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A Tutorial on Functional Program Verification TR #10-26 September 2010, revised August 2011 Yoonsik Cheon Melisa Vela Presented by Aditi Barua 1

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Functional program verification Formal program verification technique Based on Cleanroom Software Engineering Involves: Viewing program as a mathematical function (code function) Documenting function that computes the expected behavior of the code(intended function) Comparing the intended function and the code function. Introduction 2

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Advantages Requires minimal mathematical background. Reflects the way programmers verify correctness of program. Helps one to be proficient with other verification technique. 3

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Writing Intended Function & Code Function Program as mathematical function from one state to another Initial state : {x->10, sum->100} sum=sum + x; Final state :{x->10, sum->110} 4

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Concurrent Assignment Notation to express function that only states changes in input state. [x 1, x 2,…, x n := e 1, e 2, …, e n ] Each x i ’s new value is e i Evaluated concurrently at initial state Program’s variables do not appear remain same. Example: 1) sum= sum + x; [sum: = sum +x] 2) x = x + y; y = x - y;[x, y: = y, x] x = x - y; 5

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Conditional Concurrent Assignment Different functions for different conditions. Conditions are evaluated in initial state. Conditions are evaluated sequentially. If multiple conditions hold, function for first matched condition is picked. Example: [x>0 -> sign : = 1 |x sign :=-1 |else -> sign := 0] 6

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Special Symbols and keywords Identity function denoted by I [n > maxSize -> n:= maxSize| else -> I] undefined: [n > 0 -> avg:= sum/n| else -> undefined] anything [sum, i := sum + ∑ j=i…a.length-1 a[j], anything] while(i

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Verifying Correctness Verification involves showing two properties: dom of f ⊆ dom of p where f=intended function, p= code function. (p(x) = f(x) for x ∈ dom(f)) Assignment Statement Code function and intended function is often same. @//[x:=x+1] x=x+1; 8

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Verifying Correctness Sequential Composition Annotated code //@ [n > 0 → sum, avg := sum+a, (sum+a)/n] sum = sum + a; avg = sum / n; Proof of correctness [sum := sum + a]; [n != 0 → avg := sum=n] ≡ [n!= 0 → sum; avg := sum + a;(sum + a)/n] ⊑ [n > 0 → sum; avg := sum + a;(sum + a)/n] 9

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Sequential Composition(Cont.) Trace table x = x + 1; y = 2 * x; z = x * y; x = x + 1; y = 3 * x; Statementsxyz x = x + 1;x+1 y = 2 * x;2*(x+1) z = x * y;(x+1)*2(x+1) x = x + 1;x+2 y = 3 * x;3(x+2) [x, y, z := x+2, 3(x+2), 2x 2 +4x+2] 10

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Sequential Composition(Cont.) Modular Verification Annotated code //@ [f0] //@ [f1] S1; //@ [f2] S2; Proof of correctness (f1;f2 ⊑ f0). (S1 ⊑ f1) (S2 ⊑ f2) 11

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Conditional Trace table p = a * r; if (a < b) b = b - a; else b = b - p; StatementConditionpb p = a * r; if (a < b) b = b - a; a

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Case Analysis Annotated code //@ [f] if (B) S1; else S2; Proof of correctness (B ⇒ S1 ⊑ f) (¬B ⇒ S2 ⊑ f) Conditional Statement(Cont.) 13

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//@ [f1] while (B) S //@ [f1] if (B) { S while (B) S } Verifying Iteration //@ [f1] if (B) { S [f1] } More involved as there is no known algorithm to calculate code function for whole statements. Solution: Proof by Induction Intended function is the induction hypothesis. 14

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Annotated code Verifying Iteration(Cont.) Using induction to prove correctness of while statement. Proof of correctness Need to discharge following three proof obligations: 1) Termination of the loop 2) Basis step: ¬(i < a:length) ⇒ I ⊑ f1 3) Induction step: i < a:length ⇒ f2;f1 ⊑ f1 and the correctness of f2 and its code //@ [f1] if (B) { //@[f2] S [f1] } //@ [f1] while (B) //@[f2] S 15

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Initialized Loop Uninitialized loop is a Generalization of initialized loop. Loop preceded with initialization computes something useful. Example: /*@ f 1 :[sum, i := sum + ∑ j=i…a.length-1 a[j], anything]*/ while(i

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Verification of Initialized Loop Annotated code //@ [f0] //@ [f1] S1 //@ [f2] while (B) { //@ [f3] S2 } Proof of correctness Discharging the following proof obligations: 1 ) f1;f2 ⊑ f0. 2) S1 ⊑ f1. 3) while (B) S2 ⊑ f2, which requires the following subproofs. a) Termination of the loop. b) Basis step: ¬B ⇒ I ⊑ f2. c) Induction step: B ⇒ f3;f2 ⊑ f2 and S2 ⊑ f3. 17

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Exercise Annotate with intended function while (i < a.length) { if (a[i] > k) { r++; } i++; } 18

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Solution // f 1 : [r, i := r + ∑ j=i…a.length-1 (a[j] > 0 ? 1 : 0), anything] while (i < a.length) { // f 2 : [r, i := a[i] > 0 ? r + 1 : r, i + 1] // [r := a[i] > 0 ? r + 1 : r] if (a[i] > k) { [r:=r+1] r++; } [i:= i+1] i++; } 19

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Reference Yoonsik Cheon and Melisa Vela. A Tutorial on Functional Program Verification, Technical Report 10-26, Department of Computer Science, University of Texas at El Paso, El Paso, TX, September 2010. 20

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