Download presentation

Presentation is loading. Please wait.

Published byKeyla Fincher Modified over 2 years ago

1
**Hoare’s Correctness Triplets Dijkstra’s Predicate Transformers**

Axiomatic Semantics Hoare’s Correctness Triplets Dijkstra’s Predicate Transformers Calculating with programs : Reasoning reduced to symbol manipulation. Helps determine precise Boundary conditions. Formalizing intuitions. Other approaches: Denotational Semantics: Real meaning in terms of functions on N. Equivalence: f(x) = f(x) f(x) = if f(x) ==1 then 0 else 1 unsatisfiable (“non-sense”) f(x) = f(x) f(x) = if f(x) ==1 then 1 else f(x) multiple solutions (“no information”) McCarthy’s 91-function Operational Semantics: Abstract interpreter based

2
**gcd-lcm algorithm w/ invariant**

{PRE: (x = n) and (y = m)} u := x; v := y; while {INV: 2*m*n = x*v + y*u} (x <> y) do if x > y then x := x - y; u := u + v else y := y - x; v := v + u fi od {POST:(x = gcd(m, n)) and (lcm(m, n) = (u+v) div 2)} cs784(tk/pm)

3
**Goal of a program = IO Relation**

Problem Specification Properties satisfied by the input and expected of the output (usually described using “assertions”). E.g., Sorting problem Input : Sequence of numbers Output: Permutation of input that is ordered. View Point All other properties are ignored. Timing behavior Resource consumption … cs784(tk/pm)

4
**ax·i·om n.1. A self-evident or universally recognized truth; a maxim**

2. An established rule, principle, or law. 3. A self-evident principle or one that is accepted as true without proof as the basis for argument; a postulate. From a dictionary cs784(tk/pm)

5
Axiomatic Semantics Capture the semantics of the elements of the PL as axioms Capture the semantics of composition as a rule of inference. Apply the standard rules/logic of inference. Consider termination separately. cs784(tk/pm)

6
**States and Assertions States: Variables mapped to Values**

Includes all variables Files etc. are considered “global” variables. No notion of value-undefined variables At a given moment in execution An assertion is a logic formula involving program variables, arithmetic/boolean operations, etc. All assertions are attached to a control point. Assertions: States mapped to Boolean Boolean connectives: and, or, not, implies, … For-all, There-exists Special predicates defined just for use in assertions (not for use in the program). cs784(tk/pm)

7
**Hoare’s Logic Hoare Triplets: {P} S {Q}**

P, pre-condition assertion; S, statements of a PL; Q, post-condition assertion If S begins executing in a state satisfying P, upon completion of S, the resulting state satisfies Q. {P} S {Q} has no relevance if S is begun otherwise. A Hoare triplet is either true or false. Never undefined. The entire {P}S{Q} is considered true if the resulting state satisfies Q if and when S terminates. If not, the entire {P}S{Q} is false. cs784(tk/pm)

8
**Hoare Triplet Examples**

true triplets {x = 11 } x := 0 { x = 0 } we can give a weaker precondition {x = 0 } x := x + 1 { x = 1 } {y = 0} if x <> y then x:= y fi { x = 0 } {false } x := 0 { x = 111 } correct because “we cannot begin” no state satisfies false post condition can be any thing you dream {true} while true do od {x = 0} true is the weakest of all predicates correct because control never reaches post {false} while true do od {x = 0} false is the strongest of all predicates false triplet {true} if x < 0 then x:= -x fi { x > 0 } 1. False = empty set of states. Precondition unsatisfiable, so Hoare triple trivially valid. 2. Strong precondition 4. Nontermination. 5. Multipath program : Else null statement; 6. Partially correct because the IF-part of definition not met, so there are no guarantees from THEN. 7. Modify the precondition to get a valid triple. Material Implication : IF sun rises in the west THEN there will be snow in July in Mexico City. cs784(tk/pm)

9
Weaker/Stronger An assertion R is said to be weaker than assertion P if the truth of P implies the truth of R written: P→R equivalently not P or R. For arbitrary A, B we have: A and B → B This general idea is from Propositional Calculus n > 0 is of course weaker than n = 1, but this follows from Number Theory. cs784(tk/pm)

10
**Weaker/Stronger Q’ stronger P’ weaker Q’ Q P P’ States States P’ Q**

The program transforms a state into another state. (point to point map) Assertions characterize a collection of states. cs784(tk/pm)

