代入 z3 公式中的函数符号 [英] Substituting function symbols in z3 formulas
问题描述
在公式中替换函数符号(用另一个函数)的最佳方法是什么?Z3py 的 substitute
似乎只适用于表达式,我现在要做的是尝试猜测可以应用该函数的 consts/vars 的所有可能组合,然后用另一个函数的应用程序替换它们.有没有更好的方法来做到这一点?
What is the best way to substitute a function symbol (with another function) in a formula?
Z3py's substitute
seems to only work with expressions, and what I do now is I try to guess all possible combinations of consts/vars to which the function could be applied and then substitute those with an application of another function. Is there a better way to do that?
推荐答案
我们可以实现一个简单的自底向上重写器,给定一个术语 s
、一个函数 f
和term t
会将 s
中的每个 f
-application f(r_1, ..., r_n)
替换为 <代码>t[r_1, ..., r_n].我使用符号 t[r_1, ..., r_n]
来表示通过用术语 r_1 替换
, ..., t
中的自由变量而获得的术语r_n
.
We can implement a simple bottom-up rewriter that given a term s
, a function f
and term t
will replace every f
-application f(r_1, ..., r_n)
in s
with t[r_1, ..., r_n]
. I'm using the notation t[r_1, ..., r_n]
to denote the term obtained by replacing the free-variables in t
with the terms r_1
, ..., r_n
.
重写器可以实现 Z3 API.我使用 AstMap
缓存结果,使用 todo
列表存储仍需处理的表达式.
The rewriter can be implemented the Z3 API. I use an AstMap
to cache results, and a todo
list to store expressions that still have to be processed.
这是一个简单的例子,它用 g(t+1)
替换 f(t)
形式的 f
-applications in <代码>s.
Here is a simple example that replaces f
-applications of the form f(t)
with g(t+1)
in s
.
x = Var(0, IntSort())
print rewrite(s, f, g(x + 1))
这是代码和更多示例.请注意,我仅在一小部分示例中测试了代码.
Here is the code and more examples. Beware, I only tested the code in a small set of examples.
from z3 import *
def update_term(t, args):
# Update the children of term t with args.
# len(args) must be equal to the number of children in t.
# If t is an application, then len(args) == t.num_args()
# If t is a quantifier, then len(args) == 1
n = len(args)
_args = (Ast * n)()
for i in range(n):
_args[i] = args[i].as_ast()
return z3._to_expr_ref(Z3_update_term(t.ctx_ref(), t.as_ast(), n, _args), t.ctx)
def rewrite(s, f, t):
"""
Replace f-applications f(r_1, ..., r_n) with t[r_1, ..., r_n] in s.
"""
todo = [] # to do list
todo.append(s)
cache = AstMap(ctx=s.ctx)
while todo:
n = todo[len(todo) - 1]
if is_var(n):
todo.pop()
cache[n] = n
elif is_app(n):
visited = True
new_args = []
for i in range(n.num_args()):
arg = n.arg(i)
if not arg in cache:
todo.append(arg)
visited = False
else:
new_args.append(cache[arg])
if visited:
todo.pop()
g = n.decl()
if eq(g, f):
new_n = substitute_vars(t, *new_args)
else:
new_n = update_term(n, new_args)
cache[n] = new_n
else:
assert(is_quantifier(n))
b = n.body()
if b in cache:
todo.pop()
new_n = update_term(n, [ cache[b] ])
cache[n] = new_n
else:
todo.append(b)
return cache[s]
f = Function('f', IntSort(), IntSort())
a, b = Ints('a b')
s = Or(f(a) == 0, f(a) == 1, f(a+a) == 2)
# Example 1: replace all f-applications with b
print rewrite(s, f, b)
# Example 2: replace all f-applications f(t) with g(t+1)
g = Function('g', IntSort(), IntSort())
x = Var(0, IntSort())
print rewrite(s, f, g(x + 1))
# Now, f and g are binary functions.
f = Function('f', IntSort(), IntSort(), IntSort())
g = Function('g', IntSort(), IntSort(), IntSort())
# Example 3: replace all f-applications f(t1, t2) with g(t2, t1)
s = Or(f(a, f(a, b)) == 0, f(b, a) == 1, f(f(1,0), 0) == 2)
# The first argument is variable 0, and the second is variable 1.
y = Var(1, IntSort())
print rewrite(s, f, g(y, x))
# Example 4: quantifiers
s = ForAll([a], f(a, b) >= 0)
print rewrite(s, f, g(y, x + 1))
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