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Weighted One-Deterministic-Counter Automata

Authors Prince Mathew , Vincent Penelle, Prakash Saivasan, A.V. Sreejith

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Author Details

Prince Mathew
  • Indian Institute of Technology Goa, India
Vincent Penelle
  • Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800, F-33400, Talence, France
Prakash Saivasan
  • The Institute of Mathematical Sciences, HBNI, India
  • CNRS UMI ReLaX, India
A.V. Sreejith
  • Indian Institute of Technology Goa, India

Funding

  • Sreejith, A.V.: supported by SERB for the project "Probabilistic Pushdown Automata" [MTR/2021/000788].

Acknowledgements

The authors would like to thank Rahul C S for his intuitive suggestions that helped in proving Lemma 14.

Cite As Get BibTex

Prince Mathew, Vincent Penelle, Prakash Saivasan, and A.V. Sreejith. Weighted One-Deterministic-Counter Automata. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 39:1-39:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.FSTTCS.2023.39

Abstract

We introduce weighted one-deterministic-counter automata (odca). These are weighted one-counter automata (oca) with the property of counter-determinacy, meaning that all paths labelled by a given word starting from the initial configuration have the same counter-effect. Weighted odcas are a strict extension of weighted visibly ocas, which are weighted ocas where the input alphabet determines the actions on the counter. 
We present a novel problem called the co-VS (complement to a vector space) reachability problem for weighted odcas over fields, which seeks to determine if there exists a run from a given configuration of a weighted odca to another configuration whose weight vector lies outside a given vector space. We establish two significant properties of witnesses for co-VS reachability: they satisfy a pseudo-pumping lemma, and the lexicographically minimal witness has a special form. It follows that the co-VS reachability problem is in 𝖯.
These reachability problems help us to show that the equivalence problem of weighted odcas over fields is in 𝖯 by adapting the equivalence proof of deterministic real-time ocas [Stanislav Böhm and Stefan Göller, 2011] by Böhm et al. This is a step towards resolving the open question of the equivalence problem of weighted ocas. Finally, we demonstrate that the regularity problem, the problem of checking whether an input weighted odca over a field is equivalent to some weighted automaton, is in 𝖯. We also consider boolean odcas and show that the equivalence problem for (non-deterministic) boolean odcas is in PSPACE, whereas it is undecidable for (non-deterministic) boolean ocas.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automata extensions
Keywords
  • One-counter automata
  • Equivalence
  • Weighted automata
  • Reachability

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