Software Model Checking via Large-Block Encoding Dirk Beyer ∗ Alessandro Cimatti † Alberto Griggio ∗‡ Simon Fraser University FBK-irst, Trento University of Trento & Simon Fraser University Simon Fraser University University of Trento A fundamental source of inefficiency of this approach is the fact that the control-flow of the program can induce a huge number of paths (and nodes) in the ART, which are explored independently of each other. We propose a novel, broader view on ART-based software model checking, where a much more compact abstract space is used, resulting thus in a much smaller number of paths to be enumerated by the ART. Instead of using edges that represent single program operations, we encode entire parts of the program in one edge. In contrast to SBE, we call our new approach large-block encoding (LBE). In general, the new encoding may result in an exponential reduction of the number of ART nodes. The generalization from SBE to LBE has two main consequences. First, LBE requires a more general representation of abstract states than SBE. SBE is typically based on mere conjunctions of predicates. Because the LBE approach summarizes large portions of the control flow, conjunctions are not sufficient, and we need to use arbitrary Boolean combinations of predicates to represent the abstract states. Second, LBE requires a more accurate abstraction in the abstract-successor computations. Intuitively, an abstract edge represents many different paths of the program, and therefore it is necessary that the abstract-successor computations take the relationships between the predicates into account. In order to make this generalization practical, we rely on efficient solvers for satisfiability modulo theories (SMT). In particular, enabling factors are the capability of performing Boolean reasoning efficiently (e.g., [21]), the availability of effective algorithms for abstraction computation (e.g., [11], [18]), and interpolation procedures to extract new predicates [9], [12]. Considering Boolean abstraction and large-block encoding in addition to the traditional techniques, we obtain the following interesting observations: (i) whilst the SBE approach requires a large number of successor computations, the LBE approach reduces the number of successor computations dramatically (possibly exponentially); (ii) whilst Cartesian abstraction can be efficiently computed with a linear number of SMT solver queries, Boolean abstraction is expensive to compute because it requires an enumeration of all satisfiable assignments for the predicates. Therefore, two combinations of the above strategies provide an interesting tradeoff: The combination of SBE with Cartesian abstraction was successfully implemented by tools like B LAST and SLAM. We investigate the combination of LBE with Boolean abstraction, by first Abstract—Several successful approaches to software verification are based on the construction and analysis of an abstract reachability tree (ART). The ART represents unwindings of the control-flow graph of the program. Traditionally, a transition of the ART represents a single block of the program, and therefore, we call this approach single-block encoding (SBE). SBE may result in a huge number of program paths to be explored, which constitutes a fundamental source of inefficiency. We propose a generalization of the approach, in which transitions of the ART represent larger portions of the program; we call this approach large-block encoding (LBE). LBE may reduce the number of paths to be explored up to exponentially. Within this framework, we also investigate symbolic representations: for representing abstract states, in addition to conjunctions as used in SBE, we investigate the use of arbitrary Boolean formulas; for computing abstract-successor states, in addition to Cartesian predicate abstraction as used in SBE, we investigate the use of Boolean predicate abstraction. The new encoding leverages the efficiency of state-of-the-art SMT solvers, which can symbolically compute abstract large-block successors. Our experiments on benchmark C programs show that the large-block encoding outperforms the single-block encoding. I. I NTRODUCTION Software model checking is an effective technique for software verification. Several advances in the field have lead to tools that are able to verify programs of considerable size, and show significant advantages over traditional techniques in terms of precision of the analysis (e.g., SLAM [3] and B LAST [6]). However, efficiency and scalability remain major concerns in software model checking and hamper the adaptation of the techniques in industrial practice. Several successful tools for software model checking are based on the construction and analysis of an abstract reachability tree (ART), and predicate abstraction is one of the favorite abstract domains. The ART represents unwindings of the control-flow graph of the program. The search is usually guided by the control flow of the program. Nodes of the ART typically consist of the control-flow location, the call stack, and formulas that represent the data states. During the refinement process, the ART nodes are incrementally refined. In the traditional ART approach, each program operation (assignment operation, assume operation, function call, function return) is represented by a single edge in the ART. Therefore, we call this approach single-block encoding (SBE). ∗ Supported in part by the Canadian NSERC grant RGPIN 341819-07 and by the SFU grant PRG 06-3. † Supported in part by the European Commission under project FP7-2007-IST-1-217069 COCONUT. ‡ Supported in part by SRC/GRC under Custom Research Project 2009-TJ-1880 WOLFLING and by MIUR under PRIN project 20079E5KM8 002. 