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Alternating Demon Space Is Closed Under Complement and Other Simulations for Sublogarithmic Space

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Developments in Language Theory (DLT 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9840))

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Abstract

We present new simulations for \(\text {ASpace}^{\mathrm {dm}}(s(n))\), the class of languages that can be accepted by alternating Turing machines starting with s(n) worktape cells delimited initially. Under weak constructibility assumptions, not excluding monotone functions below \(\log n\), we show: (i) \(\text {ASpace}^{\mathrm {dm}}(s(n))\subseteq \text {DTime}(n\!\cdot \!2^{O(s(n))})\). This extends, to sublogarithmic space, the classical simulation of alternating space by deterministic time. (ii) \(\text {ASpace}^{\mathrm {dm}}(s(n))\subseteq \text {NTimeSpace}(n\!\cdot \!2^{O(s(n))},2^{O(s(n))})\), a simulation with simultaneous bounds on time and space. This improves the known inclusion, stating that \(\text {ASpace}^{\mathrm {dm}}(s(n))\subseteq \text {NSpace}(2^{O(s(n))})\). (iii) \(\text {ASpace}^{\mathrm {dm}}(s(n))=\text {co-}\text {ASpace}^{\mathrm {dm}}(s(n)))\), i.e., the alternating space is closed under complement. This simulation does not depend on whether s(n) is above \(\log n\) nor on whether the original machine gets into infinite loops, which solves a long-standing open problem.

Supported by the Slovak grant contracts VEGA 1/0142/15 and APVV-15-0091.

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Notes

  1. 1.

    By \(X\text {Space}^{\mathrm {dm}}(s(n))\), for \(X\in \{\text {D},\text {N},\text {A}\}\), we denote the classes of languages accepted by deterministic, nondeterministic, and alternating Turing machines starting with a worktape consisting of \(\lfloor {s(n)}\rfloor \) blank cells delimited by endmarkers (here \(\lfloor {x}\rfloor \) denotes the largest integer satisfying \(i\le x\), for the given real value x), as opposed to the more common complexity classes \(X\text {Space}(s(n))\) where the worktape is initially empty and the machine must use its own computational power to make sure that it respects, along each computation path on each input of length n, the space bound of s(n). The notation “dm” derives from “Demon” Turing Machines [5].

  2. 2.

    It is known that \(\text {ASpace}(o(\log \log n))\) contains only regular languages [15]. However, it is still possible to accept some nonregular languages, if \(\lfloor {s(n)}\rfloor \le o(\log \log n)\) worktape cells are delimited automatically at the very beginning. As an example [2], take \(\mathcal {L}= \{1^{n}:\,n\mod \lceil {\log \log \log n}\rceil =0\}\), contained in \(\text {DSpace}^{\mathrm {dm}}(\log \log \log \log n)\).

  3. 3.

    For \(i\in \{0,n+1\}\), we take \(a_{i}\in \{\rhd ,\lhd \}\), two new symbols representing the respective endmarkers.

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Authors and Affiliations

  1. Department of Computer Science, P.J. Šafárik University, Jesenná 5, 04001, Košice, Slovakia

    Viliam Geffert

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  1. Viliam Geffert
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Correspondence to Viliam Geffert .

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  1. Université du Québec à Montréal , Montreal, Québec, Canada

    Srečko Brlek

  2. Dept Mathematiques, Univ du Quebec Montreal Dept Mathematiques, Montreal, Québec, Canada

    Christophe Reutenauer

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Geffert, V. (2016). Alternating Demon Space Is Closed Under Complement and Other Simulations for Sublogarithmic Space. In: Brlek, S., Reutenauer, C. (eds) Developments in Language Theory. DLT 2016. Lecture Notes in Computer Science(), vol 9840. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53132-7_16

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  • DOI: https://doi.org/10.1007/978-3-662-53132-7_16

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