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Complement for two-way alternating automata

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Abstract

We consider the problem of converting a two-way alternating finite automaton (2AFA) with n states to a 2AFA accepting the complement of its language. Complementing is trivial for halting 2AFAs, by swapping the roles of existential and universal decisions and the roles of accepting and rejecting states. However, since 2AFAs do not have resources to detect infinite loops by counting executed steps, it was not known whether the cost of complementing is polynomial in n in the general case. Here we shall show that 2AFAs can be complemented by using \(O(n^7)\) states.

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Notes

  1. Throughout the paper, m denotes the length of the input and n the number of states.

  2. Sometimes, we call this tree “standard”, since other trees induced by the same configuration graph will also be used later.

  3. This means that \({\mathsf {A}}\) must be in an accepting state when it halts; the position of the input head at this moment is not relevant; such configuration has no sons; and the entire subtree degenerates into a single node.

  4. Both \(\langle {}d,r{}\rangle \in D{\times }Q\) and \(dr\in D{\cdot }Q\) represent an ordered pair satisfying \(d\in D\) and \(r\in Q\). We introduce a superfluous binary operator in order not to complicate notation more than necessary.

  5. The chronological order in which \(\kappa _{\scriptscriptstyle {j_{\pi }}},\ldots ,\kappa _{\scriptscriptstyle {j_1}}\) are determined as accepting does not correspond to the order in which \({\mathsf {A}}'\) traverses from these configurations to \(\kappa \). For example, in Fig. 1, the configuration \(\langle {}q_{\scriptscriptstyle {0}},3{}\rangle \) is determined as accepting earlier than \(\langle {}q_{\scriptscriptstyle {0}},5{}\rangle \), but \(\langle {}q_{\scriptscriptstyle {1}},4{}\rangle \) is visited from \(\langle {}q_{\scriptscriptstyle {0}},5{}\rangle \) earlier than from \(\langle {}q_{\scriptscriptstyle {0}},3{}\rangle \).

  6. It should be pointed out that \(\kappa \) can also be visited in the mode “\(\mathsf {{}from}\_{\mathsf {succ}}\)” by traversals from configurations that are not its valid sons. However, by (a.1), such visits do not modify the read–write contents in the cell.

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Authors and Affiliations

  1. Department of Computer Science, P. J. Šafárik University, Kosice, Slovakia

    Viliam Geffert

  2. Department of Computer Science, Carnegie Mellon University in Qatar, Doha, Qatar

    Christos A. Kapoutsis & Mohammad Zakzok

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  1. Viliam Geffert
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  2. Christos A. Kapoutsis
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  3. Mohammad Zakzok
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Correspondence to Viliam Geffert.

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Supported by the Slovak Grant Agency for Science (VEGA) under contract 1/0056/18 “Descriptional and Computational Complexity of Automata and Algorithms” and by the Slovak Research and Development Agency under Contract APVV-15-0091 “Efficient Algorithms, Automata, and Data Structures”.

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Geffert, V., Kapoutsis, C.A. & Zakzok, M. Complement for two-way alternating automata. Acta Informatica 58, 463–495 (2021). https://doi.org/10.1007/s00236-020-00373-8

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