On the Complexity of Intersection Non-emptiness for Star-Free Language Classes

Authors Emmanuel Arrighi , Henning Fernau , Stefan Hoffmann , Markus Holzer , Ismaël Jecker , Mateus de Oliveira Oliveira , Petra Wolf



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Author Details

Emmanuel Arrighi
  • University of Bergen, Norway
Henning Fernau
  • Fachbereich IV, Informatikwissenschaften, Universität Trier, Germany
Stefan Hoffmann
  • Fachbereich IV, Informatikwissenschaften, Universität Trier, Germany
Markus Holzer
  • Institut für Informatik, Universität Giessen, Germany
Ismaël Jecker
  • Institute of Science and Technology, Klosterneuburg, Austria
Mateus de Oliveira Oliveira
  • University of Bergen, Norway
Petra Wolf
  • Fachbereich IV, Informatikwissenschaften, Universität Trier, Germany

Acknowledgements

We like to thank Lukas Fleischer and Michael Wehar for our discussions. This work started at the Schloss Dagstuhl Event 20483 Moderne Aspekte der Komplexitätstheorie in der Automatentheorie https://www.dagstuhl.de/20483.

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Emmanuel Arrighi, Henning Fernau, Stefan Hoffmann, Markus Holzer, Ismaël Jecker, Mateus de Oliveira Oliveira, and Petra Wolf. On the Complexity of Intersection Non-emptiness for Star-Free Language Classes. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 34:1-34:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.FSTTCS.2021.34

Abstract

In the Intersection Non-emptiness problem, we are given a list of finite automata A_1, A_2,… , A_m over a common alphabet Σ as input, and the goal is to determine whether some string w ∈ Σ^* lies in the intersection of the languages accepted by the automata in the list. We analyze the complexity of the Intersection Non-emptiness problem under the promise that all input automata accept a language in some level of the dot-depth hierarchy, or some level of the Straubing-Thérien hierarchy. Automata accepting languages from the lowest levels of these hierarchies arise naturally in the context of model checking. We identify a dichotomy in the dot-depth hierarchy by showing that the problem is already NP-complete when all input automata accept languages of the levels B_0 or B_{1/2} and already PSPACE-hard when all automata accept a language from the level B_1. Conversely, we identify a tetrachotomy in the Straubing-Thérien hierarchy. More precisely, we show that the problem is in AC^0 when restricted to level L_0; complete for L or NL, depending on the input representation, when restricted to languages in the level L_{1/2}; NP-complete when the input is given as DFAs accepting a language in L_1 or L_{3/2}; and finally, PSPACE-complete when the input automata accept languages in level L_2 or higher. Moreover, we show that the proof technique used to show containment in NP for DFAs accepting languages in L_1 or L_{3/2} does not generalize to the context of NFAs. To prove this, we identify a family of languages that provide an exponential separation between the state complexity of general NFAs and that of partially ordered NFAs. To the best of our knowledge, this is the first superpolynomial separation between these two models of computation.

Subject Classification

ACM Subject Classification
  • Theory of computation → Regular languages
  • Theory of computation → Problems, reductions and completeness
Keywords
  • Intersection Non-emptiness Problem
  • Star-Free Languages
  • Straubing-Thérien Hierarchy
  • dot-depth Hierarchy
  • Commutative Languages
  • Complexity

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