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Predicate Abstraction and Canonical Abstraction for Singly - linked Lists Roman Manevich Mooly Sagiv Tel Aviv University Eran Yahav G. Ramalingam IBM T.J. Watson

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2 Motivating Example 1: CEGAR curr = head; while (curr != tail) { assert (curr != null); curr = curr.n; } curr n nnn tail … Counterexample-guided abstraction refinement generates following predicates: curr.n ≠ null, curr.n.n ≠ null, … after i refinement steps: curr(.n) i ≠ null head

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3 Motivating Example 1: CEGAR curr = head; while (curr != tail) { assert (curr != null); curr = curr.n; } n nnn tail … In general, problem is undecidable: V. Chakaravathy [POPL 2003] State-of-the-art canonical abstractions can prove assertion curr head

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4 Motivating Example 2 // @pre cyclic(x) t = null; y = x; while (t != x && y.data < low) { t = y.n; y = t; } z = y; while (z != x && z.data < high) { t = z.n; z = t; } t = null; if (y != z) { y.n = null; y.n = z; } // @post cyclic(x)

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5 Motivating Example 2 x n n n n n nn y z x n n n nn y z @pre cyclic(x)@post cyclic(x)

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6 Existing Canonical Abstraction x,cn rx,ry,rz n n y,cn rx,ry,rz z,cn rx,ry,rz n n n n n x n n n n n nn y z cn rx,ry,rz concreteabstract order between variables lost! cannot establish @post cyclic(x)

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7 Overview and Main Results Current predicate abstraction refinement methods not adequate for analyzing heaps Predicate abstraction can simulate arbitrary finite abstract domains Often requires too many predicates New family of abstractions for lists Bounded number of sharing patterns Handles cycles more precisely than existing canonical abstractions Encode abstraction with two methods: Canonical abstraction Polynomial predicate abstraction

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8 Outline New abstractions for lists Observations on concrete shapes Static naming scheme Encoding via predicate abstraction Encoding via canonical abstraction Controlling the number of predicates via heap-sharing depth parameter Experimental results Related work Conclusion

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9 Concrete Shapes Assume the following class of (list-) heaps Heap contains only singly-linked lists No garbage (easy to handle) A heap can be decomposed into Basic shape (sharing pattern) List lengths

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10 Concrete Shapes x y class SLL { Object value; SLL n; } n nnnnnnnnn nnn n

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11 Interrupting Nodes x y Interruption: node pointed-to by a variable or shared by n fields #interruptions ≤ 2 · #variables (bounded number of sharing patterns) n nnnnnnnnn nnn n

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12 Maximal Uninterrupted Lists x y Maximal uninterrupted list: maximal list segment between two interruptions not containing interruptions in-between n nnnnnnnnn nnn n

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13 Maximal Uninterrupted Lists x y max. uninterrupted 2max. uninterrupted 1 max. uninterrupted 3 max. uninterrupted 4 n nnnnnnnnn nnn n

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14 Maximal Uninterrupted Lists x y n nnnnnnnnn nnn n 24 4 4 number of links

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15 Maximal Uninterrupted Lists x y n nnnnnnnnn nnn n 2>2 Abstract lengths: {1,2,>2}

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16 Using Static Names Goal: name all sharing patterns Prepare static names for interruptions Derive predicates for canonical abstraction Prepare static names for max. uninterrupted lists Derive predicates for predicate abstraction All names expressed by FO TC formulae

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17 Naming Interruptions x y x1 y1 x2 y2 We name interruptions by adding auxiliary variables For every variable x : x1, …,xk (k=#variables) x y n nnnnnnnnn nnnn

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18 Naming Max. Uninterrupted Lists x y x1 y1 x2 y2 x y [x1,x2] [x1,y2] [y1,x2] [y1,y2] [x,x1] [x,y1] [x2,x2] [x2,y2] [y2,x2] [y2,y2] [y,x1] [y,y1] n nnnnnnnnn nnnn

