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Axiomatic Semantics Semantics and Application to Program Verification Axiomatic semantics Semantics and Application to Program Verification Antoine Min´e Ecole´ normale sup´erieure, Paris year 2013{2014 Course 4 12 March 2014 Course 4 Axiomatic semantics Antoine Min´e p. 1 / 55 Introduction Operational semantics Models precisely program execution as low-level transitions between internal states (transition systems, execution traces, big-step semantics) Denotational semantics Maps programs into objects in a mathematical domain (higher level, compositional, domain oriented) Aximoatic semantics (today) Prove properties about programs programs are annotated with logical assertions a rule-system defines the validity of assertions (logical proofs) clearly separates programs from specifications (specification ' user-provided abstraction of the behavior, it is not unique) enables the use of logic tools (partial automation) Course 4 Axiomatic semantics Antoine Min´e p. 2 / 55 Overview Specifications (informal examples) Floyd{Hoare logic Dijkstra's predicate transformers (weakest precondition, strongest postcondition) Verification conditions (partially automated program verification) Advanced topics Auxiliary variables Non-determinism Total correctness (termination) Arrays Course 4 Axiomatic semantics Antoine Min´e p. 3 / 55 Specifications Specifications Course 4 Axiomatic semantics Antoine Min´e p. 4 / 55 express the intended behavior of the function (returned value) add requirements for the function to actually behave as intended (a requires/ensures pair is a function contract) strengthen the requirements to ensure termination Specifications Example: function specification example in C + ACSL int mod(intA, intB) { int Q = 0; int R = A; while (R >= B) { R = R - B; Q = Q + 1; } returnR; } Course 4 Axiomatic semantics Antoine Min´e p. 5 / 55 add requirements for the function to actually behave as intended (a requires/ensures pair is a function contract) strengthen the requirements to ensure termination Specifications Example: function specification example in C + ACSL //@ ensures nresult == A mod B; int mod(intA, intB) { int Q = 0; int R = A; while (R >= B) { R = R - B; Q = Q + 1; } returnR; } express the intended behavior of the function (returned value) Course 4 Axiomatic semantics Antoine Min´e p. 5 / 55 strengthen the requirements to ensure termination Specifications Example: function specification example in C + ACSL //@ requires A>=0 && B>=0; //@ ensures nresult == A mod B; int mod(intA, intB) { int Q = 0; int R = A; while (R >= B) { R = R - B; Q = Q + 1; } returnR; } express the intended behavior of the function (returned value) add requirements for the function to actually behave as intended (a requires/ensures pair is a function contract) Course 4 Axiomatic semantics Antoine Min´e p. 5 / 55 Specifications Example: function specification example in C + ACSL //@ requires A>=0 && B>0; //@ ensures nresult == A mod B; int mod(intA, intB) { int Q = 0; int R = A; while (R >= B) { R = R - B; Q = Q + 1; } returnR; } express the intended behavior of the function (returned value) add requirements for the function to actually behave as intended (a requires/ensures pair is a function contract) strengthen the requirements to ensure termination Course 4 Axiomatic semantics Antoine Min´e p. 5 / 55 Specifications Example: program annotations example with full assertions //@ requires A>=0 && B>0; //@ ensures nresult == A mod B; int mod(int A, int B) { int Q = 0; int R = A; //@ assert A>=0 && B>0 && Q=0 && R==A; while (R >= B) { //@ assert A>=0 && B>0 && R>=B && A==Q*B+R; R = R - B; Q = Q + 1; } //@ assert A>=0 && B>0 && R>=0 && R<B && A==Q*B+R; return R; } Assertions give detail about the internal computations why and how contracts are fulfilled (Note: r = a mod b means 9q: a = qb + r ^ 0 ≤ r < b) Course 4 Axiomatic semantics Antoine Min´e p. 6 / 55 Specifications Example: ghost variables example with ghost variables //@ requires A>=0 && B>0; //@ ensures nresult == A mod B; int mod(int A, int B) { int R = A; while (R >= B) { R = R - B; } // 9Q: A = QB + R and 0 ≤ R < B return R; } Program annotations can be more complex than the program Course 4 Axiomatic semantics Antoine Min´e p. 7 / 55 Specifications Example: ghost variables example with ghost variables //@ requires A>=0 && B>0; //@ ensures nresult == A mod B; int mod(int A, int B) { //@ ghost int q = 0; int R = A; //@ assert A>=0 && B>0 &&q=0 && R==A; while (R >= B) { //@ assert A>=0 && B>0 && R>=B && A==q*B+R; R = R - B; //@ ghost q = q + 1; } //@ assert A>=0 && B>0 && R>=0 && R<B && A==q*B+R; return R; } Program annotations can be