Calculate minimum value of Bitvector in Z3

Question

I have the following piece of code: (declare-const L4 (_ BitVec 6)) (declare-const L1 (_ BitVec 6)) (declare-const L0 (_ BitVec 6)) (declare-const l2 (_ BitVec 6))

(assert (= l2 (_ bv8 6)))

;; All is encoding the set that contains {0, 1, 2, 3, 4, 5}
(define-const All (_ BitVec 6) #b111111) 
;; Empty is encoding the empty set
(define-const Empty (_ BitVec 6) #b000000) 

(define-fun LT_l ((S (_ BitVec 6)) (l (_ BitVec 6))) Bool
    ;; True if for all x in S x < l
    (= (bvand (bvshl All l) S) Empty))

(define-fun is_in ((e (_ BitVec 6)) (S (_ BitVec 6))) Bool
   ;; True if e is an element of the "set" S.
   (not (= (bvand (bvshl (_ bv1 6) e) S) Empty))) 

(define-fun is_minimal ((e (_ BitVec 6)) (S (_ BitVec 6))) Bool
    ;; True if e is the minimal element of S
    (and (is_in e S) ;; S contains e
         ;; (1 << e) - 1 represents the set of elements that are smaller than e
         (= (bvand (bvsub (bvshl (_ bv1 6) e) (_ bv1 6)) S) Empty))) 

;; To encode that forall x in L0 and forall y in L1. x < y
(define-fun LT ((L0 (_ BitVec 6)) (L1 (_ BitVec 6))) Bool
    ; True if forall x in L0 and forall y in L1, x < y
    (or (= L0 Empty)
        (= L1 Empty)
        (exists ((min (_ BitVec 6))) (and (is_minimal min L1) (LT_l L0 min)))))

(assert (not (= L0 Empty)))
(assert (not (= L1 Empty)))
(assert (not (= L4 Empty)))
(assert (LT_l L4 l2))
(assert (LT L0 L1))
(check-sat)
(get-model)
(assert (LT L1 L0))
(check-sat)

When I run this code I get the model is: sat (model (define-fun min!0 () (_ BitVec 6) #b000011) (define-fun L1 () (_ BitVec 6) #b001000) (define-fun L0 () (_ BitVec 6) #b000100) (define-fun L4 () (_ BitVec 6) #b000100) (define-fun l2 () (_ BitVec 6) #b001000) ) unsat

Why is the result of min is:

(define-fun min!0 () (_ BitVec 6)
    #b000011)

and not b001000 since the smallest value of L1 is this and not b000011.

Someone can explain me?

Finally, I define the function Lt_l that checks if for all x in S x < l, but now I wanted to do GT_l that checks if for all x in S l < x. I have the following code:

(define-fun GT_l ((S (_ BitVec 6)) (l (_ BitVec 6))) Bool
    (= (bvand (bvneg (bvshl (_ bv0 6) l)) S) Empty))

But this does not work why?

Thanks

Answer 1

In your example, you are representing sets using bit-vectors. For example, the bit-vector #b101000 represents the set {5, 3}. The output (define-fun L1 () (_ BitVec 6) #b001000) is essentially saying that L1 is the "set" {3}. One possible confusion is that bit-vectors are being used to represent sets and elements. The bit-vector min!0 represents an element. The output (define-fun min!0 () (_ BitVec 6) #b000011) is saying that min!0 is the value 3, and it is indeed the "minimal value" in L1.