978-1-4244-4966-8/09/$25.00 ©2009 IEEE M. Erkan Keremoglu ∗ Roberto Sebastiani ‡ 25 Authorized licensed use limited to: Universitat Passau. Downloaded on May 31,2010 at 14:00:24 UTC from IEEE Xplore. Restrictions apply. with the generation of verification conditions as performed by static program verifiers like S PEC # [4] or C ALYSTO [1]. formally defining LBE in terms of a summarization of the control-flow automaton for the program, and then implementing this LBE approach together with a Boolean predicate abstraction. We evaluate the performance and precision by comparing it with the model checker B LAST and with an own implementation of the traditional approach. Our own implementation of the SBE and LBE approach is integrated as a new component into CPACHECKER [8]1 . The experiments show that, despite the simplicity of the idea underlying LBE, our new approach outperforms the previous approach. Structure. Section II provides the necessary background. Section III explains our contribution in detail. We experimentally evaluate our novel approach in Sect. IV. In Sect. V, we draw some conclusions and outline directions for future research. II. BACKGROUND A. Programs and Control-Flow Automata We restrict the presentation to a simple imperative programming language, where all operations are either assignments or assume operations, and all variables range over integers.2 We represent a program by a control-flow automaton (CFA). A CFA A = (L, G) consists of a set L of program locations, which model the program counter l , and a set G ⊆ L×Ops×L of control-flow edges, which model the operations that are executed when control flows from one program location to another. The set of variables that occur in operations from Ops is denoted by X. A program P = (A, l0 , lE ) consists of a CFA A = (L, G) (which models the control flow of the program), an initial program location l0 ∈ L (which models the program entry) such that G does not contain any edge (·, ·, l0 ), and a target program location lE ∈ L (which models the error location). A concrete data state of a program is a variable assignment c : X → Z that assigns to each variable an integer value. The set of all concrete data states of a program is denoted by C. A set r ⊆ C of concrete data states is called region. We represent regions using first-order formulas (with free variables from X): a formula ϕ represents the set S of all data states c that imply ϕ (i.e., S = {c | c |= ϕ}). A concrete state of a program is a pair (l, c), where l ∈ L is a program location and c is a concrete data state. A pair (l, ϕ) represents the following set of all concrete states: {(l, c) | c |= ϕ}. The concrete semantics of an operation op ∈ Ops is defined by the strongest postcondition operator SPop : for a formula ϕ, SPop (ϕ) represents the set of data states that are reachable from any of the states in the region represented by ϕ after the execution of op. Given a formula ϕ that represents a set of concrete data states, for an assignment operation s := e, we have s : ϕ[s→ ∧ (s = e[s→ ); and for an assume SPs:=e (ϕ) = ∃ s] s] operation assume(p), we have SPassume(p) (ϕ) = ϕ ∧ p. A path σ is a sequence (op 1 , l1 ), ..., (op n , ln ) of pairs of operations and locations. The path σ is called program path if for every i with 1 ≤ i ≤ n there exists a CFA edge g = (li−1 , op i , li ), i.e., σ represents a syntactical walk through the CFA. The concrete semantics for a program path σ = (op 1 , l1 ), ..., (op n , ln ) is defined as the successive application of the strongest postoperator for each operation: SPσ (ϕ) = SPop n (...SPop 1 (ϕ)...). The set of concrete states that result from running σ is represented by the pair (ln , SPσ (true)). A program path σ is feasible if Example. We illustrate the advantage of LBE over SBE on the example program in Fig. 1 (a). In SBE, each program location is modeled explicitly, and an abstract-successor computation is performed for each program operation. Figure 1 (b) shows the structure of the resulting ART. In the figure, abstract states are drawn as ellipses, and labeled with the location of the abstract state; the arrows indicate that there exists an edge from the source location to the target location in the control-flow. The ART represents all feasible program paths. For example, the leftmost program path is taking the ‘then’ branch of every ‘if’ statement. For every edge in the ART, an abstract-successor computation is performed, which potentially includes several SMT solver queries. The problems given to the SMT solver are usually very small, and the runtime sums up over a large amount of simple queries. Therefore, model checkers that are based on SBE (like B LAST) experience serious performance problems on programs with such an exploding structure (cf. the test_locks examples in Table I). In LBE, the control-flow graph is summarized, such that control-flow edges represent entire subgraphs of the original control-flow. In our example, most of the program is summarized into one control-flow edge. Figure 1 (c) shows the structure of the resulting ART, in which all feasible paths of the program are represented by one single edge. The exponential growth of the ART does not occur.  