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Path-Sensitive Analysis for Linear Arithmetic and Uninterpreted Functions SAS 2004 Sumit Gulwani George Necula EECS Department University of California, Berkeley

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1 y := 2; z := a; y := a; z := 2; u := 1; v := 1+a; t 1 := y-u; t 2 := v-z; True False Example u := a-1; v := 3; Assert(t 1 =t 2 Æ t 1 =1 Æ z=2); a=2? All 3 asserts are true a=2?

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2 y := 2; z := a; y := a; z := 2; u := 1; v := 1+a; t 1 := y-u; t 2 := v-z; True False Path-Insensitive Analysis u := a-1; v := 3; Assert(t 1 =t 2 Æ t 1 =1 Æ z=2); * Most PTIME analyses treat conditionals as non-deterministic. They will verify only t 1 =t 2 *

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3 y := 2; z := a; y := a; z := 2; u := 1; v := 1+a; t 1 := y-u; t 2 := v-z; True False Path-Sensitive Analysis u := a-1; v := 3; Assert(t 1 =t 2 Æ t 1 =1 Æ z=2); c1c1 We can do better by doing a boolean abstraction of conditionals. Each atomic predicate is abstracted to a boolean variable This will also verify t 1 =1 This is still abstract though! z=2 not verified undecidable to reason completely c1c1

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4 Outline Existing approach (MVR) vs. our approach (FCED) FCEDs for linear arithmetic FCEDs for uninterpreted function terms

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5 y := 2; z := a; y := a; z := 2; u := 1; v := 1+a; t 1 := y-u; t 2 := v-z; True False Multi-Valued ROBDDs (MVRs) c1 2a y =c2 1a-1 u = u := a-1; v := 3; Assert(t 1 =t 2 ); Assert(t 1 =1); c1c1 c2c2 |MVR(t 1 )| = |MVR(y)| £ |MVR(u)| MVR(t 1 ) does not share nodes with MVR(y) and MVR(u) Need a normal form for leaves c1 c2 1-a+3a-11 t 1 =

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6 y := 2; z := a; y := a; z := 2; u := 1; v := 1+a; t 1 := y-u; t 2 := v-z; True False Free Conditional Expression Diagrams (FCEDs) c1 2a y =c2 1a-1 u = -t 1 = u := a-1; v := 3; Assert(t 1 =t 2 ); Assert(t 1 =1); c1c1 c2c2 |FCED(t 1 )| = |FCED(y)| + |FCED(u)| FCED(t 1 ) shares nodes with FCED(y) and FCED(u) No need for normal form

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7 Outline Existing approach (MVR) vs. our approach (FCEDs) FCEDs for linear arithmetic FCEDs for uninterpreted function terms

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8 Problem Definition e = q | y | e 1 § e 2 | q £ e | if b then e 1 else e 2 b = c | b 1 Æ b 2 | b 1 Ç b 2 e: conditional linear arithmetic expression b: boolean formula y: rational variable c: boolean variable q: rational constant Construct FCED for an expression e, given FCEDs for its subexpressions. Check 2 FCEDs for equivalence

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9 FCED An FCED f is a DAG with the following kind of nodes. f := y | q | Plus(f 1,f 2 ) | Minus(f 1,f 2 ) | Times(q,f) | Choose(f 1,f 2 ) | Guard(g,f) Choose(f 1,f 2 ) means f 1 or f 2 Guard(g,f) means if g then f Boolean expressions g are represented using ROBDDs g := true | false | c | If(c,g 1,g 2 )

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10 Example c1 2a c2 1a-1 + choose guard choose guard plus R(c1) 2 R( : c1) a R(c2) 1 R( : c2) a-1 Formalization

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11 Example c1 2a c2 1a-1 + choose guard choose guard plus R(c1) 2 R( : c1) a R(c2) 1 R( : c2) a-1 Formalization

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12 FCED Construction FCED(y) = Leaf(y) FCED(q) = Leaf(q) FCED(e1+e2) = Plus (FCED(e1), FCED(e2)) FCED(q £ e) = Times(q,FCED(e)) FCED(if b then e1 else e2) = Choose(Guard(R(b),e1), Guard(R(NOT(b)),e2)

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13 FCED Construction FCED(y) = Leaf(y) FCED(q) = Leaf(q) FCED(e1+e2) = Plus (FCED(e1), FCED(e2)) FCED(q £ e) = Times(q,FCED(e)) FCED(if b then e1 else e2) = Choose(||R(b),FCED(e1)||, ||NOT R(b), FCED(e2)||)

