[英]Synthesize a loop program that preserves the loop invariant and variant
我想創建一個具有以下先決條件的程序:不變:
y = x ∗ x ∧ z = y ∗ x ∧ x ≤ n
變體:
n − x
程序結構如下:
while<cond>
<invariant specification>
<body>
程序應該如何看起來像用 frama-c 或why3 編寫的?
編輯:我通過刪除乘法並添加加法來修改您的程序。 通過這樣做,我使用了兩個循環。 我運行了我的程序,但我收到了警告。 這是程序:
#include <limits.h>
/*@ requires n < INT_MAX; // avoids overflow in computing z
*/
void f(unsigned long n) {
unsigned long x = 0, y = 0, z = 0, contor, aux_x, aux_y;
/*@ loop assigns x, y, z, contor, aux_x, aux_y;
@ loop invariant y == x * x && z == y * x && x <= n;
@ loop variant n - x;
@ */
while (x < n) {
x++;
contor = 1;
aux_x = 0;
aux_y = 0;
/* @ loop assings contor, aux_x, aux_y;
@ loop invariant 1 <= contor <= x;
@ loop variant x - contor, x, y;
@*/
while (contor <= x) {
aux_x += x;
aux_y += y;
contor++;
}
y = aux_x;
z = aux_y;
}
}
這些是警告:
[kernel] Parsing loop.c (with preprocessing)
[rte] annotating function f
[wp] loop.c:20: Warning: Missing assigns clause (assigns 'everything' instead)
[wp] 6 goals scheduled
[wp] [Alt-Ergo 2.4.1] Goal typed_f_loop_assigns_part2 : Timeout (Qed:6ms) (10s) (cached)
[wp] [Alt-Ergo 2.4.1] Goal typed_f_loop_variant_decrease : Timeout (Qed:16ms) (10s) (cached)
[wp] [Alt-Ergo 2.4.1] Goal typed_f_loop_invariant_established : Timeout (Qed:3ms) (10s)
[wp] [Alt-Ergo 2.4.1] Goal typed_f_loop_invariant_preserved : Timeout (Qed:11ms) (10s)
[wp] [Cache] found:2, updated:2
[wp] Proved goals: 2 / 6
Qed: 2 (7ms)
Alt-Ergo 2.4.1: 0 (interrupted: 4) (cached: 2)
[wp:pedantic-assigns] loop.c:5: Warning:
No 'assigns' specification for function 'f'.
Callers assumptions might be imprecise.
你能解釋一下為什么即使我為內循環指定了不變量和變體,我也會收到那些討厭的警告嗎?
目前還不清楚你想要實現什么,但這里有一個 C 程序,可以用frama-c -wp loop.c
證明,並具有適當的不變量和變體:
/*@ requires n < 2097152; // avoids overflow in computing z
*/
void f(unsigned long n) {
unsigned long x = 0, y = 0, z = 0;
/*@ loop invariant y == x * x && z == y * x && x <= n;
loop assigns x,y,z;
loop variant n - x;
*/
while (x < n) {
x++;
y = x * x;
z = y * x;
}
}
注意: requires
並不是在計算z
時避免溢出的最通用的方法,但計算2^21
比取2^64-1
的三次根更容易。
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