Related Work. The model checkers SLAM and B LAST are typical examples for the SBE approach [3], [6], both based on counterexample-guided abstraction refinement (CEGAR) [13]. The tool SATABS is also based on CEGAR, but it performs a fully symbolic search in the abstract space [15]. In contrast, our approach still follows the lazy-abstraction paradigm [17], i.e., it abstracts and refines chunks of the program “on-thefly”. The work of McMillan is also based on lazy abstraction, but instead of using predicate abstraction as abstract domain, he directly uses Craig interpolants from infeasible error paths, thus avoiding abstract-successor computations [19]. A different approach to software model checking is bounded model checking (BMC), with the most prominent example CBMC [14]. Programs are unrolled up to a given depth, and a formula is constructed which is satisfiable iff one of the considered program executions reaches a certain error location. The BMC approaches are targeted towards discovering bugs, and can not be used to prove program safety. Finally, the summarizations performed in our large-block encoding bear some similarities 1 Available at http://www.sosy-lab.org/∼dbeyer/CPAchecker 2 Our implementation is based on CPACHECKER , which operates on C programs that are given in the C IL intermediate language [20]; function calls are supported. 26 Authorized licensed use limited to: Universitat Passau. Downloaded on May 31,2010 at 14:00:24 UTC from IEEE Xplore. Restrictions apply. L1 ERROR L2 L1: L2: L3: L4: L5: L6: L7: L8: L9: L10: L11: L12: L13: ERR: if(p1) { x1 = 1; } if(p2) { x2 = 2; } if(p3) { x3 = 3; } if(p1) { if (x1 != 1) goto ERR; } if (p2) { if (x2 != 2) goto ERR; } if (p3) { if (x3 != 3) goto ERR; } return EXIT_SUCCESS; return EXIT_FAILURE; (a) Example C program Fig. 1. L4 L5 L6 L8 ERROR L7 L8 L9 L8 L9 ERROR L9 ERROR L9 L11 L10 L11 ERROR L12 L4 L5 L5 L6 L7 L7 L8 L9 L9 L8 L9 L10 L11 L10 L11 L10 L11 L12 L7 L7 L8 L9 ERROR L11 L12 L13 L10 L11 L12 L13 L13 L9 L10 L11 L13 L9 L9 ERROR L10 L11 L3 L5 L6 L7 L3 ERROR L6 L7 L7 L8 L8 L10 L11 L12 L13 L9 L11 L11 ERROR L12 ERROR L13 L12 L13 L13 L13 L13 ERROR L1 L12 L13 L13 L13 ERROR (b) ART for SBE (c) ART for LBE Example program and corresponding ARTs for SBE and LBE; this example was considered as verification challenge for ART-based approaches SPσ (true) is satisfiable. A concrete state (ln , cn ) is called reachable if there exists a feasible program path σ whose final location is ln and such that cn |= SPσ (true). A location l is reachable if there exists a concrete state c such that (l, c) is reachable. A program is safe if lE is not reachable. whose corresponding propositional variable occurs positive in the assignment. The disjunction of all such conjunctions is the Boolean predicate abstraction for ϕ. The abstract strongest postoperator SPπ for a predicate abstraction with precision π π transforms the abstract state ϕπB into its successor ϕ′ B for π a program operation op, written as ϕ′ B = SPπop (ϕπB ), if π ϕ′ B is the Boolean predicate abstraction of SPop (ϕπB ), i.e., ϕ′ πB = (SPop (ϕπB ))πB . For more details, we refer the reader to the work of Lahiri et al. [18]. B. Predicate Abstraction Let P be a set of predicates over program variables in a quantifier-free theory T . A formula ϕ is a Boolean combination of predicates from P. A precision for a formula is a finite subset π ⊂ P of predicates. C. ART-based Software Model Checking with SBE Cartesian Predicate Abstraction. Let π be a precision. The Cartesian predicate abstraction ϕπC of a formula ϕ is the strongest conjunction of predicates from π that is entailed by  {p ∈ π | ϕ ⇒ p}. Such a predicate abstraction of ϕ: ϕπC := a formula ϕ, which represents a region of concrete program states, is used as an abstract state (i.e., an abstract representation of the region) in program verification. For a formula ϕ and a precision π, the Cartesian predicate abstraction ϕπC of ϕ can be computed by |π| SMT-solver queries. The abstract strongest postoperator SPπ for a predicate abstraction with precision π π transforms the abstract state ϕπC into its successor ϕ′ C for π ′π π a program operation op, written as ϕ C = SPop (ϕC ), if π ϕ′ C is the Cartesian predicate abstraction of SPop (ϕπC ), i.e., ′π ϕ C = (SPop (ϕπC ))πC . For more details, we refer the reader to the work of Ball et al. [2]. Boolean Predicate Abstraction. Let π be a precision. The Boolean predicate abstraction ϕπB of a formula ϕ is the strongest Boolean combination of predicates from π that is entailed by ϕ. For a formula ϕ and a precision π, the Boolean predicate abstraction ϕπB of ϕ can be computed by querying an SMT solver in the following way: For each predicate pi ∈ π, we introduce a propositional variable vi . Now we ask an SMT solver to enumerateall satisfying assignments of v1 , ..., v|π| in the formula ϕ ∧ pi ∈π (pi ⇔ vi ). For each satisfying assignment, we construct a conjunction of all predicates from π The precision for a program is a function Π : L → 2P , which assigns to each program location a precision for a formula. An ART-based algorithm for software model checking takes an initial precision Π (which is typically very coarse) for the predicate abstraction, and constructs an ART for the input program and Π. An ART is a tree whose nodes are labeled with program locations and abstract states [6] (i.e., n = (l, ϕ)). For a given ART node, all children nodes are labeled with successor locations and abstract successor states, according to the strongest postoperator and the predicate abstraction. A node n = (l, ϕ) is called covered if there exists another ART node n′ = (l, ϕ′ ) that entails n (i.e., s.t. ϕ′ |= ϕ). An ART is called complete if every node is either covered or all possible abstract successor states are present in the ART as children of the node. If a complete ART is constructed and the ART does not contain any error node, then the program is considered correct [6]. If the algorithm adds an error node to the ART, then the corresponding path σ is checked to determine if σ is feasible (i.e., if the corresponding concrete program path is executable) or infeasible (i.e., if there is no corresponding program execution). In the former case the path represents a witness for a program bug. In the latter case the path is analyzed, and a refinement Π′ of Π is generated, such that the same path cannot occur again during the ART exploration. The concept of using an infeasible error path for abstraction refine- 27 Authorized licensed use limited to: Universitat Passau. Downloaded on May 31,2010 at 14:00:24 UTC from IEEE Xplore. Restrictions apply. 