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19 A Predicate Abstraction For every pair of variables x,y (regular and auxiliary) Aliased[x,y] = x and y point to same node UList1[x,y] = max. uninterrupted list of length 1 UList2[x,y] = max. uninterrupted list of length 2 UList[x,y] = max. uninterrupted list of any length For every variable x (regular and auxiliary) UList1[x,null] = max. uninterrupted list of length 1 UList2[x,null] = max. uninterrupted list of length 2 UList[x,null] = max. uninterrupted list of any length Predicates expressed by FO TC formulae

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20 Predicate Abstraction Example x y x1 y1 x2 y2 x y n nnnnnnnnn nnnn Aliased[x2,x2]Aliased[x1,x1]Aliased[y,y]Aliased[x,x] Aliased[y1,x1]Aliased[x1,y1]Aliased[y2,y2]Aliased[y1,y1] UList[x,y1]UList[x,x1]Aliased[y2,x2]Aliased[x2,y2] UList2[y1,y2]UList2[y1,x2]UList2[x1,y2]UList2[x1,x2] UList[y1,y2]UList[y1,x2]UList[x1,y2]UList[x1,x2] UList[x2,y2]UList[x2,x2]UList[y,y1]UList[y,x1] UList[y2,y2]UList[y2,x2] concrete abstract

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21 A Canonical Abstraction For every variable x (regular and auxiliary) x(v) = v is pointed-to by x cul[x](v) = uninterrupted list from node pointed-to by x to v Predicates expressed by FO TC formulae

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22 Canonical Abstraction Example x y x1 y1 x2 y2 x y cul[x] cul[y] cul[x1] cul[y1] cul[x2] cul[y2] n nnnnnnnnn nnnn concrete

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23 Canonical Abstraction Example x y x1 y1 x2 y2 x y cul[x] cul[y] cul[x1] cul[y1] cul[x2] cul[y2] n nnnnn n n n nn abstract

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24 Canonical Abstraction of Cyclic List x n n n n n nn y z concreteabstract x n n n nn y z cul[x] cul[y] cul[z] n n

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25 Canonical Abstraction of Cyclic List abstract preabstract post x n n n nn y z cul[x] cul[y] cul[z] n n x n n n nn y z cul[x] cul[y] cul[z]

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26 Heap-sharing Depth x y x1 y1 x2 y2 x y In this example the heap-sharing depth is 2 In practice depth expected to be low (≤ 1) n nnnnnnnnn nnnn

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27 Setting the Heap-sharing Depth x y x1 y1 x y Setting the heap-sharing depth parameter to 1 results in lost information about shape n nnnnnnnnn nnnn Heap-sharing depth parameter d determines number of static names : control over number of predicates

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28 Experimental Results

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29 Related Work Dor, Rode and Sagiv [SAS ’00] Checking cleanness in lists Sagiv, Reps and Wilhelm [TOPLAS ‘02] General framework + abstractions for lists Dams and Namjoshi [VMCAI ’03] Semi-automatic predicate abstraction for shape analysis Balaban, Pnueli and Zuck [VMCAI ’05] Predicate abstraction for shapes via small models Deutsch [PLDI ’94] Symbolic access paths with lengths

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30 Conclusion New abstractions for lists Observations about concrete shapes Precise for programs containing heaps with sharing and cycles, ignoring list lengths Parametric in sharing-depth d:[1…k] Encoded new abstractions via Canonical abstraction O(d·k) Polynomial predicate abstraction O(d 2 ·k 2 ) d=1 sufficient for all examples

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31 Missing from Talk Simulating abstract domains by pred. abs. Formal definition of abstractions Abstract transformers Decidable logic can be automatically derived Abstraction equivalence: for every two concrete heaps H1,H2 β PA (H1)=β PA (H2) iff β C (H1)=β C (H2) Abstractions with less predicates Cycle breaking Linear static naming scheme

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32 Merci

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33 Simulating Finite Domains by Predicate Abstraction Assume finite abstract domain of numbered elements {1,…,n} Naïve simulation Predicates {P i } i=1…n P i Holds when abstract program state is i Simulation using logarithmic number of predicates Use binary representation of numbers Predicates {P j } j=1…log n P j Holds when j-th bit of abstract program state is 1

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