A Myhill-Nerode Type Characterization of 2detLIN Languages

TL;DR

This paper characterizes the class of 2detLIN languages, recognized by deterministic linear automata with two heads, using a Myhill-Nerode type equivalence, and establishes conditions for a complete, non-crossing class-based characterization.

Contribution

It introduces a Myhill-Nerode style characterization for 2detLIN languages using prefix-suffix pairs, advancing understanding of deterministic linear automata.

Findings

Finitely many classes correspond to 2detLIN languages

Characterization requires non-crossing prefix-suffix pairs

Provides conditions for a complete language class characterization

Abstract

Linear automata are automata with two reading heads starting from the two extremes of the input, are equivalent to 5' -> 3' Watson-Crick (WK) finite automata. The heads read the input in opposite directions and the computation finishes when the heads meet. These automata accept the class LIN of linear languages. The deterministic counterpart of these models, on the one hand, is less expressive, as only a proper subset of LIN, the class 2detLIN is accepted; and on the other hand, they are also equivalent in the sense of the class of the accepted languages. Now, based on these automata models, we characterize the class of 2detLIN languages with a Myhill-Nerode type of equivalence classes. However, as these automata may do the computation of both the prefix and the suffix of the input, we use prefix-suffix pairs in our classes. Additionally, it is proven that finitely many classes in the characterization match with the 2detLIN languages, but we have some constraints on the used prefix-suffix pairs, i.e., the characterization should have the property to be complete and it must not have any crossing pairs.