1: while 1: while ment is called counterexample-guided abstraction refinement (CEGAR) [13]. The concept of iteratively constructing an ART and refining only the precisions along the considered path is called lazy abstraction [17]. Craig interpolation is a successful approach to predicate extraction during refinement [16]. After refining the precision, the algorithm continues with the next iteration, using Π′ instead of Π to construct the ART, until either a complete error-free ART is obtained, or an error is found (note that the procedure might not terminate). For more details and a more in-depth illustration of the overall ART algorithm, we refer the reader to the B LAST article [6]. In order to make the algorithm scale on practical examples, implementations such as B LAST or SLAM use the simple but coarse Cartesian abstraction, instead of the expensive but precise Boolean abstraction. Despite its potential imprecision, Cartesian abstraction has been proved successful for the verification of many real-world programs. In the SBE approach, given the large number of successor computations, the computation of the Boolean predicate abstraction is in fact too expensive, as it may require an SMT solver to enumerate an exponential number of assignments on the predicates in the precision, for each single successor computation. The reason for the success of Cartesian abstraction if used together with SBE, is that for a given program path, state overapproximations that are expressible as conjunctions of atomic predicates —for which Boolean and Cartesian abstractions are equivalent— are often good enough to prove that the error location is not reachable in the abstract space. L1: while (i>0) { [i > 0] [i ≤ 0] L2: if (x==1) { 2: if L3: z = 0; 7 [x == 1] [x = 1] } else { L4: z = 1; 3: then 4: else } z=0 z=1 L5: i = i-1; 5 i=i−1 L6: } [i > 0] [i ≤ 0] 2: if [x == 1] [x = 1] 3: then 7 4: else z=0 z=1 5 i=i−1 6 1: while [i > 0] [i ≤ 0] 2: if [x = 1] z=1 [i ≤ 0] [i ≤ 0] 2: if 7 7 [x == 1] z=0 1: while 1: while [i > 0] [x == 1] [x = 1] z=0 z=1 [i > 0] 7 [x == 1] [x = 1] z=0 z=1 i=i−1 5 i=i−1 5 i=i−1 Fig. 2. CFA Transformation: a) Program, b) CFA, c)–e) Intermediate CFAs, f) CFA-Summary. In the CFAs, assume(p) is represented as [p], op 1 ; op 2 is represented by drawing op 2 below op 1 , and op 1  op 2 by drawing op 2 beside op 1 Rule 2 (Choice). If L2 = {l1 , l2 } and A|L2 = (L2 , G2 ) is the subgraph of A with nodes l1 l1 from L2 and the set G2 of edges contains the two edges op op 2 op 1  op 2 1 (l1 , op 1 , l2 ) and (l1 , op 2 , l2 ), l2 l2 then we change the CFA A in the following way: (1) we remove the two edges (l1 , op 1 , l2 ) and (l1 , op 2 , l2 ) from G and add the edge (l1 , op 1  op 2 , l2 ) to G, where SPop 1 op 2 (ϕ) = SPop 1 (ϕ) ∨ SPop 2 (ϕ). (Note that there might be a backwards edge (l2 , ·, l1 ).) Let P = (A, l0 , lE ) be a program and let A′ be a CFA. The CFA A′ is a CFA-summary of A if A′ is obtained from A via application of Rule 1 and then stepwise application of Rules 1 and 2, and no rule can be further applied. III. L ARGE -B LOCK E NCODING A. Summarization of Control-Flow Automata The large-block encoding is achieved by a summarization of the program CFA, in which each loop-free subgraph of the CFA is replaced by a single control-flow edge with a large formula that represents the removed subgraph. This process, which we call CFA-summarization, consists of the fixpoint application of the three rewriting rules that we describe below: first we apply Rule 0 once, and then we repeatedly apply Rules 1 and 2, until no rule is applicable anymore. Let P = (A, l0 , lE ) be a program with CFA A = (L, G). Example. Figure 2 shows a program (a) and its corresponding CFA (b). The control-flow automaton (CFA) is iteratively transformed to a CFA-summary (f) as follows: Rule 1 eliminates location 6 to (c), Rule 1 eliminates location 3 and then location 4 to (d), Rule 2 replaces the two edges 2–5 to (e), Rule 1 eliminates location 2 and then location 5 to (f).  Rule 0 (Error Sink). We remove all edges (lE , ·, ·) from G, s.t., the target location lE is a sink node with no outgoing edges. Rule 1 (Sequence). If G contains an edge (l1 , op 1 , l2 ) with l1 = l2 and no other incoming edges for l2 (i.e. edges (·, ·, l2 )), and G→ l2 is l1 l1 the subset of G of outgoing edges op 1 for l2 , then we change the CFA A op 1 ; op 2 in the following way: (1) we rel2 move location l2 from L, and (2) op 1 ; op 3 op 3 we remove the edge (l1 , op 1 , l2 ) op 2 → and all edges in Gl2 from G, and l3 l4 l3 l4 for each edge (l2 , op i , li ) ∈ G→ l2 , we add the edge (l1 , op 1 ; op i , li ) to G, where SPop 1 ; op i (ϕ) = SPop i (SPop 1 (ϕ)). (Note that G→ l2 might contain an edge (l2 , ·, l1 ).) In the context of this article, we use the CFA-summary for program analysis, i.e., we want to verify if the error location of the program is reachable. The following theorem states that our summarization of a CFA is correct in this sense. (The proof can be found in our extended technical report [5].) Theorem 3.1 (Correctness of Summarization): Let P = (A, l0 , lE ) be a program and let A′ = (L′ , G′ ) be a CFAsummary of A. Then: (i) {l0 , lE } ⊆ L′ , and (ii) lE is reachable in (A′ , l0 , lE ) if and only if lE is reachable in P . The summarization can be performed in polynomial time. The time taken by Rule 0 is proportional to the number of outgoing edges for lE . Since each application of Rule 1 or Rule 2 removes at least one edge, there can be at most |G| − 1 such applications. A naive way to determine the set of