Version bump for release 0.1.8 RC 1

This adds the rule “(not F or G) === (F -> G)” to the simplification
rule tableau.

CHANGELOG.md

@@ -1,6 +1,16 @@
 # Change Log
 
-## (unreleased)
+## 0.1.8 RC 1 (2018-04-10)
+
+### Features
+
+* more, advanced simplification rules
+
+## 0.1.7 (2018-04-08)
+
+### Features
+
+* support for declaring placeholders with the `#external` directive
 
 ### Internal
 

LICENSE.md

@@ -1,6 +1,6 @@
 # The MIT License (MIT)
 
-Copyright © 2016–2017 Patrick Lühne
+Copyright © 2016–2018 Patrick Lühne
 
 Permission is hereby granted, free of charge, to any person obtaining a copy
 of this software and associated documentation files (the "Software"), to deal

app/main.cpp

@@ -70,7 +70,7 @@ int main(int argc, char **argv)
 
 	if (version)
 	{
-		std::cout << "anthem version 0.1.7-git" << std::endl;
+		std::cout << "anthem version 0.1.8-rc.1" << std::endl;
 		return EXIT_SUCCESS;
 	}
 

examples/graph-coloring.lp

@@ -1,12 +1,9 @@
-colored(V, red) :- vertex(V), not colored(V, green), not colored(V, blue).
+#external color(1).
-colored(V, green) :- vertex(V), not colored(V, red), not colored(V, blue).
+#external edge(2).
-colored(V, blue) :- vertex(V), not colored(V, red), not colored(V, green).
+#external vertex(1).
+#show color/2.
 
-:- edge(V1, V2), colored(V1, C), colored(V2, C).
+{color(V,C)} :- vertex(V), color(C).
-
+covered(V) :- color(V, _).
-vertex(a).
+:- vertex(V), not covered(V).
-vertex(b).
+:- color(V1,C), color(V2,C), edge(V1,V2).
-vertex(c).
-
-edge(a, b).
-edge(a, c).

examples/schur-numbers.lp

@@ -3,3 +3,5 @@ covered(I) :- in(I, S).
 
 :- I = 1..n, not covered(I).
 :- in(I, S), in(J, S), in(I + J, S).
+
+#show in/2.

include/anthem/ASTUtils.h

@@ -5,6 +5,7 @@
 
 #include <anthem/AST.h>
 #include <anthem/ASTVisitors.h>
+#include <anthem/Context.h>
 
 namespace anthem
 {
@@ -40,7 +41,7 @@ class VariableStack
 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);
+void collectPredicateSignatures(const Formula &formula, std::vector<PredicateSignature> &predicateSignatures, Context &context);
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 // Replacing Variables

include/anthem/Context.h

@@ -16,6 +16,14 @@ namespace anthem
 //
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 
+struct PredicateSignatureMeta
+{
+	ast::PredicateSignature predicateSignature;
+	bool used{false};
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
 struct Context
 {
 	Context() = default;
@@ -30,7 +38,8 @@ struct Context
 	bool performSimplification = false;
 	bool performCompletion = false;
 
-	std::optional<std::vector<ast::PredicateSignature>> visiblePredicateSignatures;
+	std::optional<std::vector<PredicateSignatureMeta>> visiblePredicateSignatures;
+	std::optional<std::vector<PredicateSignatureMeta>> externalPredicateSignatures;
 
 	ast::ParenthesisStyle parenthesisStyle = ast::ParenthesisStyle::Normal;
 };

