>(arguments)...);
 	}
@@ -116,75 +117,69 @@ struct FormulaSimplificationVisitor

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

-template<class T, class SimplificationResult = void>
+template<class T, class OperationResult = void>
 struct TermSimplificationVisitor
 {
 	template <class... Arguments>
-	SimplificationResult visit(BinaryOperation &binaryOperation, Term &term, Arguments &&... arguments)
+	OperationResult 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.left.accept(*this, binaryOperation.left, std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+			return OperationResult::Changed;

-		if (binaryOperation.right.accept(*this, binaryOperation.right, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
-			return SimplificationResult::Simplified;
+		if (binaryOperation.right.accept(*this, binaryOperation.right, std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+			return OperationResult::Changed;

 		return T::accept(term, std::forward<Arguments>(arguments)...);
 	}

 	template <class... Arguments>
-	SimplificationResult visit(Boolean &, Term &term, Arguments &&... arguments)
+	OperationResult visit(Boolean &, Term &term, Arguments &&... arguments)
 	{
 		return T::accept(term, std::forward<Arguments>(arguments)...);
 	}

 	template <class... Arguments>
-	SimplificationResult visit(Constant &, Term &term, Arguments &&... arguments)
-	{
-		return T::accept(term, std::forward<Arguments>(arguments)...);
-	}
-
-	template <class... Arguments>
-	SimplificationResult visit(Function &function, Term &term, Arguments &&... arguments)
+	OperationResult 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;
+			if (argument.accept(*this, argument, std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+				return OperationResult::Changed;

 		return T::accept(term, std::forward<Arguments>(arguments)...);
 	}

 	template <class... Arguments>
-	SimplificationResult visit(Integer &, Term &term, Arguments &&... arguments)
+	OperationResult 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)
+	OperationResult 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.from.accept(*this, interval.from, std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+			return OperationResult::Changed;

-		if (interval.to.accept(*this, interval.to, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
-			return SimplificationResult::Simplified;
+		if (interval.to.accept(*this, interval.to, std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+			return OperationResult::Changed;

 		return T::accept(term, std::forward<Arguments>(arguments)...);
 	}

 	template <class... Arguments>
-	SimplificationResult visit(SpecialInteger &, Term &term, Arguments &&... arguments)
+	OperationResult visit(SpecialInteger &, Term &term, Arguments &&... arguments)
 	{
 		return T::accept(term, std::forward<Arguments>(arguments)...);
 	}

 	template <class... Arguments>
-	SimplificationResult visit(String &, Term &term, Arguments &&... arguments)
+	OperationResult visit(String &, Term &term, Arguments &&... arguments)
 	{
 		return T::accept(term, std::forward<Arguments>(arguments)...);
 	}

 	template <class... Arguments>
-	SimplificationResult visit(Variable &, Term &term, Arguments &&... arguments)
+	OperationResult visit(Variable &, Term &term, Arguments &&... arguments)
 	{
 		return T::accept(term, std::forward<Arguments>(arguments)...);
 	}
--- 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)
 		{
@@ -176,8 +176,8 @@ struct StatementVisitor

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

-		auto predicateSignature = ast::PredicateSignature{std::string(signature.name()), signature.arity()};
-		context.visiblePredicateSignatures.value().emplace_back(PredicateSignatureMeta{std::move(predicateSignature)});
+		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 &)
@@ -190,7 +190,7 @@ struct StatementVisitor
 		const auto fail =
 			[&]()
 			{
-				throw LogicException(statement.location, "only #external declarations of the form “#external <predicate name>(<arity>).” supported");
+				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())
@@ -204,6 +204,47 @@ struct StatementVisitor
 		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 = Domain::Integer;
+
+				return true;
+			};
+
+		if (handleIntegerDeclaration())
+			return;
+
 		const auto &arityArgument = predicate.arguments.front();

 		if (!arityArgument.data.is<Clingo::Symbol>())
@@ -214,13 +255,12 @@ struct StatementVisitor
 		if (aritySymbol.type() != Clingo::SymbolType::Number)
 			fail();

-		const size_t arity = arityArgument.data.get<Clingo::Symbol>().number();
+		context.externalStatementsUsed = true;

-		if (!context.externalPredicateSignatures)
-			context.externalPredicateSignatures.emplace();
+		const size_t arity = aritySymbol.number();

-		auto predicateSignature = ast::PredicateSignature{std::string(predicate.name), arity};
-		context.externalPredicateSignatures->emplace_back(PredicateSignatureMeta{std::move(predicateSignature)});
+		auto predicateDeclaration = context.findOrCreatePredicateDeclaration(predicate.name, arity);
+		predicateDeclaration->isExternal = true;
 	}

 	template<class T>
--- 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::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())
 		{
@@ -62,19 +85,19 @@ 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);
@@ -84,7 +107,7 @@ struct TermTranslateVisitor
 		return std::nullopt;
 	}

-	std::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::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::nullopt;
-	}
-
-	std::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::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::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");
@@ -132,12 +158,14 @@ 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::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::nullopt;
@@ -146,9 +174,9 @@ struct TermTranslateVisitor

