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CMU CS 15213 - Lecture

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Verifying Programs with BDDsSept. 22, 2006Verifying Programs with BDDsSept. 22, 2006TopicsTopics Representing Boolean functions with Binary Decision Diagrams Application to program verificationclass08-bdd.ppt15-213“The course that gives CMU its Zip!”15-213, F’06–2–Verification ExampleVerification ExampleDo these functions produce Do these functions produce identical results?identical results?How could you find out?How could you find out?How about exhaustive testing?How about exhaustive testing?int abs(int x) {int mask = x>>31;return (x ^ mask) + ~mask + 1;}int test_abs(int x) {return (x < 0) ? -x : x; }–3–More ExamplesMore Examplesint addXY(int x, int y){return x+y;}int addYX(int x, int y){return y+x;}?=int mulXY(int x, int y){return x*y;}int mulYX(int x, int y){return y*x;}?=–4–How Can We Verify Programs?How Can We Verify Programs?TestingTesting Exhaustive testing not generally feasible Currently, programs only tested over small fraction of possible casesFormal VerificationFormal Verification Mathematical “proof” that code is correct Did Pythagoras show that a2+ b2= c2by testing?abc–5–Bit-Level Program VerificationBit-Level Program Verification View computer word as 32 separate bit values Each output becomes Boolean function of inputsabsx0x1x2•••x31y0y1y2•••y31••••••x0x1x2•••x31•••yiabsiint abs(int x) {int mask = x>>31;return (x ^ mask) + ~mask + 1;}–6–Extracting Boolean RepresentationExtracting Boolean RepresentationDo these functions produce Do these functions produce identical results?identical results?int bitOr(int x, int y){return ~(~x & ~y);}int test_bitOr(int x, int y){return x | y;}yxv1 = ~xv2 = ~yv3 = v1 & v2v4 = ~v3v5 = x | yt = v4 == v5Straight-Line Evaluation–7–Tabular Function RepresentationTabular Function Representation List every possible function valueComplexityComplexity Function with n variables00001111001100110101010100010101x1x2x3f–8–Algebraic Function RepresentationAlgebraic Function Representation f(x1, x2, x3) = (x1+ x2) · x3 Boolean AlgebraComplexityComplexity Representation Determining properties of functionz E.g., deciding whether two expressions are equivalent00001111001100110101010100010101x1x2x3fx1·x3x2·x3–9–Tree RepresentationTree RepresentationTruth Table Decision Tree Vertex represents decision Follow green (dashed) line for value 0 Follow red (solid) line for value 1 Function value determined by leaf valueComplexityComplexity0 0x30 1x3x20 1x30 1x3x2x100001111001100110101010100010101x1x2x3f–10–Ordered Binary Decision DiagramsOrdered Binary Decision DiagramsInitial Tree Reduced GraphCanonical representation of Boolean functionCanonical representation of Boolean function Two functions equivalent if and only if graphs isomorphicz Can be tested in linear time Desirable property: simplest form is canonical.x20 1x3x10 0x30 1x3x20 1x30 1x3x2x1(x1+ x2) · x3–11–Example FunctionsExample FunctionsConstantsUnique unsatisfiable functionUnique tautology10VariableTreat variableas function0 1xOdd ParityLinearrepresentationx2x3x410x4x3x2x1Typical Functionx20 1x4x1 (x1 + x2) · x4 No vertex labeled x3 independent of x3 Many subgraphs shared –12–More Complex FunctionsMore Complex Functionsb3b3a3Coutb3b2b2a2b2b2a2b3a3S3b2b1b1a1b1b1a1b2a2S2b1a0a0b1a1S1b010b0a0S0FunctionsFunctions Add 4-bit words a and b Get 4-bit sum S Carry output bit CoutShared RepresentationShared Representation Graph with multiple roots 31 nodes for 4-bit adder 571 nodes for 64-bit adder Linear growth!–13–Symbolic ExecutionSymbolic Execution(3-bit word size)xx210x110x010yy210y110y010v1 = ~xx201x101x001v2 = ~yy201y101y001–14–Symbolic Execution (cont.)Symbolic Execution (cont.)v3 = v1 & v2x2y201x1y101x0y001v4 = ~v3x2y210x1y110x0y010v5 = x | yx2y210x1y110x0y010t = v4 == v51–15–Counterexample GenerationCounterexample GenerationFind values of Find values of xx& & yyfor which for which these programs produce these programs produce different resultsdifferent resultsint bitOr(int x, int y){return ~(~x & ~y);}int bitXor(int x, int y){return x ^ y;}yxv1 = ~xv2 = ~yv3 = v1 & v2v4 = ~v3v5 = x ^ yt = v4 == v5Straight-Line Evaluation–16–Symbolic ExecutionSymbolic Executionv4 = ~v3x2y210x1y110x0y010v5 = x ^ yx2y210y2x1y110y1x0y010y0t = v4 == v5x2y2x1y1x0y001x = 111y = 001–17–Performance: GoodPerformance: Goodint addXY(int x, int y){return x+y;}int addYX(int x, int y){return y+x;}010020030040050060070080090010000 8 16 24 32Word SizeSecondsEnumerateBDD–18–Performance: BadPerformance: Badint mulXY(int x, int y){return x*y;}int mulYX(int x, int y){return y*x;}010020030040050060070080090010000 8 16 24 32Word SizeSecondsEnumerateBDD–19–Why Is Multiplication Slow?Why Is Multiplication Slow?Multiplication function intractable for BDDsz Exponential growth, regardless of variable ordering1456041881552144MultMultAddAddBitsBitsMultiplication-4Add-4Node Counts–20–What if Multiplication were Easy?What if Multiplication were Easy?int factorK(int x, int y){int K = XXXX...X;int rangeOK =1 < x && x <= y;int factorOK =x*y == K;return!(rangeOK && factorOK);}int one(int x, int y){return 1;}–21–Dealing with ConditionalsDealing with ConditionalsDuring Evaluation, Keep Track of:During Evaluation, Keep Track of: Current Context: Under what condition would code be evaluated Definedness (for each variable)z Has it been assigned a valueint abs(int x){int r;if (x < 0)r = -x;elser = x;return r;}t1 = x<0xv1 = -xr = v1v2 = rr = x11t1t11!t1000t111000t1?v1:0t1?v1:xt1?v1:xContextrdefinedrvalue–22–Dealing with LoopsDealing with LoopsUnrollUnroll Turn into bounded sequence of conditionalsz Default limit = 33 Signal runtime error if don’t complete within limitint ilog2(unsigned x){int r = -1;while (x) {r++; x >>= 1;}return r;}int ilog2(unsigned x){int r = 31;if (x) {r++; x >>= 1;} else return r;if (x) {r++; x >>= 1;} else return r;. . .if (x) {r++; x >>= 1;} else return r;error();}Unrolled–23–EvaluationEvaluationStrengthsStrengths Provides 100% guarantee of correctness Performance very good for simple arithmetic functionsWeaknessesWeaknesses Important integer functions have exponential blowup Not practical for programs that build and operate on large data structures–24–Some HistorySome HistoryOriginsOrigins Lee 1959, Akers 1976z Idea of representing Boolean function as BDD


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