Axiomatic Semantics

Axiomatic Semantics Predicate Transformers L18-9WP

Motivation Input Output • 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. • Program • Transform input to output. L18-9WP

Sorting algorithms • Bubble sort; Shell sort; • Insertion sort; Selection sort; • Merge sort; Quick sort; • Heap sort; • Axiomatic Semantics To show that a program satisfies its specification, it is convenient to have the description of the language constructs in terms of assertions characterizing the input and the corresponding output states. L18-9WP

Axiomatic Approaches • Hoare’s Proof System (partial correctness) • Dijkstra’s Predicate Transformer (total correctness) Assertion: Logic formula involving program variables, arithmetic/boolean operations, etc. Hoare Triples : {P} S {Q} pre-condition statements post-condition (assertion) (program) (assertion) L18-9WP

Swap Example { x = n and y = m } t := x; x := y; y := t; { x = m and y = n} • program variables vs ghost/logic variables • States : Variables -> Values • Assertions : States -> Boolean (= Powerset of States) L18-9WP

Partial vs Total Correctness {P} S {Q} • S is partiallycorrect for P and Qif and only if whenever S is executed in a state satisfying Pand the execution terminates,then the resulting state satisfies Q. • S is totallycorrect for P and Qif and only if whenever S is executed in a state satisfying P ,then the execution terminates, and the resulting state satisfies Q. L18-9WP

Examples • Totally correct (hence, partially correct) • { false } x := 0; { x = 111 } • { x = 11 } x := 0; { x = 0 } • { x = 0 } x := x + 1; { x = 1 } • {false} while true do; {x = 0} • {y = 0} if x <> y then x:= y; { x = 0 } • Not totally correct, but partially correct • {true} while true do; {x = 0} • Not partially correct • {true} if x < 0 then x:= -x; { x > 0 } L18-9WP

Axioms and Inference Rules • Assignment axiom {Q[e]} x := e; {Q[x]} • Inference Rule for statement composition {P} S1 {R} {R} S2 {Q} {P} S1; S2 {Q} • Example {x = y} x := x+1; {x = y+1} {x = y+1} y := y+1; {x = y} {x = y} x:=x+1; y:=y+1; {x = y} L18-9WP

Generating additional valid triples {P} S {Q} from {P’} S {Q’} P’ States States P’ Q P Q’ L18-9WP

Rule of Consequence {P’} S {Q’} and P=>P’and Q’=>Q {P} S {Q} • Strengthening the antecedent • Weakening the consequent • Example {x=0 and y=0} x:=x+1;y:=y+1; {x = y} {x=y} x:=x+1; y:=y+1; {x<=y or x=5} (+ Facts from elementary mathematics[boolean algebra + arithmetic] ) L18-9WP

Predicate Transformers • Assignment wp( x := e , Q ) = Q[x<-e] • Composition wp( S1 ; S2 , Q) = wp( S1 , wp( S2 , Q )) • Correctness {P} S {Q} = (P => wp( S , Q)) L18-9WP

Correctness Illustrated P => wp( S , Q) States States Q wp(S,Q) P L18-9WP

Correctness Proof {x=0 and y=0} x:=x+1;y:=y+1; {x = y} • wp(y:=y+1; , {x = y}) = { x = y+1 } • wp(x:=x+1; , {x = y+1}) = { x+1 = y+1 } • wp(x:=x+1;y:=y+1; , {x = y}) = { x+1 = y+1 } = { x = y } • { x = 0 and y = 0 } => { x = y } L18-9WP

Conditionals { P and B } S1 {Q} {P and not B } S2 {Q} {P} if B then S1 else S2; {Q} wp(if B then S1 else S2; , Q) = (B => wp(S1,Q)) and (not B => wp(S2,Q)) = (B and wp(S1,Q)) or (not Band wp(S2,Q)) L18-9WP

“Invariant”: Summation Program { s = i * (i + 1) / 2 } i := i + 1; s := s + i; { s = i * (i + 1) / 2 } • Intermediate Assertion ( s and i different) { s + i = i * (i + 1) / 2 } • Weakest Precondition { s+i+1 = (i+1) * (i+1+1) / 2 } L18-9WP

while-loop : Hoare’s Approach {Invand B} S {Inv} {Inv} while B do S {Invand not B} Proof of Correctness {P} while B do S {Q} = P => Invand {Inv} B {Inv} and {Invand B} S {Inv} and {Invandnot B => Q} + Loop Termination argument L18-9WP

{I} while B do S {I and not B} {I and B} S {I} 0 iterations: {I} {I and not B} notB holds 1 iteration: {I} S {I and not B} B holdsnotBholds 2 iterations: {I} S ; S {I and not B} B holds B holdsnotBholds • Infinite loop if B never becomes false. L18-9WP

Example1 : while-loop correctness { n>0 and x=1 and y=1} while (y < n) [ y++; x := x*y;] {x = n!} • Choice of Invariant • {I and not B} => Q • {I and (y >= n)} => (x = n!) • I = {(x = y!) and (n >= y)} • Precondition implies invariant { n>0 and x=1 and y=1} => { 1=1! and n>=1 } L18-9WP

Verify Invariant {I and B} => wp(S,I) wp( y++; x:=x*y; , {x=y! and n>=y}) = { x=y! and n>=y+1 } I and B = { x=y! and n>=y } and { y<n } = { x=y! and n>y } • Termination • Variant : ( n - y ) y : 1 -> 2 -> … -> n (n-y) : (n-1) -> (n-2) -> … -> 0 L18-9WP

Detailed Working wp( y++; x:=x*y; , {x=y! and n>=y}) = wp(y++,{x*y=y! and n>=y}) = wp(y++,{x=y-1! and n>=y}) = wp(y++,{x=y-1! and n>=y}) = {x=y+1-1! and n>=y+1} L18-9WP

GCD-LCM code PRE: (x = n) and (y = m) • u := x; v := y; • while (x <> y) do ASSERT: (** INVARIANT **) • begin • if x > y then x := x - y; u := u + v • else y := y - x; v := v + u • end; POST: (x = gcd(n,m)) and (lcm (n,m) = (u+v) div 2) L18-9WP

while-loop : Dijkstra’s Approach wp( while B do S , Q) = P0 or P1 or … or Pn or … = there existsk >= 0 such that Pk Pi: Set of states causingi-iterations ofwhile-loopbefore halting in a state inQ. P0 = not B and Q P1 = B and wp(S, P0) Pk+1 = B and wp(S, Pk) L18-9WP

States States ... wp Q P2 P0 P1 P0 P0 => wp(skip, Q) P0 subset Q P1 => wp(S, P0) L18-9WP

Example2 : while-loop correctness P0 = { y >= n and x = n! } Pk = B and wp(S,Pk-1) P1 = { y<n andy+1>=n and x*(y+1) = n! } Pk = y=n-kand x=(n-k)! Weakest Precondition Assertion: Wp = there existsk >= 0 such that P0 or {y = n-kand x = (n-k)!} Verification : P = n>0 and x=1 and y=1 Fori = n-1: P => Wp L18-9WP

Induction Proof Hypothesis : Pk = {y=n-kand 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)!) Valid preconditions: • { n = 4 and y = 2 and x = 2 } (k = 2) • { n = 5 and x = 5! and y = 6} (no iteration) L18-9WP

Axiomatic Semantics