arXiv subjects

Erik Paul

Publications and source records attributed to Erik Paul.

3 recordsLinked to original sources

arXiv2026

Fagin's Theorem for Semiring Turing Machines

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.

cs.CC↗

arXiv2024

Logical Characterizations of Weighted Complexity Classes

Fagin's seminal result characterizing $\mathsf{NP}$ in terms of existential second-order logic started the fruitful field of descriptive complexity theory. In recent years, there has been much interest in the investigation of quantitative (weighted) models of computations. In this paper, we start the study of descriptive complexity based on weighted Turing machines over arbitrary semirings. We provide machine-independent characterizations (over ordered structures) of the weighted complexity classes $\mathsf{NP}[\mathcal{S}], \mathsf{FP}[\mathcal{S}]$, $\mathsf{FPLOG}[\mathcal{S}]$, $\mathsf{FPSPACE}[\mathcal{S}]$, and $\mathsf{FPSPACE}_{poly}[\mathcal{S}]$ in terms of definability in suitable weighted logics for an arbitrary semiring $\mathcal{S}$. In particular, we prove weighted versions of Fagin's theorem (even for arbitrary structures, not necessarily ordered, provided that the semiring is idempotent and commutative), the Immerman--Vardi's theorem (originally for $\mathsf{P}$) and the Abiteboul--Vianu--Vardi's theorem (originally for $\mathsf{PSPACE}$). We also address a recent open problem proposed by Eiter and Kiesel.

math.LO↗

arXiv2023

Weighted HOM-Problem for Nonnegative Integers

The HOM-problem asks whether the image of a regular tree language under a given tree homomorphism is again regular. It was recently shown to be decidable by Godoy, Giménez, Ramos, and Àlvarez. In this paper, the N-weighted version of this problem is considered and its decidability is proved. More precisely, it is decidable in polynomial time whether the image of a regular N-weighted tree language under a nondeleting, nonerasing tree homomorphism is regular.

cs.FL↗