11
**Partial vs Total Correctness**

Are P and S such that termination is guaranteed? S is partially correct for P and Q iff whenever the execution terminates, the resulting state satisfies Q. S is totally correct for P and Q iff the execution is guaranteed to terminate, and the resulting state satisfies Q. Logical Implication : IF false THEN (1 = 2) is valid. cs784(tk/pm)

12
**Hoare Triplet Examples**

Totally correct (hence, partially correct) {x = 11} x := 0 {x = 0} {x = 0} x := x + 1 {x = 1} {y = 0}if x <> y then x:= y fi {x = 0} {false} while true do S od {x = 0} {false} x := 0 {x = 111} Not totally correct, but partially correct {true} while true do S od {x = 0} Not partially correct {true} if x < 0 then x:= -x fi {x > 0} False = empty set of states. Precondition unsatisfiable, so Hoare triple trivially valid. 1a. Unecessarily Strong precondition 1d and 2. Nontermination. 3. Multipath program : Else null statement; 6. Partially correct because the IF-part of definition not met, so there are no guarantees from THEN. 7. Modify the precondition to get a valid triple. Material Implication : IF sun rises in the west THEN there will be snow in July in Mexico City. cs784(tk/pm)

13
**Assignment axiom {Q(e)} x := e {Q(x)} Q(x) has free occurrences of x.**

Q(e): every free x in Q replaced with e Assumption: e has no side effects. Caveats If x is not a “whole” variable (e.g., a[2]), we need to work harder. PL is assumed to not facilitate aliasing. cs784(tk/pm)

14
**Inference Rules Rules are written as Can also be stated as:**

Hypotheses: H1, H2, H Conclusion: C1 Can also be stated as: H1 and H2 and H3 implies C1 Given H1, H2, and H3, we can conclude C1. cs784(tk/pm)

15
**Soundness and Completeness**

Soundness is about “validity” Completeness is about “deducibililty” Ideally in a formal system, we should have both. Godel’s Incompleteness Theorem: Cannot have both Inference Rules ought to be sound What we proved/ inferred/ deduced is valid Examples of Unsound Rules A and B and C not B x > y implies x > y+1 (in the context of numbers) All the rules we present from now on are sound cs784(tk/pm)

16
**Rule of Consequence Suppose: {P’} S {Q’}, P=>P’, Q’=>Q**

Conclude: {P} S {Q} Replace precondition by a stronger one postcondition by a weaker one cs784(tk/pm)

17
**Statement Composition Rule**

{P} S1 {R}, {R} S2 {Q} {P} S1;S2 {Q} Using Rule of Consequence {P} S1 {R1}, R1 R2, {R2} S2 {Q} {P} S1;S2 {Q} Reasoning turned into symbol manipulation : Substitution. Confusing with constructs such as: {??} x == x++ * 5 { x = y } cs784(tk/pm)

18
**if-then-else-fi Hoare’s Triplets**

{P and B} S1 {Q} {P and not B} S2 {Q} {P} if B then S1 else S2 fi {Q} We assumed that B is side-effect free Execution of B does not alter state cs784(tk/pm)

19
Invariants An invariant is an assertion whose truth-value does not change Recall: All assertions are attached to a control point. An Example: x > y The values of x and y may or may not change each time control reaches that point. But suppose the > relationship remains true. Then x > y is an invariant Focus on what is essential for the problem at hand rather than the weakest conditiion. Invariant = truth preserved. cs784(tk/pm)

20
**Loop Invariants Semantics of while-loop {I and B} S {I}**

{I} while B do S od {I and not B} Termination of while-loop is not included in the above. We assumed that B is side-effect free. cs784(tk/pm)

21
**Data Invariants Well-defined OOP classes**

Public methods ought to have a pre- and post-conditions defined There is a common portion across all public methods That common portion is known as the data invariant of the class. cs784(tk/pm)

22
**while-loop: Hoare’s Approach**

Wish to prove: {P} while B do S od {Q} Given: P, B, S and Q Not given: a loop invariant I The I is expected to be true just before testing B To prove {P} while B do S od {Q}, discover a strong enough loop invariant I so that P => I {I and B} S {I} I and not B => Q We used the Rule of Consequence twice Focus on what is essential for the problem at hand rather than the weakest conditiion. Invariant = truth preserved. cs784(tk/pm)

23
**A while-loop example { n > 0 and x = 1 and y = 1}**

while (n > y) do y := y + 1; x := x*y od {x = n!} Choosing invariant requires insight and is goal driven. Invariant must hold in the loop. So cannot have (n = y) or (x = n!) etc n > 0 implies n >= 1 because n is of type natural-number cs784(tk/pm)