include/anthem/Equality.h

@@ -0,0 +1,417 @@
+#ifndef __ANTHEM__EQUALITY_H
+#define __ANTHEM__EQUALITY_H
+
+#include <anthem/AST.h>
+#include <anthem/ASTUtils.h>
+
+namespace anthem
+{
+namespace ast
+{
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+//
+// Equality
+//
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+// 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);
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct FormulaEqualityVisitor
+{
+	Tristate visit(const And &and_, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<And>())
+			return Tristate::Unknown;
+
+		const auto &otherAnd = otherFormula.get<And>();
+
+		for (const auto &argument : and_.arguments)
+		{
+			const auto match = std::find_if(
+				otherAnd.arguments.cbegin(), otherAnd.arguments.cend(),
+				[&](const auto &otherArgument)
+				{
+					return equal(argument, otherArgument) == Tristate::True;
+				});
+
+			if (match == otherAnd.arguments.cend())
+				return Tristate::Unknown;
+		}
+
+		for (const auto &otherArgument : otherAnd.arguments)
+		{
+			const auto match = std::find_if(
+				and_.arguments.cbegin(), and_.arguments.cend(),
+				[&](const auto &argument)
+				{
+					return equal(otherArgument, argument) == Tristate::True;
+				});
+
+			if (match == and_.arguments.cend())
+				return Tristate::Unknown;
+		}
+
+		return Tristate::True;
+	}
+
+	Tristate visit(const Biconditional &biconditional, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<Biconditional>())
+			return Tristate::Unknown;
+
+		const auto &otherBiconditional = otherFormula.get<Biconditional>();
+
+		if (equal(biconditional.left, otherBiconditional.left) == Tristate::True
+		    && equal(biconditional.right, otherBiconditional.right) == Tristate::True)
+		{
+			return Tristate::True;
+		}
+
+		if (equal(biconditional.left, otherBiconditional.right) == Tristate::True
+		    && equal(biconditional.right, otherBiconditional.left) == Tristate::True)
+		{
+			return Tristate::True;
+		}
+
+		return Tristate::Unknown;
+	}
+
+	Tristate visit(const Boolean &boolean, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<Boolean>())
+			return Tristate::Unknown;
+
+		const auto &otherBoolean = otherFormula.get<Boolean>();
+
+		return (boolean.value == otherBoolean.value)
+			? Tristate::True
+			: Tristate::False;
+	}
+
+	Tristate visit(const Comparison &comparison, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<Comparison>())
+			return Tristate::Unknown;
+
+		const auto &otherComparison = otherFormula.get<Comparison>();
+
+		if (comparison.operator_ != otherComparison.operator_)
+			return Tristate::Unknown;
+
+		if (equal(comparison.left, otherComparison.left) == Tristate::True
+		    && equal(comparison.right, otherComparison.right) == Tristate::True)
+		{
+			return Tristate::True;
+		}
+
+		// Only = and != are commutative operators, all others don’t need to be checked with exchanged arguments
+		if (comparison.operator_ != Comparison::Operator::Equal
+		    && comparison.operator_ != Comparison::Operator::NotEqual)
+		{
+			return Tristate::Unknown;
+		}
+
+		if (equal(comparison.left, otherComparison.right) == Tristate::True
+		    && equal(comparison.right, otherComparison.left) == Tristate::True)
+		{
+			return Tristate::True;
+		}
+
+		return Tristate::Unknown;
+	}
+
+	Tristate visit(const Exists &, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<Exists>())
+			return Tristate::Unknown;
+
+		// TODO: implement stronger check
+		return Tristate::Unknown;
+	}
+
+	Tristate visit(const ForAll &, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<ForAll>())
+			return Tristate::Unknown;
+
+		// TODO: implement stronger check
+		return Tristate::Unknown;
+	}
+
+	Tristate visit(const Implies &implies, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<Implies>())
+			return Tristate::Unknown;
+
+		const auto &otherImplies = otherFormula.get<Implies>();
+
+		if (equal(implies.antecedent, otherImplies.antecedent) == Tristate::True
+		    && equal(implies.consequent, otherImplies.consequent) == Tristate::True)
+		{
+			return Tristate::True;
+		}
+
+		return Tristate::Unknown;
+	}
+
+	Tristate visit(const In &in, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<In>())
+			return Tristate::Unknown;
+
+		const auto &otherIn = otherFormula.get<In>();
+
+		if (equal(in.element, otherIn.element) == Tristate::True
+		    && equal(in.set, otherIn.set) == Tristate::True)
+		{
+			return Tristate::True;
+		}
+
+		return Tristate::Unknown;
+	}
+
+	Tristate visit(const Not &not_, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<Not>())
+			return Tristate::Unknown;
+
+		const auto &otherNot = otherFormula.get<Not>();
+
+		return equal(not_.argument, otherNot.argument);
+	}
+
+	Tristate visit(const Or &or_, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<Or>())
+			return Tristate::Unknown;
+
+		const auto &otherOr = otherFormula.get<Or>();
+
+		for (const auto &argument : or_.arguments)
+		{
+			const auto match = std::find_if(
+				otherOr.arguments.cbegin(), otherOr.arguments.cend(),
+				[&](const auto &otherArgument)
+				{
+					return equal(argument, otherArgument) == Tristate::True;
+				});
+
+			if (match == otherOr.arguments.cend())
+				return Tristate::Unknown;
+		}
+
+		for (const auto &otherArgument : otherOr.arguments)
+		{
+			const auto match = std::find_if(
+				or_.arguments.cbegin(), or_.arguments.cend(),
+				[&](const auto &argument)
+				{
+					return equal(otherArgument, argument) == Tristate::True;
+				});
+
+			if (match == or_.arguments.cend())
+				return Tristate::Unknown;
+		}
+
+		return Tristate::True;
+	}
+
+	Tristate visit(const Predicate &predicate, const Formula &otherFormula)
+	{
+		if (!otherFormula.is<Predicate>())
+			return Tristate::Unknown;
+
+		const auto &otherPredicate = otherFormula.get<Predicate>();
+
+		if (!matches(predicate, otherPredicate))
+			return Tristate::False;
+
+		assert(predicate.arguments.size() == otherPredicate.arguments.size());
+
+		for (size_t i = 0; i < predicate.arguments.size(); i++)
+			if (equal(predicate.arguments[i], otherPredicate.arguments[i]) != Tristate::True)
+				return Tristate::Unknown;
+
+		return Tristate::True;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct TermEqualityVisitor
+{
+	Tristate visit(const BinaryOperation &binaryOperation, const Term &otherTerm)
+	{
+		if (!otherTerm.is<BinaryOperation>())
+			return Tristate::Unknown;
+
+		const auto &otherBinaryOperation = otherTerm.get<BinaryOperation>();
+
+		if (binaryOperation.operator_ != otherBinaryOperation.operator_)
+			return Tristate::Unknown;
+
+		if (equal(binaryOperation.left, otherBinaryOperation.left) == Tristate::True
+		    && equal(binaryOperation.right, otherBinaryOperation.right) == Tristate::True)
+		{
+			return Tristate::True;
+		}
+
+		// Only + and * are commutative operators, all others don’t need to be checked with exchanged arguments
+		if (binaryOperation.operator_ != BinaryOperation::Operator::Plus
+		    && binaryOperation.operator_ != BinaryOperation::Operator::Multiplication)
+		{
+			return Tristate::Unknown;
+		}
+
+		if (equal(binaryOperation.left, binaryOperation.right) == Tristate::True
+		    && equal(binaryOperation.right, binaryOperation.left) == Tristate::True)
+		{
+			return Tristate::True;
+		}
+
+		return Tristate::Unknown;
+	}
+
+	Tristate visit(const Boolean &boolean, const Term &otherTerm)
+	{
+		if (!otherTerm.is<Boolean>())
+			return Tristate::Unknown;
+
+		const auto &otherBoolean = otherTerm.get<Boolean>();
+
+		return (boolean.value == otherBoolean.value)
+			? Tristate::True
+			: Tristate::False;
+	}
+
+	Tristate visit(const 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>())
+			return Tristate::Unknown;
+
+		const auto &otherFunction = otherTerm.get<Function>();
+
+		if (function.name != otherFunction.name)
+			return Tristate::False;
+
+		if (function.arguments.size() != otherFunction.arguments.size())
+			return Tristate::False;
+
+		for (size_t i = 0; i < function.arguments.size(); i++)
+			if (equal(function.arguments[i], otherFunction.arguments[i]) != Tristate::True)
+				return Tristate::Unknown;
+
+		return Tristate::True;
+	}
+
+	Tristate visit(const Integer &integer, const Term &otherTerm)
+	{
+		if (!otherTerm.is<Integer>())
+			return Tristate::Unknown;
+
+		const auto &otherInteger = otherTerm.get<Integer>();
+
+		return (integer.value == otherInteger.value)
+			? Tristate::True
+			: Tristate::False;
+	}
+
+	Tristate visit(const Interval &interval, const Term &otherTerm)
+	{
+		if (!otherTerm.is<Interval>())
+			return Tristate::Unknown;
+
+		const auto &otherInterval = otherTerm.get<Interval>();
+
+		if (equal(interval.from, otherInterval.from) != Tristate::True)
+			return Tristate::Unknown;
+
+		if (equal(interval.to, otherInterval.to) != Tristate::True)
+			return Tristate::Unknown;
+
+		return Tristate::True;
+	}
+
+	Tristate visit(const SpecialInteger &specialInteger, const Term &otherTerm)
+	{
+		if (!otherTerm.is<SpecialInteger>())
+			return Tristate::Unknown;
+
+		const auto &otherSpecialInteger = otherTerm.get<SpecialInteger>();
+
+		return (specialInteger.type == otherSpecialInteger.type)
+			? Tristate::True
+			: Tristate::False;
+	}
+
+	Tristate visit(const String &string, const Term &otherTerm)
+	{
+		if (!otherTerm.is<String>())
+			return Tristate::Unknown;
+
+		const auto &otherString = otherTerm.get<String>();
+
+		return (string.text == otherString.text)
+			? Tristate::True
+			: Tristate::False;
+	}
+
+	Tristate visit(const Variable &variable, const Term &otherTerm)
+	{
+		if (!otherTerm.is<Variable>())
+			return Tristate::Unknown;
+
+		const auto &otherVariable = otherTerm.get<Variable>();
+
+		return (variable.declaration == otherVariable.declaration)
+			? Tristate::True
+			: Tristate::False;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+Tristate equal(const Formula &lhs, const Formula &rhs)
+{
+	return lhs.accept(FormulaEqualityVisitor(), rhs);
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+Tristate equal(const Term &lhs, const Term &rhs)
+{
+	return lhs.accept(TermEqualityVisitor(), rhs);
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+}
+}
+
+#endif