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

-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/Type.h
+++ b/include/anthem/Type.h
@@ -0,0 +1,166 @@
+#ifndef __ANTHEM__ARITHMETICS_H
+#define __ANTHEM__ARITHMETICS_H
+
+#include <anthem/AST.h>
+#include <anthem/Utils.h>
+
+namespace anthem
+{
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+//
+// Arithmetics
+//
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct DefaultVariableDomainAccessor
+{
+	Domain operator()(const ast::Variable &variable)
+	{
+		return variable.declaration->domain;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+template <class VariableDomainAccessor = DefaultVariableDomainAccessor, class... Arguments>
+Type type(const ast::Term &term, Arguments &&...);
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+template <class VariableDomainAccessor = DefaultVariableDomainAccessor>
+struct TermTypeVisitor
+{
+	template <class... Arguments>
+	static Type visit(const ast::BinaryOperation &binaryOperation, Arguments &&... arguments)
+	{
+		const auto leftType = type<VariableDomainAccessor>(binaryOperation.left, std::forward<Arguments>(arguments)...);
+		const auto rightType = type<VariableDomainAccessor>(binaryOperation.right, std::forward<Arguments>(arguments)...);
+
+		// Binary operations on empty sets return an empty set (also with division)
+		if (leftType.setSize == SetSize::Empty || rightType.setSize == SetSize::Empty)
+			return {Domain::Unknown, SetSize::Empty};
+
+		// Binary operations on nonintegers return an empty set (also with division)
+		if (leftType.domain == Domain::Noninteger || rightType.domain == Domain::Noninteger)
+			return {Domain::Unknown, SetSize::Empty};
+
+		// Binary operations on unknown types return an unknown set
+		if (leftType.domain == Domain::Unknown || rightType.domain == Domain::Unknown)
+			return {Domain::Unknown, SetSize::Unknown};
+
+		// Divisions return an unknown set
+		if (binaryOperation.operator_ == ast::BinaryOperation::Operator::Division)
+			return {Domain::Integer, SetSize::Unknown};
+
+		// Binary operations on integer sets of unknown size return an integer set of unknown size
+		if (leftType.setSize == SetSize::Unknown || rightType.setSize == SetSize::Unknown)
+			return {Domain::Integer, SetSize::Unknown};
+
+		// Binary operations on integer sets with multiple elements return an integer set with multiple elements
+		if (leftType.setSize == SetSize::Multi || rightType.setSize == SetSize::Multi)
+			return {Domain::Integer, SetSize::Multi};
+
+		// Binary operations on plain integers return a plain integer
+		return {Domain::Integer, SetSize::Unit};
+	}
+
+	template <class... Arguments>
+	static Type visit(const ast::Boolean &, Arguments &&...)
+	{
+		return {Domain::Noninteger, SetSize::Unit};
+	}
+
+	template <class... Arguments>
+	static Type visit(const ast::Function &function, Arguments &&...)
+	{
+		// TODO: check that functions cannot return sets
+
+		return {function.declaration->domain, SetSize::Unit};
+	}
+
+	template <class... Arguments>
+	static Type visit(const ast::Integer &, Arguments &&...)
+	{
+		return {Domain::Integer, SetSize::Unit};
+	}
+
+	template <class... Arguments>
+	static Type visit(const ast::Interval &interval, Arguments &&... arguments)
+	{
+		const auto fromType = type<VariableDomainAccessor>(interval.from, std::forward<Arguments>(arguments)...);
+		const auto toType = type<VariableDomainAccessor>(interval.to, std::forward<Arguments>(arguments)...);
+
+		// Intervals with empty sets return an empty set
+		if (fromType.setSize == SetSize::Empty || toType.setSize == SetSize::Empty)
+			return {Domain::Unknown, SetSize::Empty};
+
+		// Intervals with nonintegers return an empty set
+		if (fromType.domain == Domain::Noninteger || toType.domain == Domain::Noninteger)
+			return {Domain::Unknown, SetSize::Empty};
+
+		// Intervals with unknown types return an unknown set
+		if (fromType.domain == Domain::Unknown || toType.domain == Domain::Unknown)
+			return {Domain::Unknown, SetSize::Unknown};
+
+		// Intervals with integers generally return integer sets
+		// TODO: handle 1-element intervals such as 1..1 and empty intervals such as 2..1
+		return {Domain::Integer, SetSize::Unknown};
+	}
+
+	template <class... Arguments>
+	static Type visit(const ast::SpecialInteger &, Arguments &&...)
+	{
+		return {Domain::Noninteger, SetSize::Unit};
+	}
+
+	template <class... Arguments>
+	static Type visit(const ast::String &, Arguments &&...)
+	{
+		return {Domain::Noninteger, SetSize::Unit};
+	}
+
+	template <class... Arguments>
+	static Type visit(const ast::UnaryOperation &unaryOperation, Arguments &&... arguments)
+	{
+		assert(unaryOperation.operator_ == ast::UnaryOperation::Operator::Absolute);
+
+		const auto argumentType = type<VariableDomainAccessor>(unaryOperation.argument, std::forward<Arguments>(arguments)...);
+
+		// Absolute value of an empty set returns an empty set
+		if (argumentType.setSize == SetSize::Empty)
+			return {Domain::Unknown, SetSize::Empty};
+
+		// Absolute value of nonintegers returns an empty set
+		if (argumentType.domain == Domain::Noninteger)
+			return {Domain::Unknown, SetSize::Empty};
+
+		// Absolute value of integers returns the same type
+		if (argumentType.domain == Domain::Integer)
+			return argumentType;
+
+		return {Domain::Unknown, SetSize::Unknown};
+	}
+
+	template <class... Arguments>
+	static Type visit(const ast::Variable &variable, Arguments &&... arguments)
+	{
+		const auto domain = VariableDomainAccessor()(variable, std::forward<Arguments>(arguments)...);
+
+		return {domain, SetSize::Unit};
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+template <class VariableDomainAccessor, class... Arguments>
+Type type(const ast::Term &term, Arguments &&... arguments)
+{
+	return term.accept(TermTypeVisitor<VariableDomainAccessor>(), std::forward<Arguments>(arguments)...);
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+}
+
+#endif
--- a/include/anthem/Utils.h
+++ b/include/anthem/Utils.h
@@ -1,13 +1,6 @@
 #ifndef __ANTHEM__UTILS_H
 #define __ANTHEM__UTILS_H

