## Reachability Analysis of Pushdown Automata: Application to Model-Checking (1997)

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### BibTeX

@MISC{Bouajjani97reachabilityanalysis,

author = {Ahmed Bouajjani and Javier Esparza and Oded Maler},

title = {Reachability Analysis of Pushdown Automata: Application to Model-Checking},

year = {1997}

}

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### Abstract

We apply the symbolic analysis principle to pushdown systems. We represent (possibly infinite) sets of configurations of such systems by means of finite-state automata. In order to reason in a uniform way about analysis problems involving both existential and universal path quantification (like model-checking for branching-time logics), we consider the more general class of alternating pushdown systems and use alternating finite-state automata as a representation structure for their sets of configurations. We give a simple and natural procedure to compute sets of predecessors for this representation structure. We apply this procedure and the automata-theoretic approach to model-checking to define new model-checking algorithms for pushdown systems and both linear and branching-time properties. From these results we derive upper bounds for several model-checking problems, and we also provide matching lower bounds, using reductions based on some techniques introduced by Walukiewicz.

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Citation Context ... and then add the transition s ;!i+1 q to Ai+1. The computation time of the set is quadratic in the number of states of Ai (which is equal to the number of states of A) and linear in the length of w (=-=[1]-=-, Theorem 9.5). Thus, the construction of Ai+1 from Ai takes time O(jAj2 jPj2 ). Now, the sequence A0� A1�:::must reach the xpoint Apre after at most O(jAj2 jPj) steps, because this is an upper bound ... |

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Citation Context ...d in the literature. For example, in the case of timed automata, special kinds of polyhedra (regions) are used to represent in nite sets of states (vectors of reals corresponding to clock valuations) =-=[3]-=-. Polyhedra are also used for linear hybrid systems. However, in this case, there is no algorithm for computing a nite representation of the exact set of predecessors (the reachability problem is unde... |

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Citation Context ...logics. More precisely, weshowhow to construct AMA's accepting the set of all con gurations satisfying !-regular properties of linear-time temporal logics (including all properties expressible in LTL =-=[18]-=- or the linear-time -calculus [19]), or properties expressed as formulas of the alternation-free modal -calculus. A rst version of these results appeared in [8] (where the logic CTL [13] is considered... |

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Citation Context ...the case of linear hybrid systems, the procedure for calculating the set of predecessors for these structures is not guaranteed to terminate. Finally, notice that symbolic representations (e.g. BDD's =-=[10]-=-) are also used in the nite-state case in order to overcome the state-explosion problem [17]. In this paper we consider pushdown systems,as well as the more general class of alternating pushdown syste... |

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Citation Context ...this case, there is no algorithm for computing a nite representation of the exact set of predecessors (the reachability problem is undecidable), but upper approximations of this set can be calculated =-=[2]-=-. In [5], representation structures called QDD's are introduced for FIFO-channel systems. These structures are nite-state automata representing sets of queue contents. As in the case of linear hybrid ... |

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Citation Context ...sed in linear time temporal logics like LTL or the linear-time -calculus are !-regular sets over the alphabet . Moreover, there exist algorithms which construct Buchi automata to recognize these sets =-=[21, 20]-=-. This is all we needtoknow about these logics in this paper in order to give model checking algorithms for PDS's. Let P =(P� ;� ) be a PDS, and let : P ! be a labelling function, which associates a s... |

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Citation Context ... these structures is not guaranteed to terminate. Finally, notice that symbolic representations (e.g. BDD's [10]) are also used in the nite-state case in order to overcome the state-explosion problem =-=[17]-=-. In this paper we consider pushdown systems,as well as the more general class of alternating pushdown systems, i.e., pushdown systems with both existential and universal nondeterminism (see [20] for ... |

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Citation Context ...nsition relation of Ai. For every i 0, Ai+1 is obtained from Ai by conserving the same states and transitions, and adding for every transition rule 3 The idea is inspired by the construction given in =-=[6]-=-, pages 91-93, of a nite-state automaton recognizing the closure of a regular language under the rewriting relation induced by amonadic string-rewriting system.s(pj� ) ,! (pk k w �w) and every state q... |

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Citation Context ...the model-checking problems mentioned above are in DEXPTIME, and that the model-checking problem for pushdown systems and a subset of CTL cansbe solved in PSPACE. Using a technique due to Walukiewicz =-=[22]-=-, we complement these results with matching lower bounds, i.e., we show that all these problems are complete for their corresponding complexity classes. The paper is structured as follows. In Section ... |

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Citation Context ...izations of global model-checking algorithms for branching-time logics and nite-state systems. An approach similar to ours, based on automata representation of the stack contents, has been adopted in =-=[15]-=-. However, the techniques used there are di erent from ours, and the branching-time properties are expressed there in a logic (CTL ) which is incomparable with the alternation-free modal -calculus. We... |