include/anthem/Simplification.h

@@ -12,6 +12,14 @@ namespace anthem
 //
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 
+enum class SimplificationResult
+{
+	Simplified,
+	Unchanged,
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
 void simplify(ast::Formula &formula);
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////

include/anthem/SimplificationVisitors.h

@@ -0,0 +1,198 @@
+#ifndef __ANTHEM__SIMPLIFICATION_VISITORS_H
+#define __ANTHEM__SIMPLIFICATION_VISITORS_H
+
+#include <anthem/AST.h>
+#include <anthem/Simplification.h>
+
+namespace anthem
+{
+namespace ast
+{
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+//
+// Simplification Visitor
+//
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+template<class T>
+struct FormulaSimplificationVisitor
+{
+	template <class... Arguments>
+	SimplificationResult visit(And &and_, Formula &formula, Arguments &&... arguments)
+	{
+		for (auto &argument : and_.arguments)
+			if (argument.accept(*this, argument, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+				return SimplificationResult::Simplified;
+
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Biconditional &biconditional, Formula &formula, Arguments &&... arguments)
+	{
+		if (biconditional.left.accept(*this, biconditional.left, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		if (biconditional.right.accept(*this, biconditional.right, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Boolean &, Formula &formula, Arguments &&... arguments)
+	{
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Comparison &, Formula &formula, Arguments &&... arguments)
+	{
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Exists &exists, Formula &formula, Arguments &&... arguments)
+	{
+		if (exists.argument.accept(*this, exists.argument, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(ForAll &forAll, Formula &formula, Arguments &&... arguments)
+	{
+		if (forAll.argument.accept(*this, forAll.argument, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Implies &implies, Formula &formula, Arguments &&... arguments)
+	{
+		if (implies.antecedent.accept(*this, implies.antecedent, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		if (implies.consequent.accept(*this, implies.consequent, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(In &, Formula &formula, Arguments &&... arguments)
+	{
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Not &not_, Formula &formula, Arguments &&... arguments)
+	{
+		if (not_.argument.accept(*this, not_.argument, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Or &or_, Formula &formula, Arguments &&... arguments)
+	{
+		for (auto &argument : or_.arguments)
+			if (argument.accept(*this, argument, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+				return SimplificationResult::Simplified;
+
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Predicate &, Formula &formula, Arguments &&... arguments)
+	{
+		return T::accept(formula, std::forward<Arguments>(arguments)...);
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+template<class T, class SimplificationResult = void>
+struct TermSimplificationVisitor
+{
+	template <class... Arguments>
+	SimplificationResult visit(BinaryOperation &binaryOperation, Term &term, Arguments &&... arguments)
+	{
+		if (binaryOperation.left.accept(*this, binaryOperation.left, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		if (binaryOperation.right.accept(*this, binaryOperation.right, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		return T::accept(term, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Boolean &, Term &term, Arguments &&... arguments)
+	{
+		return T::accept(term, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Constant &, Term &term, Arguments &&... arguments)
+	{
+		return T::accept(term, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Function &function, Term &term, Arguments &&... arguments)
+	{
+		for (auto &argument : function.arguments)
+			if (argument.accept(*this, argument, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+				return SimplificationResult::Simplified;
+
+		return T::accept(term, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Integer &, Term &term, Arguments &&... arguments)
+	{
+		return T::accept(term, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Interval &interval, Term &term, Arguments &&... arguments)
+	{
+		if (interval.from.accept(*this, interval.from, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		if (interval.to.accept(*this, interval.to, std::forward<Arguments>(arguments)...) == SimplificationResult::Simplified)
+			return SimplificationResult::Simplified;
+
+		return T::accept(term, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(SpecialInteger &, Term &term, Arguments &&... arguments)
+	{
+		return T::accept(term, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(String &, Term &term, Arguments &&... arguments)
+	{
+		return T::accept(term, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	SimplificationResult visit(Variable &, Term &term, Arguments &&... arguments)
+	{
+		return T::accept(term, std::forward<Arguments>(arguments)...);
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+}
+}
+
+#endif