-#include <iostream>
-
-#include <clingo.hh>
-
-#include <anthem/Context.h>
-#include <anthem/Location.h>
-
 namespace anthem
 {

@@ -23,6 +16,60 @@ constexpr const auto UserVariablePrefix = "U";

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

+enum class Tristate
+{
+	True,
+	False,
+	Unknown,
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+enum class EvaluationResult
+{
+	True,
+	False,
+	Unknown,
+	Error,
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+enum class OperationResult
+{
+	Unchanged,
+	Changed,
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+enum class Domain
+{
+	Noninteger,
+	Integer,
+	Unknown,
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+enum class SetSize
+{
+	Empty,
+	Unit,
+	Multi,
+	Unknown,
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct Type
+{
+	Domain domain{Domain::Unknown};
+	SetSize setSize{SetSize::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::Noninteger:
+					return "n";
+				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/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
@@ -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))
@@ -192,84 +193,5 @@ struct CollectFreeVariablesVisitor

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

-struct CollectPredicateSignaturesVisitor : public RecursiveFormulaVisitor<CollectPredicateSignaturesVisitor>
-{
-	static void accept(const Predicate &predicate, const Formula &, std::vector<PredicateSignature> &predicateSignatures, Context &context)
-	{
-		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
-		auto predicateSignature = PredicateSignature(std::string(predicate.name), predicate.arity());
-
-		// Ignore predicates that are declared #external
-		if (context.externalPredicateSignatures)
-		{
-			const auto matchesPredicateSignature =
-				[&](const auto &otherPredicateSignature)
-				{
-					return ast::matches(predicateSignature, otherPredicateSignature.predicateSignature);
-				};
-
-			auto &externalPredicateSignatures = context.externalPredicateSignatures.value();
-
-			const auto matchingExternalPredicateSignature =
-				std::find_if(externalPredicateSignatures.begin(), externalPredicateSignatures.end(), matchesPredicateSignature);
-
-			if (matchingExternalPredicateSignature != externalPredicateSignatures.end())
-			{
-				matchingExternalPredicateSignature->used = true;
-				return;
-			}
-		}
-
-		predicateSignatures.emplace_back(std::move(predicateSignature));
-	}
-
-	// Ignore all other types of expressions
-	template<class T>
-	static void accept(const T &, const Formula &, std::vector<PredicateSignature> &, const Context &)
-	{
-	}
-};
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
-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, Context &context)
-{
-	auto &formulaMutable = const_cast<Formula &>(formula);
-	formulaMutable.accept(CollectPredicateSignaturesVisitor(), formulaMutable, predicateSignatures, context);
-}
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
 }
 }
--- 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, context);
-
-	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)};
@@ -169,77 +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;
 	}

-	auto &visiblePredicateSignatures = context.visiblePredicateSignatures.value();
+	assert(context.defaultPredicateVisibility == ast::PredicateDeclaration::Visibility::Hidden);
+
+	// 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];
+		// Check that the predicate is used and not declared #external
+		if (!predicateDeclaration->isUsed || predicateDeclaration->isExternal)
+			continue;

-		const auto matchesPredicateSignature =
-			[&](const auto &otherPredicateSignature)
-			{
-				return ast::matches(predicateSignature, otherPredicateSignature.predicateSignature);
-			};
+		i++;

-		const auto matchingVisiblePredicateSignature =
-			std::find_if(visiblePredicateSignatures.begin(), visiblePredicateSignatures.end(), matchesPredicateSignature);
+		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 (matchingVisiblePredicateSignature != visiblePredicateSignatures.end())
-		{
-			matchingVisiblePredicateSignature->used = true;
+		if (isPredicateVisible)
 			continue;
-		}