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14 Normalize Guard Operator Inputs: guard g, FCED f Output: FCED f s.t. f ´ f 8 guard nodes Guard(g,f) in f, BV(g) < BV(f) ||g,f|| = Guard(g,f), if BV(g) < BV(f) ||g, Plus(f 1,f 2 ) = Plus(||g,f 1 ||, ||g, f 2 ||) ||g, Choose(f1,f2) = Choose(||g,f1||, ||g, f2||) ||g 1, Guard(g 2,f )|| = Guard(|| INTERSECT(g 1,g 2 ),f ||) …

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15 guard R(c1) guard R(c1) guard R(c1) Example: Normalize Guard Operator plus choose guard R(c2) z R( : c2) 6 Given f, construct ||R(c1),f|| guard choose guard R(c1)R( : c1) 3 2 choose guard R( : c1) 3 guard R(c1) 2 R(c1 Æ c1) guard 2 R ( : c1 Æ c1) guard 3 choose

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16 Randomized Equivalence Testing for FCEDs Assign hash values to nodes of FCEDs in bottom-up manner V: FCED Node ! Integer V(Leaf(q)) = q V(Leaf(y)) = r y V(Plus(f 1,f 2 )) = V(f 1 ) + V(f 2 ) V(Choose(f 1,f 2 )) = V(f 1 ) + V(f 2 ) V(Guard(g,f)) = H(g) £ V(f) H: Guard ! Integer H(true) = 1, H(false) = 0 H(c) = r c H(If(c,g 1,g 2 )) = r c £ H(g 1 ) + (1-r c ) £ H(g 2 )

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17 Randomized Equivalence Testing for FCEDs Completeness f 1 ´ f 2 ) V(f 1 ) = V(f 2 ) Soundness f 1 ´ f 2 ) Pr[V(f 1 ) = V(f 2 )] · s/t s: maximum # of nodes in a FCED t: size of set from which random values are chosen Proof: Poly: FCED ! Polynomials such that V(f) is the value of Poly(f)

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18 Outline Existing approach (MVR) vs. our approach (FCEDs) FCEDs for linear arithmetic FCEDs for uninterpreted function terms

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19 Problem Definition e = y | F(e 1,e 2 ) | if b then e 1 else e 2 b = c | b 1 Æ b 2 | b 1 Ç b 2 e: conditional uninterpreted function term b: boolean formula y: variable c: boolean variable Construct FCED for an expression e, given FCEDs for its subexpressions. Check 2 FCEDs for equivalence

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20 FCED An FCED f is a DAG with the following kind of nodes. f := y | F(f 1,f 2 ) | Choose(f 1,f 2 ) | Guard(g,f) Choose(f 1,f 2 ) means f 1 or f 2 Guard(g,f) means if g then f Boolean expressions g are represented using ROBDDs g := true | false | c | If(c,g 1,g 2 )

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21 FCED Construction FCED(y) = Leaf(y) FCED(F(e 1,e 2 )) = F(FCED(e 1 ), FCED(e 2 )) FCED(if b then e 1 else e 2 ) = Choose(||R(b),FCED(e 1 )||, ||NOT R(b), FCED(e 2 )||)

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22 Randomized Equivalence Testing of FCEDs Assign hash values to nodes of FCEDs in bottom-up manner V: FCED Node ! Tuple of k integers K ¸ depth of any FCED V(y) = [r y,…r y ] V(Choose(f 1,f 2 )) = V(f 1 ) + V(f 2 ) V(Guard(g,f)) = H(g) £ V(f) V(F(f 1,f 2 )) = V(f 1 ) £ M + V(f 2 ) £ N M, N: random k £ k matrices

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23 Randomized Equivalence Testing for FCEDs Completeness f 1 ´ f 2 ) V(f 1 ) = V(f 2 ) Soundness f 1 ´ f 2 ) Pr[V(f 1 ) = V(f 2 )] · s: maximum # of nodes in a FCED t: size of set from which random values are chosen Proof: more involved

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24 Conclusion and Future Work Randomization can help achieve simplicity and efficiency at the expense of making soundness probabilistic. Integrate randomized techniques with symbolic algorithms Few interesting possible extensions: –Combination of uninterpreted functions with arithmetic –Partially interpreted functions like commutative and/or associative functions –Model memory

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