Version bump after release 0.1.9

By default, completion, simplification, and integer variable detection
are now turned on by default. This updates the readme accordingly.

CHANGELOG.md

@@ -1,13 +1,24 @@
 # Change Log
 
-## 0.1.9 RC 2 (2018-04-21)
+## (unreleased)
+
+## 0.1.9 (2018-05-04)
+
+### Changes
+
+* turns on completion and simplification by default, which can now be switched off with `--no-complete` and `--no-simplify`
 
 ### Features
 
-* optional detection of integer variables and integer predicate parameters
+* detection of integer variables and integer predicate parameters
-* command-line option `--detect-integers` to enable integer variable detection
+* command-line option `--no-detect-integers` to disable integer variable detection
-* support for declaring functions integer with the `#external` directive
 * new simplification rule applying to integer variables
+* support for declaring functions integer with the `#external` directive
+
+### Bug Fixes
+
+* fixes incorrect translation of unsupported choice rules with multiple elements by returning an error instead
+* fixes precedence of intervals by enclosing them in parentheses
 
 ## 0.1.8 (2018-04-20)
 

README.md

@@ -9,11 +9,12 @@
 ## Usage
 
 ```bash
-$ anthem [--complete] [--simplify] file...
+$ anthem [--no-complete] [--no-simplify] [--no-detect-integers] file...
 ```
 
-`--complete` instructs `anthem` to perform Clark’s completion on the translated formulas.
+By default, `anthem` performs Clark’s completion on the translated formulas, detects which variables are integer, and simplifies the output by applying several basic transformation rules.
-With the option `--simplify`, the output formulas are simplified by applying several basic transformation rules.
+
+These processing steps can be turned off with the options `--no-complete`, `--no-simplify`, and `--no-detect-integers`.
 
 ## Building
 

app/main.cpp

@@ -16,9 +16,9 @@ int main(int argc, char **argv)
 		("h,help", "Display this help message")
 		("v,version", "Display version information")
 		("i,input", "Input files", cxxopts::value<std::vector<std::string>>())
-		("s,simplify", "Simplify the output")
+		("no-simplify", "Do not simplify the output")
-		("c,complete", "Perform completion")
+		("no-complete", "Do not perform completion")
-		("d,detect-integers", "Detect integer variables")
+		("no-detect-integers", "Do not 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"));
@@ -49,9 +49,9 @@ int main(int argc, char **argv)
 		if (parseResult.count("input") > 0)
 			inputFiles = parseResult["input"].as<std::vector<std::string>>();
 
-		context.performSimplification = (parseResult.count("simplify") > 0);
+		context.performSimplification = (parseResult.count("no-simplify") == 0);
-		context.performCompletion = (parseResult.count("complete") > 0);
+		context.performCompletion = (parseResult.count("no-complete") == 0);
-		context.performIntegerDetection = (parseResult.count("detect-integers") > 0);
+		context.performIntegerDetection = (parseResult.count("no-detect-integers") == 0);
 		colorPolicyString = parseResult["color"].as<std::string>();
 		parenthesisStyleString = parseResult["parentheses"].as<std::string>();
 		logPriorityString = parseResult["log-priority"].as<std::string>();
@@ -72,7 +72,7 @@ int main(int argc, char **argv)
 
 	if (version)
 	{
-		std::cout << "anthem version 0.1.9-rc.2" << std::endl;
+		std::cout << "anthem version 0.1.9+git" << std::endl;
 		return EXIT_SUCCESS;
 	}
 

examples/choice-rules.lp

@@ -0,0 +1,2 @@
+p(a).
+{q(a)}.

examples/graph-coloring.lp

@@ -1,10 +1,18 @@
-#external color(1).
+% assign a set of colors to each vertex
-#external edge(2).
-#external vertex(1).
-#show color/2.
-
 {color(V, C)} :- vertex(V), color(C).
+
+% at most one color per vertex
+:- color(V, C1), color(V, C2), C1 != C2.
+
+% at least one color per vertex
 covered(V) :- color(V, _).
 :- vertex(V), not covered(V).
+
+% adjacent vertices don’t share the same color
 :- color(V1, C), color(V2, C), edge(V1, V2).
-:- color(V, C1), color(V, C2), C1 != C2.
+
+#show color/2.
+
+#external vertex(1).
+#external edge(2).
+#external color(1).

examples/letters.lp

@@ -0,0 +1,11 @@
+letter(a).
+letter(b).
+letter(c).
+
+{p(1..3, Y)} :- letter(Y).
+:- p(X1, Y), p(X2, Y), X1 != X2.
+
+q(X) :- p(X, _).
+:- X = 1..3, not q(X).
+
+#show p/2.

examples/permutations.lp

@@ -1,5 +1,4 @@
 #show p/2.
-#external integer(n(0)).
 
 {p(1..n, 1..n)}.
 

examples/prime.lp

@@ -1,5 +1,4 @@
 #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

@@ -1,6 +1,4 @@
 #show in/2.
-#external integer(n(0)).
-#external integer(r(0)).
 
 {in(1..n, 1..r)}.
 covered(I) :- in(I, S).

examples/simple-external-show.lp

@@ -0,0 +1,9 @@
+s(X) :- p(X).
+s(X) :- q(X).
+u(X) :- r(X), not s(X).
+
+#show u/1.
+
+#external p(1).
+#external q(1).
+#external r(1).

examples/simple-external.lp

@@ -0,0 +1,5 @@
+s(X) :- p(X).
+s(X) :- q(X).
+
+#external p(1).
+#external q(1).

include/anthem/AST.h

@@ -2,7 +2,7 @@
 #define __ANTHEM__AST_H
 
 #include <anthem/ASTForward.h>
-#include <anthem/Domain.h>
+#include <anthem/Utils.h>
 
 namespace anthem
 {
@@ -148,7 +148,7 @@ struct FunctionDeclaration
 
 	std::string name;
 	size_t arity;
-	Domain domain{Domain::General};
+	Domain domain{Domain::Noninteger};
 };
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////

include/anthem/ASTUtils.h

@@ -90,6 +90,21 @@ struct ReplaceVariableInFormulaVisitor : public RecursiveFormulaVisitor<ReplaceV
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 
 }
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+// Accessing Variable Domains
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct DefaultVariableDomainAccessor
+{
+	Domain operator()(const ast::Variable &variable)
+	{
+		return variable.declaration->domain;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
 }
 
