The Rely-Guarantee Method for Verifying Shared Variable Concurrent Program 正确性公理

文章目录

• proof system
• Assignment axiom
• Await axiom
• Consequence rule
• Sequential composition rule
• Conditional rule
• Iteration rule
• Auxiliary variable rule

proof system

Assignment axiom

$pre \to post[ \bar e/ \bar x] \\ (pre \land \lceil \bar x'= \bar e \rceil ) \to guar \\ pre \ \underline {stable \ when} \ rely \\ { post \ \underline {stable \ when} \ rely }\\ \overline{ \bar x:= \bar e \ \underline{sat} \ (pre, rely ,guar ,post)}$
$where \lceil \bar x = \bar e \rceil \ \overset {def}{=} (\bar x' = \bar e \lor x'=x)\land \forall z \in (y- \bar x). z' =z$

An example :$x:=10 \ \underline {sat} \ (true ,x>0 \to \bar x \ge x ,true ,x \ge 10)$
证：$1、pre \to post[ \bar e/ \bar x] \vDash true \to x\ge 10[10/ \bar x] =true\ 成立 \\ 2、pre \land \lceil \bar x'= \bar e \rceil ) \to guar \vDash true \land x'=10 \to true 成立 \\ 3、 pre\ \underline {stable \ when} \ rely \vDash true \land x \gt 0 \to true\ 成立 \\ 4、 post \ \underline {stable \ when} \ rely \vDash x \ge 10 \land x \gt 0 \to x' \ge x \Rightarrow x' \ge 10 成立 \\ x:=10 \ \underline {sat} \ (true ,x>0 \to \bar x \ge x ,true ,x \ge 10) \\$

Await axiom

$pre\ \underline {stable\ when} \ rely \\ post\ \underline {stable\ when }\ rely \\ \underline{ P\ \underline {sat}\ (pre \land b \land y=v_0,y' =y,true,guar [ {v_0/y,y/y'}] \land post)} \\ { await\ b\ then\ P\ end\ \underline {sat} (pre,rely,guar ,post) }$

An example : $await \ x\gt 0\ then\ x:=x-1\ \ end$
satisfies $(x \ge 0,x \ge0 \to x' \ge 0,x' \le x,x \ge 0)$

Consequence rule

$pre \to pre_1,rely \to rely_1, \\guar_1\to guar, post_1 \to post \\ \underline {P{sat}\ (pre_1,rely_1,guar_1,post_1) } \\P\ \underline{sat}\ (pre,rely,guar,post)$

An example: $x:=10\ \underline{sat}\ (x=-2,x \gt 0 \to x' \ge x,,true,x \ge10 \lor x=-6)$

Sequential composition rule

$P\ \underline {sat}\ (pre,rely,guar,mid) \\ Q\ \underline{sat}\ (mid,rely,guar,post) \\ \overline{P;Q \underline {sat}\ (pre,rely,guar,post)}$

An example:
$x:=x+1\ \underline {sat}\ (x \ge x_0,x_0 \le x \to x \le x',x' \ge x,x \ge x_0+1) \\ x:=x+1\ \underline{sat}\ (x \ge x_0+1,x_0 \le x \to x \le x' ,x' \ge x,x \ge x_0+2 ) \\ x:=x+1;x:=x+1\ \underline{sat}\ (x \ge x_0,x_0 \le x \to x \le x',x' \ge x,x \ge x_0+2)$

Conditional rule

$pre\ \underline{stable\ when }\ rely \\ P_i\ \underline{sat}\ (pre \land b_i,rely,guar,post) \\ skip\ \underline{sat}\ (pre \land \lnot(b_1 \lor \dots \lor b_n),rely ,guar,post) \\ \overline{if\ b_1 \rightarrow p_1 \square \dots \square b_n \to P_n\ fi\ \underline {sat} \ (pre ,rely,guar,post)}$

An example: $if\ x \lt10\ \to x:=10\ fi\ \underline{sat}\ (true,x \le x',x \lt x',x \ge10)$ `

$x:=10\ \underline{sat}\ (x \lt 10,x \le x',x \le x',x \ge 10) \\ and\ skip\ \underline{sat}\ (x \ge 10 ,x \lt x',x \lt x',x \ge 10)$

Iteration rule

$pre\ \underline {stable \ when}\ rely \\ pre \land \lnot b \to post \\ post\ \underline{ stable \ when}\ rely \\ \bm P\ \underline{sat} \ (pre \land b ,rely ,guar,pre) \\ \overline{ \bm {while}\ b\ \bm {do}\ P\ \bm {od}\ \underline{sat}\ (pre,rely,guar,post)}$

An example:
$\bm {while}\ b\ \bm{do}\ P\ \bm {od}\ \underline{sat}\ (true,x \le x' ,x \le x' ,x \gt 10)$
## Parallel rule
$(rely \lor guar_1) \to rely_2 \\ (rely \lor guar_2) \to rely_1 \\ (guar_1 \lor guar_2) \to guar \\ \bm p\ \underline{sat}\ (pre ,rely_1,guar_1,post_1) \\ \bm Q\ \underline{sat}\ (pre ,rely_2,guar_2,post_2) \\ \overline {P || Q \ \underline {sat}\ (pre,rely ,guar ,post_1 \land post_2)}$

Auxiliary variable rule

$\exists z. pre_1(y,z,y_0) \\ \exists z'.rely_1((y,z).(y',z'),y_0) \\ \underline{\bm P\ \underline {sat}\ (pre \land pre1,rely \land rely_1,guar,post)}\\ \bm Q \underline{sat}\ (pre,rely,guar,post)$

An example:$x:=x+1 || x:=x+1$

来源：CSDN

作者：初九-第二人生

链接：https://blog.csdn.net/qq_22017379/article/details/103282295