Z3的C API中“Z3_mk_forall”和“Z3_mk _forall_const”之间的差异

Question

我被这两个函数弄糊涂了。它们似乎采用了相同的一组参数（一个可以直接转换为另一个），并且每个参数都返回一个AST。函数做同样的事情吗？如果没有，我什么时候需要每个？

签名2:

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做同样的事情。主要区别在于，Z3_mk_forall_const采用了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特定应用程序。特别是，在我看来，当有少量固定数量的符号需要构建量词时，Z3_mk_forall_const会更自然。相反，在量词中符号数量可能不同的情况下，Z3_mk_forall可能会更自然，在这种情况下，生成一个绑定变量数组是很自然的，您将用索引来处理这些变量。

还有其他的优点和缺点。例如，请参阅此问题："如何在Z3_mk_forall_const中声明用作绑定变量的常量？"在这个问题中，询问者希望避免在其全局上下文中引入大量变量，这对于使用Z3_mk_forall_const是必要的。回答者（Christoph）建议使用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在回答中还透露，在内部，基于Z3_mk_forall_const的方法被翻译成相当于Z3_mk_forall的方法，所以在引擎盖下确实没有区别。然而，API的差异可能会给程序员带来很大的不同。

在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 << "nn";
}
/**
 * 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("%snn", 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("%sn", 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：

• "对于中的所有量词Z3"
  
• "C API用于量化器"
  
• "Z3量词支持"
  

Z3源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和src/api/z3_api.h中。