Strengthening vs. Incremental Proof

# Temporal Verification of Reactive Systems: Safety

This preview shows pages 1–10. Sign up to view the full content.

CS256/Winter 2007 — Lecture #7 Zohar Manna 7-1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Strengthening vs. Incremental Proof Comparing the Strategies We want to prove 0 q , but q is not inductive. We have two options: 1 Strengthening Strengthen it to q ϕ . Prove 0 ( q ϕ ) and deduce 0 q . 2 Incremental First prove 0 ϕ and then prove 0 q relative to ϕ . Resulting verifcation conditions: 1 I1. Θ q ϕ I2. { q ϕ } T { q ϕ } 2 I1’. Θ ϕ I2’. { ϕ } T { ϕ } 0 ϕ I1”. Θ q I2”. { q ϕ } T { q } 0 q 7-2
Strengthening vs. Incremental Proof (Con’t) 1 is strictly more powerful than 2 . 2 implies 1 since ρ τ ϕ ϕ 0 | {z } I2’ ρ τ q ϕ q 0 | {z } I2” [ ρ τ q ϕ q 0 ϕ 0 | {z } I2 ] In practice, 2 is often more useful than 1 allows breaking down the proof in more manage- able pieces smaller veriFcation conditions more intuitive 7-3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Strengthening vs. Incremental Proof (Con’t) Example: local x : integer where x = 1 0 : loop forever do h 1 : x := x + 1 i Show q 1 : at - 0 x > 0 q 2 : at - 1 x > 0 both are P -valid neither of them is inductive but q 1 q 2 is inductive! 7-4
Combining the Strategies Rule inc-inv : (incremental invariance) For assertions q , ϕ , χ 1 ,. . . , χ k I0. P q 0 χ 1 , . . . , 0 χ k I1. P q ( k ^ i =1 χ i ) ϕ q I2. P q Θ ϕ I3. P q n ( k ^ i =1 χ i ) ϕ o T { ϕ } P q 0 q If ϕ satis±es I2 and I3, we say that ϕ is inductive relative to χ 1 , . . . χ k 7-5

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Combining the Strategies (Con’t) Note that Θ must be stronger than all the χ i ’s (i.e., P q Θ χ i ) and so P q k ^ i =1 χ i Θ ϕ if P q Θ ϕ From now on, we usually omit “ P q ” and “ P q ”. 7-6
Detecting Trivial Verifcation Conditions { ϕ } T { ϕ } Don’t check every τ ∈ T . Ignore { ϕ } τ I { ϕ } always true Ignore { ϕ } τ { ϕ } if τ does not modify any variable in ϕ For { ϕ } τ { ϕ } where ϕ : p q ρ τ p q | {z } ϕ p 0 q 0 | {z } ϕ 0 Consider only τ ’s that validate p or falsify q 7-7

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Finding Inductive Assertions Two methods: 1. Bottom-up: based on the program text only algorithmic guaranteed to produce an inductive invariant 2. Top-down: guided by the property we want to prove heuristic not guaranteed to produce an inductive invariant 7-8
Finding Inductive Assertions Bottom-Up Approach Transition-validated assertions: 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 34

Strengthening vs. Incremental Proof - CS256/Winter 2007...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online