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Dual Adjunction Between $Ω$-Automata and Wilke Algebra Quotients

Anton Chernev, Helle Hvid Hansen, Clemens Kupke

TL;DR

Lasso semigroups are introduced as a generalisation of Wilke algebras that mirrors how lasso automata generalise $\Omega$-automata, and it is shown that finite lasso semigroups characterise regular lasso languages.

Abstract

$Ω$-automata and Wilke algebras are formalisms for characterising $ω$-regular languages via their ultimately periodic words. $Ω$-automata read finite representations of ultimately periodic words, called lassos, and they are a subclass of lasso automata. We introduce lasso semigroups as a generalisation of Wilke algebras that mirrors how lasso automata generalise $Ω$-automata, and we show that finite lasso semigroups characterise regular lasso languages. We then show a dual adjunction between lasso automata and quotients of the free lasso semigroup with a recognising set, and as our main result we show that this dual adjunction restricts to one between $Ω$-automata and quotients of the free Wilke algebra with a recognising set.

Dual Adjunction Between $Ω$-Automata and Wilke Algebra Quotients

TL;DR

Lasso semigroups are introduced as a generalisation of Wilke algebras that mirrors how lasso automata generalise -automata, and it is shown that finite lasso semigroups characterise regular lasso languages.

Abstract

-automata and Wilke algebras are formalisms for characterising -regular languages via their ultimately periodic words. -automata read finite representations of ultimately periodic words, called lassos, and they are a subclass of lasso automata. We introduce lasso semigroups as a generalisation of Wilke algebras that mirrors how lasso automata generalise -automata, and we show that finite lasso semigroups characterise regular lasso languages. We then show a dual adjunction between lasso automata and quotients of the free lasso semigroup with a recognising set, and as our main result we show that this dual adjunction restricts to one between -automata and quotients of the free Wilke algebra with a recognising set.
Paper Structure (16 sections, 9 theorems, 44 equations, 1 figure)

This paper contains 16 sections, 9 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Languages of Infinite Words
  4. Lasso Automata and $\Omega$-Automata
  5. Wilke Algebras
  6. Algebraic Recognition of Lasso Languages
  7. Lasso Semigroups
  8. From Lasso Semigroups to Lasso Automata
  9. From Lasso Automata to Lasso Semigroups
  10. Finite Lasso Semigroups Recognise Regular Lasso Languages
  11. Dual Adjunction Between Lasso Automata and Lasso Semigroups
  12. Categories of Lasso Automata and Lasso Semigroups
  13. Functoriality of Aut and Alg
  14. Lasso Adjunction
  15. Restricting the Adjunction to Omega-Automata and Wilke Algebras
  16. ...and 1 more sections

Key Result

proposition 1

For every lasso automaton $A = (X, Y, q, \rho, \sigma, \xi, F)$:

Figures (1)

  • Figure 1: Examples of lasso automata. The dotted arrows are $\sigma$-transitions.

Theorems & Definitions (46)

  • definition 1: Lasso automaton CianciaVenema2012StreamAutomataAreCoalgebras
  • remark 1
  • definition 2: $\Omega$-automaton CianciaVenema2012StreamAutomataAreCoalgebras
  • definition 3: Wilke algebra Wilke93AlgTheoryForRegLanguagesFinInf
  • definition 4: Lasso semigroup
  • remark 2
  • proof
  • definition 5: Extended lasso semigroup
  • remark 3
  • definition 6: $\Aut$
  • ...and 36 more