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The Complexity of Debate Checking

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Abstract

We study probabilistic debate checking, where a silent resource-bounded verifier reads a dialogue about the membership of a given string in the language under consideration between a prover and a refuter. We consider debates of partial and zero information, where the prover is prevented from seeing some or all of the messages of the refuter, as well as those of complete information. This model combines and generalizes the concepts of one-way interactive proof systems, games of possibly incomplete information, and probabilistically checkable debate systems. We give full characterizations of the classes of languages with debates checkable by verifiers operating under simultaneous bounds of O(1) space and O(1) random bits. It turns out such verifiers are strictly more powerful than their deterministic counterparts, and allowing a logarithmic space bound does not add to this power. PSPACE and EXPTIME have zero- and partial-information debates, respectively, checkable by constant-space verifiers for any desired error bound when we omit the randomness bound. No amount of randomness can give a verifier under only a fixed time bound a significant performance advantage over its deterministic counterpart. However, randomness does seem to help verifiers with simultaneous bounds on space and time. In the case of logspace and polynomial-time verifiers, we show that logarithmic randomness is sufficient to check complete- and partial-information debates for all languages in PSPACE. Any such language can be reduced to the quantified max word problem for matrices, which allows us to present a nonapproximability result for the optimization version of this problem.

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Notes

  1. This may be likened to a debate where the prover is deaf, and can “hear” the refuter only when the refuter chooses to write her message on a board for him to read.

  2. The work by Feigenbaum et al. [20] is notable in the sense that the “prover ” (in our terminology) is also allowed to hide its messages from the “refuter.”

  3. The relationship between the probabilistically checkable debate systems of [11] and our model is very similar to the one between probabilistically checkable proofs and interactive proof systems.

  4. The truthful player can be either P1 or P0 depending on the membership situation of a specific input.

  5. See [7] for a more detailed and formal treatment.

  6. A quantum version was introduced recently in [42].

  7. The simplification is the elimination of negating states and multiple work tapes.

  8. We defined ATMs with a single work tape in Section 2.2.1. Versions with multiple work tapes can be defined in the usual manner.

  9. See, for instance, [37] for a review of the power of oneway-IPSs under various resource bounds.

  10. Note that the last configuration presented in such a round must be accepting, since we assume that P1 must be honest if P0 is violating the protocol.

  11. Since P1 is truthful in this case, a $ will be encountered after at most s(n) P1 symbols are read.

  12. Note that, in this section, there is no need to distinguish the debates checkable for some error bound and those checkable for any error bound, since our definition in Section 2.1.3 ensures that all time-bounded verifiers halt with probability 1, and therefore can be made to run repeatedly to achieve any desired error bound.

  13. Notice the similar contrast between I P(log-space, poly-time) = PSPACE [9, 38] and oneway-I P(log-space, poly-time)= N P [10].

  14. Note that what we call the “reading configurations” are named “communication configurations” in [10].

  15. In [8], the class in question is denoted as ∀BC-SPACE(log).

  16. See [1] and [40] for the formal definitions.

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Acknowledgements

We are grateful to the anonymous reviewer for his helpful comments.

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Authors and Affiliations

  1. Department of Computer Science, University of Chicago, Chicago, IL, USA

    H. Gökalp Demirci

  2. Department of Computer Engineering, BoCemğaziçi University, Bebek, 34342, İstanbul, Turkey

    A. C. Cem Say

  3. Faculty of Computing, University of Latvia, Raina bulv. 19, LV-1586, Rīga, Latvia

    Abuzer Yakaryılmaz

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  1. H. Gökalp Demirci
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Correspondence to H. Gökalp Demirci.

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Abuzer Yakaryılmaz was partially supported by ERC Advanced Grant MQC.

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Demirci, H.G., Say, A.C.C. & Yakaryılmaz, A. The Complexity of Debate Checking. Theory Comput Syst 57, 36–80 (2015). https://doi.org/10.1007/s00224-014-9547-7

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