24
**while-loop: Choose the Invariant**

Invariant I should be such that I and not B Q I and not (n > y) (x = n!) Choose (n ≥ y and x = y!) as our I Precondition Invariant n > 0 and x=1 and y=1 n ≥ 1 and 1=1! Choosing invariant requires insight and is goal driven. Invariant must hold in the loop. So cannot have (n = y) or (x = n!) etc n > 0 implies n >= 1 because n is of type natural-number cs784(tk/pm)

25
**while-loop: Verify Invariant**

I === n ≥ y and x = y! Verify: {I and n > y} y:= y + 1; x:=x*y {I} {I and n > y} y:= y + 1 {n ≥ y and x*y = y!} {I and n > y} y:= y + 1 {n ≥ y and x= (y-1)!} (I and n > y) (n ≥ y+1 and x= (y+1-1)!) (I and n > y) (n > y and x= y!) (n ≥ y and x = y! and n > y) (n > y and x= y!) QED Hoare’s triplets, but also using wp(). cs784(tk/pm)

26
**while-loop: I and not B Q**

I === n ≥ y and x = y! n ≥ y and x = y! and not (n > y) x = n! n = y and x = y! x = n! QED Hoare’s triplets, but also using wp(). cs784(tk/pm)

27
**while-loop: Termination**

Termination is not part of Hoare’s Triplets General technique: Find a quantity that decreases in every iteration. And, has a lower bound The quantity may or may not be computed by the algorithm For our example: Consider n – y values of y: 1, 2, …, n-1, n values of n - y: n-1, n-2, …, 1, 0 Hoare’s triplets, but also using wp(). cs784(tk/pm)

28
**Weakest Preconditions**

We want to determine minimally what must be true immediately before executing S so that assertion Q is true after S terminates. S is guaranteed to terminate The Weakest-Precondition of S is a mathematical function mapping any post condition Q to the "weakest" precondition Pw. Pw is a condition on the initial state ensuring that execution of S terminates in a final state satisfying R. Among all such conditions Pw is the weakest wp(S, Q) = Pw cs784(tk/pm)

29
**Dijkstra’s wp(S, Q) Let Pw = wp(S, Q)**

Def of wp(S, Q): Weakest precondition such that if S is started in a state satisfying Pw, S is guaranteed to terminate and Q holds in the resulting state. Consider all predicates Pi so that {Pi}S{Q}. Discard any Pi that does not guarantee termination of S. Among the Pi remaining, choose the weakest. This is Pw. {P} S {Q} versus P => wp(S, Q) {Pw} S {Q} is true. But, the semantics of {Pw} S {Q} does not include termination. If P => wp(S, Q) then {P}S{Q} also, and furthermore S terminates. cs784(tk/pm)

30
**Properties of wp Law of the Excluded Miracle wp(S, false) = false**

Distributivity of Conjunction wp(S, P and Q) = wp(S,P) and wp(S,Q) Law of Monotonicity (Q→R) → (wp(S,Q)→wp(S,R)) Distributivity of Disjunction wp(S,P) or wp(S, Q) → wp(S,P or Q) cs784(tk/pm)

31
**Predicate Transformers**

Assignment wp(x := e, Q(x)) = Q(e) Composition wp(S1;S2, Q) = wp(S1, wp(S2,Q)) For programs without loops (and recursion), Hoare’s and Dijkstra’s approach converge. Hoare’s approach generates triples while Dijkstra’s approach tries to capture the semantics Unaddressed Questions: Why weakest precondition rather than strongest post-condition? Expressiveness in FOL for while? Application: Distributed computing program proofs. cs784(tk/pm)

32
**A Correctness Proof {x=0 and y=0} x:=x+1;y:=y+1 {x = y}**

wp(x:=x+1;y:=y+1, x = y) wp(x:=x+1, wp(y:=y+1, x = y)) === wp(x:=x+1, x = y+1) === x+1 = y+1 === x = y x = 0 and y = 0 x = y use a better example === at the meta level, = part of assertion language syntax cs784(tk/pm)

33
**if-then-else-fi in Dijkstra’s wp**

wp(if B then S1 else S2 fi, Q) === (B wp(S1,Q)) and (not B wp(S2,Q)) === (B and wp(S1,Q)) or (not B and wp(S2,Q)) cs784(tk/pm)

