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The Containment Problem for Unambiguous Register Automata and Unambiguous Timed Automata

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Abstract

We investigate the complexity of the containment problem “Does \(L(\mathcal {A})\subseteq L({\mathscr{B}})\) hold?” for register automata and timed automata, where \({\mathscr{B}}\) is assumed to be unambiguous and \(\mathcal {A}\) is arbitrary. We prove that the problem is decidable in the case of register automata over \((\mathbb N,=)\), in the case of register automata over \((\mathbb Q,<)\) when \({\mathscr{B}}\) has a single register, and in the case of timed automata when \({\mathscr{B}}\) has a single clock. We give a 2-EXPSPACE algorithm in the first case, whose complexity is a single exponential in the case that \({\mathscr{B}}\) has a bounded number of registers. In the other cases, we give an EXPSPACE algorithm.

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Notes

  1. We note that this coincides with the notion of support for a finite configuration, in the terminology of set theory with atoms. Similarly, some notions that we define hereafter are classical in the “atom” terminology. We choose to ignore this terminology for the sake of clarity.

  2. Note the structural similarity to the unambiguous register automaton in Fig. 1.

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Authors and Affiliations

  1. Department of Algebra, Faculty of Mathematics and Physics, Charles University in Prague, Prague, Czech Republic

    Antoine Mottet

  2. Universität Leipzig, Leipzig, Germany

    Karin Quaas

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  1. Antoine Mottet
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  2. Karin Quaas
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Correspondence to Antoine Mottet.

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This article belongs to the Topical Collection: Special Issue on Theoretical Aspects of Computer Science (2019)

Guest Editors: Rolf Niedermeier and Christophe Paul

The first author received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant Agreement No 771005, CoCoSym).

The second author was supported by Deutsche Forschungsgemeinschaft (DFG), Project 406907430

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Mottet, A., Quaas, K. The Containment Problem for Unambiguous Register Automata and Unambiguous Timed Automata. Theory Comput Syst 65, 706–735 (2021). https://doi.org/10.1007/s00224-020-09997-2

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  • DOI: https://doi.org/10.1007/s00224-020-09997-2

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