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Algebraic Language Theory with Effects

Authors Fabian Lenke , Stefan Milius , Henning Urbat , Thorsten Wißmann

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LIPIcs.ICALP.2025.165.pdf
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ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Theory of computation → Regular languages
Keywords
  • Automaton
  • Monoid
  • Monad
  • Effect
  • Algebraic language theory

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Abstract

Regular languages - the languages accepted by deterministic finite automata - are known to be precisely the languages recognized by finite monoids. This characterization is the origin of algebraic language theory. In this paper, we generalize the correspondence between automata and monoids to automata with generic computational effects given by a monad, providing the foundations of an effectful algebraic language theory. We show that, under suitable conditions on the monad, a language is computable by an effectful automaton precisely when it is recognizable by (1) an effectful monoid morphism into an effect-free finite monoid, and (2) a monoid morphism into a monad-monoid bialgebra whose carrier is a finitely generated algebra for the monad, the former mode of recognition being conceptually completely new. Our prime application is a novel algebraic approach to languages computed by probabilistic finite automata. Additionally, we derive new algebraic characterizations for nondeterministic probabilistic finite automata and for weighted finite automata over unrestricted semirings, generalizing previous results on weighted algebraic recognition over commutative rings.

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Fabian Lenke, Stefan Milius, Henning Urbat, and Thorsten Wißmann. Algebraic Language Theory with Effects. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 165:1-165:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ICALP.2025.165

Author Details

Fabian Lenke
  • Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Stefan Milius
  • Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Henning Urbat
  • Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany
Thorsten Wißmann
  • Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

Funding

Fabian Lenke and Henning Urbat are supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 470467389. Stefan Milius is supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 517924115.

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