more complex than the program and require reasoning on enriched states (ghost variables) Course 4 Axiomatic semantics Antoine Min´e p. 7 / 55 Specifications Example: class invariants example in ESC/Java public class OrderedArray { int a[]; int nb; //@invariant nb >= 0 && nb <= 20 //@invariant (nforall int i; (i >= 0 && i < nb-1) ==> a[i] <= a[i+1]) public OrderedArray() { a = new int[20]; nb = 0; } public void add(int v) { if (nb >= 20) return; int i; for (i=nb; i > 0 && a[i-1] > v; i--) a[i] = a[i-1]; a[i] = v; nb++; } } class invariant: property of the fields true outside all methods it can be temporarily broken within a method but it must be restored before exiting the method Course 4 Axiomatic semantics Antoine Min´e p. 8 / 55 Specifications Language support Contracts (and class invariants): built in few languages (Eiffel) available as a library / external tool (C, Java, C#, etc.) Contracts can be: checked dynamically checked statically (Frama-C, Why, ESC/Java) inferred statically (CodeContracts) In this course: deductive methods (logic) to check (prove) statically (at compile-time) partially automatically (with user help) that contracts hold Course 4 Axiomatic semantics Antoine Min´e p. 9 / 55 Floyd{Hoare logic Floyd{Hoare logic Course 4 Axiomatic semantics Antoine Min´e p. 10 / 55 Floyd{Hoare logic Hoare triples Hoare triple: fPg prog fQg prog is a program fragment P and Q are logical assertions over program variables def (e.g. P =( X ≥ 0 ^ Y ≥ 0) _ (X < 0 ^ Y < 0)) A triple means: if P holds before prog is executed then Q holds after the execution of prog unless prog does not terminates or encounters an error P is the precondition, Q is the postcondition fPg prog fQg expresses partial correctness (does not rule out errors and non-termination) Hoare triples serve as judgements in a proof system (introduced in [Hoare69]) Course 4 Axiomatic semantics Antoine Min´e p. 11 / 55 Floyd{Hoare logic Language stat ::= X expr (assignment) j skip (do nothing) j fail (error) j stat; stat (sequence) j if expr then stat else stat (conditional) j while expr do stat (loop) X 2 V: integer-valued variables expr: integer arithmetic expressions we assume that: expressions are deterministic (for now) expression evaluation do not cause error for instance, to avoid division by zero, we can: either define 1=0 to be a valid value, such as 0 or systematically guard divisions (e.g.: if X = 0 then fail else ··· =X ··· ) Course 4 Axiomatic semantics Antoine Min´e p. 12 / 55 Floyd{Hoare logic Hoare rules: axioms Axioms: fPg skip fPg fPg fail fQg any property true before skip is true afterwards any property is true after fail Course 4 Axiomatic semantics Antoine Min´e p. 13 / 55 Floyd{Hoare logic Hoare rules: axioms Assignment axiom: fP[e=X ]g X e fPg for P over X to be true after X e P must be true over e before the assignment P[e=X ] is P where free occurrences of X are replaced with e e must be deterministic the rule is \backwards"( P appears as a postcondition) examples: ftrueg X 5 fX = 5g fY = 5g X Y fX = 5g fX + 1 ≥ 0g X X + 1 fX ≥ 0g ffalseg X Y + 3 fY = 0 ^ X = 12g fY 2 [0; 10]g X Y + 3 fX = Y + 3 ^ Y 2 [0; 10]g Course 4 Axiomatic semantics Antoine Min´e p. 14 / 55 Floyd{Hoare logic Hoare rules: consequence Rule of consequence: P ) P0 Q0 ) Q fP0g c fQ0g fPg c fQg we can weaken a Hoare triple by: weakening its postcondition Q ( Q0 strengthening its precondition P ) P0 we assume a logic system to be available to prove formulas on assertions, such as P ) P0 (e.g., arithmetic, set theory, etc.) examples: the axiom for fail can be replaced with ftrueg fail ffalseg (as P ) true and false ) Q always hold) fX = 99 ^ Y 2 [1; 10]g X Y + 10 fX = Y + 10 ^ Y 2 [1; 10]g (as fY 2 [1; 10]g X Y + 10 fX = Y + 10 ^ Y 2 [1; 10]g and X = 99 ^ Y 2 [1; 10] ) Y 2 [1; 10]) Course 4 Axiomatic semantics Antoine Min´e p. 15 / 55 Floyd{Hoare logic Hoare rules: tests fP ^ eg s fQg fP ^ :eg t fQg Tests: fPg if e then s else t fQg to prove that Q holds after the test we prove that it holds after each branch (s, t) under the assumption that it is executed (e, :e) example: fX < 0g X −X fX > 0g fX > 0g skip fX > 0g f(X 6= 0) ^ (X < 0)g X −X fX > 0g f(X 6= 0) ^ (X ≥ 0)g skip fX > 0g fX 6= 0g if X < 0 then X −X else skip fX > 0g Course 4 Axiomatic semantics Antoine Min´e p. 16 / 55 Floyd{Hoare logic Hoare rules: sequences fPg s fRg fRg t fQg Sequences: fPg s; t fQg to prove a sequence s; t we must invent an intermediate assertion R implied by P after s, and implying Q after t (often denoted fPg s fRg t fQg) example: fX = 1 ^ Y = 1g X X + 1 fX = 2 ^ Y = 1g Y Y − 1 fX = 2 ^ Y = 0g Course 4 Axiomatic semantics Antoine Min´e p.
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