include/anthem/StatementVisitor.h

@@ -176,7 +176,8 @@ struct StatementVisitor
 
 		context.logger.log(output::Priority::Debug, statement.location) << "showing “" << signature.name() << "/" << signature.arity() << "”";
 
-		context.visiblePredicateSignatures.value().emplace_back(std::string(signature.name()), signature.arity());
+		auto predicateSignature = ast::PredicateSignature{std::string(signature.name()), signature.arity()};
+		context.visiblePredicateSignatures.value().emplace_back(PredicateSignatureMeta{std::move(predicateSignature)});
 	}
 
 	void visit(const Clingo::AST::ShowTerm &, const Clingo::AST::Statement &statement, std::vector<ast::ScopedFormula> &, Context &)
@@ -184,6 +185,44 @@ struct StatementVisitor
 		throw LogicException(statement.location, "only #show statements for atoms (not terms) are supported currently");
 	}
 
+	void visit(const Clingo::AST::External &external, const Clingo::AST::Statement &statement, std::vector<ast::ScopedFormula> &, Context &context)
+	{
+		const auto fail =
+			[&]()
+			{
+				throw LogicException(statement.location, "only #external declarations of the form “#external <predicate name>(<arity>).” supported");
+			};
+
+		if (!external.body.empty())
+			fail();
+
+		if (!external.atom.data.is<Clingo::AST::Function>())
+			fail();
+
+		const auto &predicate = external.atom.data.get<Clingo::AST::Function>();
+
+		if (predicate.arguments.size() != 1)
+			fail();
+
+		const auto &arityArgument = predicate.arguments.front();
+
+		if (!arityArgument.data.is<Clingo::Symbol>())
+			fail();
+
+		const auto &aritySymbol = arityArgument.data.get<Clingo::Symbol>();
+
+		if (aritySymbol.type() != Clingo::SymbolType::Number)
+			fail();
+
+		const size_t arity = arityArgument.data.get<Clingo::Symbol>().number();
+
+		if (!context.externalPredicateSignatures)
+			context.externalPredicateSignatures.emplace();
+
+		auto predicateSignature = ast::PredicateSignature{std::string(predicate.name), arity};
+		context.externalPredicateSignatures->emplace_back(PredicateSignatureMeta{std::move(predicateSignature)});
+	}
+
 	template<class T>
 	void visit(const T &, const Clingo::AST::Statement &statement, std::vector<ast::ScopedFormula> &, Context &)
 	{

src/anthem/ASTUtils.cpp

@@ -59,7 +59,7 @@ bool VariableStack::contains(const VariableDeclaration &variableDeclaration) con
 		};
 
 	const auto layerContainsVariableDeclaration =
-		[&variableDeclaration, &variableDeclarationMatches](const auto &layer)
+		[&variableDeclarationMatches](const auto &layer)
 		{
 			return (std::find_if(layer->cbegin(), layer->cend(), variableDeclarationMatches) != layer->cend());
 		};
@@ -194,7 +194,7 @@ struct CollectFreeVariablesVisitor
 
 struct CollectPredicateSignaturesVisitor : public RecursiveFormulaVisitor<CollectPredicateSignaturesVisitor>
 {
-	static void accept(const Predicate &predicate, const Formula &, std::vector<PredicateSignature> &predicateSignatures)
+	static void accept(const Predicate &predicate, const Formula &, std::vector<PredicateSignature> &predicateSignatures, Context &context)
 	{
 		const auto predicateSignatureMatches =
 			[&predicate](const auto &predicateSignature)
@@ -206,12 +206,35 @@ struct CollectPredicateSignaturesVisitor : public RecursiveFormulaVisitor<Collec
 			return;
 
