Sets
From CVC4
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");
|
| 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() ) );
| |
| Tuple Sort | [<Sort_1>, ..., <Sort_n>] | (Tuple <Sort_1>, ..., <Sort_n>) | CVC4::ExprManager::mkTupleType(std::vector<CVC4::Type>& types) |
X: [INT, INT];
|
(declare-fun X () (Tuple Int Int))
|
em.mkTupleType( em.integerType(), em.integerType() );
| |
| Tuple constructor | (X1, ..., Xn)
|
(mkTuple X1, ..., Xn)
|
em.mkExpr(kind::APPLY_CONSTRUCTOR, X1, ..., Xn);
|
| Transpose | TRANSPOSE(X)
|
(transpose X)
|
--
|
| Transitive Closure | TCLOSURE(X)
|
(tclosure X)
|
--
|
| Join | X JOIN Y
|
(join X Y)
|
--
|
| Product | X PRODUCT Y
|
(product X Y)
|
--
|
