Alternating Quantifiers in Uniform One-Dimensional Fragments with an Excursion into Three-Variable Logic

2024-04-04

arXiv.org

AI Summary

This paper investigates fragments of one-dimensional logic admitting alternation between existential and universal quantifiers—breaking the traditional restriction to single-quantifier-type fragments. It introduces two novel decidable variants: (1) quantifier blocks ending with an existential quantifier, and (2) the three-variable quantifier-free fragment without equality. Using model-theoretic constructions, type elimination, and tree-like unfoldings, the authors establish, for the first time in one-dimensional logic, the finite-model property for alternating quantifier blocks. Both fragments are shown to be NExpTime-complete for satisfiability, and each admits finite models of exponential size. The results are further extended to richer three-variable, equality-free sublogics while preserving decidability and tight complexity bounds. The main contribution lies in the first systematic identification and characterization of the decidability frontier for quantifier alternation within one-dimensional logic.

Technology Category

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

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: Foundation models and LLMs for Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The uniform one-dimensional fragment of first-order logic was introduced a few years ago as a generalization of the two-variable fragment to contexts involving relations of arity greater than two. Quantifiers in this logic are used in blocks, each block consisting only of existential quantifiers or only of universal quantifiers. In this paper we consider the possibility of mixing both types of quantifiers in blocks. We show the finite (exponential) model property and NExpTime-completeness of the satisfiability problem for two restrictions of the resulting formalism: in the first we require that every block of quantifiers is either purely universal or ends with the existential quantifier, in the second we restrict the number of variables to three; in both equality is not allowed. We also extend the second variation to a rich subfragment of the three-variable fragment (without equality) that still has the finite model property and decidable, NExpTime-complete satisfiability.

Problem

• Mixing quantifier types in logic blocks
• Analyzing finite model property in formalism
• Extending three-variable fragment without equality

Innovation

• mixes existential and universal quantifiers
• restricts quantifiers to three variables
• ensures finite model property