locations and edges to which to apply each rule requires 28 Authorized licensed use limited to: Universitat Passau. Downloaded on May 31,2010 at 14:00:24 UTC from IEEE Xplore. Restrictions apply. O(|V | · k) time, where k is the maximum out-degree of locations. Finally, each application of Rule 2 requires O(1) time, and each application of Rule 1 O(k) time. Therefore, a naive summarization algorithm requires O(|G| · |V | · k) time, which reduces to O(|G|·|V |) if k is bounded (i.e., if we rewrite a priori all switches into nested ifs).3 involving formulas with a complex Boolean structure. With LBE, the cost of each symbolic operation, rather than their number, becomes a critical performance factor. To this extent, LBE makes it possible to fully exploit the power and functionality of modern SMT solvers: First, the capability of modern SMT solvers to perform large amounts of Boolean reasoning allows for handling large Boolean combinations of atomic expressions, instead of simple conjunctions. Second, the capability of some SMT solvers (e.g., [10]) to perform All-SMT and interpolation allows for efficient computation of Boolean abstractions and interpolants, respectively. SMT-based Boolean abstraction and interpolation were shown to outperform previous approaches [11], [12], [18], especially when dealing with complex formulas. With SBE, instead, the use of modern SMT technology does not lead to significant improvements of the overall ART-based algorithm, because each SMT query involves only simple conjunctions. 4 B. LBE versus SBE for Software Model Checking The use of LBE instead of the standard SBE requires no modification to the general model-checking algorithm, which is still based on ART construction with CEGAR-based refinement. The main difference is that in LBE, there is no one-to-one correspondence between ART paths and syntactical program paths. A single CFA edge corresponds to a set of paths between its source and target location, and a single ART path corresponds to a set of program paths. An ART node represents an overapproximation of the data region that is reachable by following any of the program paths represented by the ART path that leads to it. This difference leads to two observations. First, LBE can lead to exponentially-smaller ARTs than SBE, and thus it can drastically reduce the number of successor computations (cf. example in Sect. I) and the number of abstraction-refinement steps for infeasible error paths. Each of these operations, however, is typically more expensive than with SBE, because more complex formulas are involved. Second, LBE requires a more general representation of abstract states. When using SBE, abstract states are typically represented as sets/conjunctions of predicates. This is sufficient for practical examples because each abstract state represents a data region reachable by a single program path, which can be encoded essentially as a conjunction of atomic formulas. With LBE, such representation would be too coarse, since each abstract state represents a data region that is reachable on several different program paths. Therefore, we need to use a representation for arbitrary (and larger) Boolean combinations of predicates. This generalization of the representation of abstract states requires a generalization of the representation of the transfers, i.e., replacing the Cartesian abstraction with a more precise form of abstraction. In this paper, we evaluate the use of the Boolean abstraction, which allows for a precise representation of arbitrary Boolean combinations of predicates. With respect to the traditional SBE approach, LBE allows us to trade part of the cost of the explicit enumeration of program paths with that of the symbolic computation of abstract successor states: rather than having to build large ARTs via SBE by performing a substantial amount of relatively cheap operations (Cartesian abstract postoperator applications along single-block edges and counterexample analysis of individual program paths), we build smaller ARTs via LBE by performing more expensive symbolic operations (Boolean abstract postoperator applications along large portions of the control flow and counterexample analysis of multiple program paths), IV. P ERFORMANCE E VALUATION Implementation. In order to evaluate the proposed verification method, we integrate our algorithm as a new component into the configurable software verification toolkit CPACHECKER [8]. This implementation is written in JAVA. All example programs are preprocessed and transformed into the simple intermediate language C IL [20]. For parsing C programs, CPACHECKER uses a library from the Eclipse C/C++ Development Kit. For efficient querying of formulas in the quantifier-free theory of rational linear arithmetic and equality with uninterpreted function symbols, we use the SMT solver M ATH SAT [10], which is integrated as a library (written in C++). We use BDDs for representing abstract-state formulas. Our benchmark programs, the source code, and an executable of our LBE implementation are available on the supplementary web site on Large-Block Encoding (http://www.sosy-lab.org/∼dbeyer/cpa-lbe). We ran all experiments on a 1.8 GHz Intel Core2 machine with 2 GB of RAM and 2 MB of cache, running GNU/Linux. We used a timeout of 1 800 s and a memory limit of 1.8 GB. Example Programs. We use three categories of benchmark programs. First, we experiment with programs that are specifically designed to cause an exponential blowup of the ART when using SBE (test_locks*, in the style of the example in Sect. I). Second, we use the device-driver programs that were previously used as benchmarks in the B LAST project. 