Automata in toposes, and general Myhill-Nerode theorems
Victor Iwaniack
TL;DR
This paper generalizes the theory of automata to the setting of toposes, establishing broad Myhill-Nerode theorems that encompass various automata models, including nominal automata, within a unified categorical framework.
Contribution
It extends the functorial automata approach to elementary toposes with natural number objects, deriving general Myhill-Nerode theorems applicable to a wide range of automata models.
Findings
Established general Myhill-Nerode theorems in topos-theoretic settings
Unified automata theory across different categorical frameworks
Reproduced known results for nominal automata within the topos framework
Abstract
We extend the functorial approach to automata by Colcombet and Petri\c{s}an [arXiv:1712.07121] from the category of sets to any elementary topos with a natural number object and establish general Myhill-Nerode theorems in our setting. As a special case we recover the result of Boja\'nczyk, Klin and Lasota [arXiv:1402.0897] for orbit-finite nominal automata by considering automata in the Myhill-Schanuel topos of nominal sets.
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Taxonomy
Topicssemigroups and automata theory · Advanced Algebra and Logic · Logic, Reasoning, and Knowledge