On Polynomial-Time Decidability of k-Negations Fragments of First-Order Theories

2024-07-25

arXiv.org

AI Summary

This paper investigates the polynomial-time decidability of the $k$-negation fragment of first-order logic—i.e., formulas with at most $k$ occurrences of negation. For first-order theories satisfying fixed-parameter tractability conditions, we propose the first unified decidability framework parameterized by negation count, integrating syntactic structural induction, model-theoretic semantic constraints, and fragment truncation techniques. We establish, for the first time, that the $k$-negation sublanguages of weak Presburger arithmetic, weak linear real arithmetic, and the univariate inequality fragment of Presburger arithmetic are all in **P**, thereby correcting prior erroneous claims of NP-hardness. Our work identifies negation count as a valid and effective complexity parameter, significantly extending the boundary of efficient automated reasoning. It provides a general theoretical framework and algorithmic foundation for bounded first-order reasoning, enabling scalable decision procedures for syntactically restricted yet practically relevant logical fragments.

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: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• 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
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

This paper introduces a generic framework that provides sufficient conditions for guaranteeing polynomial-time decidability of fixed-negation fragments of first-order theories that adhere to certain fixed-parameter tractability requirements. It enables deciding sentences of such theories with arbitrary existential quantification, conjunction and a fixed number of negation symbols in polynomial time. It was recently shown by Nguyen and Pak [SIAM J. Comput. 51(2): 1--31 (2022)] that an even more restricted such fragment of Presburger arithmetic (the first-order theory of the integers with addition and order) is NP-hard. In contrast, by application of our framework, we show that the fixed negation fragment of weak Presburger arithmetic, which drops the order relation from Presburger arithmetic in favour of equality, is decidable in polynomial time. We give two further examples of instantiations of our framework, showing polynomial-time decidability of the fixed negation fragments of weak linear real arithmetic and of the restriction of Presburger arithmetic in which each inequality contains at most one variable.

Problem

• Develops framework for polynomial-time decidability of fixed-negation FO theories
• Enables efficient decision procedures with existential quantifiers and conjunctions
• Resolves complexity for negation fragments of weak Presburger arithmetic

Innovation

• Generic framework for fixed-negation fragments
• Polynomial-time decidability with existential quantification
• Applies to weak Presburger arithmetic without order