5 Third, we solve various verification problems for the SSH client 4 For example, B LAST uses S IMPLIFY , version 1.5.4, as of October 2001, for computing abstract successor states. Experiments have shown that replacing this old S IMPLIFY version by a highly-tuned modern SMT solver does not significantly improve the performance, because B LAST does not use much power of the SMT solver. Moreover, it was shown that although the M ATH SAT SMT solver outperformed other tools in the computation of Craig interpolants for general formulas, the difference in performance is negligible on formulas generated by a standard SBE ART-based algorithm [12]. 5 The B LAST distribution contains 8 Windows driver benchmarks. However, we could not run three of them (parclass.i, mouclass.i, and serial.i), because C IL fails to parse them, making both CPACHECKER and B LAST fail. 3 In our implementation, we use a more efficient algorithm, which we do not describe here for lack of space. 29 Authorized licensed use limited to: Universitat Passau. Downloaded on May 31,2010 at 14:00:24 UTC from IEEE Xplore. Restrictions apply. TABLE II P ERFORMANCE RESULTS , PROGRAMS WITH ARTIFICIAL BUGS TABLE I P ERFORMANCE RESULTS B LAST Program (best result) test locks 5.c 4.50 7.81 test locks 6.c 13.91 test locks 7.c 25.00 test locks 8.c 46.84 test locks 9.c 94.57 test locks 10.c 204.55 test locks 11.c 529.16 test locks 12.c 1229.27 test locks 13.c >1800.00 test locks 14.c >1800.00 test locks 15.c cdaudio.i.cil.c 175.76 diskperf.i.cil.c >1800.00 floppy.i.cil.c 218.26 kbfiltr.i.cil.c 23.55 parport.i.cil.c 738.82 s3 clnt.blast.01.i.cil.c 33.01 62.65 s3 clnt.blast.02.i.cil.c 60.62 s3 clnt.blast.03.i.cil.c 63.96 s3 clnt.blast.04.i.cil.c 811.27 s3 srvr.blast.01.i.cil.c 360.47 s3 srvr.blast.02.i.cil.c 276.19 s3 srvr.blast.03.i.cil.c 175.64 s3 srvr.blast.04.i.cil.c 304.63 s3 srvr.blast.06.i.cil.c 478.05 s3 srvr.blast.07.i.cil.c 115.76 s3 srvr.blast.08.i.cil.c s3 srvr.blast.09.i.cil.c 445.21 115.10 s3 srvr.blast.10.i.cil.c 367.98 s3 srvr.blast.11.i.cil.c 304.05 s3 srvr.blast.12.i.cil.c 580.33 s3 srvr.blast.13.i.cil.c s3 srvr.blast.14.i.cil.c 303.21 115.88 s3 srvr.blast.15.i.cil.c 305.11 s3 srvr.blast.16.i.cil.c TOTAL (solved/time) 32 / 8591.12 TOTAL w/o test_locks* 23 / 6435.51 CPACHECKER B LAST Program (best result) cdaudio.BUG.i.cil.c 18.79 diskperf.BUG.i.cil.c 889.79 floppy.BUG.i.cil.c 119.60 kbfiltr.BUG.i.cil.c 46.80 parport.BUG.i.cil.c 1.67 s3 clnt.blast.01.BUG.i.cil.c 8.84 9.02 s3 clnt.blast.02.BUG.i.cil.c s3 clnt.blast.03.BUG.i.cil.c 6.64 9.78 s3 clnt.blast.04.BUG.i.cil.c s3 srvr.blast.01.BUG.i.cil.c 7.59 7.16 s3 srvr.blast.02.BUG.i.cil.c 7.42 s3 srvr.blast.03.BUG.i.cil.c 7.33 s3 srvr.blast.04.BUG.i.cil.c 39.81 s3 srvr.blast.06.BUG.i.cil.c 310.84 s3 srvr.blast.07.BUG.i.cil.c 40.51 s3 srvr.blast.08.BUG.i.cil.c 265.48 s3 srvr.blast.09.BUG.i.cil.c 40.24 s3 srvr.blast.10.BUG.i.cil.c 49.05 s3 srvr.blast.11.BUG.i.cil.c 38.66 s3 srvr.blast.12.BUG.i.cil.c 251.56 s3 srvr.blast.13.BUG.i.cil.c 39.94 s3 srvr.blast.14.BUG.i.cil.c s3 srvr.blast.15.BUG.i.cil.c 40.19 39.54 s3 srvr.blast.16.BUG.i.cil.c TOTAL (solved/time) 24 / 2296.25 SBE LBE 4.01 0.29 7.22 0.32 12.63 0.34 23.93 0.57 52.04 0.38 131.39 0.40 MO 0.70 MO 0.46 MO 0.49 MO 0.50 MO 0.56 MO 53.55 MO 232.00 MO 56.36 41.12 7.82 MO 378.04 755.81 19.51 1075.45 16.00 746.31 49.50 730.80 25.45 >1800.00 125.33 >1800.00 122.83 >1800.00 98.47 >1800.00 71.77 >1800.00 59.70 >1800.00 85.82 >1800.00 61.29 >1800.00 126.47 >1800.00 63.36 >1800.00 162.76 >1800.00 170.33 >1800.00 74.49 >1800.00 50.38 >1800.00 21.01 >1800.00 127.82 11 / 3580.71 35 / 2265.07 5 / 3349.48 24 / 2260.07 CPACHECKER SBE 74.39 26.53 36.49 75.45 14.62 1514.90 843.42 780.72 724.04 MO >1800.00 >1800.00 >1800.00 MO >1800.00 >1800.00 >1800.00 >1800.00 >1800.00 >1800.00 >1800.00 1656.54 >1800.00 >1800.00 10 / 5747.10 LBE 9.85 6.78 4.30 11.52 2.64 3.33 3.27 2.61 3.18 2.09 2.10 2.08 1.93 5.08 28.35 36.47 4.94 12.01 4.80 6.11 15.20 4.63 10.19 5.21 24 / 188.67 The configuration using -dfs -predH 7 is the winner (in terms of solved problems and total runtime) for the programs without defects, but is not able to verify four example programs (timeout) [5]. For the unsafe programs, -bfs -predH 7 performs best. All four configurations use the command-line options -craig 2 -nosimplemem -alias "", which specify that B LAST runs with lazy, Craig-interpolation-based refinement, no C IL preprocessing for memory access, and without pointer analysis. In all experiments with B LAST, we use the same interpolation procedure (M ATH SAT) as in our CPACHECKERbased implementation. 6 In the performance tables, we show the best result among the four configurations for every single instance (column best result). (The results of all four configurations are provided in our extended technical report [5].) Second, in order to separate the optimization efforts in B LAST from the conceptual essence of the traditional lazyabstraction algorithm, we developed a re-implementation of the traditional algorithms (column ’SBE’), as described in the B LAST tool article [6]. This re-implementation is integrated as component into CPACHECKER, so that the difference between SBE and LBE is only in the algorithms, not in the environment (same parser, same BDD package, same query optimization, etc.). Our SBE implementation uses a DFS algorithm. Third, we ran the experiments using our new LBE algorithm, which is also implemented within CPACHECKER (column LBE). Our LBE implementation uses a DFS algorithm. Note that the purpose of our experiments is to give evidence of the performance difference between SBE and LBE, because these two settings are closest to each other, since SBE and LBE and server software (s3_clnt* and s3_srvr*), which share the same program logic, but check different safety properties. The safety property is encoded as conditional call of a failure location and therefore reduces to the