34
**wp-semantics of while-loops**

DO == while B do S od IF == if B then S fi Let k stand for the number of iterations of S Clearly, k >= 0 If k > 0, while B do S od is the same as: if B then S fi; while B do S od cs784(tk/pm)

35
**while-loop: wp Approach**

wp(DO, Q) = P0 or there-exists k > 0: Pk States satisfying Pi cause i-iterations of while-loop before halting in a state in Q. Pi defined inductively P0 = not B and Q … cs784(tk/pm)

36
**wp(DO, Q) There exists a k, k ≥ 0, such that H(k, Q)**

H is defined as follows H(0, Q) = not B and Q H(k, Q) = H(0, Q) or wp(IF, H(k-1, Q)) cs784(tk/pm)

37
**Example (same as before)**

{ n>0 and x=1 and y=1} while (n > y) do y := y + 1; x := x*y od {x = n!} cs784(tk/pm)

38
**Example: while-loop correctness**

Pre === n>0 and x=1 and y=1 P0 === y >= n and x = n! Pk === B and wp(S, Pk-1) P1 === y > n and y+1>=n and x*(y+1) = n! Pk === y=n-k and x=(n-k)! Weakest Precondition: W === there exists k >= 0 such that P0 or Pk Verification : For k = n-1: Pre => W cs784(tk/pm)

39
**Induction Proof Hypothesis : Pk = (y=n-k and x=(n-k)!)**

Pk+1 = B and wp(S,Pk) = y<n and (y+1 = n-k) and (x*(y+1)=(n-k)!) = y<n and (y = n-k-1) and (x = (n-k-1)!) = y<n and (y = n- k+1) and (x = (n- k+1)!) = (y = n - k+1) and (x = (n - k+1)!) Examples of Valid preconditions: { n = 4 and y = 2 and x = 2 } (k = 2) { n = 5 and x = 5! and y = 6} (no iteration) cs784(tk/pm)

40
**Detailed Work wp(y:=y+1;x:=x*y, x=y!and n>=y)**

= wp(y:=y+1, x*y=y! and n>=y) = wp(y:=y+1, x=(y-1)! and n>=y) = x=(y+1-1)! and n>=y+1) = x=y! and n>y cs784(tk/pm)

41
**gcd-lcm algorithm w/ invariant**

{PRE: (x = n) and (y = m)} u := x; v := y; while {INV: 2*m*n = x*v + y*u} (x <> y) do if x > y then x := x - y; u := u + v else y := y - x; v := v + u fi od {POST:(x = gcd(m, n)) and (lcm(m, n) = (u+v) div 2)} cs784(tk/pm)

42
**gcd-lcm algorithm proof**

PRE implies Loop Invariant (x = n) and (y = m) implies 2*m*n = x*v + y*u {Invariant and B} Loop-Body {Invariant} {2*n*m = x*v + y*u and x <> y} loop-body {2*n*m = x*v + y*u} Invariant and not B implies POST 2*n*m = x*v + y*u and x == y implies (x = gcd(n,m)) and (lcm(n,m) = (u+v) div 2) cs784(tk/pm)

43
**gcd-lcm algorithm proof**

Invariant and not B implies POST 2*m*n = x*v + y*u and x == y implies (x = gcd(m, n)) and (lcm(m, n) = (u+v) div 2) Simplifying 2*m*n = x*(u + v) and x == y implies (x = gcd(m, n)) and (lcm(m, n) = (u+v) div 2) cs784(tk/pm)

44
**gcd-lcm algorithm proof**

Simplifying 2*m*n = x*(u + v) and x == y implies (x = gcd(m, n)) and (x*lcm(m, n) = m*n) cs784(tk/pm)

45
**Some Properties of lcm-gcd**

gcd() and lcm() are symmetric gcd(m, n) = gcd(n, m) lcm(m, n) = lcm(n, m) gcd(m, n) = gcd(m + k*n, n) where k is a natural number. gcd(m, n) * lcm(m, n) = m * n cs784(tk/pm)

Similar presentations

OK

Program Analysis and Verification Spring 2014 Program Analysis and Verification Lecture 4: Axiomatic Semantics I Roman Manevich Ben-Gurion University.

Program Analysis and Verification Spring 2014 Program Analysis and Verification Lecture 4: Axiomatic Semantics I Roman Manevich Ben-Gurion University.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on endangered species of birds Ppt on brand marketing company Ppt on field study 1 Ppt on the road not taken meaning Ppt on power line communication module Ppt on post abortion care Ppt on air pollution act 1981 Ppt on case study of wal mart Ppt on writing equations from tables Ppt on sea level rise in india