在Z3中有没有办法证明/显示给定模型是唯一的并且没有其他解决方案存在？

一个小例子来演示

(declare-const a1 Int)
(declare-const a2 Int)
(declare-const a3 Int)
(declare-const b1 Int)
(declare-const b2 Int)
(declare-const b3 Int)
(declare-const c1 Int)
(declare-const c2 Int)
(declare-const c3 Int)
(declare-const ra Int)
(declare-const rb Int)
(declare-const rc Int)
(declare-const r1 Int)
(declare-const r2 Int)
(declare-const r3 Int)
(assert (>= a1 0))
(assert (>= a2 0))
(assert (>= a3 0))
(assert (>= b1 0))
(assert (>= b2 0))
(assert (>= b3 0))
(assert (>= c1 0))
(assert (>= c2 0))
(assert (>= c3 0))
(assert (<= a1 9))
(assert (<= a2 9))
(assert (<= a3 9))
(assert (<= b1 9))
(assert (<= b2 9))
(assert (<= b3 9))
(assert (<= c1 9))
(assert (<= c2 9))
(assert (<= c3 9))
(assert (= ra 38))
(assert (= rb 1))
(assert (= rc 27))
(assert (= r1 55))
(assert (= r2 72))
(assert (= r3 6))
(assert (= ra (- (* a1 a2) a3)))
(assert (= rb (- (- b1 b2) b3)))
(assert (= rc (* (* c1 c2) c3)))
(assert (= r1 (- (* a1 b1) c1)))
(assert (= r2 (* (+ a2 b2) c2)))
(assert (= r3 (- (+ a3 b3) c3)))
(check-sat)
(get-model)

我知道以下模型是唯一的,但是我可以使用某些Z3选项或添加断言来确保这一点吗？

(model 
  (define-fun c3 () Int
    3)
  (define-fun c2 () Int
    9)
  (define-fun c1 () Int
    1)
  (define-fun b3 () Int
    5)
  (define-fun b2 () Int
    2)
  (define-fun b1 () Int
    8)
  (define-fun a3 () Int
    4)
  (define-fun a2 () Int
    6)
  (define-fun a1 () Int
    7)
  (define-fun r3 () Int
    6)
  (define-fun r2 () Int
    72)
  (define-fun r1 () Int
    55)
  (define-fun rc () Int
    27)
  (define-fun rb () Int
    1)
  (define-fun ra () Int
    38)
)

为了澄清,我正在使用Z3到de JAVA API

解决方法:

是的：我们的想法是断言找到的模型分配的值是唯一可能的分配,因此它是唯一的.这可以通过添加一个声明所有常量不等于其指定模型值的断言来完成.

对于您的示例,这将是：

(assert (not (and
  (= c3 3)
  (= c2 9)
  (= c1 1)
  (= b3 5)
  (= b2 2)
  (= b1 8)
  (= a3 4)
  (= a2 6)
  (= a1 7)
  (= r3 6)
  (= r2 72)
  (= r1 55)
  (= rc 27)
  (= rb 1)
  (= ra 38))))
(check-sat) ; unsat => no other model exists

如果您尝试更改任何值(例如,将c3 = 3更改为c3 = 4),这将再次变得令人满意.这是一个上升的@ fun链接到您的完整示例：http://rise4fun.com/Z3/nD5n

有关如何使用API​​以编程方式执行此操作的更多详细信息,请参阅以下问题和答案：

Z3: finding all satisfying models

(Z3Py) checking all solutions for equation

请注意,根据第二个链接答案中的注释,您不能仅使用SMT-lib前端以编程方式执行此操作,如果要自动执行此过程,则必须使用API​​来遍历找到的模型.