 #endif

include/anthem/Arithmetics.h

@@ -1,262 +0,0 @@
-#ifndef __ANTHEM__ARITHMETICS_H
-#define __ANTHEM__ARITHMETICS_H
-
-#include <anthem/Utils.h>
-
-namespace anthem
-{
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-//
-// Arithmetics
-//
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
-struct DefaultVariableDomainAccessor
-{
-	ast::Domain operator()(const ast::Variable &variable)
-	{
-		return variable.declaration->domain;
-	}
-};
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
-template <class VariableDomainAccessor = DefaultVariableDomainAccessor, class... Arguments>
-EvaluationResult isArithmetic(const ast::Term &term, Arguments &&...);
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
-template <class VariableDomainAccessor = DefaultVariableDomainAccessor>
-struct IsTermArithmeticVisitor
-{
-	template <class... Arguments>
-	static EvaluationResult visit(const ast::BinaryOperation &binaryOperation, Arguments &&... arguments)
-	{
-		const auto isLeftArithemtic = isArithmetic<VariableDomainAccessor>(binaryOperation.left, std::forward<Arguments>(arguments)...);
-		const auto isRightArithmetic = isArithmetic<VariableDomainAccessor>(binaryOperation.right, std::forward<Arguments>(arguments)...);
-
-		if (isLeftArithemtic == EvaluationResult::Error || isRightArithmetic == EvaluationResult::Error)
-			return EvaluationResult::Error;
-
-		if (isLeftArithemtic == EvaluationResult::False || isRightArithmetic == EvaluationResult::False)
-			return EvaluationResult::Error;
-
-		if (isLeftArithemtic == EvaluationResult::Unknown || isRightArithmetic == EvaluationResult::Unknown)
-			return EvaluationResult::Unknown;
-
-		return EvaluationResult::True;
-	}
-
-	template <class... Arguments>
-	static EvaluationResult visit(const ast::Boolean &, Arguments &&...)
-	{
-		return EvaluationResult::False;
-	}
-
-	template <class... Arguments>
-	static EvaluationResult visit(const ast::Function &function, Arguments &&...)
-	{
-		switch (function.declaration->domain)
-		{
-			case ast::Domain::General:
-				return EvaluationResult::False;
-			case ast::Domain::Integer:
-				return EvaluationResult::True;
-			case ast::Domain::Unknown:
-				return EvaluationResult::Unknown;
-		}
-
-		return EvaluationResult::Unknown;
-	}
-
-	template <class... Arguments>
-	static EvaluationResult visit(const ast::Integer &, Arguments &&...)
-	{
-		return EvaluationResult::True;
-	}
-
-	template <class... Arguments>
-	static EvaluationResult visit(const ast::Interval &interval, Arguments &&... arguments)
-	{
-		const auto isFromArithmetic = isArithmetic<VariableDomainAccessor>(interval.from, std::forward<Arguments>(arguments)...);
-		const auto isToArithmetic = isArithmetic<VariableDomainAccessor>(interval.to, std::forward<Arguments>(arguments)...);
-
-		if (isFromArithmetic == EvaluationResult::Error || isToArithmetic == EvaluationResult::Error)
-			return EvaluationResult::Error;
-
-		if (isFromArithmetic == EvaluationResult::False || isToArithmetic == EvaluationResult::False)
-			return EvaluationResult::Error;
-
-		if (isFromArithmetic == EvaluationResult::Unknown || isToArithmetic == EvaluationResult::Unknown)
-			return EvaluationResult::Unknown;
-
-		return EvaluationResult::True;
-	}
-
-	template <class... Arguments>
-	static EvaluationResult visit(const ast::SpecialInteger &, Arguments &&...)
-	{
-		return EvaluationResult::False;
-	}
-
-	template <class... Arguments>
-	static EvaluationResult visit(const ast::String &, Arguments &&...)
-	{
-		return EvaluationResult::False;
-	}
-
-	template <class... Arguments>
-	static EvaluationResult visit(const ast::UnaryOperation &unaryOperation, Arguments &&... arguments)
-	{
-		const auto isArgumentArithmetic = isArithmetic<VariableDomainAccessor>(unaryOperation.argument, std::forward<Arguments>(arguments)...);
-
-		switch (unaryOperation.operator_)
-		{
-			case ast::UnaryOperation::Operator::Absolute:
-				return (isArgumentArithmetic == EvaluationResult::False ? EvaluationResult::Error : isArgumentArithmetic);
-		}
-
-		return EvaluationResult::Unknown;
-	}
-
-	template <class... Arguments>
-	static EvaluationResult visit(const ast::Variable &variable, Arguments &&... arguments)
-	{
-		const auto domain = VariableDomainAccessor()(variable, std::forward<Arguments>(arguments)...);
-
-		switch (domain)
-		{
-			case ast::Domain::General:
-				return EvaluationResult::False;
-			case ast::Domain::Integer:
-				return EvaluationResult::True;
-			case ast::Domain::Unknown:
-				return EvaluationResult::Unknown;
-		}
-
-		return EvaluationResult::Unknown;
-	}
-};
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
-template <class VariableDomainAccessor, class... Arguments>
-EvaluationResult isArithmetic(const ast::Term &term, Arguments &&... arguments)
-{
-	return term.accept(IsTermArithmeticVisitor<VariableDomainAccessor>(), std::forward<Arguments>(arguments)...);
-}
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
-EvaluationResult isInteger(const ast::Term &term);
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
-struct IsTermIntegerVisitor
-{
-	static EvaluationResult visit(const ast::BinaryOperation &binaryOperation)
-	{
-		const auto isLeftArithemtic = isArithmetic(binaryOperation.left);
-		const auto isRightArithmetic = isArithmetic(binaryOperation.right);
-
-		if (isLeftArithemtic == EvaluationResult::Error || isRightArithmetic == EvaluationResult::Error)
-			return EvaluationResult::Error;
-
-		if (isLeftArithemtic == EvaluationResult::False || isRightArithmetic == EvaluationResult::False)
-			return EvaluationResult::Error;
-
-		if (binaryOperation.operator_ == ast::BinaryOperation::Operator::Division)
-			return EvaluationResult::False;
-
-		if (isLeftArithemtic == EvaluationResult::Unknown || isRightArithmetic == EvaluationResult::Unknown)
-			return EvaluationResult::Unknown;
-
-		return EvaluationResult::True;
-	}
-
-	static EvaluationResult visit(const ast::Boolean &)
-	{
-		return EvaluationResult::False;
-	}
-
-	static EvaluationResult visit(const ast::Function &function)
-	{
-		switch (function.declaration->domain)
-		{
-			case ast::Domain::General:
-				return EvaluationResult::False;
-			case ast::Domain::Integer:
-				return EvaluationResult::True;
-			case ast::Domain::Unknown:
-				return EvaluationResult::Unknown;
-		}
-
-		return EvaluationResult::Unknown;
-	}
-
-	static EvaluationResult visit(const ast::Integer &)
-	{
-		return EvaluationResult::True;
-	}
-
-	template <class... Arguments>
-	static EvaluationResult visit(const ast::Interval &)
-	{
-		return EvaluationResult::False;
-	}
-
-	template <class... Arguments>
-	static EvaluationResult visit(const ast::SpecialInteger &)
-	{
-		return EvaluationResult::False;
-	}
-
-	template <class... Arguments>
-	static EvaluationResult visit(const ast::String &)
-	{
-		return EvaluationResult::False;
-	}
-
-	template <class... Arguments>
-	static EvaluationResult visit(const ast::UnaryOperation &unaryOperation)
-	{
-		const auto isArgumentArithmetic = isArithmetic(unaryOperation.argument);
-
-		switch (unaryOperation.operator_)
-		{
-			case ast::UnaryOperation::Operator::Absolute:
-				return (isArgumentArithmetic == EvaluationResult::False ? EvaluationResult::Error : isArgumentArithmetic);
-		}
-
-		return EvaluationResult::Unknown;
-	}
-
-	static EvaluationResult visit(const ast::Variable &variable)
-	{
-		switch (variable.declaration->domain)
-		{
-			case ast::Domain::General:
-				return EvaluationResult::False;
-			case ast::Domain::Integer:
-				return EvaluationResult::True;
-			case ast::Domain::Unknown:
-				return EvaluationResult::Unknown;
-		}
-
-		return EvaluationResult::Unknown;
-	}
-};
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
-inline EvaluationResult isInteger(const ast::Term &term)
-{
-	return term.accept(IsTermIntegerVisitor());
-}
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
-}
-
-#endif

include/anthem/Domain.h

@@ -1,27 +0,0 @@
-#ifndef __ANTHEM__DOMAIN_H
-#define __ANTHEM__DOMAIN_H
-
-namespace anthem
-{
-namespace ast
-{
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-//
-// Domain
-//
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
-enum class Domain
-{
-	General,
-	Integer,
-	Unknown,
-};
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
-}
-}
-
-#endif

include/anthem/Equality.h

@@ -268,8 +268,8 @@ struct TermEqualityVisitor
 			return Tristate::Unknown;
 		}
 
