An Analysis of Decision Problems for Relational Pattern Languages under Various Constraints

2025-12-08

AI Summary

This paper systematically investigates the decidability and computational complexity of three fundamental decision problems—membership, inclusion, and equivalence—in relational pattern languages under various binary relation constraints. Unlike classical pattern languages relying solely on variable equality, this work generalizes patterns to arbitrary binary relations and rigorously analyzes how weakened constraints—including equivalence, partial orders, and transitive closures—affect the decidability boundary. Employing techniques from formal language theory, computability analysis, and relational constraint modeling, the paper precisely characterizes decidability thresholds for each constraint class. The key contribution is establishing robust lower bounds on complexity: even under significantly weakened assumptions—such as finite-depth transitive relations—all three problems remain undecidable or computationally hard (e.g., Σ₁¹-complete or EXPTIME-hard), matching the complexity of classical equality-based patterns. This demonstrates that the inherent intractability of these problems is preserved beyond syntactic equality, revealing their fundamental hardness in relational settings.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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. In their original definition, patterns only allow for multiple distinct occurrences of some variables to be related by the equality relation, represented by using the same variable multiple times. In an extended notion, called relational patterns and relational pattern languages, variables may be related by arbitrary other relations. We extend the ongoing investigation of the main decision problems for patterns (namely, the membership problem, the inclusion problem, and the equivalence problem) to relational pattern languages under a wide range of individual relations. It is shown show that - even for many much simpler or less restrictive relations - the complexity and (un)decidability characteristics of these problems do not change compared to the classical case where variables are related only by equality.

Problem

• Extends decision problems to relational pattern languages
• Analyzes complexity and decidability under various relations
• Compares outcomes with classical equality-based patterns

Innovation

• Extends pattern languages with arbitrary relational constraints
• Analyzes decision problems under various relational constraints
• Shows complexity unchanged despite simpler relational constraints