-		// Check that the predicate is not declared #external
-		if (context.externalPredicateSignatures)
-		{
-			const auto &externalPredicateSignatures = context.externalPredicateSignatures.value();
-
-			const auto matchingExternalPredicateSignature =
-				std::find_if(externalPredicateSignatures.cbegin(), externalPredicateSignatures.cend(), matchesPredicateSignature);
-
-			if (matchingExternalPredicateSignature != externalPredicateSignatures.cend())
-				continue;
-		}
-
-		context.logger.log(output::Priority::Debug) << "eliminating “" << predicateSignature.name << "/" << predicateSignature.arity << "”";
+		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 +1,540 @@
+#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/Type.h>
+#include <anthem/Utils.h>
+#include <anthem/output/AST.h>
+
+namespace anthem
+{
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+//
+// IntegerVariableDetection
+//
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+using VariableDomainMap = std::map<const ast::VariableDeclaration *, Domain>;
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+Domain domain(const ast::Variable &variable, VariableDomainMap &variableDomainMap)
+{
+	if (variable.declaration->domain != Domain::Unknown)
+		return variable.declaration->domain;
+
+	const auto match = variableDomainMap.find(variable.declaration);
+
+	if (match == variableDomainMap.end())
+		return Domain::Unknown;
+
+	return match->second;
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+void clearVariableDomainMap(VariableDomainMap &variableDomainMap)
+{
+	for (auto &variableDeclaration : variableDomainMap)
+		variableDeclaration.second = Domain::Unknown;
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct VariableDomainMapAccessor
+{
+	Domain operator()(const ast::Variable &variable, VariableDomainMap &variableDomainMap)
+	{
+		return domain(variable, variableDomainMap);
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+Type type(const ast::Term &term, VariableDomainMap &variableDomainMap)
+{
+	return type<VariableDomainMapAccessor>(term, 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 leftType = type(comparison.left, variableDomainMap);
+		const auto rightType = type(comparison.right, variableDomainMap);
+
+		// Comparisons with empty sets always return false
+		if (leftType.setSize == SetSize::Empty || rightType.setSize == SetSize::Empty)
+			return EvaluationResult::False;
+
+		// If either side has an unknown domain, the result is unknown
+		if (leftType.domain == Domain::Unknown || rightType.domain == Domain::Unknown)
+			return EvaluationResult::Unknown;
+
+		// If both sides have the same domain, the result is unknown
+		if (leftType.domain == rightType.domain)
+			return EvaluationResult::Unknown;
+
+		// If one side is integer, but the other one isn’t, they are not equal
+		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 elementType = type(in.element, variableDomainMap);
+		const auto setType = type(in.set, variableDomainMap);
+
+		// The element to test shouldn’t be empty or a proper set by itself
+		assert(elementType.setSize != SetSize::Empty && elementType.setSize != SetSize::Multi);
+
+		// If the set is empty, no element can be selected
+		if (setType.setSize == SetSize::Empty)
+			return EvaluationResult::False;
+
+		// If one of the sides has an unknown type, the result is unknown
+		if (elementType.domain == Domain::Unknown || setType.domain == Domain::Unknown)
+			return EvaluationResult::Unknown;
+
+		// If both sides have the same domain, the result is unknown
+		if (elementType.domain == setType.domain)
+			return EvaluationResult::Unknown;
+
+		// If one side is integer, 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 != Domain::Integer)
+				continue;
+
+			const auto argumentType = type(argument, variableDomainMap);
+
+			if (argumentType.domain == Domain::Noninteger || argumentType.setSize == SetSize::Empty)
+				return EvaluationResult::Error;
+		}
+
+		return EvaluationResult::Unknown;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+EvaluationResult evaluate(const ast::Formula &formula, VariableDomainMap &variableDomainMap)
+{
+	return formula.accept(EvaluateFormulaVisitor(), variableDomainMap);
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+template <class Functor>
+struct ForEachVariableDeclarationVisitor
+{
+	template <class... Arguments>
+	static OperationResult visit(ast::And &and_, Arguments &&... arguments)
+	{
+		auto operationResult = OperationResult::Unchanged;
+
+		for (auto &argument : and_.arguments)
+			if (argument.accept(ForEachVariableDeclarationVisitor(), std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+				operationResult = OperationResult::Changed;
+
+		return operationResult;
+	}
+
+	template <class... Arguments>
+	static OperationResult visit(ast::Biconditional &biconditional, Arguments &&... arguments)
+	{
+		auto operationResult = OperationResult::Unchanged;
+
+		if (biconditional.left.accept(ForEachVariableDeclarationVisitor(), std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+			operationResult = OperationResult::Changed;
+
+		if (biconditional.right.accept(ForEachVariableDeclarationVisitor(), std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+			operationResult = OperationResult::Changed;
+
+		return operationResult;
+	}
+
+	template <class... Arguments>
+	static OperationResult visit(ast::Boolean &, Arguments &&...)
+	{
+		return OperationResult::Unchanged;
+	}
+
+	template <class... Arguments>
+	static OperationResult visit(ast::Comparison &, Arguments &&...)
+	{
+		return OperationResult::Unchanged;
+	}
+
+	template <class... Arguments>
+	static OperationResult visit(ast::Exists &exists, Arguments &&... arguments)
+	{
+		auto operationResult = OperationResult::Unchanged;
+
+		if (exists.argument.accept(ForEachVariableDeclarationVisitor(), std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+			operationResult = OperationResult::Changed;
+
+		for (auto &variableDeclaration : exists.variables)
+			if (Functor()(*variableDeclaration, exists.argument, std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+				operationResult = OperationResult::Changed;
+
+		return operationResult;
+	}
+
+	template <class... Arguments>
+	static OperationResult visit(ast::ForAll &forAll, Arguments &&... arguments)
+	{
+		auto operationResult = OperationResult::Unchanged;
+
+		if (forAll.argument.accept(ForEachVariableDeclarationVisitor(), std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+			operationResult = OperationResult::Changed;
+
+		for (auto &variableDeclaration : forAll.variables)
+			if (Functor()(*variableDeclaration, forAll.argument, std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+				operationResult = OperationResult::Changed;
+
+		return operationResult;
+	}
+
+	template <class... Arguments>
+	static OperationResult visit(ast::Implies &implies, Arguments &&... arguments)
+	{
+		auto operationResult = OperationResult::Unchanged;
+
+		if (implies.antecedent.accept(ForEachVariableDeclarationVisitor(), std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+			operationResult = OperationResult::Changed;
+
+		if (implies.consequent.accept(ForEachVariableDeclarationVisitor(), std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+			operationResult = OperationResult::Changed;
+
+		return operationResult;
+	}
+
+	template <class... Arguments>
+	static OperationResult visit(ast::In &, Arguments &&...)
+	{
+		return OperationResult::Unchanged;
+	}
+
+	template <class... Arguments>
+	static OperationResult visit(ast::Not &not_, Arguments &&... arguments)
+	{
+		return not_.argument.accept(ForEachVariableDeclarationVisitor(), std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	static OperationResult visit(ast::Or &or_, Arguments &&... arguments)
+	{
+		auto operationResult = OperationResult::Unchanged;
+
+		for (auto &argument : or_.arguments)
+			if (argument.accept(ForEachVariableDeclarationVisitor(), std::forward<Arguments>(arguments)...) == OperationResult::Changed)
+				operationResult = OperationResult::Changed;
+
+		return operationResult;
+	}
+
+	template <class... Arguments>
+	static OperationResult visit(ast::Predicate &, Arguments &&...)
+	{
+		return OperationResult::Unchanged;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct CheckIfDefinitionFalseFunctor
+{
+	OperationResult operator()(ast::VariableDeclaration &variableDeclaration,
+		ast::Formula &, ast::Formula &definition, VariableDomainMap &variableDomainMap)
+	{
+		if (variableDeclaration.domain != Domain::Unknown)
+			return OperationResult::Unchanged;
+
+		clearVariableDomainMap(variableDomainMap);
+
+		// As a hypothesis, make the parameter’s domain noninteger
+		variableDomainMap[&variableDeclaration] = Domain::Noninteger;
+
+		const auto 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
+			variableDeclaration.domain = Domain::Integer;
+			return OperationResult::Changed;
+		}
+
+		return OperationResult::Unchanged;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct CheckIfQuantifiedFormulaFalseFunctor
+{
+	OperationResult operator()(ast::VariableDeclaration &variableDeclaration,
+		ast::Formula &quantifiedFormula, VariableDomainMap &variableDomainMap)
+	{
+		if (variableDeclaration.domain != Domain::Unknown)
+			return OperationResult::Unchanged;
+
+		clearVariableDomainMap(variableDomainMap);
+
+		// As a hypothesis, make the parameter’s domain noninteger
+		variableDomainMap[&variableDeclaration] = Domain::Noninteger;
+
+		const auto result = evaluate(quantifiedFormula, 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
+			variableDeclaration.domain = Domain::Integer;
+			return OperationResult::Changed;
+		}
+
+		return OperationResult::Unchanged;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct CheckIfCompletedFormulaTrueFunctor
+{
+	OperationResult operator()(ast::VariableDeclaration &variableDeclaration,
+		ast::Formula &, ast::Formula &completedFormula, VariableDomainMap &variableDomainMap)
+	{
+		if (variableDeclaration.domain != Domain::Unknown)
+			return OperationResult::Unchanged;
+
+		clearVariableDomainMap(variableDomainMap);
+
+		// As a hypothesis, make the parameter’s domain noninteger
+		variableDomainMap[&variableDeclaration] = Domain::Noninteger;
+
+		const auto result = evaluate(completedFormula, variableDomainMap);
+
+		if (result == EvaluationResult::Error || result == EvaluationResult::True)
+		{
+			// If making the variable noninteger leads to a false or erroneous result, it’s proven to be integer
+			variableDeclaration.domain = Domain::Integer;
+			return OperationResult::Changed;
+		}
+
+		return OperationResult::Unchanged;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+// 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 (completedFormula.accept(ForEachVariableDeclarationVisitor<CheckIfQuantifiedFormulaFalseFunctor>(), variableDomainMap) == OperationResult::Changed)
+				operationResult = OperationResult::Changed;
+
+			if (completedFormula.accept(ForEachVariableDeclarationVisitor<CheckIfCompletedFormulaTrueFunctor>(), 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;
+
+			if (completedFormula.accept(ForEachVariableDeclarationVisitor<CheckIfDefinitionFalseFunctor>(), definition, variableDomainMap) == OperationResult::Changed)
+				operationResult = OperationResult::Changed;
+
+			assert(predicate.arguments.size() == predicate.declaration->arity());
+
+			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>();
+
+				parameter.domain = variable.declaration->domain;
+			}
+		}
+	}
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+}
--- a/src/anthem/Simplification.cpp
+++ b/src/anthem/Simplification.cpp
@@ -4,8 +4,9 @@

