Z3 real arithmetics and data types theories integrating not that well

Question

This is related to the question I asked before at Z3 SMT 2.0 vs Z3 py implementation I implemented the full algebra for positive reals with infinity and the solver is misbehaving. I get unknown on this simple instance, when the commented constraint gives an actual solution for the constraint.

# Data type declaration
MyR = Datatype('MyR')
MyR.declare('inf');
MyR.declare('num',('re',RealSort()))

MyR = MyR.create()
inf = MyR.inf
num  = MyR.num
re  = MyR.re

# Functions declaration
#sum 
def msum(a, b):
    return If(a == inf, a, If(b == inf, b, num(re(a) + re(b))))

#greater or equal 
def mgeq(a, b):
  return If(a == inf, True, If(b == inf, False, re(a) >= re(b)))

#greater than
def mgt(a, b):
  return If(a == inf, b!=inf, If(b == inf, False, re(a) > re(b)))

#multiplication  inf*0=0 inf*inf=inf  num*num normal
def mmul(a, b):
    return If(a == inf, If(b==num(0),b,a), If(b == inf, If(a==num(0),a,b), num(re(a)*re(b))))

s0,s1,s2 = Consts('s0 s1 s2', MyR)

# Constraints add to solver
constraints =[
  s2==mmul(s0,s1),
  s0!=inf,
  s1!=inf
]
#constraints =[s2==mmul(s0,s1),s0==num(1),s1==num(2)]

sol1= Solver()
sol1.add(constraints)

set_option(rational_to_decimal=True)

if sol1.check()==sat:
  m = sol1.model()
  print m
else:
  print sol1.check()

I don't know whether this is surprising or expected. Is there a way to make it work?

score 4 · Answer 1 · edited May 23 '17 at 12:23

Your problem is nonlinear. The new (and complete) nonlinear arithmetic solver (nlsat) in Z3 is not integrated with other theories such as algebraic datatypes. See the posts:

This is a limitation in the current version. Future versions will address this issue.

In the meantime, you can workaround the problem by using a different encoding. If you use only real arithmetic and Booleans, the problem will be in the scope of nlsat. One possibility is to encode MyR as a Python pair: Z3 Boolean expression and Z3 Real expression.

Here is a partial encoding. I did not encode all operators. The example is also available online at http://rise4fun.com/Z3Py/EJLq

from z3 import *

# Encoding MyR as pair (Z3 Boolean expression, Z3 Real expression)
# We use a class to be able to overload +, *, <, ==
class MyRClass:
    def __init__(self, inf, val):
        self.inf = inf
        self.val = val
    def __add__(self, other):
        other = _to_MyR(other)
        return MyRClass(Or(self.inf, other.inf), self.val + other.val)
    def __radd__(self, other):
        return self.__add__(other)
    def __mul__(self, other):
        other = _to_MyR(other)
        return MyRClass(Or(self.inf, other.inf), self.val * other.val)
    def __rmul(self, other):
        return self.__mul__(other)
    def __eq__(self, other):
        other = _to_MyR(other)
        return Or(And(self.inf, other.inf),
                  And(Not(self.inf), Not(other.inf), self.val == other.val))
    def __ne__(self, other):
        return Not(self.__eq__(other))

    def __lt__(self, other):
        other = _to_MyR(other)
        return And(Not(self.inf),
                   Or(other.inf, self.val < other.val))

def MyR(name):
    # A MyR variable is encoded as a pair of variables name.inf and name.var
    return MyRClass(Bool('%s.inf' % name), Real('%s.val' % name))

def MyRVal(v):
    return MyRClass(BoolVal(False), RealVal(v))

def Inf():
    return MyRClass(BoolVal(True), RealVal(0))

def _to_MyR(v):
    if isinstance(v, MyRClass):
        return v
    elif isinstance(v, ArithRef):
        return MyRClass(BoolVal(False), v)
    else:
        return MyRVal(v)

s0 = MyR('s0')
s1 = MyR('s1')
s2 = MyR('s2')

sol = Solver()
sol.add( s2 == s0*s1,
         s0 != Inf(),
         s1 != Inf(),
         s0 == s1,
         s2 == 2,
         )
print sol
print sol.check()
print sol.model()

Z3 real arithmetics and data types theories integrating not that well

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