 		// TODO: avoid copies
-		predicateSignatures.emplace_back(std::string(predicate.name), predicate.arity());
+		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> &)
+	static void accept(const T &, const Formula &, std::vector<PredicateSignature> &, const Context &)
 	{
 	}
 };
@@ -240,10 +263,10 @@ bool matches(const PredicateSignature &lhs, const PredicateSignature &rhs)
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 
 // TODO: remove const_cast
-void collectPredicateSignatures(const Formula &formula, std::vector<PredicateSignature> &predicateSignatures)
+void collectPredicateSignatures(const Formula &formula, std::vector<PredicateSignature> &predicateSignatures, Context &context)
 {
 	auto &formulaMutable = const_cast<Formula &>(formula);
-	formulaMutable.accept(CollectPredicateSignaturesVisitor(), formulaMutable, predicateSignatures);
+	formulaMutable.accept(CollectPredicateSignaturesVisitor(), formulaMutable, predicateSignatures, context);
 }
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////

src/anthem/Completion.cpp

@@ -165,7 +165,7 @@ std::vector<ast::Formula> complete(std::vector<ast::ScopedFormula> &&scopedFormu
 
 	// Get a list of all predicates
 	for (const auto &scopedFormula : scopedFormulas)
-		ast::collectPredicateSignatures(scopedFormula.formula, predicateSignatures);
+		ast::collectPredicateSignatures(scopedFormula.formula, predicateSignatures, context);
 
 	std::sort(predicateSignatures.begin(), predicateSignatures.end(),
 		[](const auto &lhs, const auto &rhs)

src/anthem/HiddenPredicateElimination.cpp

@@ -194,41 +194,40 @@ void eliminateHiddenPredicates(const std::vector<ast::PredicateSignature> &predi
 		return;
 	}
 
-	const auto &visiblePredicateSignatures = context.visiblePredicateSignatures.value();
+	auto &visiblePredicateSignatures = context.visiblePredicateSignatures.value();
-
-	// Check for undeclared predicates that are requested to be shown
-	for (const auto &visiblePredicateSignature : visiblePredicateSignatures)
-	{
-		const auto matchesPredicateSignature =
-			[&](const auto &predicateSignature)
-			{
-				return ast::matches(predicateSignature, visiblePredicateSignature);
-			};
-
-		const auto matchingPredicateSignature =
-			std::find_if(predicateSignatures.cbegin(), predicateSignatures.cend(), matchesPredicateSignature);
-
-		if (matchingPredicateSignature == predicateSignatures.cend())
-			context.logger.log(output::Priority::Warning) << "cannot show undeclared predicate “" << visiblePredicateSignature.name << "/" << visiblePredicateSignature.arity <<"”";
-	}
 
 	// Replace all occurrences of hidden predicates
 	for (size_t i = 0; i < predicateSignatures.size(); i++)
 	{
 		auto &predicateSignature = predicateSignatures[i];
 
-		const auto matchesVisiblePredicateSignature =
+		const auto matchesPredicateSignature =
-			[&](const auto &visiblePredicateSignature)
+			[&](const auto &otherPredicateSignature)
 			{
-				return ast::matches(predicateSignature, visiblePredicateSignature);
+				return ast::matches(predicateSignature, otherPredicateSignature.predicateSignature);
 			};
 
-		const auto matchingPredicateSignature =
+		const auto matchingVisiblePredicateSignature =
-			std::find_if(visiblePredicateSignatures.cbegin(), visiblePredicateSignatures.cend(), matchesVisiblePredicateSignature);
+			std::find_if(visiblePredicateSignatures.begin(), visiblePredicateSignatures.end(), matchesPredicateSignature);
 
 		// If the predicate ought to be visible, don’t eliminate it
-		if (matchingPredicateSignature != visiblePredicateSignatures.cend())
+		if (matchingVisiblePredicateSignature != visiblePredicateSignatures.end())
+		{
+			matchingVisiblePredicateSignature->used = true;
 			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 << "”";
 

src/anthem/Simplification.cpp

@@ -3,7 +3,9 @@
 #include <optional>
 
 #include <anthem/ASTCopy.h>
-#include <anthem/ASTVisitors.h>
+#include <anthem/Equality.h>
+#include <anthem/output/AST.h>
+#include <anthem/SimplificationVisitors.h>
 
 namespace anthem
 {
@@ -97,18 +99,65 @@ struct ReplaceVariableInFormulaVisitor : public ast::RecursiveFormulaVisitor<Rep
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 
-// Simplifies exists statements by using the equivalence “exists X (X = t and F(X))” == “F(t)”
+template<class SimplificationRule>
-// The exists statement has to be of the form “exists <variables> <conjunction>”
+SimplificationResult simplify(ast::Formula &formula)
-void simplify(ast::Exists &exists, ast::Formula &formula)
 {
-	// Simplify formulas like “exists X (X = Y)” to “#true”
+	return SimplificationRule::apply(formula);
-	// TODO: check that this covers all cases
+}
-	if (exists.argument.is<ast::Comparison>())
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+template<class FirstSimplificationRule, class SecondSimplificationRule, class... OtherSimplificationRules>
+SimplificationResult simplify(ast::Formula &formula)
+{
+	if (simplify<FirstSimplificationRule>(formula) == SimplificationResult::Simplified)
+		return SimplificationResult::Simplified;
+
+	return simplify<SecondSimplificationRule, OtherSimplificationRules...>(formula);
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleExistsWithoutQuantifiedVariables
+{
+	static constexpr const auto Description = "exists () (F) === F";
+
+	static SimplificationResult apply(ast::Formula &formula)
 	{
+		if (!formula.is<ast::Exists>())
+			return SimplificationResult::Unchanged;
+
+		auto &exists = formula.get<ast::Exists>();
+
+		if (!exists.variables.empty())
+			return SimplificationResult::Unchanged;
+
+		formula = std::move(exists.argument);
+
+		return SimplificationResult::Simplified;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleTrivialAssignmentInExists
+{
+	static constexpr const auto Description = "exists X (X = Y) === #true";
+
+	static SimplificationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::Exists>())
+			return SimplificationResult::Unchanged;
+
+		const auto &exists = formula.get<ast::Exists>();
+
+		if (!exists.argument.is<ast::Comparison>())
+			return SimplificationResult::Unchanged;
+
 		const auto &comparison = exists.argument.get<ast::Comparison>();
 