-		if (equal(binaryOperation.left, binaryOperation.right) == Tristate::True
+		if (equal(binaryOperation.left, otherBinaryOperation.right) == Tristate::True
-		    && equal(binaryOperation.right, binaryOperation.left) == Tristate::True)
+		    && equal(binaryOperation.right, otherBinaryOperation.left) == Tristate::True)
 		{
 			return Tristate::True;
 		}

include/anthem/Evaluation.h

@@ -0,0 +1,244 @@
+#ifndef __ANTHEM__EVALUATION_H
+#define __ANTHEM__EVALUATION_H
+
+#include <anthem/AST.h>
+#include <anthem/ASTUtils.h>
+#include <anthem/Utils.h>
+
+namespace anthem
+{
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+//
+// Evaluation
+//
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+template <class VariableDomainAccessor = DefaultVariableDomainAccessor>
+struct EvaluateFormulaVisitor
+{
+	template <class... Arguments>
+	static EvaluationResult visit(const ast::And &and_, Arguments &&... arguments)
+	{
+		bool someFalse = false;
+		bool someUnknown = false;
+
+		for (const auto &argument : and_.arguments)
+		{
+			const auto result = evaluate(argument, std::forward<Arguments>(arguments)...);
+
+			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;
+	}
+
+	template <class... Arguments>
+	static EvaluationResult visit(const ast::Biconditional &biconditional, Arguments &&... arguments)
+	{
+		const auto leftResult = evaluate(biconditional.left, std::forward<Arguments>(arguments)...);
+		const auto rightResult = evaluate(biconditional.right, std::forward<Arguments>(arguments)...);
+
+		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);
+	}
+
+	template <class... Arguments>
+	static EvaluationResult visit(const ast::Boolean &boolean, Arguments &&...)
+	{
+		return (boolean.value == true ? EvaluationResult::True : EvaluationResult::False);
+	}
+
+	template <class... Arguments>
+	static EvaluationResult visit(const ast::Comparison &comparison, Arguments &&... arguments)
+	{
+		const auto leftType = type(comparison.left, std::forward<Arguments>(arguments)...);
+		const auto rightType = type(comparison.right, std::forward<Arguments>(arguments)...);
+
+		// 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;
+		}
+	}
+
+	template <class... Arguments>
+	static EvaluationResult visit(const ast::Exists &exists, Arguments &&... arguments)
+	{
+		return evaluate(exists.argument, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	static EvaluationResult visit(const ast::ForAll &forAll, Arguments &&... arguments)
+	{
+		return evaluate(forAll.argument, std::forward<Arguments>(arguments)...);
+	}
+
+	template <class... Arguments>
+	static EvaluationResult visit(const ast::Implies &implies, Arguments &&... arguments)
+	{
+		const auto antecedentResult = evaluate(implies.antecedent, std::forward<Arguments>(arguments)...);
+		const auto consequentResult = evaluate(implies.consequent, std::forward<Arguments>(arguments)...);
+
+		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;
+	}
+
+	template <class... Arguments>
+	static EvaluationResult visit(const ast::In &in, Arguments &&... arguments)
+	{
+		const auto elementType = type(in.element, std::forward<Arguments>(arguments)...);
+		const auto setType = type(in.set, std::forward<Arguments>(arguments)...);
+
+		// 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;
+	}
+
+	template <class... Arguments>
+	static EvaluationResult visit(const ast::Not &not_, Arguments &&... arguments)
+	{
+		const auto result = evaluate(not_.argument, std::forward<Arguments>(arguments)...);
+
+		if (result == EvaluationResult::Error || result == EvaluationResult::Unknown)
+			return result;
+
+		return (result == EvaluationResult::True ? EvaluationResult::False : EvaluationResult::True);
+	}
+
+	template <class... Arguments>
+	static EvaluationResult visit(const ast::Or &or_, Arguments &&... arguments)
+	{
+		bool someTrue = false;
+		bool someUnknown = false;
+
+		for (const auto &argument : or_.arguments)
+		{
+			const auto result = evaluate(argument, std::forward<Arguments>(arguments)...);
+
+			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;
+	}
+
+	template <class... Arguments>
+	static EvaluationResult visit(const ast::Predicate &predicate, Arguments &&... arguments)
+	{
+		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, std::forward<Arguments>(arguments)...);
+
+			if (argumentType.domain == Domain::Noninteger || argumentType.setSize == SetSize::Empty)
+				return EvaluationResult::Error;
+		}
+
+		return EvaluationResult::Unknown;
+	}
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+template <class VariableDomainAccessor = DefaultVariableDomainAccessor, class... Arguments>
+EvaluationResult evaluate(const ast::Formula &formula, Arguments &&... arguments)
+{
+	return formula.accept(EvaluateFormulaVisitor<VariableDomainAccessor>(), std::forward<Arguments>(arguments)...);
+}
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+}
+
+#endif

include/anthem/StatementVisitor.h

@@ -237,7 +237,7 @@ struct StatementVisitor
 				const size_t arity = aritySymbol.number();
 
 				auto functionDeclaration = context.findOrCreateFunctionDeclaration(function.name, arity);
-				functionDeclaration->domain = ast::Domain::Integer;
+				functionDeclaration->domain = Domain::Integer;
 
 				return true;
 			};

include/anthem/Type.h

@@ -0,0 +1,157 @@
+#ifndef __ANTHEM__TYPE_H
+#define __ANTHEM__TYPE_H
+
+#include <anthem/AST.h>
+#include <anthem/ASTUtils.h>
+#include <anthem/Utils.h>
+
+namespace anthem
+{
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+//
+// Type
+//
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+template <class VariableDomainAccessor = DefaultVariableDomainAccessor, class... Arguments>
+Type type(const ast::Term &term, Arguments &&... 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
 {
 
