Recent advances in SMT

Programming Z3

System Overview

SAT

  from z3 import *
  Tie, Shirt = Bools('Tie Shirt')
  s = Solver()
  s.add(Or(Tie, Shirt), 
        Or(Not(Tie), Shirt), 
        Or(Not(Tie), Not(Shirt)))
  print(s.check())
  print(s.model())

SMT

  Z = IntSort()
  f = Function('f', Z, Z)
  x, y, z = Ints('x y z')
  A = Array('A',Z,Z)
  fml = Implies(x + 2 == y, f(Store(A, x, 3)[y - 2]) == f(y - x + 1))
  solve(Not(fml))

SMT-LIB and Python

 (set-logic QF_LIA)
 (declare-const x Int)
 (declare-const y Int)
 (assert (> (+ (mod x 4) (* 3 (div y 2))) (- x y)))
 (check-sat)

Python version:

 solve((x % 4) + 3 * (y / 2) > x - y)

Python and SMT-LIB

It is also possible to extract an SMT-LIB2 representation of a solver state.

from z3 import *
x, y = Ints('x y')
s = Solver()
s.add((x % 4) + 3 * (y / 2) > x - y)
print(s.sexpr())

produces the output

(declare-fun y () Int)
(declare-fun x () Int)
(assert (> (+ (mod x 4) (* 3 (div y 2))) (- x y)))

Accessing Expressions

x = Int('x')
y = Int('y')
n = x + y >= 3
print("num args: ", n.num_args())
print("children: ", n.children())
print("1st child:", n.arg(0))
print("2nd child:", n.arg(1))
print("operator: ", n.decl())
print("op name:  ", n.decl().name())

('num args: ', 2)
('children: ', [x + y, 3])
('1st child:', x + y)
('2nd child:', 3)
('operator: ', >=)
('op name:  ', '>=')

Quantifiers

solve(y == x + 1, ForAll([y], Implies(y <= 0, x < y)))

• Is this satisfiable?

Lambdas

m, m1 = Array('m', Z, Z), Array('m1', Z, Z)
def memset(lo,hi,y,m):
    return Lambda([x], If(And(lo <= x, x <= hi), y, Select(m, x)))
solve([m1 == memset(1, 700, z, m), Select(m1, 6) != z])

Q = Array('Q', Z, B)
prove(Implies(ForAll(Q, Implies(Select(Q, x), Select(Q, y))), x == y))

SMT Theories

SAT + Theories

• Mainstream SMT solver architecture based on co-processor architecture to CDCL.

• Alternatives:

  • Model Constructing SAT - instantiate values of non-Booleans
  • MIP(T) - MIP solver is main engine
  • Super-position

CDCL(T) by example

CDCL(T) by example (2)

CDCL(T) by example (3)

CDCL(T) by example (4)

CDCL(T) by example (5)

CDCL(T) by example (6)