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Hoare-style program verification K. Rustan M. Leino Guest lecturer Rob DeLines CSE 503, Software Engineering University of Washington 26 Apr 2004

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Is this program correct? How do we know? int Find(float[] a, int m, int n, float x) { while (m < n) { int j = (m+n) / 2 ; if (a[j] < x) { m = j+ 1 ; } else if (x < a[j]) { n = j-1; } else { return j; } } return - 1 ; }

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Making sense of programs Program semantics defines programming language – e.g., Hoare logic, Dijkstra's weakest preconditions Specifications record design decisions – bridge intent and code Tools amplify human effort – manage details – find inconsistencies – ensure quality

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State predicates A predicate is a boolean function on the program state Examples: – x = 8 – x < y – m n (j | 0j<a.length a[j] NaN) – true – false

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Hoare triples For any predicates P and Q and any program S, {P} S {Q} says that if S is started in (a state satisfying) P, then it terminates in Q postcondition precondition

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Examples {true} x := 12 {x = 12 } {x < 40 } x := 12 { 10 x} {x < 40 } x := x+ 1 {x 40 } {m n} j :=(m+n)/ 2 {m j n} {0 m < n a.length a[m] = x} r := Find(a, m, n, x) {m r} {false} S {x n + y n = z n }

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Precise triples If{P} S {Q} and {P} S {R}, then does {P} S {Q R} hold?

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Precise triples If{P} S {Q} and {P} S {R}, then does {P} S {Q R} hold? The most precise Q such that {P} S {Q} is called the strongest postcondition of S with respect to P. yes

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Weakest preconditions If{P} S {R} and {Q} S {R}, then {P Q} S {R} holds. The most general P such that {P} S {R} is called the weakest precondition of S with respect to R, written wp(S, R)

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Triples and wp {P} S {Q} if and only if P wp(S, Q)

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Program semantics skip no-op wp(skip, R) R wp(skip, x n + y n = z n ) x n + y n = z n

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Program semantics assignment evaluate E and change value of w to E wp(w := E, R) R[w := E] wp(x := x + 1, x 10)x+1 10x < 10 wp(x := 15, x 10)15 10false wp(y := x + 3*y, x 10) x 10 wp(x,y := y,x, x < y)y < x replace w by E in R

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Program semantics assert if P holds, do nothing, else don't terminate wp(assert P, R) P R wp(assert x < 10, 0 x)0 x < 10 wp(assert x = y*y, 0 x)x = y*y 0 xx = y*y wp(assert false, x 10)false

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Program compositions If {P} S {Q} and {Q} T {R}, then {P} S ; T {R} If {P B} S {R} and {P B} T {R}, then {P} if B then S else T end {R}

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wp(S;T, R) wp(S, wp(T, R)) wp(x := x+1 ; assert x y, 0 < x)wp(x := x+1, wp(assert x y, 0 < x))wp(x := x+1, 0 < x y) 0 < x+1 y0 x < y wp(y := y+1 ; x := x + 3*y, y 10 3 x)wp(y := y+1, wp(x := x+3*y, y 10 3 x))wp(y := y+1, y 10 3 x+3*y)y+1 10 3 x+3*(y+1)y < 10 3 x + 3*y + 3y < 10 0 x + 3*y Program semantics sequential composition

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wp(if B then S else T end, R) (B wp(S, R)) (B wp(T, R)) (B wp(S, R)) (B wp(T, R)) wp(if x < y then z := y else z := x end, 0 z)(x < y wp(z := y, 0 z)) ((x < y) wp(z := x, 0 z))(x < y 0 y) (y x 0 x)0 y 0 x wp(if x10 then x := x+1 else x := x + 2 end, x 10)(x10 wp(x := x+1, x 10)) ((x10) wp(x := x+2, x 10))(x10 x+1 10) (x=10 x+2 10)(x10 x < 10) falsex < 10 Program semantics conditional composition

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Example if (x != null) { n = x.f; } else { n = z-1; z++; } a = new char[n]; true n >= 0 z-1 >= 0 x != null && x.f >= 0 (x != null ==> x != null && x.f >= 0) && (x == null ==> z-1 >= 0)

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A good exercise Define change w such that P by giving its weakest precondition

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Loops To prove {P} while B do S end {Q} find invariant J and well-founded variant function vf such that: – invariant holds initially: P J – invariant is maintained: {J B} S {J} – invariant is sufficient: J B Q – variant function is bounded: J B 0 vf – variant function decreases: {J B vf=VF} S {vf<VF}

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