Difference between revisions of "Sets"

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Most of the above operators (e.g. set union, intersection, difference) have semantics that are straightforward. The semantics for the universe set is more subtle and is explained in the following.
 
Most of the above operators (e.g. set union, intersection, difference) have semantics that are straightforward. The semantics for the universe set is more subtle and is explained in the following.
  
−
The universe set (as univset (Set T)) is not  
+
The universe set (as univset (Set T)) is *not* interpreted as the set containing all elements of type T. Instead it is defined as any set such that all sets of type (Set T) are interpreted as subsets of it. 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 () (uniont (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.
  
−
For all sets X of type (Set T), the complement operator is such that:  
+
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)
 
   (complement X) = (setminus (as univset (Set T)) X)
 
holds in all models.
 
holds in all models.
  
−
The motivation for these semantics is to ensure that the universe set and applications of set complement can always be interpreted as a finite set in (quantifier-free) inputs.
+
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.
  
 
=Finite Relations=
 
=Finite Relations=

Revision as of 11:34, 12 March 2019

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.

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.

Semantics

Most of the above operators (e.g. set union, intersection, difference) have semantics that are straightforward. The semantics for the universe set is 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 is defined as any set such that all sets of type (Set T) are interpreted as subsets of it. 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 () (uniont (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.

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);
Retrieved from "http://cvc4.cs.stanford.edu/w/index.php?title=Sets&oldid=5742"