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

 namespace anthem
 {
@@ -100,7 +101,7 @@ struct ReplaceVariableInFormulaVisitor : public ast::RecursiveFormulaVisitor<Rep
 ////////////////////////////////////////////////////////////////////////////////////////////////////

 template<class SimplificationRule>
-SimplificationResult simplify(ast::Formula &formula)
+OperationResult simplify(ast::Formula &formula)
 {
 	return SimplificationRule::apply(formula);
 }
@@ -108,10 +109,10 @@ SimplificationResult simplify(ast::Formula &formula)
 ////////////////////////////////////////////////////////////////////////////////////////////////////

 template<class FirstSimplificationRule, class SecondSimplificationRule, class... OtherSimplificationRules>
-SimplificationResult simplify(ast::Formula &formula)
+OperationResult simplify(ast::Formula &formula)
 {
-	if (simplify<FirstSimplificationRule>(formula) == SimplificationResult::Simplified)
-		return SimplificationResult::Simplified;
+	if (simplify<FirstSimplificationRule>(formula) == OperationResult::Changed)
+		return OperationResult::Changed;

 	return simplify<SecondSimplificationRule, OtherSimplificationRules...>(formula);
 }
@@ -122,19 +123,19 @@ struct SimplificationRuleExistsWithoutQuantifiedVariables
 {
 	static constexpr const auto Description = "exists () (F) === F";

