Sets

Finite Sets

As of July 2014 (CVC4 v1.4), we include support for theory of finite sets. The simplest way to get a sense of the syntax is to look at an example:

• CVC language example
• SMT language example
• API example: tutorial, source code

For reference, below is a short summary of the sorts, constants, functions and predicates.

	CVC language	SMTLIB language	C++ API
Logic string	Not needed	append "FS" for finite sets	append "FS" for finite sets
		(set-logic QF_UFLIAFS)	smt.setLogic("QF_UFLIAFS");
Sort	SET OF <Element Sort>	(Set <Element Sort>)	CVC4::ExprManager::mkSetType(CVC4::Type elementType)
	X: SET OF INT;	(declare-fun X () (Set Int))	em.mkSetType( em.integerType() );
Union	X | Y	(union X Y)	em.mkExpr(kind::UNION, X, Y);
Intersection	X & Y	(intersection X Y)	em.mkExpr(kind::INTERSECTION, X, Y);
Set subtraction	X – Y	(setminus X Y)	em.mkExpr(kind::SETMINUS, X, Y);
Membership	x IS_IN X	(member x X)	em.mkExpr(kind::MEMBER, x, X);
Subset	X <= Y	(subset X Y)	em.mkExpr(kind::SUBSET, X, Y);
Empty set	{} :: <Type Ascription>	(as emptyset <Type Ascription>)	CVC4::EmptySet(CVC4::SetType setType)
	{} :: SET OF INT	(as emptyset (Set Int))	em.mkConst(EmptySet(em.mkSetType(em.integerType())));
Singleton set	{1}	(singleton 1)	em.mkExpr(kind::SINGLETON, oneExpr);
Cardinality	CARD( X )	(card X)	em.mkExpr(kind::CARD, X);
Insert/finite sets	{1, 2, 3, 4}	(insert 1 2 3 (singleton 4))	em.mkExpr(kind::INSERT, c1, c2, c3, sgl4);
Complement	~ X	(complement X)	em.mkExpr(kind::COMPLEMENT, X);
Universe set	UNIVERSE :: <Type Ascription>	(as univset <Type Ascription>)
	UNIVERSE :: SET OF INT	(as univset (Set Int))	em.mkNullaryOperator(em.mkSetType(em.integerType()),kind::UNIVERSE_SET);

Operator precedence for CVC language: & | – IS_IN <= =. For example, A - B | A & C <= D is read as ( A - ( B | (A & C) ) ) <= D.

Finite Relations

	CVC language	SMTLIB language	C++ API
Logic string	Not needed	(set-logic QF_ALL)	smt.setLogic("QF_ALL");
Tuple Sort	[<Sort_1>, ..., <Sort_n>]	(Tuple <Sort_1>, ..., <Sort_n>)	CVC4::ExprManager::mkTupleType(std::vector<CVC4::Type>& types)
	t: [INT, INT];	(declare-fun t () (Tuple Int Int))	std::vector<Type> types; types.push_back(em.mkIntegerType());</br> types.push_back(em.mkIntegerType());</br> em.mkTupleType( types );
Tuple constructor	(t1, ..., tn)	(mkTuple t1, ..., tn)	DatatypeType tt = em.mkTupleType(types); const Datatype& dt = tt.getDatatype(); Expr c = dt[0].getConstructor(); em.mkExpr(kind::APPLY_CONSTRUCTOR, c, t1, ..., tn);
Tuple selector	t.i	((_ tupSel i) t)	DatatypeType tt = em.mkTupleType(types); const Datatype& dt = tt.getDatatype(); Expr s = dt[0][i].getSelector(); em.mkExpr(kind::APPLY_SELECTOR, s, t);
Relation Sort	SET OF [<Sort_1>, ..., <Sort_n>]	(Set (Tuple <Sort_1>, ..., <Sort_n>))	CVC4::ExprManager::mkSetType(CVC4::Type elementType)
	X: SET OF [INT, INT];	(declare-fun X () (Set (Tuple Int Int)))	em.mkSetType( em.mkTupleType( em.integerType(), em.integerType() ) );
Transpose	TRANSPOSE(X)	(transpose X)	em.mkExpr(kind::TRANSPOSE, X);
Transitive Closure	TCLOSURE(X)	(tclosure X)	em.mkExpr(kind::TCLOSURE, X);
Join	X JOIN Y	(join X Y)	em.mkExpr(kind::JOIN, X);
Product	X PRODUCT Y	(product X Y)	em.mkExpr(kind::PRODUCT, X);