reachability of a certain error location. All benchmark programs from the B LAST web page are preprocessed with C IL. For the second and third groups of programs, we also performed experiments with artificial defects introduced. Experimental Configurations. For a careful and fair performance comparison, we performed experiments using three different configurations. First, we use B LAST, version 2.5, which is a highly optimized state-of-the-art software model checker. B LAST is implemented in the programming language OC AML. We ran B LAST using all four combinations of breadth-first search (-bfs) versus depth-first search (-dfs), both with and without heuristics for improving the predicate discovery. B LAST provides five different levels of heuristics for predicate discovery, and we use only the lowest (-predH 0) and the highest option (-predH 7). Interestingly, every combination is best for some particular example programs, with considerable differences in runtime and memory consumption. 6 We tried also to use M ATH SAT instead of S IMPLIFY for computing abstract successor states, but this did not improve the performance of B LAST. 30 Authorized licensed use limited to: Universitat Passau. Downloaded on May 31,2010 at 14:00:24 UTC from IEEE Xplore. Restrictions apply. TABLE III D ETAILED COMPARISON BETWEEN SBE AND LBE; ENTRIES MARKED WITH (*) DENOTE PARTIAL STATISTICS FOR ANALYSES THAT TERMINATED UNSUCCESSFULLY ( IF AVAILABLE ) Program test locks 5.c test locks 6.c test locks 7.c test locks 8.c test locks 9.c test locks 10.c test locks 11.c test locks 12.c test locks 13.c test locks 14.c test locks 15.c cdaudio.i.cil.c diskperf.i.cil.c floppy.i.cil.c kbfiltr.i.cil.c parport.i.cil.c s3 clnt.blast.01.i.cil.c s3 clnt.blast.02.i.cil.c s3 clnt.blast.03.i.cil.c s3 clnt.blast.04.i.cil.c s3 srvr.blast.01.i.cil.c s3 srvr.blast.02.i.cil.c s3 srvr.blast.03.i.cil.c s3 srvr.blast.04.i.cil.c s3 srvr.blast.06.i.cil.c s3 srvr.blast.07.i.cil.c s3 srvr.blast.08.i.cil.c s3 srvr.blast.09.i.cil.c s3 srvr.blast.10.i.cil.c s3 srvr.blast.11.i.cil.c s3 srvr.blast.12.i.cil.c s3 srvr.blast.13.i.cil.c s3 srvr.blast.14.i.cil.c s3 srvr.blast.15.i.cil.c s3 srvr.blast.16.i.cil.c ART size 1344 2301 3845 6426 10926 19091 24779(*) 28119(*) 31739(*) 35178(*) 38777(*) 53323(*) – 31079(*) 19640 26188(*) 122678 354132 196599 172444 232195(*) 254667(*) – – 295698(*) – 279991(*) 189541(*) 307671(*) – 258546(*) 167333(*) 318982(*) 279319(*) 346185(*) SBE # ref # steps Tot 50 10 72 12 98 14 128 16 162 18 200 20 242(*) 22(*) 288(*) 24(*) 338(*) 26(*) 392(*) 28(*) 450(*) 30(*) 445(*) 147(*) – – 301(*) 79(*) 153 53 360(*) 143(*) 557 59 532 55 534 55 538 55 774(*) 70(*) 745(*) 79(*) – – – – 576(*) 63(*) – – 549(*) 57(*) 720(*) 72(*) 597(*) 55(*) – – 563(*) 57(*) 682(*) 70(*) 643(*) 65(*) 579(*) 58(*) 596(*) 59(*) predicates Avg Max 3 10 4 12 5 14 6 16 7 18 8 20 9(*) 22(*) 10(*) 24(*) 10(*) 26(*) 11(*) 28(*) 12(*) 30(*) 9(*) 78(*) – – 7(*) 35(*) 5 27 4(*) 41(*) 20 59 19 55 19 55 19 55 20(*) 70(*) 19(*) 78(*) – – – – 14(*) 63(*) – – 15(*) 57(*) 16(*) 71(*) 16(*) 55(*) – – 15(*) 57(*) 18(*) 69(*) 13(*) 64(*) 15(*) 58(*) 12(*) 58(*) ART size 4 4 4 4 4 4 4 4 4 4 4 6909 4890 9668 1577 38488 36 36 39 36 101 109 91 103 94 92 89 193 91 48 99 90 92 71 98 LBE # ref # predicates steps Tot Avg Max 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 140 79 5 16 145 56 6 21 176 58 4 13 47 18 2 6 474 168 4 17 5 47 11 47 5 51 12 51 5 75 18 75 5 47 11 47 6 88 22 88 7 75 18 75 6 85 21 85 7 82 20 82 6 84 21 84 5 85 21 85 5 88 22 88 4 72 18 72 5 79 19 79 6 69 17 69 6 94 23 94 5 81 20 81 6 83 20 83 4 71 17 71 6 86 21 86 The first group of experiments in Table I shows that the time complexity of SBE (and B LAST) can grow exponentially in the number of nested conditional statements, as expected. Table III explains why the SBE approach exhausts the memory: the number of abstract nodes in the reachability tree grows exponentially in the number of nested conditional statements. Therefore, SBE does not scale. The LBE approach reduces the loop-free part of the branching control-flow structure to a few edges (cf. example in the introduction), and the size of the ART is constant for this example program, because only the structure inside the body of the loop changes. There are no refinement steps necessary in the LBE approach, because the edges to the error location are infeasible. Therefore, no predicates are used. The runtime of the LBE approach slightly increases with the size of the program, because the size of the formulas that are sent to the SMT solver is slightly increasing. Although in principle the complexity of the SMT problem grows exponentially in the size of the formulas, the heuristics used by SMT solvers avoid the exponential enumeration that we observe in the case of SBE. differ only in the CFA summarization and Boolean abstraction. The first column is provided in Tables I and II to give evidence that the new approach beats the highly-optimized traditional implementation B LAST. We actually configured and ran experiments with all four combinations: SBE versus LBE, and Cartesian versus Boolean abstraction. The experimentation clearly showed that SBE does not benefit from Boolean abstraction in terms of precision, with substantial degrade in performance: the only programs for which it terminated successfully were the first five instances of the test_locks group. Similarly, the combination of LBE with Cartesian abstraction fails to solve any of the experiments, due to loss of precision. Thus, we report only on the two successful configurations, i.e., SBE in combination with Cartesian abstraction, and LBE with Boolean abstraction. Discussion of Evaluation Results. Tables I and II present performance results of our experiments, for the safe and unsafe programs