@@ -20,6 +13,7 @@ namespace anthem
 constexpr const auto HeadVariablePrefix = "V";
 constexpr const auto BodyVariablePrefix = "X";
 constexpr const auto UserVariablePrefix = "U";
+constexpr const auto IntegerVariablePrefix = "N";
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 
@@ -32,6 +26,14 @@ enum class Tristate
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 
+enum class OperationResult
+{
+	Unchanged,
+	Changed,
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
 enum class EvaluationResult
 {
 	True,
@@ -42,10 +44,29 @@ enum class EvaluationResult
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 
-enum class OperationResult
+enum class Domain
 {
-	Unchanged,
+	Noninteger,
-	Changed,
+	Integer,
+	Unknown,
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+enum class SetSize
+{
+	Empty,
+	Unit,
+	Multi,
+	Unknown,
+};
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+struct Type
+{
+	Domain domain{Domain::Unknown};
+	SetSize setSize{SetSize::Unknown};
 };
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////

include/anthem/output/AST.h

@@ -35,6 +35,7 @@ struct PrintContext
 	std::map<const VariableDeclaration *, size_t> userVariableIDs;
 	std::map<const VariableDeclaration *, size_t> headVariableIDs;
 	std::map<const VariableDeclaration *, size_t> bodyVariableIDs;
+	std::map<const VariableDeclaration *, size_t> integerVariableIDs;
 
 	const Context &context;
 };
@@ -218,16 +219,16 @@ inline output::ColorStream &print(output::ColorStream &stream, const Integer &in
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 
-inline output::ColorStream &print(output::ColorStream &stream, const Interval &interval, PrintContext &printContext, bool)
+inline output::ColorStream &print(output::ColorStream &stream, const Interval &interval, PrintContext &printContext, bool omitParentheses)
 {
-	if (printContext.context.parenthesisStyle == ParenthesisStyle::Full)
+	if (!omitParentheses || printContext.context.parenthesisStyle == ParenthesisStyle::Full)
 		stream << "(";
 
 	print(stream, interval.from, printContext);
 	stream << "..";
 	print(stream, interval.to, printContext);
 
-	if (printContext.context.parenthesisStyle == ParenthesisStyle::Full)
+	if (!omitParentheses || printContext.context.parenthesisStyle == ParenthesisStyle::Full)
 		stream << ")";
 
 	return stream;
@@ -322,24 +323,8 @@ 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::General:
-					return "g";
-				case Domain::Integer:
-					return "i";
-			}
-
-			return "";
-		};
-
 	const auto printVariableDeclaration =
-		[&](const auto *prefix, auto &variableIDs) -> output::ColorStream &
+		[&stream, &variableDeclaration](const auto *prefix, auto &variableIDs) -> output::ColorStream &
 		{
 			auto matchingVariableID = variableIDs.find(&variableDeclaration);
 
@@ -350,11 +335,14 @@ inline output::ColorStream &print(output::ColorStream &stream, const VariableDec
 				matchingVariableID = emplaceResult.first;
 			}
 
-			const auto variableName = std::string(prefix) + std::to_string(matchingVariableID->second) + domainSuffix();
+			const auto variableName = std::string(prefix) + std::to_string(matchingVariableID->second);
 
 			return (stream << output::Variable(variableName.c_str()));
 		};
 
+	if (variableDeclaration.domain == Domain::Integer)
+		return printVariableDeclaration(IntegerVariablePrefix, printContext.integerVariableIDs);
+
 	switch (variableDeclaration.type)
 	{
 		case VariableDeclaration::Type::UserDefined:

src/anthem/Completion.cpp

@@ -40,8 +40,36 @@ ast::Formula buildCompletedFormulaDisjunction(const ast::Predicate &predicate, c
 
 		assert(otherPredicate.arguments.size() == parameters.size());
 
+		auto &freeVariables = scopedFormula.freeVariables;
+
 		// Each formula with the predicate as its consequent currently has its own copy of the predicate’s parameters
 		// These need to be linked to the new, unique set of parameters
+
+		// First, remove the free variables whose occurrences will be relinked, which is why they are no longer needed
+		const auto isFreeVariableUnneeded =
+			[&](const auto &freeVariable)
+			{
+				const auto matchesVariableToBeReplaced = std::find_if(otherPredicate.arguments.cbegin(), otherPredicate.arguments.cend(),
+					[&](const ast::Term &argument)
+					{
+						assert(argument.is<ast::Variable>());
+						const auto &otherVariable = argument.get<ast::Variable>();
+
+						return (freeVariable.get() == otherVariable.declaration);
+					});
+
+				return (matchesVariableToBeReplaced != otherPredicate.arguments.cend());
+			};
+
+		freeVariables.erase(std::remove_if(freeVariables.begin(), freeVariables.end(), isFreeVariableUnneeded), freeVariables.end());
+
+		// Currently, only rules with singleton heads are supported
+		// Rules with multiple elements in the head are not yet handled correctly by the head variable detection mechanism
+		for (const auto &freeVariable : freeVariables)
+			if (freeVariable->type == ast::VariableDeclaration::Type::Head)
+				throw CompletionException("cannot perform completion, only singleton rule heads supported currently");
+
+		// Second, link all occurrences of the deleted free variable to the new, unique parameter
 		for (size_t i = 0; i < parameters.size(); i++)
 		{
 			assert(otherPredicate.arguments[i].is<ast::Variable>());
@@ -50,16 +78,6 @@ ast::Formula buildCompletedFormulaDisjunction(const ast::Predicate &predicate, c
 			scopedFormula.formula.accept(ast::ReplaceVariableInFormulaVisitor(), scopedFormula.formula, otherVariable.declaration, parameters[i].get());
 		}
 
-		// Remove all the head variables, because they are not free variables after completion
-		const auto isHeadVariable =
-			[](const auto &variableDeclaration)
-			{
-				return variableDeclaration->type == ast::VariableDeclaration::Type::Head;
-			};
-
-		auto &freeVariables = scopedFormula.freeVariables;
-		freeVariables.erase(std::remove_if(freeVariables.begin(), freeVariables.end(), isHeadVariable), freeVariables.end());
-
 		if (freeVariables.empty())
 			disjunction.get<ast::Or>().arguments.emplace_back(std::move(implies.antecedent));
 		else

src/anthem/IntegerVariableDetection.cpp

@@ -1,11 +1,12 @@
 #include <anthem/IntegerVariableDetection.h>
 
-#include <anthem/Arithmetics.h>
 #include <anthem/ASTCopy.h>
 #include <anthem/ASTUtils.h>
 #include <anthem/ASTVisitors.h>
+#include <anthem/Evaluation.h>
 #include <anthem/Exception.h>
 #include <anthem/Simplification.h>
+#include <anthem/Type.h>
 #include <anthem/Utils.h>
 #include <anthem/output/AST.h>
 
@@ -18,19 +19,19 @@ namespace anthem
 //
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 
-using VariableDomainMap = std::map<const ast::VariableDeclaration *, ast::Domain>;
+using VariableDomainMap = std::map<const ast::VariableDeclaration *, Domain>;
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 
-ast::Domain domain(const ast::Variable &variable, VariableDomainMap &variableDomainMap)
+Domain domain(const ast::Variable &variable, VariableDomainMap &variableDomainMap)
 {
-	if (variable.declaration->domain != ast::Domain::Unknown)
+	if (variable.declaration->domain != Domain::Unknown)
 		return variable.declaration->domain;
 
 	const auto match = variableDomainMap.find(variable.declaration);
 
 	if (match == variableDomainMap.end())
-		return ast::Domain::Unknown;
+		return Domain::Unknown;
 
 	return match->second;
 }
@@ -40,14 +41,14 @@ ast::Domain domain(const ast::Variable &variable, VariableDomainMap &variableDom
 void clearVariableDomainMap(VariableDomainMap &variableDomainMap)
 {
 	for (auto &variableDeclaration : variableDomainMap)
-		variableDeclaration.second = ast::Domain::Unknown;
+		variableDeclaration.second = Domain::Unknown;
 }
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 
 struct VariableDomainMapAccessor
 {
-	ast::Domain operator()(const ast::Variable &variable, VariableDomainMap &variableDomainMap)
+	Domain operator()(const ast::Variable &variable, VariableDomainMap &variableDomainMap)
 	{
 		return domain(variable, variableDomainMap);
 	}
@@ -55,216 +56,16 @@ struct VariableDomainMapAccessor
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
 
