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By Dirk Beyer, Alessandro Cimatti, Alberto Griggio, Erkan Keremoglu and Roberto Sebastiani Simon Fraser University (Spring 09) Presentation By: Pashootan Vaezipoor

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A successful approach to model checking is through construction and analysis of an abstract reachability tree (ART) + predicate abstraction Simon Fraser University (Spring 09) Presentation By: Pashootan Vaezipoor Unwind

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ART nodes consist of Control-Flow Location Call stack Data State formulas In Single-Block Encoding (SBE) each program op is represented by a single edge in ART Huge number of paths and nodes But in Large-Block Encoding (LBE) entire part of the program is represented by an edge Smaller number of paths are enumerated in ART Exponential reduction in number of states (maybe) Simon Fraser University (Spring 09) Presentation By: Pashootan Vaezipoor

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Simon Fraser University (Spring 09) Presentation By: Pashootan Vaezipoor We use Satisfiability Modulo Theories (SMT) SBELBE (more general representation of abstract states) Conjunction of PredicatesArbitrary Boolean Combination of Predicates More Accurate Abstract Successor Computation SBE + Cartesian Abs (B LAST, SLAM) LBE + Boolean Abstraction (CPA CHECKER ) Large number of successor computationsReduced number of successor computations Efficient computation of Cartesian abstraction by SMT Boolean abstraction is expensive tradeoff

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Simon Fraser University (Spring 09) Presentation By: Pashootan Vaezipoor SBE LBE

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We work on a simple imperative PL Assume Op Assignment Just integers Program is presented by a Control Flow Automaton (CFA) CFA: A(L, G) Program: P = (A, l 0, l E ) A concrete data state of the program is a variable assignment like c that assigns to each variable an integer value A formula φ represents the set S of states c that: S = {c | c |= φ} SP OP (φ): represents the set of data states that are reachable from states in region φ after applying OP Simon Fraser University (Spring 09) Presentation By: Pashootan Vaezipoor

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We define precision (like π) as a finite subset from the universal predicate set of the program Cartesian Predicate Abstraction: A CartPA φ c π of a formula φ is the strongest conjunction of predicates from π entailed by φ This is used as an Abstract State Boolean Predicate Abstraction: A BoolPA φ B π of a formula is the strongest combination of predicates from π entailed by φ Simon Fraser University (Spring 09) Presentation By: Pashootan Vaezipoor

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Simon Fraser University (Spring 09) Presentation By: Pashootan Vaezipoor Cartesian AbstractionBoolean Abstraction SimpleComplex EfficientExpensive ImprecisePrecise tradeoff

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The Precision function assigns to each program location, a precision formula The nodes of ART are like n=(l, φ) The tree is complete when there are no uncovered nodes, or all possible abstract successor states are present in the ART as the children of the node If the final ART does not have any error nodes, then we are done Else the error path is checked for feasibility If feasible: the error is reported If not feasible: refinement! For practical reasons, SBEs use Cartesian abstraction Simon Fraser University (Spring 09) Presentation By: Pashootan Vaezipoor

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Each large control-flow subgraph that is free of loops is replaced with a single control-flow edge with a large formula This is done with applying the following rules: Rule 0 (Error Sink): make all error points, a sink Rule 1 (Sequence): remove intermediate nodes and go directly to successor nodes Rule 2 (Choice): If there are two edges btw two nodes we should replace that with a single edge Simon Fraser University (Spring 09) Presentation By: Pashootan Vaezipoor

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Simon Fraser University (Spring 09) Presentation By: Pashootan Vaezipoor Rule 1 Rule 2

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Simon Fraser University (Spring 09) Presentation By: Pashootan Vaezipoor

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LBE: Possibly exponentially smaller ARTs Less abstract refinement steps Each step is more expensive than SBE More expressive representation of abstract states Simon Fraser University (Spring 09) Presentation By: Pashootan Vaezipoor

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In the paper, B LAST is used for the model checking phase All four configs are tested: ▪ bfs ▪ dfs ▪ predH 0 ▪ predH 7 The config –dfs –predH 7 is the winner for programs without defects For unsafe programs –bfs –predH 7 is winner Simon Fraser University (Spring 09) Presentation By: Pashootan Vaezipoor

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Simon Fraser University (Spring 09) Presentation By: Pashootan Vaezipoor

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In the experiments, all four combinations of LBE vs. SBE and Cartesian vs. Boolean abstraction are tested Results: SBE doesn’t benefit from Boolean Abstraction Combination of LBE with Cartesian Abstraction failed to solve any experiments due to the loss of precision SBE + CartAbs is OK LBE + BoolAbs is OK Simon Fraser University (Spring 09) Presentation By: Pashootan Vaezipoor

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