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We consider two-variable first-order logic 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. Deciding the presence or absence of such a pattern in a given automaton is in NL. In particular, this leads to NL algorithms for deciding the levels of the 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 NL-hard and, hence, NL-complete.

We consider two-variable first-order logic FO2 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 NL algorithms for deciding the levels of the FO2 quantifier alternation hierarchies. This applies to both full and half levels, each over finite and infinite words. Moreover, we show that these problems are NL-hard and, hence, NL-complete.

No abstract

For every fixed class of regular languages, there is a natural hierarchy of increasingly more general problems: Firstly, the membership problem asks whether a given language belongs to the fixed class of languages. Secondly, the separation problem asks for two given languages whether they can be separated by a language from the fixed class. And thirdly, the covering problem is a generalization of separation problem to more than two given languages. Most instances of such problems were solved by the connection of regular languages and finite monoids. Both the membership problem and the separation problem were also extended to ordered monoids. The computation of pointlikes can be interpreted as the algebraic counterpart of the covering problem. In this paper, we extend the computation of pointlikes to ordered monoids. This leads to the notion of conelikes for the corresponding algebraic framework.We apply this framework to the Trotter-Weil hierarchy and both the full and the half levels of the FO 2 quantifier alternation hierarchy. As a consequence, we solve the covering problem for the resulting subvarieties of DA. An important combinatorial tool are uniform ranker characterizations for all subvarieties under consideration; these characterizations stem from order comparisons of ranker positions.