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Citation Context ...th quanti cation must be considered, like model-checking for branching-time temporal logics (see Section 5) and also about synthesis problems, such as nding winning strategies for 2-player games (see =-=[4]-=-). A state (we use rather the word \con guration") of a pushdown system is a pair hp� wi where p is control location and w is a sequence of stack symbols (the stack contents). As a representation stru... |

82 | P.: Symbolic verification of communication protocols with infinite state spaces using QDDs - Boigelot, Godefroid - 1999 |

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Citation Context ...to construct AMA's accepting the set of all con gurations satisfying !-regular properties of linear-time temporal logics (including all properties expressible in LTL [18] or the linear-time -calculus =-=[19]-=-), or properties expressed as formulas of the alternation-free modal -calculus. A rst version of these results appeared in [8] (where the logic CTL [13] is considered instead of the more expressive al... |

78 | Model checking for context-free processes - Burkart, Steffen |

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Citation Context .... We start by considering the case where all the -subformulas of ' are -formulas. We construct an APDS AP which is, roughly speaking, the product of P and the alternating automaton corresponding to ' =-=[14]-=-� we then reduce the problemsof computing ['] to computing the value of preAP for a certain regular set of con gurations. Intuitively, a con guration h[p� ]�wi belongs to this set if is a basic formul... |

30 | Verifying Temporal Properties of Systems. Birkhauser - Bradfield - 1992 |

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Citation Context ...mplete, but polynomial in the size of thessystem. Since the properties of systems one wishes to check can be usually encoded into short formulas, model-checkers based on linear-time logics, like SPIN =-=[16]-=-, have proved to be useful in practice. Theorem 3.2 shows that the complexity ofmodelchecking for PDS's is worse than the complexity for nite-state systems, but not much worse: it remains polynomial i... |

6 |
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Citation Context ...ssible in LTL [18] or the linear-time -calculus [19]), or properties expressed as formulas of the alternation-free modal -calculus. A rst version of these results appeared in [8] (where the logic CTL =-=[13]-=- is considered instead of the more expressive alternation-free modal -calculus). Moreover, our approach allows us to obtain a number of complexity results: we show that the model-checking problems men... |

3 |
Verifying Temporal Properties of Systems. Birkhauser
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(Show Context)
Citation Context ...t of con gurations for every variable X free in ', then Ct is also a regular set of con gurations. The following result can be easily proved using standard techniques based on the notion of signature =-=[9]-=-: Proposition 5.1 Let AP be the APDS obtained from P and ' using the construction above. A con guration hp� wi of P belongs to ['] iff the con guration h[p� ']�wi of AP belongs to pre AP (Ct). Applyin... |

2 |
Symbolic Veri cation of Communication Protocols with In nite State Spaces Using QDDs
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(Show Context)
Citation Context ...e, there is no algorithm for computing a nite representation of the exact set of predecessors (the reachability problem is undecidable), but upper approximations of this set can be calculated [2]. In =-=[5]-=-, representation structures called QDD's are introduced for FIFO-channel systems. These structures are nite-state automata representing sets of queue contents. As in the case of linear hybrid systems,... |

2 |
Alternating automata and program veri cation
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(Show Context)
Citation Context ...blem [17]. In this paper we consider pushdown systems,as well as the more general class of alternating pushdown systems, i.e., pushdown systems with both existential and universal nondeterminism (see =-=[20]-=- for a survey on alternating automata). This general setting allows to reason in a uniform way aboutanalysis problems where existential and universal path quanti cation must be considered, like model-... |

1 |
Ste en. Model Checking for Context-Free Processes
- Burkart, B
- 1992
(Show Context)
Citation Context ...ans of some reductions based on Walukiewicz's ideas [22]. The model-checking problem for pushdown systems and the modal -calculus (or its alternation-free fragment) has been studied in several papers =-=[11, 12, 22]-=-. The main advantage of our approach (apart from an homogeneous treatment of both branching-time and linear-time logics) is the simplicity of our algorithms: only well known concepts from automata the... |

1 |
Ste en. Composition, Decomposition and Model-Checking of Pushdown Processes
- Burkart, B
- 1995
(Show Context)
Citation Context ...ans of some reductions based on Walukiewicz's ideas [22]. The model-checking problem for pushdown systems and the modal -calculus (or its alternation-free fragment) has been studied in several papers =-=[11, 12, 22]-=-. The main advantage of our approach (apart from an homogeneous treatment of both branching-time and linear-time logics) is the simplicity of our algorithms: only well known concepts from automata the... |