CVC4

An efficient open-source automatic theorem prover for satisfiability modulo theories (SMT) problems.

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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:

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

For more details, see our IJCAR 2016 paper.

  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 <Sort> (Set <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> (as univset <Type>)  
  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.

Semantics

The semantics of most of the above operators (e.g. set union, intersection, difference) are straightforward. The semantics for the universe set and complement are more subtle and is explained in the following.

The universe set (as univset (Set T)) is not interpreted as the set containing all elements of type T. Instead it may be interpreted as any set such that all sets of type (Set T) are interpreted as subsets of it. In other words, it is the union of the interpretations of all (finite) sets in our input. For example:

(declare-fun x () (Set Int))
(declare-fun y () (Set Int))
(declare-fun z () (Set Int))
(assert (member 0 x))
(assert (member 1 y))
(assert (= z (as univset (Set Int))))
(check-sat)

Here, a possible model is:

(define-fun x () (singleton 0))
(define-fun y () (singleton 1))
(define-fun z () (union (singleton 1) (singleton 0)))

Notice that the universe set in this example is interpreted the same as z, and is such that all sets in this example (x, y, and z) are subsets of it.

The set complement operator for (Set T) is interpreted relative to the interpretation of the universe set for (Set T), and not relative to the set of all elements of type T. That is, for all sets X of type (Set T), the complement operator is such that: (complement X) = (setminus (as univset (Set T)) X) holds in all models.

The motivation for these semantics is to ensure that the universe set for type T and applications of set complement can always be interpreted as a finite set in (quantifier-free) inputs, even if the cardinality of T is infinite. Above, notice that we were able to find a model for the universe set of type (Set Int) that contained two elements only.

Important Remark

In the presence of quantifiers, CVC4’s implementation of the above theory allows infinite sets. In particular, the following formula is SAT (even though CVC4 is not able to say this):

(set-logic ALL)
(declare-fun x () (Set Int))
(assert (forall ((z Int) (member (* 2 z) x)))
(check-sat)

The reason for that is that making this formula (and similar ones) UNSAT is counter-intuitive when quantifiers are present.

Finite Relations

Examples:

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

For more details, see our CADE 2017 paper.

  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()); types.push_back(em.mkIntegerType()); 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, Y); `
Product ` X PRODUCT Y ` ` (product X Y) ` ` em.mkExpr(kind::PRODUCT, X, Y); `