A Myhill-Nerode Type Characterization of 2detLIN Languages

2025-07-21

AI Summary

This paper addresses the lack of an algebraic characterization for the class of languages recognized by two-way deterministic linear automata (2detLIN). We introduce, for the first time, a Myhill–Nerode–type equivalence relation based on prefix–suffix pairs, requiring both cross-freeness and completeness to precisely capture the recognition mechanism of bidirectional read heads in deterministic linear computation. By integrating concepts from WK finite automata and linear automata models, we establish a rigorous characterization: a language belongs to 2detLIN if and only if it admits a finite, cross-free, and complete partition of prefix–suffix equivalence classes. This result provides the first algebraic decidability criterion for 2detLIN and furnishes a novel theoretical framework for analyzing the structural properties of languages accepted by bidirectional deterministic automata.

Technology Category

• Knowledge Representation and Reasoning: Knowledge Representation Languages
• Natural Language Processing: Lexical Semantics and Morphology
• Planning, Routing, and Scheduling: Deterministic Planning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• Characterize 2detLIN languages using Myhill-Nerode equivalence classes
• Analyze prefix-suffix pairs in deterministic linear automata models
• Ensure finite equivalence classes with no crossing pairs constraints

Innovation

• Two-head linear automata for 2detLIN languages
• Myhill-Nerode equivalence with prefix-suffix pairs
• Constraints on complete non-crossing pair characterization