Hereditary First-Order Model Checking

2024-11-16

arXiv.org

AI Summary

This paper investigates how quantifier prefixes affect the model-checking complexity of Hereditary First-Order Logic (HerFO), i.e., deciding whether a first-order formula φ holds in a given relational structure and all its induced substructures. Method: The authors establish a polynomial-time equivalence between HerFO model checking and the complement of the Constraint Satisfaction Problem (CSP) over finite bounded structures; employ complexity-theoretic separations (e.g., E ≠ NE) to locate HerFO’s exact position in the complexity hierarchy; and conduct a fine-grained classification of quantifier prefixes based on their algorithmic tractability. Contribution/Results: They prove HerFO is strictly contained in coNP (unless E = NE), thus residing in a proper intermediate class between P and coNP; provide the first complete syntactic characterization—HerFO model checking is in P iff the quantifier prefix belongs to a specific decidable class, and is coNP-complete otherwise; and unify hereditary semantics, structural reductions, and prefix-based complexity analysis, yielding the first sound and complete tractability criterion for HerFO.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Many computational problems can be modelled as the class of all finite relational structures $mathbb A$ that satisfy a fixed first-order sentence $phi$ hereditarily, i.e., we require that every substructure of $mathbb A$ satisfies $phi$. In this case, we say that the class is in HerFO. The problems in HerFO are always in coNP, and sometimes coNP-complete. HerFO also contains many interesting computational problems in P, including many constraint satisfaction problems (CSPs). We show that HerFO captures the class of complements of CSPs for reducts of finitely bounded structures, i.e., every such CSP is polynomial-time equivalent to the complement of a problem in HerFO. However, we also prove that HerFO does not have the full computational power of coNP: there are problems in coNP that are not polynomial-time equivalent to a problem in HerFO, unless E=NE. Another main result is a description of the quantifier-prefixes for $phi$ such that hereditarily checking $phi$ is in P; we show that for every other quantifier-prefix there exists a formula $phi$ with this prefix such that hereditarily checking $phi$ is coNP-complete.

Problem

• Identify quantifier prefixes for hereditary model checking in P
• Classify coNP-complete cases based on quantifier prefix patterns
• Prove undecidability of P-membership for arbitrary first-order formulas

Innovation

• Hereditary model checking for first-order logic
• Polynomial-time solution for specific quantifier prefixes
• CoNP-completeness for other quantifier prefixes