Version bump for release 0.1.9 RC 3

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.

CHANGELOG.md

@@ -1,6 +1,13 @@
 # Change Log
 
-## (unreleased)
+## 0.1.9 RC 3 (2018-04-22)
+
+### Features
+
+* 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)
 

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+git" << std::endl;
+		std::cout << "anthem version 0.1.9-rc.3" << std::endl;
 		return EXIT_SUCCESS;
 	}
 

examples/permutations.lp

@@ -1,3 +1,6 @@
+#show p/2.
+%#external integer(n(0)).
+
 {p(1..n, 1..n)}.
 
 :- p(X, Y1), p(X, Y2), Y1 != Y2.
@@ -8,5 +11,3 @@ q2(Y) :- p(_, Y).
 
 :- not q1(X), X = 1..n.
 :- not q2(Y), Y = 1..n.
-
-#show p/2.

examples/prime.lp

@@ -1,4 +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).

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.

include/anthem/AST.h

@@ -2,6 +2,7 @@
 #define __ANTHEM__AST_H
 
 #include <anthem/ASTForward.h>
+#include <anthem/Utils.h>
 
 namespace anthem
 {
@@ -103,32 +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)}
 	{
 	}
@@ -138,12 +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
 {
@@ -204,13 +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)}
 	{
 	}
@@ -220,35 +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};
 };
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
@@ -360,6 +381,7 @@ struct VariableDeclaration
 	VariableDeclaration &operator=(VariableDeclaration &&other) = default;
 
 	Type type;
+	Domain domain{Domain::Unknown};
 	std::string name;
 };
 

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);

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;
@@ -35,6 +35,7 @@ struct Interval;
 struct Not;
 struct Or;
 struct Predicate;
+struct PredicateDeclaration;
 struct SpecialInteger;
 struct String;
 struct UnaryOperation;
@@ -63,7 +64,6 @@ using Formula = Clingo::Variant<
 using Term = Clingo::Variant<
 	BinaryOperation,
 	Boolean,
-	Constant,
 	Function,
 	Integer,
 	Interval,

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

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)
 	{

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;

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;
 };

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())

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;

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);
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 

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

include/anthem/Simplification.h

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

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)...);
 	}

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>

include/anthem/Term.h

@@ -65,13 +65,13 @@ ast::UnaryOperation::Operator translate(Clingo::AST::UnaryOperator unaryOperator
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 
-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);
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 
 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())
 		{
@@ -85,19 +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);
@@ -107,7 +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);
@@ -120,35 +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::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::UnaryOperation &unaryOperation, const Clingo::AST::Term &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)
 	{
 		const auto operator_ = translate(unaryOperation.unary_operator, term);
-		auto argument = translate(unaryOperation.argument, ruleContext, variableStack);
+		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, const ast::VariableStack &variableStack)
+	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, variableStack);
-		auto right = translate(interval.right, ruleContext, 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");
@@ -157,12 +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;
@@ -171,9 +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");

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

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

include/anthem/output/AST.h

@@ -46,12 +46,12 @@ output::ColorStream &print(output::ColorStream &stream, const BinaryOperation &b
 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 Constant &constant, 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);
@@ -167,16 +167,9 @@ inline output::ColorStream &print(output::ColorStream &stream, const Comparison
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 
-inline output::ColorStream &print(output::ColorStream &stream, const Constant &constant, PrintContext &, bool)
-{
-	return (stream << constant.name);
-}
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
 inline output::ColorStream &print(output::ColorStream &stream, const Function &function, PrintContext &printContext, bool)
 {
-	stream << function.name;
+	stream << function.declaration->name;
 
 	if (function.arguments.empty())
 		return stream;
@@ -191,7 +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 << ")";
@@ -244,7 +237,7 @@ inline output::ColorStream &print(output::ColorStream &stream, const Interval &i
 
 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;
@@ -266,6 +259,13 @@ inline output::ColorStream &print(output::ColorStream &stream, const Predicate &
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 
+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)
@@ -322,8 +322,24 @@ inline output::ColorStream &print(output::ColorStream &stream, const Variable &v
 
 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);
 
@@ -334,7 +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()));
 		};

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));
 }
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
@@ -293,11 +286,6 @@ struct FixDanglingVariablesInTermVisitor
 	{
 	}
 
-	template <class... Arguments>
-	void visit(Constant &, Arguments &&...)
-	{
-	}
-
 	template <class... Arguments>
 	void visit(Function &function, Arguments &&... arguments)
 	{

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)
@@ -197,84 +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);
-}
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
 }
 }

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;
 }

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;
 		}
 

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;
+			}
+		}
+	}
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+}

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 +461,7 @@ struct SimplificationRuleImplicationFromDisjunction
 
 		formula = ast::Formula::make<ast::Implies>(std::move(negativeSideArgument), std::move(positiveSide));
 
-		return SimplificationResult::Simplified;
+		return OperationResult::Changed;
 	}
 };
 
@@ -470,15 +471,15 @@ struct SimplificationRuleNegatedComparison
 {
 	static constexpr const auto Description = "(not F [comparison] G) === (F [negated comparison] 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::Comparison>())
-			return SimplificationResult::Unchanged;
+			return OperationResult::Unchanged;
 
 		auto &comparison = not_.argument.get<ast::Comparison>();
 
@@ -506,7 +507,35 @@ struct SimplificationRuleNegatedComparison
 
 		formula = std::move(comparison);
 
-		return SimplificationResult::Simplified;
+		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;
 	}
 };
 
@@ -526,7 +555,8 @@ const auto simplifyWithDefaultRules =
 		SimplificationRuleSubsumptionInBiconditionals,
 		SimplificationRuleDeMorganForConjunctions,
 		SimplificationRuleImplicationFromDisjunction,
-		SimplificationRuleNegatedComparison
+		SimplificationRuleNegatedComparison,
+		SimplificationRuleIntegerSetInclusion
 	>;
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
@@ -535,7 +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);
 	}
@@ -545,7 +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);
 }
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////

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;
+		}
+	}
 }
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////