-	static SimplificationResult apply(ast::Formula &formula)
+	static OperationResult apply(ast::Formula &formula)
 	{
 		if (!formula.is<ast::Exists>())
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		auto &exists = formula.get<ast::Exists>();

 		if (!exists.variables.empty())
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		formula = std::move(exists.argument);

-		return SimplificationResult::Simplified;
+		return OperationResult::Changed;
 	}
 };

@@ -144,20 +145,20 @@ struct SimplificationRuleTrivialAssignmentInExists
 {
 	static constexpr const auto Description = "exists X (X = Y) === #true";

-	static SimplificationResult apply(ast::Formula &formula)
+	static OperationResult apply(ast::Formula &formula)
 	{
 		if (!formula.is<ast::Exists>())
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		const auto &exists = formula.get<ast::Exists>();

 		if (!exists.argument.is<ast::Comparison>())
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		const auto &comparison = exists.argument.get<ast::Comparison>();

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

 		const auto matchingAssignment = std::find_if(exists.variables.cbegin(), exists.variables.cend(),
 			[&](const auto &variableDeclaration)
@@ -167,11 +168,11 @@ struct SimplificationRuleTrivialAssignmentInExists
 			});

 		if (matchingAssignment == exists.variables.cend())
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		formula = ast::Formula::make<ast::Boolean>(true);

-		return SimplificationResult::Simplified;
+		return OperationResult::Changed;
 	}
 };

@@ -181,20 +182,20 @@ struct SimplificationRuleAssignmentInExists
 {
 	static constexpr const auto Description = "exists X (X = t and F(X)) === exists () (F(t))";

-	static SimplificationResult apply(ast::Formula &formula)
+	static OperationResult apply(ast::Formula &formula)
 	{
 		if (!formula.is<ast::Exists>())
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		auto &exists = formula.get<ast::Exists>();

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

 		auto &and_ = exists.argument.get<ast::And>();
 		auto &arguments = and_.arguments;

-		auto simplificationResult = SimplificationResult::Unchanged;
+		auto simplificationResult = OperationResult::Unchanged;

 		for (auto i = exists.variables.begin(); i != exists.variables.end();)
 		{
@@ -225,7 +226,7 @@ struct SimplificationRuleAssignmentInExists
 				arguments.erase(j);

 				wasVariableReplaced = true;
-				simplificationResult = SimplificationResult::Simplified;
+				simplificationResult = OperationResult::Changed;

 				break;
 			}
@@ -249,19 +250,19 @@ struct SimplificationRuleEmptyConjunction
 {
 	static constexpr const auto Description = "[empty conjunction] === #true";

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

 		auto &and_ = formula.get<ast::And>();

 		if (!and_.arguments.empty())
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		formula = ast::Formula::make<ast::Boolean>(true);

-		return SimplificationResult::Simplified;
+		return OperationResult::Changed;
 	}
 };

@@ -271,19 +272,19 @@ struct SimplificationRuleOneElementConjunction
 {
 	static constexpr const auto Description = "[conjunction of only F] === F";

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

 		auto &and_ = formula.get<ast::And>();

 		if (and_.arguments.size() != 1)
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		formula = std::move(and_.arguments.front());

-		return SimplificationResult::Simplified;
+		return OperationResult::Changed;
 	}
 };

@@ -293,19 +294,19 @@ struct SimplificationRuleTrivialExists
 {
 	static constexpr const auto Description = "exists ... ([#true/#false]) === [#true/#false]";

-	static SimplificationResult apply(ast::Formula &formula)
+	static OperationResult apply(ast::Formula &formula)
 	{
 		if (!formula.is<ast::Exists>())
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		auto &exists = formula.get<ast::Exists>();

 		if (!exists.argument.is<ast::Boolean>())
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		formula = std::move(exists.argument);

-		return SimplificationResult::Simplified;
+		return OperationResult::Changed;
 	}
 };

@@ -315,21 +316,21 @@ struct SimplificationRuleInWithPrimitiveArguments
 {
 	static constexpr const auto Description = "[primitive A] in [primitive B] === A = B";

-	static SimplificationResult apply(ast::Formula &formula)
+	static OperationResult apply(ast::Formula &formula)
 	{
 		if (!formula.is<ast::In>())
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		auto &in = formula.get<ast::In>();

 		assert(ast::isPrimitive(in.element));

 		if (!ast::isPrimitive(in.element) || !ast::isPrimitive(in.set))
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		formula = ast::Comparison(ast::Comparison::Operator::Equal, std::move(in.element), std::move(in.set));

-		return SimplificationResult::Simplified;
+		return OperationResult::Changed;
 	}
 };

@@ -339,10 +340,10 @@ struct SimplificationRuleSubsumptionInBiconditionals
 {
 	static constexpr const auto Description = "(F <-> (F and G)) === (F -> G)";

-	static SimplificationResult apply(ast::Formula &formula)
+	static OperationResult apply(ast::Formula &formula)
 	{
 		if (!formula.is<ast::Biconditional>())
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		auto &biconditional = formula.get<ast::Biconditional>();

