Z3 的 C API 中 `Z3_mk_forall` 和 `Z3_mk_forall_const` 之间的区别？

Question

我对这两个功能感到困惑。他们似乎采用了相同的参数集（一个可以直接转换为另一个）并且每个 returns 一个 AST。这些功能做同样的事情吗？如果没有，我什么时候需要每个？

2人签名：

Z3_ast Z3_mk_forall (Z3_context c,
                     unsigned weight,
                     unsigned num_patterns,
                     Z3_pattern const patterns[],
                     unsigned num_decls,
                     Z3_sort const sorts[],
                     Z3_symbol const decl_names[],
                     Z3_ast body)

Z3_ast Z3_mk_forall_const (Z3_context c,
                           unsigned weight,
                           unsigned num_bound,
                           Z3_app const bound[],
                           unsigned num_patterns,
                           Z3_pattern const patterns[],
                           Z3_ast body)

Answer 1

是的，Z3 团队提供了多种方法来完成同一件事。主要区别在于 Z3_mk_forall_const 采用使用正常机制定义的常量列表，而 Z3_mk_forall 需要使用 Z3_mk_bound.

创建的绑定变量列表

哪种机制更容易使用取决于您的具体应用。特别是，在我看来，当您要在其上构建量词的符号数量较少且固定时， Z3_mk_forall_const 会更自然。相反，Z3_mk_forall 在量词中的符号数量可能不同的情况下可能更自然，在这种情况下，生成一个绑定变量数组是很自然的，您将使用索引对其进行寻址。

还有其他优点和缺点。例如，看到这个问题： "How to declare constants to use as bound variables in Z3_mk_forall_const?" 在那个问题中，提问者希望避免在他们的全局上下文中引入很多变量，这将是使用 Z3_mk_forall_const 所必需的。回答者 (Christoph) 建议改用 Z3_mk_forall ，但这也不理想，因为对于嵌套的量词，这将导致每个量词的索引不同。 Christoph 还在那个答案中透露，在内部，基于 Z3_mk_forall_const 的方法被翻译成等同于 Z3_mk_forall 的方法，所以在底层实际上没有区别。然而，API 差异对程序员来说可能会有很大的不同。

C++ API 中还为程序员提供了一种（更简单的）机制，如果您能够使用它的话。以下是使用三种不同方法的示例：

// g++ --std=c++11 z3-quantifier-support.cpp -I../src/api/ -I../src/api/c++/ libz3.so

#include <stdio.h>
#include "z3.h"

#include <iostream>
#include "z3++.h"

using namespace z3;

/**
 * This is by far the most concise and easiest to use if the C++ API is available to you.
 */
void example_cpp_forall() {
    context c;
    expr a = c.int_const("a");
    expr b = c.int_const("b");
    expr x = c.int_const("x");
    expr axiom = forall(x, implies(x <= a, x < b));
    std::cout << "Result obtained using the C++ API with forall:\n" << axiom << "\n\n";
}

/**
 * Example using Z3_mk_forall_const. Not as clean as the C++ example, but this was still
 * significantly easier for me to get working than the example using Z3_mk_forall().
 */
void example_c_Z3_mk_forall_const() {
    // Get the context
    Z3_config cfg;
    Z3_context ctx;
    cfg = Z3_mk_config();
    ctx = Z3_mk_context(cfg);

    // Declare integers a, b, and x
    Z3_sort I = Z3_mk_int_sort(ctx);
    Z3_symbol a_S = Z3_mk_string_symbol(ctx, "a");
    Z3_symbol b_S = Z3_mk_string_symbol(ctx, "b");
    Z3_symbol x_S = Z3_mk_string_symbol(ctx, "x");
    Z3_ast a_A = Z3_mk_const(ctx, a_S, I);
    Z3_ast b_A = Z3_mk_const(ctx, b_S, I);
    Z3_ast x_A = Z3_mk_const(ctx, x_S, I);

    // Build the AST (x <= a) --> (x < b)
    Z3_ast x_le_a = Z3_mk_le(ctx, x_A, a_A);
    Z3_ast x_lt_b = Z3_mk_lt(ctx, x_A, b_A);
    Z3_ast f = Z3_mk_implies(ctx, x_le_a, x_lt_b);
    Z3_app vars[] = {(Z3_app) x_A};
    Z3_ast axiom = Z3_mk_forall_const(ctx, 0, 1, vars, 0, 0, f);

    printf("Result obtained using the C API with Z3_mk_forall_const:\n");
    printf("%s\n\n", Z3_ast_to_string(ctx, axiom));
}

/**
 * Example using Z3_mk_forall. For the example, this is the most cumbersome.
 */
void example_c_Z3_mk_forall() {

    // Get the context
    Z3_config cfg;
    Z3_context ctx;
    cfg = Z3_mk_config();
    ctx = Z3_mk_context(cfg);

    // Declare integers a and b
    Z3_sort I = Z3_mk_int_sort(ctx);
    Z3_symbol a_S = Z3_mk_string_symbol(ctx, "a");
    Z3_symbol b_S = Z3_mk_string_symbol(ctx, "b");
    Z3_ast a_A = Z3_mk_const(ctx, a_S, I);
    Z3_ast b_A = Z3_mk_const(ctx, b_S, I);

    // Declare bound variables, in this case, just x
    Z3_symbol x_S = Z3_mk_string_symbol(ctx, "x");
    Z3_ast x_A = Z3_mk_bound(ctx, 0, I);

    // Z3_mk_forall requires all names, types, and bound variables to be provided in
    // arrays. In this example, where there is only one quantified variable, this seems a
    // bit cumbersome. If we were dealing with an varying number of quantified variables,
    // then this would seem more reasonable.
    const unsigned sz = 1;
    const Z3_sort types[] = {I};
    const Z3_symbol names[] = {x_S};
    const Z3_ast xs[] = {x_A};

    // Build the AST (x <= a) --> (x < b)
    Z3_ast x_le_a = Z3_mk_le(ctx, x_A, a_A);
    Z3_ast x_lt_b = Z3_mk_lt(ctx, x_A, b_A);
    Z3_ast f = Z3_mk_implies(ctx, x_le_a, x_lt_b);

    // In the Z3 docs for Z3_mk_pattern, the following sentence appears: "If a pattern is
    // not provided for a quantifier, then Z3 will automatically compute a set of
    // patterns for it." So I tried supplying '0' for the number of patterns, and 'NULL'
    // for the list of patterns, and Z3_mk_forall still seems to function.
    Z3_ast axiom = Z3_mk_forall(ctx, 0, 0, NULL, sz, types, names, f);

    printf("Result obtained using the C API with Z3_mk_forall:\n");
    printf("%s\n", Z3_ast_to_string(ctx, axiom));
}

int main() {
    example_cpp_forall();
    example_c_Z3_mk_forall_const();
    example_c_Z3_mk_forall();
}

我还发现这些问题很有帮助：

Z3 源代码中提供的示例和注释也很有帮助，特别是在 examples/c/test_capi.c, examples/c++/example.cpp, and src/api/z3_api.h。

Z3 的 C API 中 `Z3_mk_forall` 和 `Z3_mk_forall_const` 之间的区别？

Difference between `Z3_mk_forall` and `Z3_mk_forall_const` in C API for Z3?

api

quantifiers

smt

z3