Very basically, this tool will help us to find values for variables that need to satisfy some conditions and calculating them by hand will be so annoying. Therefore, you can indicate to Z3 the conditions the variables need to satisfy and it will find some values (if possible).
#pip3 install z3-solverfrom z3 import*s =Solver()#The solver will be given the conditionsx =Bool("x")#Declare the symbos x, y and zy =Bool("y")z =Bool("z")# (x or y or !z) and ys.add(And(Or(x,y,Not(z)),y))s.check()#If response is "sat" then the model is satifable, if "unsat" something is wrongprint(s.model())#Print valid values to satisfy the model
Ints/Simplify/Reals
from z3 import*x =Int('x')y =Int('y')#Simplify a "complex" ecuationprint(simplify(And(x +1>=3, x**2+ x**2+ y**2+2>=5)))#And(x >= 2, 2*x**2 + y**2 >= 3)#Note that Z3 is capable to treat irrational numbers (An irrational algebraic number is a root of a polynomial with integer coefficients. Internally, Z3 represents all these numbers precisely.)
#so you can get the decimals you need from the solutionr1 =Real('r1')r2 =Real('r2')#Solve the ecuationprint(solve(r1**2+ r2**2==3, r1**3==2))#Solve the ecuation with 30 decimalsset_option(precision=30)print(solve(r1**2+ r2**2==3, r1**3==2))
Printing Model
from z3 import*x, y, z =Reals('x y z')s =Solver()s.add(x >1, y >1, x + y >3, z - x <10)s.check()m = s.model()print ("x = %s"% m[x])for d in m.decls():print("%s = %s"% (d.name(), m[d]))
Machine Arithmetic
Modern CPUs and main-stream programming languages use arithmetic over fixed-size bit-vectors. Machine arithmetic is available in Z3Py as Bit-Vectors.
from z3 import*x =BitVec('x', 16)#Bit vector variable "x" of length 16 bity =BitVec('y', 16)e =BitVecVal(10, 16)#Bit vector with value 10 of length 16bitsa =BitVecVal(-1, 16)b =BitVecVal(65535, 16)print(simplify(a == b))#This is True!a =BitVecVal(-1, 32)b =BitVecVal(65535, 32)print(simplify(a == b))#This is False
Signed/Unsigned Numbers
Z3 provides special signed versions of arithmetical operations where it makes a difference whether the bit-vector is treated as signed or unsigned. In Z3Py, the operators <, <=, >, >=, /, % and >> correspond to the signed versions. The corresponding unsigned operators are ULT, ULE, UGT, UGE, UDiv, URem and LShR.
from z3 import*# Create to bit-vectors of size 32x, y =BitVecs('x y', 32)solve(x + y ==2, x >0, y >0)# Bit-wise operators# & bit-wise and# | bit-wise or# ~ bit-wise notsolve(x & y ==~y)solve(x <0)# using unsigned version of < solve(ULT(x, 0))
Functions
Interpreted functions such as arithmetic where the function + has a fixed standard interpretation (it adds two numbers). Uninterpreted functions and constants are maximally flexible; they allow any interpretation that is consistent with the constraints over the function or constant.
Example: f applied twice to x results in x again, but f applied once to x is different from x.
from z3 import*x =Int('x')y =Int('y')f =Function('f', IntSort(), IntSort())s =Solver()s.add(f(f(x)) == x, f(x) == y, x != y)s.check()m = s.model()print("f(f(x)) =", m.evaluate(f(f(x))))print("f(x) =", m.evaluate(f(x)))print(m.evaluate(f(2)))s.add(f(x) ==4)#Find the value that generates 4 as responses.check()print(m.model())
Examples
Sudoku solver
# 9x9 matrix of integer variablesX = [ [ Int("x_%s_%s"% (i+1, j+1))for j inrange(9) ]for i inrange(9) ]# each cell contains a value in {1, ..., 9}cells_c = [ And(1<= X[i][j], X[i][j] <=9)for i inrange(9)for j inrange(9) ]# each row contains a digit at most oncerows_c = [ Distinct(X[i])for i inrange(9) ]# each column contains a digit at most oncecols_c = [ Distinct([ X[i][j] for i inrange(9) ])for j inrange(9) ]# each 3x3 square contains a digit at most oncesq_c = [ Distinct([ X[3*i0 + i][3*j0 + j]for i inrange(3) for j inrange(3) ])for i0 inrange(3)for j0 inrange(3) ]sudoku_c = cells_c + rows_c + cols_c + sq_c# sudoku instance, we use '0' for empty cellsinstance = ((0,0,0,0,9,4,0,3,0), (0,0,0,5,1,0,0,0,7), (0,8,9,0,0,0,0,4,0), (0,0,0,0,0,0,2,0,8), (0,6,0,2,0,1,0,5,0), (1,0,2,0,0,0,0,0,0), (0,7,0,0,0,0,5,2,0), (9,0,0,0,6,5,0,0,0), (0,4,0,9,7,0,0,0,0))instance_c = [ If(instance[i][j] ==0,True, X[i][j] == instance[i][j])for i inrange(9)for j inrange(9) ]s =Solver()s.add(sudoku_c + instance_c)if s.check()== sat: m = s.model() r = [ [ m.evaluate(X[i][j])for j inrange(9) ]for i inrange(9) ]print_matrix(r)else:print"failed to solve"
