Deciding FO2 Alternation for Automata over Finite and Infinite Words

TL;DR

This paper characterizes the quantifier alternation levels of two-variable first-order logic over finite and infinite words using forbidden automata patterns, providing efficient algorithms for their decision.

Contribution

It introduces subword-patterns for automata, enabling NL algorithms to decide FO^2 alternation levels over finite and infinite words, and establishes their NL-completeness.

Findings

Forbidden patterns characterize automata levels.

NL algorithms decide FO^2 alternation levels.

Problems are NL-complete.

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.