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On the complexity of ω-pushdown automata

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Abstract

Finite automata over infinite words (called ω-automata) play an important role in the automatatheoretic approach to system verification. Different types of ω-automata differ in their succinctness and complexity of their emptiness problems, as a result, theory of ω-automata has received considerable research attention. Pushdown automata over infinite words (called ω-PDAs), a generalization of ω-automata, are a natural model of recursive programs. Our goal in this paper is to conduct a relatively complete investigation on the complexity of the emptiness problems for variants of ω-PDAs. For this purpose, we consider ω-PDAs of five standard acceptance types: Büchi, Parity, Rabin, Streett and Muller acceptances. Based on the transformation for ω- automata and the efficient algorithm proposed by Esparza et al. in CAV’00 for verifying the emptiness problem of ω-PDAs with Büchi acceptance, it is trivial to check the emptiness problem of other ω-PDAs. However, this naive approach is not optimal. In this paper, we propose novel algorithms for the emptiness problem of ω-PDAs based on the observations of the structure of accepting runs. Our algorithms outperform algorithms that go through Büchi PDAs. In particular, the space complexity of the algorithm for Streett acceptance that goes through Büchi acceptance is exponential, while ours is polynomial. The algorithm for Parity acceptance that goes through Büchi acceptance is in O(k 3 n 2 m) time and O(k2 nm) space, while ours is in O(kn 2 m) time and O(nm) space, where n (resp. m and k) is the number of control states (resp. transitions and index). Finally, we show that our algorithms yield a better solution for the pushdown model checking problem against linear temporal logic with fairness.

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Acknowledgments

This work was partially supported by National Natural Science Foundation of China (Grant Nos. 61402179, 61532019, 61103012), ChenGuang Project of the Shanghai Municipal Education Commission (SHMEC) and Shanghai Education Development Foundation (SHEDF) (Grant No. 13CG21), and Open Project of Shanghai Key Laboratory of Trustworthy Computing (Grant No. 07dz22304201404).

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Correspondence to Fu Song or Min Zhang.

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Lei, Y., Song, F., Liu, W. et al. On the complexity of ω-pushdown automata. Sci. China Inf. Sci. 60, 112102 (2017). https://doi.org/10.1007/s11432-016-9026-x

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Keywords

  • pushdown automata
  • emptiness
  • ω-words
  • model checking
  • linear temporal logic