Navigational hierarchies of regular languages
Thomas Place, Marc Zeitoun

TL;DR
This paper introduces alternative hierarchies for star-free languages using a temporal logic-based operator, explores their relationships with concatenation hierarchies, and proves decidability results for membership and separation problems.
Contribution
It develops new navigational hierarchies for star-free languages, analyzes their properties, and establishes decidability results for key language class problems.
Findings
Navigational hierarchies are strictly intertwined with concatenation hierarchies.
Decidability of separation implies decidability of membership at level two.
Level two for certain classes corresponds to variants of two-variable logic.
Abstract
We study the class of star-free languages. A long-standing goal is to classify them by the complexity of their descriptions. The most influential research effort involves concatenation hierarchies, which measure alternations between ``complement'' and ``union plus concatenation''. We explore alternative hierarchies that also stratify star-free languages. They are built with an operator . From an input class , it produces a larger one , consisting of all languages definable in a variant of unary temporal logic, where temporal modalities depend on . Level in the navigational hierarchy of basis is constructed by applying this operator times to . As bases , we focus on group languages and natural extensions thereof, denoted . We prove that the navigational hierarchies of bases and are strictly intertwined and conduct a…
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Taxonomy
Topicssemigroups and automata theory · DNA and Biological Computing · Cellular Automata and Applications
