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arXiv · 2105.09291

Deciding FO2 Alternation for Automata over Finite and Infinite Words

Viktor Henriksson·Manfred Kufleitner
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Abstract

We consider two-variable first-order logic $\text{FO}^2$ and its quantifier alternation hierarchies over both finite and infinite words. Our main results are forbidden patterns for deterministic automata (finite words) and for Carton-Michel automata (infinite words). In order to give concise patterns, we allow the use of subwords on paths in finite graphs. This concept is formalized as subword-patterns. For certain types of subword-patterns there exists a non-deterministic logspace algorithm to decide their presence or absence in a given automaton. In particular, this leads to $\mathbf{NL}$ algorithms for deciding the levels of the $\text{FO}^2$ quantifier alternation hierarchies. This applies to both full and half levels, each over finite and infinite words. Moreover, we show that these problems are $\mathbf{NL}$-hard and, hence, $\mathbf{NL}$-complete.

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Viktor Henriksson, Manfred Kufleitner. 2021-05-19. Deciding FO2 Alternation for Automata over Finite and Infinite Words. https://arxiv.org/abs/2105.09291

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