Z3py returns unknown for equation using pow() function

问题

Z3py returns unknown for the following simple problem using pow() function.

import z3;

goal = z3.Goal();
goal = z3.Then('purify-arith','nlsat');
solver = goal.solver();

x = z3.Real('x');
solver.add(x <= 1.8);
solver.add(x >= 0);

z = 10**x;
#z = pow(10,x) returns same result

solver.add(z >= 0, z <= 1.8);
print solver.check()

returns

unknown

Obviously x = 0, z = 1 is a satisfactory solution. Any suggestions in changing the way the equations are constructed, or modifying the tactics are appreciated.

回答1:

There may be some bug, but the following returns a model of x = 0 and z = 1, even though it gives unknown. Assuming the Python API also shows this model, you could create a simple hack to extract the model and add it to the assertions to check, similar to how this prevents future models from reusing old model values: Z3: finding all satisfying models

Here's the example (rise4fun link: http://rise4fun.com/Z3/dPnI ):

(declare-const x Real)
(declare-const z Real)

(assert (>= x 0.0))
(assert (<= x 1.8))
(assert (= z (^ 10.0 x)))
(assert (<= z 1.8))
(assert (>= z 0.0))

(apply (repeat (then purify-arith simplify ctx-simplify ctx-solver-simplify nlsat qfnra-nlsat))) ; gives:
; (goals
;(goal
;  (>= x 0.0)
;  (<= x (/ 9.0 5.0))
;  (<= (^ 10.0 x) (/ 9.0 5.0))
;  (>= (^ 10.0 x) 0.0)
;  :precision precise :depth 44)
;)

; (apply (repeat (then (repeat purify-arith) (repeat simplify) (repeat ctx-simplify) (repeat ctx-solver-simplify)  (repeat nlsat)  (repeat qfnra-nlsat))))

(check-sat) ; unknown
;(check-sat-using qfnra-nlsat) ; unknown

(get-model) ; gives x = 0.0 and z = 1.0 even though unknown from check-sat

; could extract 0 and 1 in python API and check whether it's sat:
(assert (= x 0))
(assert (= z 1))

(check-sat) ; sat

You might also be interested in this related post: Z3 support for square root

For completeness, here's the model extraction idea in Python that seems to work (using 4.3.3, probably a build from unstable, but a while ago likely):

import z3;

print z3.get_version_string();

goal = z3.Goal();
goal = z3.Then('purify-arith','nlsat');
#solver = goal.solver();

solver = z3.Solver();

x = z3.Real('x');
z = z3.Real('z');
solver.add(x <= 1.8);
solver.add(x >= 0);
solver.add(z == 10.0 ** x);

# z = 10**x;
#z = pow(10,x) returns same result

solver.add(z >= 0, z <= 1.8);

print solver

print solver.check()
print solver.model()

m = solver.model()

solver.add(x == m[x])
solver.add(z == m[z])

print solver

print solver.check()

This gives:

D:\>python exponent.py
4.3.3
[x <= 9/5, x >= 0, z == 10**x, z >= 0, z <= 9/5]
unknown
[x = 0, z = 1]
[x <= 9/5,
 x >= 0,
 z == 10**x,
 z >= 0,
 z <= 9/5,
 x == 0,
 z == 1]
sat

来源：https://stackoverflow.com/questions/30038609/z3py-returns-unknown-for-equation-using-pow-function

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