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On Cascades of Reset Automata

Authors Roberto Borelli , Luca Geatti , Marco Montali , Angelo Montanari

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Author Details

Roberto Borelli
  • University of Udine, Italy
Luca Geatti
  • University of Udine, Italy
Marco Montali
  • Free University of Bozen-Bolzano, Italy
Angelo Montanari
  • University of Udine, Italy

Funding

Luca Geatti and Angelo Montanari acknowledge the support from the 2024 Italian INdAM-GNCS project "Certificazione, monitoraggio, ed interpretabilità in sistemi di intelligenza artificiale", ref. no. CUP E53C23001670001 and the support from the Interconnected Nord-Est Innovation Ecosystem (iNEST), which received funding from the European Union Next-GenerationEU (PIANO NAZIONALE DI RIPRESA E RESILIENZA (PNRR) – MISSIONE 4 COMPONENTE 2, INVESTIMENTO 1.5 -– D.D. 1058 23/06/2022, ECS00000043). In addition, Marco Montali and Angelo Montanari acknowledges the support from the MUR PNRR project FAIR - Future AI Research (PE00000013) also funded by the European Union Next-GenerationEU.

Acknowledgements

The authors would like to thank Alessio Mansutti for his valuable comments during the preparation of this paper.

Cite As Get BibTex

Roberto Borelli, Luca Geatti, Marco Montali, and Angelo Montanari. On Cascades of Reset Automata. In 42nd International Symposium on Theoretical Aspects of Computer Science (STACS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 327, pp. 20:1-20:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.STACS.2025.20

Abstract

The Krohn-Rhodes decomposition theorem is a pivotal result in automata theory. It introduces the concept of cascade product, where two semiautomata, that is, automata devoid of initial and final states, are combined in a feed-forward fashion. The theorem states that any semiautomaton can be decomposed into a sequence of permutation-reset semiautomata. For the counter-free case, this decomposition consists entirely of reset components with two states each. This decomposition has significantly impacted recent research in various areas of computer science, including the identification of a class of transformer encoders equivalent to star-free languages and the conversion of Linear Temporal Logic formulas into past-only expressions (pastification).
The paper revisits the cascade product in the context of reset automata, thus considering each component of the cascade as a language acceptor. First, we give regular expression counterparts of cascades of reset automata. We then establish several expressiveness results, identifying hierarchies of languages based on the restriction of the height (number of components) of the cascade or of the number of states in each level. We also show that any cascade of reset automata can be transformed, with a quadratic increase in height, into a cascade that only includes two-state components. Finally, we show that some fundamental operations on cascades, like intersection, union, negation, and concatenation with a symbol to the left, can be directly and efficiently computed by adding a two-state component.

Subject Classification

ACM Subject Classification
  • Theory of computation → Regular languages
  • Theory of computation → Automata extensions
Keywords
  • Automata
  • Cascade products
  • Regular expressions
  • Krohn-Rhodes theory

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