Logic and Computation through the Lens of Semirings

TL;DR

This paper explores the expressivity and computational complexity of first-order logic under semiring semantics, linking logical frameworks with algebraic circuit models across various semirings.

Contribution

It provides a logical characterization of constant-depth arithmetic circuits using an extended first-order logic applicable to all commutative, positive semirings.

Findings

Characterizes model checking complexity via semiring-based machine models.

Defines arithmetic circuit complexity in logical terms.

Extends first-order logic to capture constant-depth arithmetic circuits.

Abstract

We study the expressivity and computational aspects of first-order logic and its extensions in the semiring semantics developed by Gr\"adel and Tannen. We characterize the complexity of model checking and data complexity of first-order logic both in terms of a generalization of Blum-Shub-Smale machines and arithmetic circuits defined over a semiring. In particular, we give a logical characterization of constant-depth arithmetic circuits by an extension of first-order logic that holds for any semiring that is both commutative and positive.