 		if (comparison.operator_ != ast::Comparison::Operator::Equal)
-			return;
+			return SimplificationResult::Unchanged;
 
 		const auto matchingAssignment = std::find_if(exists.variables.cbegin(), exists.variables.cend(),
 			[&](const auto &variableDeclaration)
@@ -117,107 +166,331 @@ void simplify(ast::Exists &exists, ast::Formula &formula)
 					|| matchesVariableDeclaration(comparison.right, *variableDeclaration);
 			});
 
-		if (matchingAssignment != exists.variables.cend())
+		if (matchingAssignment == exists.variables.cend())
-			formula = ast::Formula::make<ast::Boolean>(true);
+			return SimplificationResult::Unchanged;
 
-		return;
+		formula = ast::Formula::make<ast::Boolean>(true);
+
+		return SimplificationResult::Simplified;
 	}
+};
 
-	if (!exists.argument.is<ast::And>())
+////////////////////////////////////////////////////////////////////////////////////////////////////
-		return;
 
-	auto &conjunction = exists.argument.get<ast::And>();
+struct SimplificationRuleAssignmentInExists
-	auto &arguments = conjunction.arguments;
+{
+	static constexpr const auto Description = "exists X (X = t and F(X)) === exists () (F(t))";
 
-	// Simplify formulas of type “exists X (X = t and F(X))” to “F(t)”
+	static SimplificationResult apply(ast::Formula &formula)
-	for (auto i = exists.variables.begin(); i != exists.variables.end();)
 	{
-		const auto &variableDeclaration = **i;
+		if (!formula.is<ast::Exists>())
+			return SimplificationResult::Unchanged;
 
-		bool wasVariableReplaced = false;
+		auto &exists = formula.get<ast::Exists>();
 
-		// TODO: refactor
+		if (!exists.argument.is<ast::And>())
-		for (auto j = arguments.begin(); j != arguments.end(); j++)
+			return SimplificationResult::Unchanged;
+
+		auto &and_ = exists.argument.get<ast::And>();
+		auto &arguments = and_.arguments;
+
+		auto simplificationResult = SimplificationResult::Unchanged;
+
+		for (auto i = exists.variables.begin(); i != exists.variables.end();)
 		{
-			auto &argument = *j;
+			const auto &variableDeclaration = **i;
-			// Find term that is equivalent to the given variable
-			auto assignedTerm = extractAssignedTerm(argument, variableDeclaration);
 
-			if (!assignedTerm)
+			bool wasVariableReplaced = false;
-				continue;
 
-			// Replace all occurrences of the variable with the equivalent term
+			// TODO: refactor
-			for (auto k = arguments.begin(); k != arguments.end(); k++)
+			for (auto j = arguments.begin(); j != arguments.end(); j++)
 			{
-				if (k == j)
+				auto &argument = *j;
+				// Find term that is equivalent to the given variable
+				auto assignedTerm = extractAssignedTerm(argument, variableDeclaration);
+
+				if (!assignedTerm)
 					continue;
 
-				auto &otherArgument = *k;
+				// Replace all occurrences of the variable with the equivalent term
-				otherArgument.accept(ReplaceVariableInFormulaVisitor(), otherArgument, variableDeclaration, assignedTerm.value());
+				for (auto k = arguments.begin(); k != arguments.end(); k++)
+				{
+					if (k == j)
+						continue;
+
+					auto &otherArgument = *k;
+					otherArgument.accept(ReplaceVariableInFormulaVisitor(), otherArgument, variableDeclaration, assignedTerm.value());
+				}
+
+				arguments.erase(j);
+
+				wasVariableReplaced = true;
+				simplificationResult = SimplificationResult::Simplified;
+
+				break;
 			}
 
-			arguments.erase(j);
+			if (wasVariableReplaced)
-			wasVariableReplaced = true;
+			{
-			break;
+				i = exists.variables.erase(i);
+				continue;
+			}
+
+			i++;
 		}
 
-		if (wasVariableReplaced)
+		return simplificationResult;
-		{
-			i = exists.variables.erase(i);
-			continue;
-		}
-
-		i++;
 	}
+};
 
-	// If there are no arguments left, we had a formula of the form “exists X1, ..., Xn (X1 = Y1 and ... and Xn = Yn)”
+////////////////////////////////////////////////////////////////////////////////////////////////////
-	// Such exists statements are useless and can be safely replaced with “#true”
+
-	if (arguments.empty())
+struct SimplificationRuleEmptyConjunction
+{
+	static constexpr const auto Description = "[empty conjunction] === #true";
+
+	static SimplificationResult apply(ast::Formula &formula)
 	{
+		if (!formula.is<ast::And>())
+			return SimplificationResult::Unchanged;
+
+		auto &and_ = formula.get<ast::And>();
+
+		if (!and_.arguments.empty())
+			return SimplificationResult::Unchanged;
+
 		formula = ast::Formula::make<ast::Boolean>(true);
-		return;
+
+		return SimplificationResult::Simplified;
 	}
+};
 
-	// If the argument now is a conjunction with just one element, directly replace the input formula with the argument
+////////////////////////////////////////////////////////////////////////////////////////////////////
-	if (arguments.size() == 1)
-		exists.argument = std::move(arguments.front());
 
