Kleene Theorems for Lasso Languages and $ω$-Languages

TL;DR

Rational lasso languages and expressions are introduced, a Kleene theorem for lasso languages is shown and the connection between rational lasso and $\omega$-expressions is explored, which yields a Kleene theorem for $\omega$-languages with respect to saturated lasso automata.

Abstract

Automata operating on pairs of words were introduced as an alternative way of capturing acceptance of regular $ω$-languages. Families of DFAs and lasso automata operating on such pairs followed, giving rise to minimisation algorithms, a Myhill-Nerode theorem and language learning algorithms. Yet Kleene theorems for such a well-established class are still missing. We introduce rational lasso languages and expressions, show a Kleene theorem for lasso languages and explore the connection between rational lasso and $ω$-expressions, which yields a Kleene theorem for $ω$-languages with respect to saturated lasso automata. For one direction of the Kleene theorems, we also provide a Brzozowski construction for lasso automata from rational lasso expressions.

Figures (4)

• Figure 1: Diagram showing our main contributions as dashed arrows.
• Figure 2: A lasso automaton accepting $\{(a^k,ba^j)\mid k,j\in \mathbb{N}\}$.
• Figure 3: A saturated lasso automaton ($\Omega$-automaton).
• Figure 4: A finite lasso automaton accepting $\llbracket b(a^\ast b^\circ)\rrbracket_\circ$.

Theorems & Definitions (72)

• Definition 2.1
• Proposition 2.2
• Proposition 2.3: Lasso Representation Lemma
• Definition 2.4: ciancia:2019:omegaAutomata
• Definition 2.5: ciancia:2019:omegaAutomata
• Example 2.6
• Definition 2.7
• Example 2.8
• Definition 3.1
• Definition 3.2