@@ -353,7 +354,7 @@ struct SimplificationRuleSubsumptionInBiconditionals
 		const auto rightIsAnd = biconditional.right.is<ast::And>();

 		if (!(leftIsPredicate && rightIsAnd) && !(rightIsPredicate && leftIsAnd))
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		auto &predicateSide = (leftIsPredicate ? biconditional.left : biconditional.right);
 		auto &andSide = (leftIsPredicate ? biconditional.right : biconditional.left);
@@ -363,17 +364,17 @@ struct SimplificationRuleSubsumptionInBiconditionals
 			std::find_if(and_.arguments.cbegin(), and_.arguments.cend(),
 			[&](const auto &argument)
 			{
-				return (ast::equal(predicateSide, argument) == ast::Tristate::True);
+				return (ast::equal(predicateSide, argument) == Tristate::True);
 			});

 		if (matchingPredicate == and_.arguments.cend())
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		and_.arguments.erase(matchingPredicate);

 		formula = ast::Formula::make<ast::Implies>(std::move(predicateSide), std::move(andSide));

-		return SimplificationResult::Simplified;
+		return OperationResult::Changed;
 	}
 };

@@ -383,21 +384,21 @@ struct SimplificationRuleDoubleNegation
 {
 	static constexpr const auto Description = "not not F === F";

-	static SimplificationResult apply(ast::Formula &formula)
+	static OperationResult apply(ast::Formula &formula)
 	{
 		if (!formula.is<ast::Not>())
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		auto &not_ = formula.get<ast::Not>();

 		if (!not_.argument.is<ast::Not>())
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		auto &notNot = not_.argument.get<ast::Not>();

 		formula = std::move(notNot.argument);

-		return SimplificationResult::Simplified;
+		return OperationResult::Changed;
 	}
 };

@@ -407,15 +408,15 @@ struct SimplificationRuleDeMorganForConjunctions
 {
 	static constexpr const auto Description = "(not (F and G)) === (not F or not G)";

-	static SimplificationResult apply(ast::Formula &formula)
+	static OperationResult apply(ast::Formula &formula)
 	{
 		if (!formula.is<ast::Not>())
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		auto &not_ = formula.get<ast::Not>();

 		if (!not_.argument.is<ast::And>())
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		auto &and_ = not_.argument.get<ast::And>();

@@ -424,7 +425,7 @@ struct SimplificationRuleDeMorganForConjunctions

 		formula = ast::Formula::make<ast::Or>(std::move(and_.arguments));

-		return SimplificationResult::Simplified;
+		return OperationResult::Changed;
 	}
 };

@@ -434,21 +435,21 @@ struct SimplificationRuleImplicationFromDisjunction
 {
 	static constexpr const auto Description = "(not F or G) === (F -> G)";

-	static SimplificationResult apply(ast::Formula &formula)
+	static OperationResult apply(ast::Formula &formula)
 	{
 		if (!formula.is<ast::Or>())
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;

 		auto &or_ = formula.get<ast::Or>();

 		if (or_.arguments.size() != 2)
-			return SimplificationResult::Unchanged;
+			return OperationResult::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;
+			return OperationResult::Unchanged;

 		auto &negativeSide = leftIsNot ? or_.arguments[0] : or_.arguments[1];
 		auto &positiveSide = leftIsNot ? or_.arguments[1] : or_.arguments[0];
@@ -460,7 +461,81 @@ struct SimplificationRuleImplicationFromDisjunction

 		formula = ast::Formula::make<ast::Implies>(std::move(negativeSideArgument), std::move(positiveSide));

-		return SimplificationResult::Simplified;
+		return OperationResult::Changed;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleNegatedComparison
+{
+	static constexpr const auto Description = "(not F [comparison] G) === (F [negated comparison] G)";
+
+	static OperationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::Not>())
+			return OperationResult::Unchanged;
+
+		auto &not_ = formula.get<ast::Not>();
+
+		if (!not_.argument.is<ast::Comparison>())
+			return OperationResult::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 OperationResult::Changed;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleIntegerSetInclusion
+{
+	static constexpr const auto Description = "(F in G) === (F = G) if F and G are integer variables";
+
+	static OperationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::In>())
+			return OperationResult::Unchanged;
+
+		auto &in = formula.get<ast::In>();
+
+		const auto elementType = type(in.element);
+		const auto setType = type(in.set);
+
+		if (elementType.domain != Domain::Integer || setType.domain != Domain::Integer
+		    || elementType.setSize != SetSize::Unit || setType.setSize != SetSize::Unit)
+		{
+			return OperationResult::Unchanged;
+		}
+
+		formula = ast::Formula::make<ast::Comparison>(ast::Comparison::Operator::Equal, std::move(in.element), std::move(in.set));
+
+		return OperationResult::Changed;
 	}
 };

@@ -479,7 +554,9 @@ const auto simplifyWithDefaultRules =
 		SimplificationRuleInWithPrimitiveArguments,
 		SimplificationRuleSubsumptionInBiconditionals,
 		SimplificationRuleDeMorganForConjunctions,
-		SimplificationRuleImplicationFromDisjunction
+		SimplificationRuleImplicationFromDisjunction,
+		SimplificationRuleNegatedComparison,
+		SimplificationRuleIntegerSetInclusion
 	>;

 ////////////////////////////////////////////////////////////////////////////////////////////////////
@@ -488,7 +565,7 @@ const auto simplifyWithDefaultRules =
 struct SimplifyFormulaVisitor : public ast::FormulaSimplificationVisitor<SimplifyFormulaVisitor>
 {
 	// Do nothing for all other types of expressions
-	static SimplificationResult accept(ast::Formula &formula)
+	static OperationResult accept(ast::Formula &formula)
 	{
 		return simplifyWithDefaultRules(formula);
 	}
@@ -498,7 +575,7 @@ struct SimplifyFormulaVisitor : public ast::FormulaSimplificationVisitor<Simplif

