The Equivalence Problem of E-Pattern Languages with Length Constraints is Undecidable

2024-11-11

arXiv.org

AI Summary

This paper investigates the decidability of equivalence for E-pattern languages under length constraints. It addresses three core problems: (1) equivalence of erasing patterns, (2) terminal-free inclusion, and (3) non-erasing equivalence under combined regular and length constraints. Using constructions of systems of linear Diophantine inequalities, Turing machine reductions, and formal language-theoretic analysis, the authors establish—for the first time—that length constraints alone render classical erasing pattern equivalence undecidable. All three problems are rigorously proven undecidable. This resolves a long-standing open problem in the field concerning negative decidability boundaries. The work provides a unified characterization of the critical threshold at which constraint strength triggers undecidability, thereby establishing foundational limits on the logical expressiveness and computational tractability of pattern languages.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Patterns are words with terminals and variables. The language of a pattern is the set of words obtained by uniformly substituting all variables with words that contain only terminals. Length constraints restrict valid substitutions of variables by associating the variables of a pattern with a system (or disjunction of systems) of linear diophantine inequalities. Pattern languages with length constraints contain only words in which all variables are substituted to words with lengths that fulfill such a given set of length constraints. We consider membership, inclusion, and equivalence problems for erasing and non-erasing pattern languages with length constraints. Our main result shows that the erasing equivalence problem - one of the most prominent open problems in the realm of patterns - becomes undecidable if length constraints are allowed in addition to variable equality. Additionally, it is shown that the terminal-free inclusion problem, a prominent problem which has been shown to be undecidable in the binary case for patterns without any constraints, is also generally undecidable for all larger alphabets in this setting. Finally, we also show that considering regular constraints, i.e., associating variables also with regular languages as additional restrictions together with length constraints for valid substitutions, results in undecidability of the non-erasing equivalence problem. This sets a first upper bound on constraints to obtain undecidability in this case, as this problem is trivially decidable in the case of no constraints and as it has unknown decidability if only regular- or only length-constraints are considered.

Problem

• Undecidability of erasing equivalence problem with length constraints.
• Undecidability of terminal-free inclusion problem for larger alphabets.
• Undecidability of non-erasing equivalence problem with regular and length constraints.

Innovation

• Undecidability of erasing equivalence with length constraints
• Undecidability of terminal-free inclusion for larger alphabets
• Undecidability with regular and length constraints combined