Exclusion and inclusion in Z3 - set

Question

I am trying to model inclusion and exclusion of elements in sets with Z3. In particular inclusion of elements with distinct values, and exclusion of elements not already in a target set. So basically I want to have a set U and have Z3 find a set U_d that only contains elements of U with distinct values.
My current approach uses quantifiers, but I'm having trouble understanding how to state that I want to always include elements in U_d if they appear in U.
( set-option :produce-models true)
;;; Two simple sorts.
;;; Sets and Zs.
( declare-sort Z 0 )
( declare-sort Set 0 )
;;; A set can contain a Z or not.
;;; Zs can have a value.
( declare-fun contains (Set Z) bool )
( declare-fun value (Z) Int )
;;; Two sets and two Z instances for use in the example.
( declare-const set Set )
( declare-const distinct_set Set )
( declare-const A Z )
( declare-const B Z )
;;; The elements and sets are distinct.
( assert ( distinct A B ) )
( assert ( distinct set distinct_set ) )
;;; Set 'set' contains A but not B
( assert ( = ( contains set A ) true ) )
( assert ( = ( contains set B ) false ) )
;;; Assert that all elements contained by distinct_set have different values unless they're the same variable.
( assert
( forall ( (x Z) (y Z) )
( =>
( and
( contains distinct_set x )
( contains distinct_set y )
( = ( value x ) ( value y ) ) )
( = x y ) )))
;;; distinct_set can contain only elements that appear in set.
;;; In other words, distinct_set is a proper set.
( assert
( forall ( ( x Z ) )
( =>
( contains distinct_set x )
( contains set x ))))
;;; Give elements some values.
( assert ( = (value A) 0 ) )
( assert ( = (value B) 1 ) )
( push )
( check-sat )
( get-value (( contains distinct_set A )))
( get-value (( contains distinct_set B )))
( pop )
The assignments it produces are:
sat
((( contains distinct_set A ) false))
((( contains distinct_set B ) false))
The assignments I would like are:
sat
((( contains distinct_set A ) true))
((( contains distinct_set B ) false))
I understand that an assignment of false to both A and B is a logically correct assignment, but I don't know how to state things in such a way as to rule those sorts of cases out.
Perhaps I'm not thinking about the problem correctly.
Any advice would be much appreciated. :)

Answer 1

What do you think of the following assertion?
(assert
(forall ((x Z))
(=> (contains set x)
(exists ((y Z))
(and (= (value x) (value y))
(contains set y)
(contains distinct_set y))))))
It says that for every element x of set (i.e., U), there is a y s.t.
value of y is equal to value of x
y is also an element of set
y is an element of distinct_set (i.e., U_d)
This assertion makes sure that if there are two elements in set with the same value, then one and only one of them is an element of distinct_set. Is that what you want?
Note that, if we just add this assertion, Z3 will still produce a model where
((( contains distinct_set A ) false))
((( contains distinct_set B ) false))
If we inspect the model produced by Z3 using (get-model), we will notice that set contains another element different from A. So, to force set to contain only the element A, we have to assert
(assert
(forall ((x Z))
(= (contains set x) (= x A))))
After you add this assertion, the following two assertions become redundant:
( assert ( = ( contains set A ) true ) )
( assert ( = ( contains set B ) false ) )
Now, let us consider the case where set contains two values: A and C, and they both have the same value. The following script also asks questions such as: is there a model where
distinct_set does not contain A
distinct_set does not contain A nor C
distinct_set contains A and C
Script:
( set-option :produce-models true)
;;; Two simple sorts.
;;; Sets and Zs.
( declare-sort Z 0 )
( declare-sort Set 0 )
;;; A set can contain a Z or not.
;;; Zs can have a value.
( declare-fun contains (Set Z) bool )
( declare-fun value (Z) Int )
;;; Two sets and two Z instances for use in the example.
( declare-const set Set )
( declare-const distinct_set Set )
( declare-const A Z )
( declare-const B Z )
( declare-const C Z )
;;; The elements and sets are distinct.
( assert ( distinct A B C) )
( assert ( distinct set distinct_set ) )
;;; set contains only A and C
(assert
(forall ((x Z))
(= (contains set x) (or (= x A) (= x C)))))
;;; Assert that all elements contained by distinct_set have different values unless they're the same variable.
( assert
( forall ( (x Z) (y Z) )
( =>
( and
( contains distinct_set x )
( contains distinct_set y )
( = ( value x ) ( value y ) ) )
( = x y ) )))
;;; distinct_set can contain only elements that appear in set.
;;; In other words, distinct_set is a proper set.
( assert
( forall ( ( x Z ) )
( =>
( contains distinct_set x )
( contains set x ))))
;;; Give elements some values.
( assert ( = (value A) 0 ) )
( assert ( = (value B) 1 ) )
( assert ( = (value C) 0 ) )
(assert
(forall ((x Z))
(=> (contains set x)
(exists ((y Z))
(and (= (value x) (value y))
(contains set y)
(contains distinct_set y))))))
( push )
( check-sat )
( get-model )
( get-value (( contains distinct_set A )))
( get-value (( contains distinct_set B )))
( get-value (( contains distinct_set C )))
( echo "Is there another model where A is not in distinct_set")
( assert (not ( contains distinct_set A )))
(check-sat)
( get-value (( contains distinct_set A )))
( get-value (( contains distinct_set B )))
( get-value (( contains distinct_set C )))
( echo "Is there another model where A and C are not in distinct_set")
( assert (not ( contains distinct_set C )))
(check-sat)
( pop ) ;; retracting the last two assertions
( push )
( echo "Is there a model where A and C are in distinct_set")
( assert ( contains distinct_set A ))
( assert ( contains distinct_set C ))
( check-sat )