A Tighter Bound for the Determinization of Visibly Pushdown Automata

Axel Legay and Azadeh Farzan (Eds.): International Workshop on Verification of Infinite-State Systems (INFINITY 2009) EPTCS 10, 2009, pp. 62–76, doi:10.4204/EPTCS.10.5 A Tighter Bound for the Determinization of Visibly Pushdown Automata∗ Nguyen Van Tang Research Center for Verification and Semantics National Institute of Advanced Industrial Science and Technology Toyonaka, Osaka, 560-0083 Japan t.nguyen@aist.go.jp Visibly pushdown automata (VPA), introduced by Alur and Madhusuan in 2004, is a subclass of pushdown automata whose stack behavior is completely determined by the input symbol according to a fixed partition of the input alphabet. Since its introduce, VPAs have been shown to be useful in various context, e.g., as specification formalism for verification and as automaton model for process- ing XML streams. Due to high complexity, however, implementation of formal verification based on VPA framework is a challenge. In this paper we consider the problem of implementing VPA- based model checking algorithms. For doing so, we first present an improvement on upper bound for determinization of VPA. Next, we propose simple on-the-fly algorithms to check universality and in- clusion problems of this automata class. Then, we implement the proposed algorithms in a prototype tool. Finally, we conduct experiments on randomly generated VPAs. The experimental results show that the proposed algorithms are considerably faster than the standard ones. 1 Introduction Visibly pushdown automata [1] are pushdown automata whose stack behavior (i.e. whether to execute a push, a pop, or no stack operation) is completely determined by the input symbol according to a fixed partition of the input alphabet. As shown in [1], this class of visibly pushdown automata enjoys many good properties similar to those of the class of finite automata. The main reason for this being that is, each nondeterministic VPA can be transformed into an equivalent deterministic one. Therefore, checking context-free properties of pushdown models is feasible as long as the calls and returns are made visible. As a result, visibly pushdown automata have turned out to be useful in various context, e.g. as specification formalism for verification and synthesis problem for pushdown systems [3, 4, 11], as automaton model for processing XML streams [10, 9], and as AOP protocols for component-based systems [12, 13]. As each nondeterministic VPA can be determinized, all problems that concern the accepted languages such as universality and inclusion problems are decidable. To check universality for a nondeterminis- tic VPA M over its alphabet Σ (that is, to check if L(M) = Σ∗), the standard method is first to make it complete, determinize it, complement it, and then checks for emptiness. To check the inclusion prob- lem L(M) ⊆L(N), the standard method computes the complement of N, takes its intersection with M and then, check for emptiness. This is costly as computing the complement necessitates a full deter- minization. This explosion is in some sense unavoidable, because determinization for VPAs requires exponential time blowup [1]. Therefore, one of the questions raised is that whether one can implement ∗This research is partially supported by a COE-project N. V. Tang 63 efficiently operations like determinization as well as decision procedures like universality ( or. inclusion) checking for VPAs. During the recent years, a new approach called antichain method has been proposed to imple- ment efficiently operations like universality or inclusion checking on nondeterministic word or tree au- tomata [14, 6]. Unfortunately, the antichain technique cannot be directly used for checking universality and inclusion of VPA. This is because the set of configurations of a VPA is infinite and thus, computing the set of antichains may not terminate. In this paper, we focus on the problem of checking universality and inclusion for VPAs. We make the following contributions towards to this overall goal. • First, we present an improvement on upper bound for determinization of VPA. In [1], Alur and Madhusudan showed that any nondeterministic VPA with n states can be translated into a deter- ministic one with at most 2n2+n states. Here, we show that this upper bound can be made tighter. More precisely, we optimize Alur-Madhusudan’s determinization procedure, and show that any nondeterministic VPA with n states can be transformed into a deterministic one with at most 2n2 states. • Second, we apply the standard method to check universality and inclusion problems for nondeter- ministic VPA. This method includes two main steps: determinization and reachability checking for non-accepting configurations. For determinization, we use the Alur-Madhusudan’s procedure [1]. For reachability checking, we apply the symbolic technique P-automata [7, 8] to compute the sets of all reachable configurations of a VPA. • Third, we present an on-the-fly method to check universality of VPA. The idea is very simple that we

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