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Correctness Proofs. Correctness Proofs. Formal mathematical argument that an algorithm meets its specification, which means that it always produces the correct output for any permitted input. Correctness Proofs. - PowerPoint PPT Presentation

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  • Correctness Proofs

  • Correctness ProofsFormal mathematical argument that an algorithm meets its specification, which means that it always produces the correct output for any permitted input.

  • Correctness ProofsIs Important to understand what a deteiled formal correctness proof looks like, because otherwise you wont know what somebody is really saying with an informal correctness argument.

  • Invariants, Preconditions and Posconditions.P holds in the initial state.P holds after step k if it hods before step k.If P holds when the algorithm terminates, then the output of the algorithm is correct.

  • Hoare LogicAttach to each statement of a program a precondition and a postcondition.

    Precondition, Statement and Postcondition, form a Hoare triple.

  • Hoare Logic{ x is an integer }x := 2*x{ x is even }{ P: x is an integer }x := 2*x{ Q: x is even }x := x+1{R: x is odd }{ x is even }x := x + 1{ x is odd }

  • Hoare Logic{P} S1 {Q} {Q} S2 {R}{P} S1:S2 {R}Composition Axiom

  • Hoare Logic AxiomsRules like before, which define what new propositions can be deduced from old ones, are called Axioms.

  • Pre-strengthening AxiomMaking the precondition stronger doesnt change the truth of a Hoare triple.

    {Q} S {R} P Q{P} S {R}

  • Pre strengthening AxiomMostly used to sneak in extra facts that dont appear explicitly in our original precondition.

    If whenever Q is true,P P Qis also true.

  • Post weakening AxiomMaking the postcondition weaker is also allowed.

    {P} S {Q} Q R{P} S {R}

  • Post weakening AxiomTypically used for getting rid of bits of a postcondition we dont care about.

    The direction of the implications is important. Pre weakening and post strengthening do NOT produce valid proofs.

  • Assignment Axiom{P[x/t]} x := t {P}

    If P is true with x replaced by t before the assignment, it is true without the replacement afterwards.

    {0 = 0} x := {x=0}{x+5 < 12} x:= x+5 {x < 12}{x < 7} x:= x+5 {x < 12}

  • Baggage LemmaUsed to carry along extra baggage that you will need later.

    { } S {x = A}

    But you also need know that y is unchanged.

    {y = B} S {y = B x = A}

  • StrategyWrite down the algorithm.

    Precondition and postcondition for each statements.

    Prove for each statement that its postcondition follow from its precondition.

  • Proofs for if/then/else statements{P}if B then{P and B}do something{Q}else{P and Not B}do something{Q}end if{Q}

  • Proofs for if/then/else statements{P B} S1 {Q} {P B} S2 {Q}{P} if B then S1 else S2 end if {Q}

  • Proofs for Loops{P}while B do{R and B}body{R}end while{Q}

  • Proofs for Loops{A is an array with indices 0..n-1}i := nWhile i 0 do{A[j] = 0 for all j >= i}i := i 1{A[j] = 0 for all j >= i+1}A[i] := 0{A[j] = 0 for all j >= i}end while{A[0] .. A[n-1] are all equal to zero}

  • Total vs Partial CorrectnessWe will want to show that an algorithm produce the right output in a reasonable amount of time, tipically bounded by some function of the size of the input.

  • Proofs for Recursive ProceduresProcedure Euclid(x,y: integer) return integer{}if y = 0 then{y = 0}gcd := x{gcd = gcd(x,y)}else{y 0}gcd := Euclid(y,x mod y){gcd = gcd(x,y)}endif{gcd = gcd(x,y)}return gcdend procedure{return value = gcd(x,y)