MoL-2022-10: Topics in Ω-Automata: A Journey through Lassos, Algebra, Coalgebra and Expressions

MoL-2022-10: Cruchten, Mike (2022) Topics in Ω-Automata: A Journey through Lassos, Algebra, Coalgebra and Expressions. [Report]

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In recent years, automata theory has been brought into the realm of category theory which allowed the generalisation of results and provided new perspectives. With the introduction of the Ω-automaton, an ω-automaton which arises as a coalgebra, there is a wide range of directions to explore. Unlike other types of ω-automata, Ω-automata are deterministic and their acceptance is local. Like deterministic finite automata, they can be minimised and as they are coalgebraic, we can use categorical techniques to study them.
We investigate Ω-automata from multiple perspectives and introduce several new tools along the way. Lassos are studied in more depth giving rise to the Lasso Representation Lemma which specifies the exact relation between lassos and ultimately periodic words. We also partially establish the connection between Wilke algebras and Ω-automata, highlighting the relationship between language acceptance and language recognition. Using the already existing Myhill-Nerode theorem for ω-regular languages, we carry out a strengthening which gives a lower bound for Ω-automata based on the index of the Nerode congruence we define. Furthermore, we present a Brzozowski construction for Ω-automata using lasso expressions. This gives new insights into more effective ways of constructing Ω-automata and brings us closer to the question of size constraints. Finally, we discuss some minimisation procedures for Ω-automata, in particular the Brzozowski minimisation algorithm and an algorithm exploiting a dual equivalence.

Item Type: Report
Report Nr: MoL-2022-10
Series Name: Master of Logic Thesis (MoL) Series
Year: 2022
Subjects: Logic
Depositing User: Dr Marco Vervoort
Date Deposited: 29 Jul 2022 20:50
Last Modified: 29 Jul 2022 20:50

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