respectively. All runtimes are given in seconds of processor time, ‘>1800.00’ indicates a timeout, ‘MO’ indicates an out-of-memory. Table III shows statistics about the algorithms for SBE and LBE only. 31 Authorized licensed use limited to: Universitat Passau. Downloaded on May 31,2010 at 14:00:24 UTC from IEEE Xplore. Restrictions apply. Furthermore, we plan to investigate the use of adjustable precision-based techniques for the construction of the large blocks on-the-fly (instead of the current preprocessing step). This would enable a dynamic adjustment of the size of the large blocks, and thus we could fine-tune the amount of work that is delegated to the SMT solver. Also, we plan to explore other techniques for computing abstract successors which are more precise than Cartesian abstraction but less expensive than Boolean abstraction. For the two other classes of experiments, we see that LBE is able to successfully complete all benchmarks, and shows significant performance gains over SBE. SBE is able to solve only about one third of all benchmarks, and for the ones that complete, it is clearly outperformed by LBE. In Table III, we see that SBE has in general a much larger ART. In Table I we observe that LBE performs significantly better than any B LAST configuration. LBE performed best also in finding the error paths (cf. Table II), outperforming both SBE and B LAST. In summary, the experiments show that the LBE approach outperforms the SBE approach, both for correct and defective programs. This provides evidence of the benefits of a “more symbolic” analysis as performed in the LBE approach. One might argue that our CPACHECKER-based SBE implementation might be sub-optimal although it uses the same implementation and execution environment as LBE; in fact, both implementations currently suffer from some inefficiencies and have room for several optimizations. Therefore, we compare also with B LAST. By looking at Tables I and II, we see that LBE outperforms also B LAST, despite the fact that the latter is the result of several years of fine-tuning. B LAST in turn is much more efficient than SBE. However, the performance gap between B LAST and SBE highly depends on the commandline options used for B LAST. We conclude the section by discussing the scope of the experimental evaluation. The LBE techniques proposed in this paper bear substantial similarities to the SSA-based encodings used in tools like SATABS [15], C ALYSTO [1] or S PEC # [4]. For lack of space, we chose to not include a comparison with such tools; rather, we focussed on the more relevant issue of the impact of LBE on ART-based model checking. V. C ONCLUSION AND Acknowledgments. We thank Roman Manevich for interesting discussions about B LAST’s performance bottlenecks. R EFERENCES [1] D. Babic and A. J. Hu, “C ALYSTO: Scalable and precise extended static checking,” in Proc. ICSE. ACM, 2008, pp. 211–220. [2] T. Ball, A. Podelski, and S. K. Rajamani, “Boolean and cartesian abstractions for model checking C programs,” in Proc. TACAS, ser. LNCS 2031. Springer, 2001, pp. 268–283. [3] T. Ball and S. K. Rajamani, “The S LAM project: Debugging system software via static analysis,” in Proc. POPL. ACM, 2002, pp. 1–3. [4] M. Barnett and K. R. M. Leino, “Weakest-precondition of unstructured programs,” in Proc. PASTE. ACM, 2005, pp. 82–87. [5] D. Beyer, A. Cimatti, A. Griggio, M. E. Keremoglu, and R. Sebastiani, “Software model checking via large-block encoding, Tech. 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McMillan, “Abstractions from proofs,” in Proc. POPL. ACM, 2004, pp. 232–244. [17] T. A. Henzinger, R. Jhala, R. Majumdar, and G. Sutre, “Lazy abstraction,” in Proc. POPL. ACM, 2002, pp. 58–70. [18] S. K. Lahiri, R. Nieuwenhuis, and A. Oliveras, “SMT techniques for fast predicate abstraction,” in Proc. CAV, ser. LNCS 4144. Springer, 2006, pp. 424–437. [19] K. L. McMillan, “Lazy abstraction with interpolants,” in Proc. CAV, ser. LNCS 4144. Springer, 2006, pp. 123–136. [20] G. C. Necula, S. McPeak, S. P. Rahul, and W. Weimer, “C IL: Intermediate language and tools for analysis and transformation of C programs,” in Proc. CC, ser. LNCS 2304. Springer, 2002, pp. 213–228. [21] R. Sebastiani, “Lazy satisfiability modulo theories,” J. Satisfiability, Boolean Modeling and Computation, vol. 3, 2007. F UTURE W ORK We have proposed LBE as an alternative to the SBE modelchecking approach, based on the idea that transitions in the abstract space should represent larger fragments of the program. Our novel approach results in significantly smaller ARTs, where abstract successor computations are more involved, and thus trading cost of many explicit enumerations of program paths with the cost of symbolic successor computations. A thorough experimental evaluation shows the advantages of LBE against both our implementation of SBE and the stateof-the-art B LAST system. The existing experimental results can now be summarized as follows: (i) the combination of Cartesian predicate abstraction with SBE is successful on many practical programs [3], [6], but is not efficient on programs with nested conditional branching, (ii) the combination of Boolean predicate abstraction with SBE is intractably expensive [2], (iii) the combination of Cartesian predicate abstraction with joining paths is too imprecise [7], and (iv) the combination of Boolean predicate abstraction with LBE is the most promising combination for ART-based predicate abstraction (Tables I and II). In our future work, we plan to implement McMillan’s interpolation-based lazy-abstraction approach [19], and experiment with SBE versus LBE versions of his algorithm. 32 Authorized licensed use limited to: Universitat Passau. Downloaded on May 31,2010 at 14:00:24 UTC from IEEE Xplore. Restrictions apply.