2026-10-10T21:17:37Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:27699 2026-10-10T19:48:19Z ddc:004 open_access

The Σ-Chain Product: A Succinct Model of Automata (De)Composition Borelli, Roberto Bresolin, Davide Geatti, Luca Montanari, Angelo Zavatteri, Matteo Automata Cascade Product Formal Languages Krohn-Rhodes Theory The cascade product is a fundamental construction in automata theory, enabling hierarchical composition of automata and playing a central role in decomposition results such as the Krohn–Rhodes theorem. However, its use is limited by the exponential size required to represent cascades, which stems from the fact that each component may depend on all preceding ones, leading to exponentially large alphabets. To address this issue, we introduce the Σ-chain product, a restricted variant in which each component depends only on the input alphabet and the component immediately preceding it. We show that Σ-chains achieve linear-size representations and can be exponentially more succinct than cascades. We prove that Σ-chains and cascades are expressively equivalent even when restricting the components to specific classes of automata, such as permutation-reset automata. As a consequence, we derive that a language is regular if and only if it is recognized by a Σ-chain of permutation-reset automata. Finally, we analyze structural properties of Σ-chains of reset automata, including a relation with well-known subclasses of star-free languages. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Roberto Borelli and Davide Bresolin and Luca Geatti and Angelo Montanari and Matteo Zavatteri 2026 Is Part Of OASIcs, Volume 146, 33rd International Symposium on Temporal Representation and Reasoning (TIME 2026) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/OASIcs.TIME.2026.3 urn:nbn:de:0030-drops-276999 https://drops.dagstuhl.de/entities/document/10.4230/OASIcs.TIME.2026.3 eng https://creativecommons.org/licenses/by/4.0/legalcode