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Shape Analysis by Graph Decomposition R. Manevich M. Sagiv Tel Aviv University G. Ramalingam MSR India J. Berdine B. Cook MSR Cambridge

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2 Motivation Challenge: precise and efficient shape analyses Prove properties of dynamically allocated linked data structures Observation: often many correlations irrelevant for proving shape properties Our approach: develop a flexible abstraction that takes advantage of this

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3 h1t1... h2t2... h1t1h2t2 Example program – 2 lists // @assume h1!=null && h1==t1 && h1.n==null && // h2!=null && h2==t2 && h2.n==null // // @loop_invariant Reach(h1,t1) && // Reach(h2,t2) && // DisjointLists(h1,h2) EnqueueEvents() { L1: while (...) { List temp = new List(getEvent()); if (nondet()) { t1.n = temp; t1 = temp; } else { t2.n = temp; t2 = temp; } } } Correlation between two lists irrelevant for proving loop invariant

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4 size>2 size=2size=1 size>2 size=2size=1 Abstract states - full heaps [VMCAI’05] h1 >1 t1 h2t2 1 h2t2 h1t1 >1 h2t2 1 h1t1 >1 h2t2 >1 h1t1 1 h2t2 1 h1t1 1 h2t2 >1 h1t1 1 h2t2 h1t1 >1 h2t2 h1t1 h1t1 h2t2

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5 Graph decomposition 1 h2t2 1 h1t1 >1 h2t2 1 h1t1 h1 >1 t1 h2t2 >1 h2t2 >1 h1t1 1 h2t2 >1 h1t1 1 h2t2 h1t1 1 h2t2 h1t1 >1 h2t2 h1t1 h1t1 h2t2

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6 Connected component 1 Connected component 2 Graph decomposition 1 h2t2 1 h1t1 Connected components by undirected reachability 1 h2t2 1 h1t1 decompose

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7 Abstract states – decomposed heaps h1t1 h1 1 t1 h1 >1 t1 h2t2 h2 1 t2 h2 >1 t2 For k lists: full heap abstraction generates 3 k abstract states decomposed heap abstraction generates 3×k abstract states Coarser abstraction precise enough to prove invariant but generates fewer states

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8 Overall view h1t1... h2t2... h1t1 h2t2 h1t1 h2t2 h1t1 h2t2 >1 1 1 h1t1 h2t2 h1t1 h2t2 >1 1 1 Concrete domain: concrete heaps Full heaps domain: shape graphs Decomposed heaps domain: shape subgraphs FH FH GD GD Shape graphs track ALL correlations Shape subgraphs track SOME correlations

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9 Main results New abstraction for shape analysis reduces exponential factors by: Connected component decomposition Abstracting away null-value correlations Sound and sufficiently precise transformers Most precise transformers are FNP-complete Polynomial time efficient transformers Sufficiently precise Implementation and empirical results Sufficiently precise on set of benchmarks, including Windows device driver models State space/time reduced by factor of 33/212

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10 Outline Full heap abstraction [VMCAI’05] Reference abstraction Further abstraction by decomposition Connected component decomposition Abstracting away null-value correlations (details in paper) Abstract transformers Concretization by composition Experimental results

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11 Full heap abstraction [VMCAI’05] h1t1... h2t2... h1t1 h2t2 h1t1 h2t2 h1t1 h2t2 >1 1 1 h1t1 h2t2 h1t1 h2t2 >1 1 1 Concrete domain: concrete heaps Full heaps domain: shape graphs Decomposed heaps domain: shape subgraphs FH FH GD GD

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12 Full heap abstraction [VMCAI’05] Abstraction for singly-linked lists Basic concepts: Interruptions (bounded number of) Uninterrupted list segments (bounded number of) Abstraction keeps interruptions and abstracts segment lengths to {1,>1} Result is a shape graph x y Concrete heap x y 1 >1 Shape graph β FH FH by point-wise extension

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13 Graph decomposition abstraction h1t1... h2t2... h1t1 h2t2 h1t1 h2t2 h1t1 h2t2 >1 1 1 h1t1 h2t2 h1t1 h2t2 >1 1 1 Concrete domain: concrete heaps Full heaps domain: shape graphs Decomposed heaps domain: shape subgraphs FH FH GD GD

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14 Graph decomposition abstraction Abstraction of shape graphs Further abstraction over shape graphs Decouples connected components Intuitively different components = different logical data structures Result = set of shape subgraphs

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15 Connected components decomposition 1 h2t2 h1t1 h1 >1 t1 h2t2 GD h1t1 h2 1 t2 h1 >1 t1 h2t2

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16 Abstracting null-value correlations Actual shape graph representation captures null-value correlations (null node not shown in other slides) Abstraction reduces exponential factor due to null-value correlations Details in paper y >1 null x1 x2 xn … Null-value correlations abstraction GD y >1 null … x1 null xn

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17 Concretization GD h1t1... h2t2... h1t1 h2t2 h1t1 h2t2 h1t1 h2t2 >1 1 1 h1t1 h2t2 h1t1 h2t2 >1 1 1 Concrete domain: concrete heaps Full heaps domain: shape graphs Decomposed heaps domain: shape subgraphs FH FH GD GD

