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Unary Coded PSPACE-Complete Languages in ASPACE(loglog n)

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Abstract

We study the class of binary coded versions of unary languages that can be accepted by alternating machines with loglog n space. We show that there exists a binary PSpace-complete language \(\mathcal {L}\) such that the unary coded version of \(\mathcal {L}\) is in ASpace(loglog n). Consequently, the standard translation between unary languages accepted with loglog n space and binary languages accepted with log n space works for alternating machines if and only ifP = PSpace. In general, if a binary language is accepted deterministically in 2n⋅nO(1) time and, simultaneously, in nO(1) space—which covers many PSpace-complete problems—then its unary coded version is accepted by an alternating Turing machine using an initially delimited worktape of size loglog n. This unexpected power follows from the fact that, with an auxiliary worktape of size O(loglog n) on a unary input 1n, an alternating machine can simulate a stack with log n bits, representing the contents of the stack by its input head position. The standard push/pop operations on the stack are implemented by moving the head along the input.

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Notes

  1. XSpace(s(n)) ⊆ XSpacedm(s(n)) and these two space complexity classes are equal, if s(n) is fully space constructible. The notation “dm” derives from Demon Turing Machines [5].

  2. Throughout the paper, DTimeSpace(2n⋅nO(1),nO(1)) will be used as a shorthand notation representing \(\bigcup _{k^{\prime }\ge 0} \bigcup _{k^{\prime \prime }\ge 0}\)DTimeSpace\((2^{n}{\cdot }n^{k^{\prime }}\!,n^{k^{\prime \prime }})\). The same kind of notation will be applied to other complexity classes and other growth rates.

  3. This ensures that the same number cannot be represented by two different binary strings, using a different number of leading zeros. However, this way we exclude a binary representation of zero.

  4. This is based on the following facts. First, it is quite trivial to see that the machine can compute mi = (N + 1) mod pi for any given prime pi ≤ O(log N), by counting modulo pi while traversing across the unary input tape with ⊩ 1N ⊣. Thus, the machine has a read-only access to (m1,m2,m3,…), the first O(log N/ log log N) remainders in the Chinese Residual Representation of N + 1. With access to these remainders, the ℓ-th bit in the binary representation of N + 1 can be computed by using O(log log N) worktape space. This was shown in [7], building on ideas presented in [6, 9]. (See also [1, Theorem 4.5]. Related topics and some other applications can be found, among others, in [2, 8, 23].)

  5. If ℓ = 0,the loop is not iterated at all, i.e., the number of iterations is zero.

  6. A standard implementation by recursive calls of test_bit would require a stack forN + 2nested levels, each level with its own copy of the used variables, which would giveΩ(N ⋅ log ℓ)spacein total.

  7. For N < 2,we can take \(s \overset {\text {df.}}{=} 0\).Such short inputs can be accepted or rejected directly by a singleinitial scan across the input tape, according to their membership in\(\mathcal {L}_{2\rightarrow 1}\).

  8. Recall that A simulates an alternating auxiliary stack machine that—in our case—accepts the binarylanguage QBF2.This auxiliary stack machine keeps at most O(log m)symbols on the worktape and at most m bits in the stack.

  9. It is also quite easy to construct a polynomial-space algorithm for\(\textsc {QBF}_{2}^{\prime }\)directly, which we leave to a reader.

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Acknowledgements

The author would like to thank the reviewers and the Program Committee of CSR 2017 for their suggestions, especially for sending a summary of PC discussions which gave inspiration for several improvements, and the anonymous reviewer of ToCS for helping to simplify the proof of Lemma 4.

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  1. Department of Computer Science, P. J. Šafárik University, Jesenná 5, 04154, Košice, Slovakia

    Viliam Geffert

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Correspondence to Viliam Geffert.

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This article is part of the Topical Collection on Computer Science Symposium in Russia

A preliminary and weaker version of this work was presented at the 12th International Computer Science Symposium in Russia (CSR 2017), June 8–12, 2017, Kazan, Russia [Lect. Notes Comput. Sci., vol. 10304, pp. 141–53, Springer-Verlag (2017)].

Supported by the Slovak Grant Agency for Science under contract VEGA 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. Unary Coded PSPACE-Complete Languages in ASPACE(loglog n). Theory Comput Syst 63, 688–714 (2019). https://doi.org/10.1007/s00224-018-9844-7

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