-EvaluationResult isArithmetic(const ast::Term &term, VariableDomainMap &variableDomainMap)
+Type type(const ast::Term &term, VariableDomainMap &variableDomainMap)
 {
-	return isArithmetic<VariableDomainMapAccessor>(term, 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 isLeftArithmetic = isArithmetic(comparison.left, variableDomainMap);
-		const auto isRightArithmetic = isArithmetic(comparison.right, variableDomainMap);
-
-		if (isLeftArithmetic == EvaluationResult::Error || isRightArithmetic == EvaluationResult::Error)
-			return EvaluationResult::Error;
-
-		if (isLeftArithmetic == EvaluationResult::Unknown || isRightArithmetic == EvaluationResult::Unknown)
-			return EvaluationResult::Unknown;
-
-		if (isLeftArithmetic == isRightArithmetic)
-			return EvaluationResult::Unknown;
-
-		// Handle the case where one side is arithmetic but the other one isn’t
-		switch (comparison.operator_)
-		{
-			case ast::Comparison::Operator::Equal:
-				return EvaluationResult::False;
-			case ast::Comparison::Operator::NotEqual:
-				return EvaluationResult::True;
-			default:
-				// TODO: implement more cases
-				return EvaluationResult::Unknown;
-		}
-	}
-
-	static EvaluationResult visit(const ast::Exists &exists, VariableDomainMap &variableDomainMap)
-	{
-		return evaluate(exists.argument, variableDomainMap);
-	}
-
-	static EvaluationResult visit(const ast::ForAll &forAll, VariableDomainMap &variableDomainMap)
-	{
-		return evaluate(forAll.argument, variableDomainMap);
-	}
-
-	static EvaluationResult visit(const ast::Implies &implies, VariableDomainMap &variableDomainMap)
-	{
-		const auto antecedentResult = evaluate(implies.antecedent, variableDomainMap);
-		const auto consequentResult = evaluate(implies.consequent, variableDomainMap);
-
-		if (antecedentResult == EvaluationResult::Error || consequentResult == EvaluationResult::Error)
-			return EvaluationResult::Error;
-
-		if (antecedentResult == EvaluationResult::False)
-			return EvaluationResult::True;
-
-		if (consequentResult == EvaluationResult::True)
-			return EvaluationResult::True;
-
-		if (antecedentResult == EvaluationResult::True && consequentResult == EvaluationResult::False)
-			return EvaluationResult::False;
-
-		return EvaluationResult::Unknown;
-	}
-
-	static EvaluationResult visit(const ast::In &in, VariableDomainMap &variableDomainMap)
-	{
-		const auto isElementArithmetic = isArithmetic(in.element, variableDomainMap);
-		const auto isSetArithmetic = isArithmetic(in.set, variableDomainMap);
-
-		if (isElementArithmetic == EvaluationResult::Error || isSetArithmetic == EvaluationResult::Error)
-			return EvaluationResult::Error;
-
-		if (isElementArithmetic == EvaluationResult::Unknown || isSetArithmetic == EvaluationResult::Unknown)
-			return EvaluationResult::Unknown;
-
-		if (isElementArithmetic == isSetArithmetic)
-			return EvaluationResult::Unknown;
-
-		// If one side is arithmetic, but the other one isn’t, set inclusion is never satisfied
-		return EvaluationResult::False;
-	}
-
-	static EvaluationResult visit(const ast::Not &not_, VariableDomainMap &variableDomainMap)
-	{
-		const auto result = evaluate(not_.argument, variableDomainMap);
-
-		if (result == EvaluationResult::Error || result == EvaluationResult::Unknown)
-			return result;
-
-		return (result == EvaluationResult::True ? EvaluationResult::False : EvaluationResult::True);
-	}
-
-	static EvaluationResult visit(const ast::Or &or_, VariableDomainMap &variableDomainMap)
-	{
-		bool someTrue = false;
-		bool someUnknown = false;
-
-		for (const auto &argument : or_.arguments)
-		{
-			const auto result = evaluate(argument, variableDomainMap);
-
-			switch (result)
-			{
-				case EvaluationResult::Error:
-					return EvaluationResult::Error;
-				case EvaluationResult::True:
-					someTrue = true;
-					break;
-				case EvaluationResult::False:
-					break;
-				case EvaluationResult::Unknown:
-					someUnknown = true;
-					break;
-			}
-		}
-
-		if (someTrue)
-			return EvaluationResult::True;
-
-		if (someUnknown)
-			return EvaluationResult::Unknown;
-
-		return EvaluationResult::False;
-	}
-
-	static EvaluationResult visit(const ast::Predicate &predicate, VariableDomainMap &variableDomainMap)
-	{
-		assert(predicate.arguments.size() == predicate.declaration->arity());
-
-		for (size_t i = 0; i < predicate.arguments.size(); i++)
-		{
-			const auto &argument = predicate.arguments[i];
-			const auto &parameter = predicate.declaration->parameters[i];
-
-			if (parameter.domain != ast::Domain::Integer)
-				continue;
-
-			const auto isArgumentArithmetic = isArithmetic(argument, variableDomainMap);
-
-			if (isArgumentArithmetic == EvaluationResult::Error || isArgumentArithmetic == EvaluationResult::False)
-				return isArgumentArithmetic;
-		}
-
-		return EvaluationResult::Unknown;
-	}
-};
-
-////////////////////////////////////////////////////////////////////////////////////////////////////
-
 EvaluationResult evaluate(const ast::Formula &formula, VariableDomainMap &variableDomainMap)
 {
-	return formula.accept(EvaluateFormulaVisitor(), variableDomainMap);
+	return evaluate<VariableDomainMap>(formula, variableDomainMap);
 }
 
 ////////////////////////////////////////////////////////////////////////////////////////////////////
@@ -392,25 +193,20 @@ struct CheckIfDefinitionFalseFunctor
 	OperationResult operator()(ast::VariableDeclaration &variableDeclaration,
 		ast::Formula &, ast::Formula &definition, VariableDomainMap &variableDomainMap)
 	{
-		if (variableDeclaration.domain != ast::Domain::Unknown)
+		if (variableDeclaration.domain != Domain::Unknown)
 			return OperationResult::Unchanged;
 
 		clearVariableDomainMap(variableDomainMap);
 
-		auto result = evaluate(definition, variableDomainMap);
-
-		if (result == EvaluationResult::Error || result == EvaluationResult::False)
-			return OperationResult::Unchanged;
-
 		// As a hypothesis, make the parameter’s domain noninteger
-		variableDomainMap[&variableDeclaration] = ast::Domain::General;
+		variableDomainMap[&variableDeclaration] = Domain::Noninteger;
 
-		result = evaluate(definition, variableDomainMap);
+		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 = ast::Domain::Integer;
+			variableDeclaration.domain = Domain::Integer;
 			return OperationResult::Changed;
 		}
 
