Z3Py 学习
一、数据类型
- integer 整型 ToInt(arg)转换为整型
创建对象的方式：
x = Int('x')
x,y = Ints('x y')
- real 实数 ToReal(arg)转换为有理数
x = Real('x')
x ，y = Reals('x y')
例：
x = Real('x')
solve(3*x==1)
# 启用精度
set_option(rational_to_decimal=True)
solve(3*x==1)
# 设置精度
set_option(precision=30)
solve(3*x==1)
输出：
[x = 1/3]
[x = 0.3333333333?]
[x = 0.333333333333333333333333333333?]
- bool 布尔值
p = Bool('p')
p，q = Bool('p q')
Z3支持布尔运算符：And，Or，Not，Implies（蕴涵）， If（if-then-else）。 使用等式 == 表示双重含义。
p,q,r = Bools('p q r')
solve(Implies(p, q), r == Not(q), Or(Not(p), r))
- Solver 求解器
例：
x = Int('x')
y = Int('y')
s = Solver()
print s # []
s.add(x > 10, y == x + 2)
print s # [x > 10, y == x + 2]
print "Solving constraints in the solver s ..."
print s.check() #sat [ returns : sat unsat unknown ]
print "Create a new scope..."
s.push()
s.add(y < 11)
print s # [x > 10, y == x + 2, y < 11]
print "Solving updated set of constraints..."
print s.check() # unsat
print "Restoring state..."
s.pop()
print s # [x > 10, y == x + 2]
print "Solving restored set of constraints..."
print s.check() # sat
以下示例说明如何遍历声明为求解器的约束，以及如何收集check方法的性能统计信息。
x = Real('x')
y = Real('y')
s = Solver()
s.add(x > 1, y > 1, Or(x + y > 3, x - y < 2))
print "asserted constraints..."
for c in s.assertions():
print c
# 输出：
# x > 1
# y > 1
# Or(x + y > 3, x - y < 2)
print s.check() # sat
print "statistics for the last check method..."
print s.statistics()
# (:add-rows 1
# :assert-lower 4
# :decisions 2
# :final-checks 1
# :max-memory 2.91
# :memory 2.27
# :mk-bool-var 9
# :num-allocs 208770603
# :pivots 1
# :rlimit-count 3042)
# Traversing statistics
for k, v in s.statistics():
print "%s : %s" % (k, v)
以下示例显示了检查模型的基本方法:
x, y, z = Reals('x y z')
s = Solver()
s.add(x > 1, y > 1, x + y > 3, z - x < 10)
print s.check() # sat
m = s.model()
print "x = %s" % m[x] # x=2
print "y = %s" % m[y] # y=2
print "traversing model..."
for d in m.decls():
print "%s = %s" % (d.name(), m[d])
输出：
z = 0
y = 2
x = 2
- 算术
x = Real('x')
y = Int('y')
a, b, c = Reals('a b c')
s, r = Ints('s r')
print x + y + 1 + (a + s) # x + ToReal(y) + 1 + a + ToReal(s)
print ToReal(y) + c # ToReal(y) + c
a, b, c = Ints('a b c')
d, e = Reals('d e')
solve(a > b + 2,
a == 2*c + 10,
c + b <= 1000,
d >= e)
# [b = 0, c = 0, e = 0, d = 0, a = 10]
simplify化简：
x, y = Reals('x y')
# Put expression in sum-of-monomials form
t = simplify((x + y)**3, ':som', True)
# t = simplify((x + y)**3,som=True)
print t
# Use power operator
t = simplify(t, mul_to_power=True)
print t
# x*x*x + 3*x*x*y + 3*x*y*y + y*y*y
# x**3 + 3*x**2*y + 3*x*y**2 + y**3
- 可满足性和有效性
p, q = Bools('p q')
demorgan = And(p, q) == Not(Or(Not(p), Not(q)))
print demorgan # And(p, q) == Not(Or(Not(p), Not(q)))
def prove(f):
s = Solver()
s.add(Not(f))
if s.check() == unsat:
print "proved"
else:
print "failed to prove"
print "Proving demorgan..."
prove(demorgan)
输出：
# Proving demorgan...
# proved