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18 1 h2t2 h1t1 h1 >1 t1 h2t2 GD Abstracting correlations GD 1 h2t2 h1t1 h1 >1 t1 h2t2 h1t1 h2 t2 h2 1 t2 h1 >1 t1 h1t1 h2 1 t2 h1 >1 t1 h2t2

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19 Abstract transformers Need transformers for program statements x=new List() x=null x=y x=y.n x.n=y assume(x!=y) assume(x==y) …

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20 Abstract transformers outline Induced transformers by concretization (from subgraphs and shape graphs) Problem: concretization introduces exponential space blow-up Most precise transformers by partial concretization Avoids exponential space blow-up Requires oracle to test strong feasibility Strong feasibility test NP-complete Conservative transformers Give up on strong feasibility test Avoids exponential time blow-up

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21 Most precise transformer [CC’77] h1t1... h2t2... h1t1 h2t2 Concrete domain: concrete heaps Full heaps domain: shape graphs Decomposed heaps domain: shape subgraphs FH FH GD GD st Problem: concretization is exponential space in worst-case

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22 Partial concretization Compose weakly-feasible subgraphs Subgraphs that do not share any variables Compose only subgraphs in footprint of statement Compose at most any 2 or 3 subgraphs h1t1 h2 1 t2 h1 >1 t1 h2 1 t2 h1t1h1t1 h1 >1 t1 h1t1

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23 Transformer example temp h1t1 h1 1 t1 h2t2 t1.n = temp temp h1 1 t1 t1.n = temp temp h1 1 t1 1 t1.n = temp h2t2 t1.n = temp h2t2 temp h1 1 t1 temp h1t1

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24 Most precise transformer xz wx ywy z Can we extend to have variable w? M1M1 M2M2 M3M3 M4M4 M5M5 xzy Most precise requires strong feasibility test Check that subgraphs can be extended to include all variables

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25 Most precise transformer Inconsistency: shared variable x xz wx ywy z M1M1 M2M2 M3M3 M4M4 M5M5 xzy Most precise requires strong feasibility test Check that subgraphs can be extended to include all variables

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26 Most precise transformer Inconsistency: shared variable y Conclusion: can’t extend with w M 1 and M 4 are weakly-feasible but not strongly-feasible in {M 1,…,M 5 } Strong feasibility NP-complete Therefore most precise transformer FNP- complete xzy xz wx ywy z M1M1 M2M2 M3M3 M4M4 M5M5

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27 Making the transformers efficient Vanilla transformer inefficient in practice Incremental transformers Reuse results of previous iterations Details in paper Engineering optimizations Avoid unnecessarily composing subgraphs … Optimized transformers linear time in practice

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28 Prototype implementation Implemented in Java Supports assertions assertReach(x,y) assertDisjointLists(x,y) assertAcyclicList(x) assertCyclicList(x) assert(x==y)assert(x!=y) Check cleanness properties Absence of null derefs Absence of memory leaks No misuse of dangling pointers

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29 Experiments – precision Precision lost in just 2/21 benchmarks getLast Unable to prove x points to last cell Due to imprecise transformer Can be avoided by simple and efficient heuristics queue_2_stack Intentionally constructed Loss of correlations important to prove property Same precision as full heap analysis on other benchmarks

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30 Experiments – “standard” suite Programs operating on 1-2 lists insert, delete, reverse, merge… New analysis slightly less efficient But running times < 0.6 seconds so…

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31 Experiments – multiple lists (89,430 / 7,733) number of shape graphs number of subgraphs x

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32 Experiments – multiple lists full shape graph analysis time graph decomposition analysis time x (552.6 / 2.6)

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33 Properties of the abstraction No loss of precision when connected components represent completely independent lists Reduces state space exponentially Loss of precision when mixing abstract states GD (X 1 X 2 ) GD (X 1 ) GD (X 2 ) So where is this technique useful?

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34 Related work Partial isomorphism join [Manevich et al. SAS’04] Applied in more generic context but does not reduce exponential blow-ups addressed in this paper Heap analysis by separation [Yahav et al. PLDI’04] [Hackett et al. POPL’05] Decompose verification problem itself and conservatively approximate contexts Heap decomposition for interprocedural analysis [Rinetzky et al. POPL’05] [Rinetzky et al. SAS’05] [Gotsman et al. SAS’06] [Gotsman et al. PLDI’07] Decompose/compose at procedure boundaries Predicate/variable clustering [Clark et al. CAV’00] Statically-determined decomposition

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35 Conclusions New abstraction scheme to control precision/cost trade-off for shape analyses Efficient algorithms for abstract domain operations Abstraction Partial concretization Transformers … Applicable beyond singly-linked lists E.g., class of graphs supported by Lev-Ami et al. [CAV’06] Doubly-linked lists Trees …

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36 Ongoing work Extension for concurrent program analysis Future work: Tune abstraction by counterexample-guided refinement

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37 Questions?

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38 Conservative transformer Computes superset of subgraph computed by most precise transformer Algorithm sketch: Compose components in footprint of statement Apply local st on footprint and decompose result Test consistency instead of strong feasibility Pass other components as is Time( st ) polynomial in #vars in st x=null : linear x.n=y: quadratic assume(x==y) : cubic

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39 Concretization GD Maps sets of shape subgraphs to sets of full shape graphs Mathematically: GD (XG) = {G | β(G) XG} Algorithmically: by composing weakly-feasible subgraphs Subgraphs that do not share any variables Full shape graph includes all program variables

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