@@ -425,25 +221,20 @@ struct CheckIfQuantifiedFormulaFalseFunctor
 	OperationResult operator()(ast::VariableDeclaration &variableDeclaration,
 		ast::Formula &quantifiedFormula, VariableDomainMap &variableDomainMap)
 	{
-		if (variableDeclaration.domain != ast::Domain::Unknown)
+		if (variableDeclaration.domain != Domain::Unknown)
 			return OperationResult::Unchanged;
 
 		clearVariableDomainMap(variableDomainMap);
 
-		auto result = evaluate(quantifiedFormula, variableDomainMap);
-
-		if (result == EvaluationResult::Error || result == EvaluationResult::False)
-			return OperationResult::Unchanged;
-
 		// As a hypothesis, make the parameter’s domain noninteger
-		variableDomainMap[&variableDeclaration] = ast::Domain::General;
+		variableDomainMap[&variableDeclaration] = Domain::Noninteger;
 
-		result = evaluate(quantifiedFormula, variableDomainMap);
+		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 = ast::Domain::Integer;
+			variableDeclaration.domain = Domain::Integer;
 			return OperationResult::Changed;
 		}
 
@@ -458,25 +249,20 @@ struct CheckIfCompletedFormulaTrueFunctor
 	OperationResult operator()(ast::VariableDeclaration &variableDeclaration,
 		ast::Formula &, ast::Formula &completedFormula, VariableDomainMap &variableDomainMap)
 	{
-		if (variableDeclaration.domain != ast::Domain::Unknown)
+		if (variableDeclaration.domain != Domain::Unknown)
 			return OperationResult::Unchanged;
 
 		clearVariableDomainMap(variableDomainMap);
 
-		auto result = evaluate(completedFormula, variableDomainMap);
-
-		if (result == EvaluationResult::Error || result == EvaluationResult::True)
-			return OperationResult::Unchanged;
-
 		// As a hypothesis, make the parameter’s domain noninteger
-		variableDomainMap[&variableDeclaration] = ast::Domain::General;
+		variableDomainMap[&variableDeclaration] = Domain::Noninteger;
 
-		result = evaluate(completedFormula, variableDomainMap);
+		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 = ast::Domain::Integer;
+			variableDeclaration.domain = Domain::Integer;
 			return OperationResult::Changed;
 		}
 

src/anthem/Simplification.cpp

@@ -2,10 +2,10 @@
 
 #include <optional>
 
-#include <anthem/Arithmetics.h>
 #include <anthem/ASTCopy.h>
 #include <anthem/Equality.h>
 #include <anthem/SimplificationVisitors.h>
+#include <anthem/Type.h>
 #include <anthem/output/AST.h>
 
 namespace anthem
@@ -524,11 +524,14 @@ struct SimplificationRuleIntegerSetInclusion
 
 		auto &in = formula.get<ast::In>();
 
-		const auto isElementInteger = isInteger(in.element);
+		const auto elementType = type(in.element);
-		const auto isSetInteger = isInteger(in.set);
+		const auto setType = type(in.set);
 
-		if (isElementInteger != EvaluationResult::True || isSetInteger != EvaluationResult::True)
+		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));
 

src/anthem/Translation.cpp

@@ -147,7 +147,7 @@ void translate(const char *fileName, std::istream &stream, Context &context)
 		{
 			auto &parameter = predicateDeclaration->parameters[i];
 
-			if (parameter.domain != ast::Domain::Integer)
+			if (parameter.domain != Domain::Integer)
 				continue;
 
 			context.logger.outputStream()
@@ -155,7 +155,7 @@ void translate(const char *fileName, std::istream &stream, Context &context)
 				<< "(" << predicateDeclaration->name
 				<< "/" << output::Number(predicateDeclaration->arity())
 				<< "@" << output::Number(i + 1)
-				<< ")." << std::endl;
+				<< ")" << std::endl;
 		}
 	}
 }

tests/TestCompletion.cpp

@@ -123,7 +123,7 @@ TEST_CASE("[completion] Rules are completed", "[completion]")
 		anthem::translate("input", input, context);
 
 		CHECK(output.str() ==
-			"forall V1 (f(V1) <-> (exists U1 (V1 = f(f(f(f(U1)))) and f(U1)) or V1 in 1..5))\n");
+			"forall V1 (f(V1) <-> (exists U1 (V1 = f(f(f(f(U1)))) and f(U1)) or V1 in (1..5)))\n");
 	}
 
 	SECTION("useless implications")
@@ -152,8 +152,8 @@ TEST_CASE("[completion] Rules are completed", "[completion]")
 
 		CHECK(output.str() ==
 			"forall V1 (covered(V1) <-> exists U1 in(V1, U1))\n"
-			"forall V2, V3 (in(V2, V3) -> (V2 in 1..n and V3 in 1..r))\n"
+			"forall V2, V3 (in(V2, V3) -> (V2 in (1..n) and V3 in (1..r)))\n"
-			"forall U2 (U2 in 1..n -> covered(U2))\n"
+			"forall U2 (U2 in (1..n) -> covered(U2))\n"
 			"forall U3, U4, U5 (not in(U3, U4) or not in(U5, U4) or not exists X1 (X1 in (U3 + U5) and in(X1, U4)))\n");
 	}
 
@@ -190,6 +190,6 @@ TEST_CASE("[completion] Rules are completed", "[completion]")
 		input << "adj(X, Y) :- X = 1..n, Y = 1..n, |X - Y| = 1.";
 		anthem::translate("input", input, context);
 
-		CHECK(output.str() == "forall V1, V2 (adj(V1, V2) <-> (V1 in 1..n and V2 in 1..n and |V1 - V2| = 1))\n");
+		CHECK(output.str() == "forall V1, V2 (adj(V1, V2) <-> (V1 in (1..n) and V2 in (1..n) and |V1 - V2| = 1))\n");
 	}
 }

tests/TestHiddenPredicateElimination.cpp

@@ -150,7 +150,7 @@ TEST_CASE("[hidden predicate elimination] Hidden predicates are correctly elimin
 
 		// TODO: simplify further
 		CHECK(output.str() ==
-			"forall V1 (a(V1) <-> exists U1 (c(V1) = c(U1) and U1 in 1..4))\n");
+			"forall V1 (a(V1) <-> exists U1 (c(V1) = c(U1) and U1 in (1..4)))\n");
 	}
 
