Logical Approaches to Non-deterministic Polynomial Time over Semirings

2025-09-30

AI Summary

Establishing a logical characterization of the nondeterministic polynomial-time (NP) class within the Blum–Shub–Smale (BSS) model over semirings and identifying canonical complete problems. Method: We introduce existential second-order logic (∃SO) interpreted over semiring semantics as a logical framework for capturing computational complexity, thereby generalizing the classical logical characterization of NP to semiring-based computation. Through rigorous formal analysis and completeness-preserving reductions, we establish the satisfiability problem for propositional logic over semirings as complete for this semiring-based NP class. Contribution/Results: We prove that the Boolean fragment of this logic precisely captures the complexity of the true existential first-order theory over the reals. This work unifies algebraic semantics, logical definability, and real-number computational complexity, providing a foundational logical framework for computation over semirings within the BSS model.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Statistical Relational/Logic Learning

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
• Security and Privacy: Security and privacy of machine learning and AI applications

Abstract

We provide a logical characterization of non-deterministic polynomial time defined by BSS machines over semirings via existential second-order logic interpreted in the semiring semantics developed by Grädel and Tannen. Furthermore, we show that, similarly to the classical setting, the satisfiability problem of propositional logic in the semiring semantics is the canonical complete problem for this version of NP. Eventually, we prove that the true existential first-order theory of the semiring is a complete problem for the so-called Boolean part of this version of NP.

Problem

• Logical characterization of NP over semirings using second-order logic
• Satisfiability problem in semiring semantics as canonical NP-complete
• First-order theory of semiring as complete problem for Boolean NP

Innovation

• Logical characterization of NP via semiring semantics
• Satisfiability as canonical NP-complete problem
• First-order theory defines Boolean NP fragment