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On the Power of Counting the Total Number of Computation Paths of NPTMs

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Theory and Applications of Models of Computation (TAMC 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14637))

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Abstract

In this paper, we define and study variants of several complexity classes of decision problems that are defined via some criteria on the number of accepting paths of an NPTM. In these variants, we modify the acceptance criteria so that they concern the total number of computation paths, instead of the number of accepting ones. This direction reflects the relationship between the counting classes \(\#\textsf{P}\) and \(\textsf{TotP}\), which are the classes of functions that count the number of accepting paths and the total number of paths of NPTMs, respectively. The former is the well-studied class of counting versions of \(\textsf{NP}\) problems, introduced by Valiant (1979). The latter contains all self-reducible counting problems in \(\#\textsf{P}\) whose decision version is in \(\textsf{P}\), among them prominent \(\#\textsf{P}\)-complete problems such as Non-negative Permanent, #PerfMatch and #DNF-Sat.

We show that almost all classes introduced in this work coincide with their ‘# accepting paths’-definable counterparts, thus providing an alternative model of computation for the classes \(\mathsf {\oplus P}\), \(\mathsf {Mod_k P}\), \(\textsf{SPP}\), \(\textsf{WPP}\), \(\mathsf {C_=P}\), and \(\textsf{PP}\). Moreover, for each of these classes, we present a novel family of complete problems which are defined via problems that are \(\textsf{TotP}\)-complete under parsimonious reductions. This way, we show that all the aforementioned classes have complete problems that are defined via counting problems whose existence version is in \(\textsf{P}\), in contrast to the standard way of obtaining completeness results via counting versions of \(\textsf{NP}\)-complete problems. To the best of our knowledge, prior to this work, such results were known only for \(\mathsf {\oplus P}\) and \(\mathsf {C_=P}\).

We also build upon a result by Curticapean, to exhibit yet another way to obtain complete problems for \(\textsf{WPP}\) and \(\textsf{PP}\), namely via the difference of values of the \(\textsf{TotP}\) function #PerfMatch on pairs of graphs. Finally, for the so defined \(\textsf{WPP}\)-complete problem, we provide an exponential lower bound under the randomized Exponential Time Hypothesis, showcasing the hardness of the class.

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Aknowledgements

Aggeliki Chalki has been funded by the project “Mode(l)s of Verification and Monitorability” (MoVeMnt) (grant no 217987). Sotiris Kanellopoulos and Aris Pagourtzis have been partially supported for this work by project MIS 5154714 of the National Recovery and Resilience Plan Greece 2.0 funded by the European Union under the NextGeneration EU Program.

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Authors and Affiliations

  1. School of Electrical and Computer Engineering, National Technical University of Athens, Athens, Greece

    Eleni Bakali, Sotiris Kanellopoulos, Aris Pagourtzis & Stathis Zachos

  2. Department of Computer Science, Reykjavik University, Reykjavik, Iceland

    Aggeliki Chalki

  3. Archimedes Research Unit, Athena RC, 15125, Marousi, Greece

    Sotiris Kanellopoulos & Aris Pagourtzis

Authors
  1. Eleni Bakali
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  2. Aggeliki Chalki
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  4. Aris Pagourtzis
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  5. Stathis Zachos
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Correspondence to Aggeliki Chalki .

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Editors and Affiliations

  1. Chinese Academy of Sciences, Beijing, China

    Xujin Chen

  2. The Hong Kong Polytechnic University, Hong Kong, China

    Bo Li

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Bakali, E., Chalki, A., Kanellopoulos, S., Pagourtzis, A., Zachos, S. (2024). On the Power of Counting the Total Number of Computation Paths of NPTMs. In: Chen, X., Li, B. (eds) Theory and Applications of Models of Computation. TAMC 2024. Lecture Notes in Computer Science, vol 14637. Springer, Singapore. https://doi.org/10.1007/978-981-97-2340-9_18

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  • DOI: https://doi.org/10.1007/978-981-97-2340-9_18

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