-	// If there are still remaining variables, simplification is over
+struct SimplificationRuleOneElementConjunction
-	if (!exists.variables.empty())
+{
-		return;
+	static constexpr const auto Description = "[conjunction of only F] === F";
 
-	assert(!arguments.empty());
+	static SimplificationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::And>())
+			return SimplificationResult::Unchanged;
 
-	// If there is more than one element in the conjunction, replace the input formula with the conjunction
+		auto &and_ = formula.get<ast::And>();
-	formula = std::move(exists.argument);
+
-}
+		if (and_.arguments.size() != 1)
+			return SimplificationResult::Unchanged;
+
+		formula = std::move(and_.arguments.front());
+
+		return SimplificationResult::Simplified;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleTrivialExists
+{
+	static constexpr const auto Description = "exists ... ([#true/#false]) === [#true/#false]";
+
+	static SimplificationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::Exists>())
+			return SimplificationResult::Unchanged;
+
+		auto &exists = formula.get<ast::Exists>();
+
+		if (!exists.argument.is<ast::Boolean>())
+			return SimplificationResult::Unchanged;
+
+		formula = std::move(exists.argument);
+
+		return SimplificationResult::Simplified;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleInWithPrimitiveArguments
+{
+	static constexpr const auto Description = "[primitive A] in [primitive B] === A = B";
+
+	static SimplificationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::In>())
+			return SimplificationResult::Unchanged;
+
+		auto &in = formula.get<ast::In>();
+
+		assert(ast::isPrimitive(in.element));
+
+		if (!ast::isPrimitive(in.element) || !ast::isPrimitive(in.set))
+			return SimplificationResult::Unchanged;
+
+		formula = ast::Comparison(ast::Comparison::Operator::Equal, std::move(in.element), std::move(in.set));
+
+		return SimplificationResult::Simplified;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleSubsumptionInBiconditionals
+{
+	static constexpr const auto Description = "(F <-> (F and G)) === (F -> G)";
+
+	static SimplificationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::Biconditional>())
+			return SimplificationResult::Unchanged;
+
+		auto &biconditional = formula.get<ast::Biconditional>();
+
+		const auto leftIsPredicate = biconditional.left.is<ast::Predicate>();
+		const auto rightIsPredicate = biconditional.right.is<ast::Predicate>();
+
+		const auto leftIsAnd = biconditional.left.is<ast::And>();
+		const auto rightIsAnd = biconditional.right.is<ast::And>();
+
+		if (!(leftIsPredicate && rightIsAnd) && !(rightIsPredicate && leftIsAnd))
+			return SimplificationResult::Unchanged;
+
+		auto &predicateSide = (leftIsPredicate ? biconditional.left : biconditional.right);
+		auto &andSide = (leftIsPredicate ? biconditional.right : biconditional.left);
+		auto &and_ = andSide.get<ast::And>();
+
+		const auto matchingPredicate =
+			std::find_if(and_.arguments.cbegin(), and_.arguments.cend(),
+			[&](const auto &argument)
+			{
+				return (ast::equal(predicateSide, argument) == ast::Tristate::True);
+			});
+
+		if (matchingPredicate == and_.arguments.cend())
+			return SimplificationResult::Unchanged;
+
+		and_.arguments.erase(matchingPredicate);
+
+		formula = ast::Formula::make<ast::Implies>(std::move(predicateSide), std::move(andSide));
+
+		return SimplificationResult::Simplified;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleDoubleNegation
+{
+	static constexpr const auto Description = "not not F === F";
+
+	static SimplificationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::Not>())
+			return SimplificationResult::Unchanged;
+
+		auto &not_ = formula.get<ast::Not>();
+
+		if (!not_.argument.is<ast::Not>())
+			return SimplificationResult::Unchanged;
+
+		auto &notNot = not_.argument.get<ast::Not>();
+
+		formula = std::move(notNot.argument);
+
+		return SimplificationResult::Simplified;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleDeMorganForConjunctions
+{
+	static constexpr const auto Description = "(not (F and G)) === (not F or not G)";
+
+	static SimplificationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::Not>())
+			return SimplificationResult::Unchanged;
+
+		auto &not_ = formula.get<ast::Not>();
+
+		if (!not_.argument.is<ast::And>())
+			return SimplificationResult::Unchanged;
+
+		auto &and_ = not_.argument.get<ast::And>();
+
+		for (auto &argument : and_.arguments)
+			argument = ast::Formula::make<ast::Not>(std::move(argument));
+
+		formula = ast::Formula::make<ast::Or>(std::move(and_.arguments));
+
+		return SimplificationResult::Simplified;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct SimplificationRuleImplicationFromDisjunction
+{
+	static constexpr const auto Description = "(not F or G) === (F -> G)";
+
+	static SimplificationResult apply(ast::Formula &formula)
+	{
+		if (!formula.is<ast::Or>())
+			return SimplificationResult::Unchanged;
+
+		auto &or_ = formula.get<ast::Or>();
+
+		if (or_.arguments.size() != 2)
+			return SimplificationResult::Unchanged;
+
+		const auto leftIsNot = or_.arguments[0].is<ast::Not>();
+		const auto rightIsNot = or_.arguments[1].is<ast::Not>();
+
+		if (leftIsNot == rightIsNot)
+			return SimplificationResult::Unchanged;
+
+		auto &negativeSide = leftIsNot ? or_.arguments[0] : or_.arguments[1];
+		auto &positiveSide = leftIsNot ? or_.arguments[1] : or_.arguments[0];
+
+		assert(negativeSide.is<ast::Not>());
+		assert(!positiveSide.is<ast::Not>());
+
+		auto &negativeSideArgument = negativeSide.get<ast::Not>().argument;
+
+		formula = ast::Formula::make<ast::Implies>(std::move(negativeSideArgument), std::move(positiveSide));
+
+		return SimplificationResult::Simplified;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+const auto simplifyWithDefaultRules =
+	simplify
+	<
+		SimplificationRuleDoubleNegation,
+		SimplificationRuleTrivialAssignmentInExists,
+		SimplificationRuleAssignmentInExists,
+		SimplificationRuleEmptyConjunction,
+		SimplificationRuleTrivialExists,
+		SimplificationRuleOneElementConjunction,
+		SimplificationRuleExistsWithoutQuantifiedVariables,
+		SimplificationRuleInWithPrimitiveArguments,
+		SimplificationRuleSubsumptionInBiconditionals,
+		SimplificationRuleDeMorganForConjunctions,
+		SimplificationRuleImplicationFromDisjunction
+	>;
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 
 // Performs the different simplification techniques
-struct SimplifyFormulaVisitor : public ast::RecursiveFormulaVisitor<SimplifyFormulaVisitor>
+struct SimplifyFormulaVisitor : public ast::FormulaSimplificationVisitor<SimplifyFormulaVisitor>
 {
-	// Forward exists statements to the dedicated simplification function
-	static void accept(ast::Exists &exists, ast::Formula &formula)
-	{
-		simplify(exists, formula);
-	}
-
-	// Simplify formulas of type “A in B” to “A = B” if A and B are primitive
-	static void accept(ast::In &in, ast::Formula &formula)
-	{
-		assert(ast::isPrimitive(in.element));
-
-		if (!ast::isPrimitive(in.element) || !ast::isPrimitive(in.set))
-			return;
-
-		formula = ast::Comparison(ast::Comparison::Operator::Equal, std::move(in.element), std::move(in.set));
-	}
-
 	// Do nothing for all other types of expressions
-	template<class T>
+	static SimplificationResult accept(ast::Formula &formula)
-	static void accept(T &, ast::Formula &)
 	{
+		return simplifyWithDefaultRules(formula);
 	}
 };
 
