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A polynomial-time approximation algorithm for counting words accepted by an NFA (invited paper)

Authors:
Marcelo Arenas
PUC, Chile / IMFD, Chile
,
Luis Alberto Croquevielle
PUC, Chile / IMFD, Chile
,
Rajesh Jayaram
Carnegie Mellon University, USA
,
Cristian Riveros
PUC, Chile / IMFD, Chile
Authors Info & Claims
Published: 15 June 2021 Publication History

Abstract

Counting the number of words of a certain length accepted by a non-deterministic finite automaton (NFA) is a fundamental problem, which has many applications in different areas such as graph databases, knowledge compilation, and information extraction. Along with this, generating such words uniformly at random is also a relevant problem, particularly in scenarios where returning varied outputs is a desirable feature.
The previous problems are formalized as follows. The input of #NFA is an NFA N and a length k given in unary (that is, given as a string 0^k), and then the task is to compute the number of strings of length k accepted by N. The input of GEN-NFA is the same as #NFA, but now the task is to generate uniformly, at random, a string accepted by N of length k.
It is known that #NFA is #P-complete, so an efficient algorithm to compute this function exactly is not expected to exist. However, this does not preclude the existence of an efficient approximation algorithm for it. In this talk, we will show that #NFA admits a fully polynomial-time randomized approximation scheme (FPRAS). Prior to our work, it was open whether #NFA admits an FPRAS; in fact, the best randomized approximation scheme known for #NFA ran in time n^O(log(n)).
Besides, we will mention some consequences and applications of our results. In particular, from well-known results on counting and uniform generation, we obtain that GEN-NFA admits a fully polynomial-time almost uniform generator. Moreover, as #NFA is SpanL-complete under polynomial-time parsimonious reductions, we obtain that every function in the complexity class SpanL admits an FPRAS.

References

[1]
Marcelo Arenas, Luis Alberto Croquevielle, Rajesh Jayaram, and Cristian Riveros. 2019. Efficient Logspace Classes for Enumeration, Counting, and Uniform Generation. In Proceedings of the 38th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS 2019, Amsterdam, The Netherlands, June 30 - July 5, 2019. Pages 59–73. https://doi.org/10.1145/3294052.3319704
[2]
Marcelo Arenas, Luis Alberto Croquevielle, Rajesh Jayaram, and Cristian Riveros. 2019. Efficient Logspace Classes for Enumeration, Counting, and Uniform Generation. CoRR, abs/1906.09226, 2019.

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cover image ACM Conferences
STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing
June 2021
1797 pages
ISBN:9781450380539
DOI:10.1145/3406325
Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the Owner/Author.

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Association for Computing Machinery

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Publication History

Published: 15 June 2021

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Author Tags

  1. Non-deterministic finite automata
  2. approximate counting
  3. regular languages
  4. uniform generation

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  • ANID - Millennium Science Initiative Program - Chile

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STOC '21
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Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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References

[1]
Marcelo Arenas, Luis Alberto Croquevielle, Rajesh Jayaram, and Cristian Riveros. 2019. Efficient Logspace Classes for Enumeration, Counting, and Uniform Generation. In Proceedings of the 38th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems, PODS 2019, Amsterdam, The Netherlands, June 30 - July 5, 2019. Pages 59–73. https://doi.org/10.1145/3294052.3319704
[2]
Marcelo Arenas, Luis Alberto Croquevielle, Rajesh Jayaram, and Cristian Riveros. 2019. Efficient Logspace Classes for Enumeration, Counting, and Uniform Generation. CoRR, abs/1906.09226, 2019.