Fagin's Theorem for Semiring Turing Machines

TL;DR

This paper establishes a logical characterization of semiring-weighted Turing machines, proving a Fagin Theorem linking computational complexity over semirings to weighted existential second-order logic.

Contribution

It introduces an improved SRTM model enabling a Fagin Theorem for semiring-based complexity classes, clarifying the relationship between computation and logical expressiveness.

Findings

Proved a Fagin Theorem for the SRTM model over semirings.

Demonstrated the improved SRTM model's advantages over the original.

Reclaimed key results from previous flawed models.

Abstract

In recent years, quantitative complexity over semirings has been intensively investigated. In this context, Eiter and Kiesel (Semiring Reasoning Frameworks in AI and Their Computational Complexity, J. Artif. Intell. Res., 2023) introduced non-deterministic Turing Machines with semiring-weighted transitions (SRTMs) to capture the complexity of a manifold of semiring frameworks. Beyond computational complexity, they posed the question of how we can relate the computational power of SRTMs to logical expressiveness. While this question was partially addressed for a more limited machine model by Badia et al.\ (Logical characterizations of weighted complexity classes, MFCS, 2024), the full question remained open. To answer it, we present an improved version of Eiter and Kiesel's SRTM model of computation. First and foremost, this enables us to prove a Fagin Theorem for the SRTM model, i.e., we show that the quantitative complexity class $\text{NP}_\infty(R)$, which comprises non-deterministic polynomial time computability in the improved SRTM model over a commutative semiring $R$, is captured by a version of weighted existential second-order logic that allows for predicates interpreted as semiring-annotated relations over $R$. Furthermore, we argue that the new SRTM model is preferable over the original one and show that it reclaims some important results from Eiter and Kiesel (2023) that were flawed with respect to the latter.