@@ -225,7 +498,7 @@ struct SimplifyFormulaVisitor : public ast::RecursiveFormulaVisitor<SimplifyForm
 
 void simplify(ast::Formula &formula)
 {
-	formula.accept(SimplifyFormulaVisitor(), formula);
+	while (formula.accept(SimplifyFormulaVisitor(), formula) == SimplificationResult::Simplified);
 }
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////

src/anthem/Translation.cpp

@@ -70,6 +70,9 @@ void translate(const char *fileName, std::istream &stream, Context &context)
 		if (context.visiblePredicateSignatures)
 			context.logger.log(output::Priority::Warning) << "#show statements are ignored because completion is not enabled";
 
+		if (context.externalPredicateSignatures)
+			context.logger.log(output::Priority::Warning) << "#external statements are ignored because completion is not enabled";
+
 		for (const auto &scopedFormula : scopedFormulas)
 		{
 			ast::print(context.logger.outputStream(), scopedFormula.formula, printContext);
@@ -82,6 +85,26 @@ 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";
+
+	// 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";
+
 	// Simplify output if specified
 	if (context.performSimplification)
 		for (auto &completedFormula : completedFormulas)

tests/TestCompletion.cpp

@@ -152,9 +152,9 @@ TEST_CASE("[completion] Rules are completed", "[completion]")
 
 		CHECK(output.str() ==
 			"forall V1 (covered(V1) <-> exists U1 in(V1, U1))\n"
-			"forall V2, V3 (in(V2, V3) <-> (V2 in 1..n and V3 in 1..r and in(V2, V3)))\n"
+			"forall V2, V3 (in(V2, V3) -> (V2 in 1..n and V3 in 1..r))\n"
-			"forall U2 not (U2 in 1..n and not covered(U2))\n"
+			"forall U2 (U2 in 1..n -> covered(U2))\n"
-			"forall U3, U4, U5 not (in(U3, U4) and in(U5, U4) and exists X1 (X1 in (U3 + U5) and in(X1, U4)))\n");
+			"forall U3, U4, U5 (not in(U3, U4) or not in(U5, U4) or not exists X1 (X1 in (U3 + U5) and in(X1, U4)))\n");
 	}
 
 	SECTION("binary operations with multiple variables")

tests/TestHiddenPredicateElimination.cpp

@@ -103,9 +103,10 @@ TEST_CASE("[hidden predicate elimination] Hidden predicates are correctly elimin
 			"#show a/1.";
 		anthem::translate("input", input, context);
 
+		// TODO: simplify further
 		CHECK(output.str() ==
 			"forall V1 (a(V1) <-> not d(V1))\n"
-			"forall V2 (d(V2) <-> not not d(V2))\n"
+			"forall V2 (d(V2) <-> d(V2))\n"
 			"forall V3 (e(V3) <-> e(V3))\n");
 	}
 
@@ -164,12 +165,11 @@ TEST_CASE("[hidden predicate elimination] Hidden predicates are correctly elimin
 			"#show t/0.";
 		anthem::translate("input", input, context);
 
-		// TODO: simplify further
 		CHECK(output.str() ==
-			"(s <-> (not #false and s))\n"
+			"(s -> not #false)\n"
-			"(t <-> (not #false and t))\n"
+			"(t -> not #false)\n"
-			"not (s and not t)\n"
+			"(s -> t)\n"
-			"not (not #false and not #false and #false)\n");
+			"(#false or #false or not #false)\n");
 	}
 
 	SECTION("predicate with more than one argument is hidden correctly")

tests/TestPlaceholders.cpp

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