Computer Science > Formal Languages and Automata Theory
[Submitted on 19 May 2021 (v1), last revised 1 Sep 2021 (this version, v3)]
Title:Deciding FO2 Alternation for Automata over Finite and Infinite Words
View PDFAbstract: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.
Submission history
From: Viktor Henriksson [view email][v1] Wed, 19 May 2021 17:51:00 UTC (34 KB)
[v2] Sat, 5 Jun 2021 09:18:24 UTC (34 KB)
[v3] Wed, 1 Sep 2021 04:47:08 UTC (34 KB)
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