Question

I would like to find c and t coefficients in simple “result=x*t+c” formula for some given result/x pairs:

from z3 import *

x=Int('x')
c=Int('c')
t=Int('t')

s=Solver()

f = Function('f', IntSort(), IntSort())

# x*t+c = result
# x, result = [(1,55), (12,34), (13,300)]

s.add (f(x)==(x*t+c))
s.add (f(1)==55, f(12)==34, f(13)==300)

t=s.check()
if t==sat:
    print s.model()
else:
   print t

… but the result is obviously wrong. I probably need to find out how to map function arguments.

How should I define function correctly?

Answer 1

The assertion f(x) == x*t + c is not defining the function f for all x. It is just saying that the value of f for the given x is x*t + c. Z3 supports universal quantifiers. However, they are very expensive, and Z3 is not complete when a set of constraints contains universal quantifiers since the problem becomes undecidable. That is, Z3 may return unknown for this kind of problem.

Note that f is essentially a “macro” in your script. Instead of using a Z3 function for encoding this “macro”, we can create a Python function that does the trick. That is, a Python function that, given a Z3 expression, returns a new Z3 expression. Here is a new script. The script is also available online at: http://rise4fun.com/Z3Py/Yoi
Here is another version of the script where c and t are Real instead of Int: http://rise4fun.com/Z3Py/uZl

from z3 import *

c=Int('c')
t=Int('t')

def f(x):
    return x*t + c

# data is a list of pairs (x, r)
def find(data):
    s=Solver()
    s.add([ f(x) == r for (x, r) in data ])
    t = s.check()
    if s.check() == sat:
        print s.model()
    else:
        print t

find([(1, 55)])
find([(1, 55), (12, 34)])
find([(1, 55), (12, 34), (13, 300)])

Remark: In the SMT 2.0 front-end, macros can be defined using the command define-fun.