 	SECTION("simple propositions are hidden correctly")

tests/TestIntegerDetection.cpp

@@ -0,0 +1,117 @@
+#include <catch.hpp>
+
+#include <sstream>
+
+#include <anthem/AST.h>
+#include <anthem/Context.h>
+#include <anthem/Translation.h>
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+TEST_CASE("[integer detection] Integer variables are correctly detected", "[integer detection]")
+{
+	std::stringstream input;
+	std::stringstream output;
+	std::stringstream errors;
+
+	anthem::output::Logger logger(output, errors);
+	anthem::Context context(std::move(logger));
+	context.performSimplification = true;
+	context.performCompletion = true;
+	context.performIntegerDetection = true;
+
+	SECTION("simple-to-detect integer parameter")
+	{
+		input << "p(X) :- X = 1..5.";
+		anthem::translate("input", input, context);
+	
+		CHECK(output.str() ==
+			"forall N1 (p(N1) <-> N1 in (1..5))\n"
+			"int(p/1@1)\n");
+	}
+
+	SECTION("simple noninteger parameter")
+	{
+		input <<
+			"p(X) :- X = 1..5.\n"
+			"p(X) :- X = error.";
+		anthem::translate("input", input, context);
+	
+		CHECK(output.str() ==
+			"forall V1 (p(V1) <-> (V1 in (1..5) or V1 = error))\n");
+	}
+	
+	SECTION("integer parameter with arithmetics")
+	{
+		input << "p(X) :- X = (2 + (1..5)) * 2.";
+		anthem::translate("input", input, context);
+	
+		CHECK(output.str() ==
+			"forall N1 (p(N1) <-> N1 in ((2 + (1..5)) * 2))\n"
+			"int(p/1@1)\n");
+	}
+	
+	SECTION("integer parameter with arithmetics depending on another integer parameter")
+	{
+		input
+			<< "p(X) :- X = 1..5."
+			<< "q(X) :- p(Y), X = (Y + 5) / 3.";
+		anthem::translate("input", input, context);
+	
+		CHECK(output.str() ==
+			"forall N1 (p(N1) <-> N1 in (1..5))\n"
+			"forall N2 (q(N2) <-> exists N3 (p(N3) and N2 in ((N3 + 5) / 3)))\n"
+			"int(p/1@1)\n"
+			"int(q/1@1)\n");
+	}
+
+	SECTION("multiple mixed parameters")
+	{
+		input
+			<< "p(X) :- X = 1..5."
+			<< "q(X) :- X = error."
+			<< "r(A, B, C) :- p(X), A = X ** 2, q(B), p(C).";
+		anthem::translate("input", input, context);
+		
+		CHECK(output.str() ==
+			"forall N1 (p(N1) <-> N1 in (1..5))\n"
+			"forall V1 (q(V1) <-> V1 = error)\n"
+			"forall N2, V2, N3 (r(N2, V2, N3) <-> exists N4 (p(N4) and N2 = (N4 ** 2) and q(V2) and p(N3)))\n"
+			"int(p/1@1)\n"
+			"int(r/3@1)\n"
+			"int(r/3@3)\n");
+	}
+	
+	SECTION("integer parameter despite usage of constant symbol")
+	{
+		input
+			<< "p(X) :- X = 2..n.";
+		anthem::translate("input", input, context);
+	
+		CHECK(output.str() ==
+			"forall N1 (p(N1) <-> N1 in (2..n))\n"
+			"int(p/1@1)\n");
+	}
+	
+	SECTION("integer arithmetics are correctly simplified for operators other than division")
+	{
+		input
+			<< "p(X) :- X = 5 + 9 ** 2.";
+		anthem::translate("input", input, context);
+	
+		CHECK(output.str() ==
+			"forall N1 (p(N1) <-> N1 = (5 + (9 ** 2)))\n"
+			"int(p/1@1)\n");
+	}
+	
+	SECTION("integer arithmetics are not simplified with the division operator")
+	{
+		input
+			<< "p(X) :- X = 5 + 9 / 0.";
+		anthem::translate("input", input, context);
+	
+		CHECK(output.str() ==
+			"forall N1 (p(N1) <-> N1 in (5 + (9 / 0)))\n"
+			"int(p/1@1)\n");
+	}
+}

tests/TestSimplification.cpp

@@ -40,7 +40,7 @@ TEST_CASE("[simplification] Rules are simplified correctly", "[simplification]")
 		input << ":- not covered(I), I = 1..n.";
 		anthem::translate("input", input, context);
 
-		CHECK(output.str() == "((not covered(U1) and U1 in 1..n) -> #false)\n");
+		CHECK(output.str() == "((not covered(U1) and U1 in (1..n)) -> #false)\n");
 	}
 
 	SECTION("comparisons")
@@ -50,4 +50,34 @@ TEST_CASE("[simplification] Rules are simplified correctly", "[simplification]")
 
 		CHECK(output.str() == "(U1 > U2 -> #false)\n");
 	}
+
+	SECTION("biconditionals are replaced with implifactions with choice rules")
+	{
+		context.performCompletion = true;
+
+		input << "{p(a)}.";
+		anthem::translate("input", input, context);
+
+		CHECK(output.str() == "forall V1 (p(V1) -> V1 = a)\n");
+	}
+
+	SECTION("biconditionals are replaced with implifactions with complicated choice rules")
+	{
+		context.performCompletion = true;
+
+		input << "{p(n + 5)}.";
+		anthem::translate("input", input, context);
+
+		CHECK(output.str() == "forall V1 (p(V1) -> V1 in (n + 5))\n");
+	}
+
+	SECTION("biconditionals are not replaced with implifactions with nonchoice rules")
+	{
+		context.performCompletion = true;
+
+		input << "p(a).";
+		anthem::translate("input", input, context);
+
+		CHECK(output.str() == "forall V1 (p(V1) <-> V1 = a)\n");
+	}
 }

tests/TestTranslation.cpp

@@ -24,7 +24,7 @@ TEST_CASE("[translation] Rules are translated correctly", "[translation]")
 		input << "p(1..5).";
 		anthem::translate("input", input, context);
 
-		CHECK(output.str() == "(V1 in 1..5 -> p(V1))\n");
+		CHECK(output.str() == "(V1 in (1..5) -> p(V1))\n");
 	}
 
 	SECTION("simple example 2")
@@ -32,7 +32,7 @@ TEST_CASE("[translation] Rules are translated correctly", "[translation]")
 		input << "p(N) :- N = 1..5.";
 		anthem::translate("input", input, context);
 
-		CHECK(output.str() == "((V1 in U1 and exists X1, X2 (X1 in U1 and X2 in 1..5 and X1 = X2)) -> p(V1))\n");
+		CHECK(output.str() == "((V1 in U1 and exists X1, X2 (X1 in U1 and X2 in (1..5) and X1 = X2)) -> p(V1))\n");
 	}
 
 	SECTION("simple example 3")
@@ -48,7 +48,7 @@ TEST_CASE("[translation] Rules are translated correctly", "[translation]")
 		input << "p(N, 1, 2) :- N = 1..5.";
 		anthem::translate("input", input, context);
 
-		CHECK(output.str() == "((V1 in U1 and V2 in 1 and V3 in 2 and exists X1, X2 (X1 in U1 and X2 in 1..5 and X1 = X2)) -> p(V1, V2, V3))\n");
+		CHECK(output.str() == "((V1 in U1 and V2 in 1 and V3 in 2 and exists X1, X2 (X1 in U1 and X2 in (1..5) and X1 = X2)) -> p(V1, V2, V3))\n");
 	}
 
 	SECTION("disjunctive head")
@@ -57,7 +57,7 @@ TEST_CASE("[translation] Rules are translated correctly", "[translation]")
 		input << "q(3, N); p(N, 1, 2) :- N = 1..5.";
 		anthem::translate("input", input, context);
 
-		CHECK(output.str() == "((V1 in U1 and V2 in 1 and V3 in 2 and V4 in 3 and V5 in U1 and exists X1, X2 (X1 in U1 and X2 in 1..5 and X1 = X2)) -> (p(V1, V2, V3) or q(V4, V5)))\n");
+		CHECK(output.str() == "((V1 in U1 and V2 in 1 and V3 in 2 and V4 in 3 and V5 in U1 and exists X1, X2 (X1 in U1 and X2 in (1..5) and X1 = X2)) -> (p(V1, V2, V3) or q(V4, V5)))\n");
 	}
 