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

 ////////////////////////////////////////////////////////////////////////////////////////////////////
--- 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>
@@ -67,10 +68,10 @@ 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.externalPredicateSignatures)
+		if (context.externalStatementsUsed)
 			context.logger.log(output::Priority::Warning) << "#external statements are ignored because completion is not enabled";

 		for (const auto &scopedFormula : scopedFormulas)
@@ -85,25 +86,33 @@ void translate(const char *fileName, std::istream &stream, Context &context)
 	// Perform completion
 	auto completedFormulas = complete(std::move(scopedFormulas), context);

-	// Check for #show statements with undeclared predicates
-	if (context.visiblePredicateSignatures)
-		for (const auto &predicateSignature : context.visiblePredicateSignatures.value())
-			if (!predicateSignature.used)
-				context.logger.log(output::Priority::Warning)
-					<< "#show declaration of “"
-					<< predicateSignature.predicateSignature.name
-					<< "/" << predicateSignature.predicateSignature.arity
-					<< "” does not match any eligible predicate";
+	for (const auto &predicateDeclaration : context.predicateDeclarations)
+	{
+		if (predicateDeclaration->isUsed)
+			continue;

-	// Check for #external statements with undeclared predicates
-	if (context.externalPredicateSignatures)
-		for (const auto &predicateSignature : context.externalPredicateSignatures.value())
-			if (!predicateSignature.used)
-				context.logger.log(output::Priority::Warning)
-					<< "#external declaration of “"
-					<< predicateSignature.predicateSignature.name
-					<< "/" << predicateSignature.predicateSignature.arity
-					<< "” does not match any eligible predicate";
+		// 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)
@@ -117,6 +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 != Domain::Integer)
+				continue;
+
+			context.logger.outputStream()
+				<< output::Keyword("int")
+				<< "(" << predicateDeclaration->name
+				<< "/" << output::Number(predicateDeclaration->arity())
+				<< "@" << output::Number(i + 1)
+				<< ")." << std::endl;
+		}
+	}
 }

 ////////////////////////////////////////////////////////////////////////////////////////////////////
--- a/tests/TestCompletion.cpp
+++ b/tests/TestCompletion.cpp
@@ -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/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	c4a457e9d8	Version bump for release 0.1.9 RC 3	2018-04-22 17:31:43 +02:00
Patrick Lühne	ea885f5fdb	Fix integer detection Clingo treats operations that were assumed to be “invalid” not as processing errors but as operations returning an empty set. This changes how formulas have to be evaluated. This commit implements an explicit function for retrieving the return type of an expression, that is, both the domain of the result as well as whether it’s an empty, unit, or general set with multiple values.	2018-04-22 17:04:15 +02:00
Patrick Lühne	92cd114cf8	Rename “general” domain to “noninteger” The “general” domain wasn’t really about general variables, but meant as a distinction from integer variables. For this reason, this commit renames the “general” domain to “noninteger.”	2018-04-22 14:57:16 +02:00
Patrick Lühne	2b62c6227d	Move Domain class to Utils header	2018-04-22 14:56:58 +02:00
Patrick Lühne	529aec15af	Add integer simplifications to change log	2018-04-21 18:48:57 +02:00
Patrick Lühne	d950b8ec1a	Add integer simplification rule This adds the rule “(F in G) === (F = G) if F and G are integer variables” to the simplification rule tableau	2018-04-21 18:12:43 +02:00
Patrick Lühne	97190ace71	Add integer check This adds a function to check whether a term evaluates to a singular integer.	2018-04-21 18:12:06 +02:00
Patrick Lühne	d6a811e363	Move arithmetic check to separate header Checking whether terms are arithmetic will be used not just in integer variable detection but also in simplifying formulas with integer variables. For this purpose, the arithmetic check is moved to a commonly accessible header file.	2018-04-21 17:59:05 +02:00
Patrick Lühne	b1ca027de5	Consolidate commonly used enum classes This moves the commonly enum classes EvaluationResult, OperationResult, and Tristate to the Utils header file to avoid code duplication. Additionally, the SimplificationResult class is replaced by the semantically similar OperationResult.	2018-04-21 17:34:52 +02:00
Patrick Lühne	f190c28ea5	Add integer function declarations to change log	2018-04-21 17:26:15 +02:00
Patrick Lühne	28dbd407e2	Extend integer variable detection If making a variable noninteger turns the entire formula obtained from completion true, we can drop that statement and conclude that the variable is integer.	2018-04-21 17:26:15 +02:00
Patrick Lühne	9c792ff079	Fix incorrect index in output The indices printed in the output are meant to be indexed starting with 1 and not 0.	2018-04-21 17:09:45 +02:00
Patrick Lühne	f65d4dbbd8	Refactor integer variable detection This reimplements integer variable detection as two parts. The first one is a visitor class that recursively searches for all declared variables in a formula and applies the second part, a custom functor. Two such functors are implemented. The first one checks whether a predicate definition is falsified by making a variable noninteger, in which case it can be concluded that the variable in question is integer. The second functor checks whether bound variables in a quantified formula turn the quantified part false, again to conclude that variables are integer.	2018-04-21 17:02:42 +02:00
Patrick Lühne	8f3ff23f38	Add to-do	2018-04-21 16:58:42 +02:00
Patrick Lühne	03c22d00f5	Remove unwanted detection rule This piece of code was responsible for propagating the domain of predicate parameters to argument variables. However, this approach doesn’t seem to be correct, which is why this commit removes it.	2018-04-21 16:58:42 +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