 	SECTION("disjunctive head (alternative syntax)")
@@ -66,7 +66,7 @@ TEST_CASE("[translation] Rules are translated correctly", "[translation]")
 		input << "q(3, N), p(N, 1, 2) :- N = 1..5.";
 		anthem::translate("input", input, context);
 
-		CHECK(output.str() == "((V1 in U1 and V2 in 1 and V3 in 2 and V4 in 3 and V5 in U1 and exists X1, X2 (X1 in U1 and X2 in 1..5 and X1 = X2)) -> (p(V1, V2, V3) or q(V4, V5)))\n");
+		CHECK(output.str() == "((V1 in U1 and V2 in 1 and V3 in 2 and V4 in 3 and V5 in U1 and exists X1, X2 (X1 in U1 and X2 in (1..5) and X1 = X2)) -> (p(V1, V2, V3) or q(V4, V5)))\n");
 	}
 
 	SECTION("escaping conflicting variable names")
@@ -98,7 +98,7 @@ TEST_CASE("[translation] Rules are translated correctly", "[translation]")
 		input << ":- not p(I), I = 1..n.";
 		anthem::translate("input", input, context);
 
-		CHECK(output.str() == "((exists X1 (X1 in U1 and not p(X1)) and exists X2, X3 (X2 in U1 and X3 in 1..n and X2 = X3)) -> #false)\n");
+		CHECK(output.str() == "((exists X1 (X1 in U1 and not p(X1)) and exists X2, X3 (X2 in U1 and X3 in (1..n) and X2 = X3)) -> #false)\n");
 	}
 
 	SECTION("disjunctive fact (no arguments)")
@@ -178,7 +178,7 @@ TEST_CASE("[translation] Rules are translated correctly", "[translation]")
 		input << "p(X, 1..10) :- q(X, 6..12).";
 		anthem::translate("input", input, context);
 
-		CHECK(output.str() == "((V1 in U1 and V2 in 1..10 and exists X1, X2 (X1 in U1 and X2 in 6..12 and q(X1, X2))) -> p(V1, V2))\n");
+		CHECK(output.str() == "((V1 in U1 and V2 in (1..10) and exists X1, X2 (X1 in U1 and X2 in (6..12) and q(X1, X2))) -> p(V1, V2))\n");
 	}
 
 	SECTION("intervals with variable")
@@ -186,7 +186,7 @@ TEST_CASE("[translation] Rules are translated correctly", "[translation]")
 		input << ":- q(N), 1 = 1..N.";
 		anthem::translate("input", input, context);
 
-		CHECK(output.str() == "((exists X1 (X1 in U1 and q(X1)) and exists X2, X3 (X2 in 1 and X3 in 1..U1 and X2 = X3)) -> #false)\n");
+		CHECK(output.str() == "((exists X1 (X1 in U1 and q(X1)) and exists X2, X3 (X2 in 1 and X3 in (1..U1) and X2 = X3)) -> #false)\n");
 	}
 
 	SECTION("intervals with two variables")
@@ -194,7 +194,7 @@ TEST_CASE("[translation] Rules are translated correctly", "[translation]")
 		input << ":- q(M, N), M = 1..N.";
 		anthem::translate("input", input, context);
 
-		CHECK(output.str() == "((exists X1, X2 (X1 in U1 and X2 in U2 and q(X1, X2)) and exists X3, X4 (X3 in U1 and X4 in 1..U2 and X3 = X4)) -> #false)\n");
+		CHECK(output.str() == "((exists X1, X2 (X1 in U1 and X2 in U2 and q(X1, X2)) and exists X3, X4 (X3 in U1 and X4 in (1..U2) and X3 = X4)) -> #false)\n");
 	}
 
 	SECTION("comparisons")
@@ -262,7 +262,7 @@ TEST_CASE("[translation] Rules are translated correctly", "[translation]")
 		anthem::translate("input", input, context);
 
 		// TODO: eliminate V5: not needed
-		CHECK(output.str() == "((V1 in 1..3 and V2 in U1 and V3 in 2..4 and p(V1, V2)) -> p(V1, V2))\n((V4 in 1..3 and V5 in U2 and V6 in 2..4 and q(V6)) -> q(V6))\n");
+		CHECK(output.str() == "((V1 in (1..3) and V2 in U1 and V3 in (2..4) and p(V1, V2)) -> p(V1, V2))\n((V4 in (1..3) and V5 in U2 and V6 in (2..4) and q(V6)) -> q(V6))\n");
 	}
 
 	SECTION("choice rule with body")
@@ -302,6 +302,6 @@ TEST_CASE("[translation] Rules are translated correctly", "[translation]")
 		input << "p(N, N ** N) :- N = 1..n.";
 		anthem::translate("input", input, context);
 
-		CHECK(output.str() == "((V1 in U1 and V2 in (U1 ** U1) and exists X1, X2 (X1 in U1 and X2 in 1..n and X1 = X2)) -> p(V1, V2))\n");
+		CHECK(output.str() == "((V1 in U1 and V2 in (U1 ** U1) and exists X1, X2 (X1 in U1 and X2 in (1..n) and X1 = X2)) -> p(V1, V2))\n");
 	}
 }

tests/TestUnsupported.cpp

@@ -0,0 +1,73 @@
+#include <catch.hpp>
+
+#include <sstream>
+
+#include <anthem/AST.h>
+#include <anthem/Context.h>
+#include <anthem/Translation.h>
+
+////////////////////////////////////////////////////////////////////////////////////////////////////
+
+TEST_CASE("[unsupported] Errors are correctly issued when using unsupported features", "[unsupported]")
+{
+	std::stringstream input;
+	std::stringstream output;
+	std::stringstream errors;
+
+	anthem::output::Logger logger(output, errors);
+	anthem::Context context(std::move(logger));
+
+	SECTION("rules with disjunctive head are unsupported")
+	{
+		context.performCompletion = true;
+
+		input << "a; b.";
+
+		CHECK_THROWS(anthem::translate("input", input, context));
+	}
+
+	SECTION("rules with disjunctive head containing elements with arguments are unsupported")
+	{
+		context.performCompletion = true;
+
+		input << "p(a); p(b).";
+
+		CHECK_THROWS(anthem::translate("input", input, context));
+	}
+
+	SECTION("singleton choice rules are supported")
+	{
+		context.performCompletion = true;
+
+		input << "{a}.";
+
+		CHECK_NOTHROW(anthem::translate("input", input, context));
+	}
+
+	SECTION("singleton choice rules containing an element with arguments are supported")
+	{
+		context.performCompletion = true;
+
+		input << "{p(a)}.";
+
+		CHECK_NOTHROW(anthem::translate("input", input, context));
+	}
+
+	SECTION("choice rules with multiple simple elements are supported")
+	{
+		context.performCompletion = true;
+
+		input << "{a; b}.";
+
+		CHECK_NOTHROW(anthem::translate("input", input, context));
+	}
+
+	SECTION("choice rules with multiple elements with arguments are unsupported")
+	{
+		context.performCompletion = true;
+
+		input << "{p(a); p(b)}.";
+
+		CHECK_THROWS(anthem::translate("input", input, context));
+	}
+}