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arXiv · 2502.12939

Logic and Computation through the Lens of Semirings

Timon Barlag·Nicolas Fröhlich·Teemu Hankala·Miika Hannula·Minna Hirvonen·Vivian Holzapfel·Juha Kontinen·Arne Meier·Laura Strieker
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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.

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Timon Barlag, Nicolas Fröhlich, Teemu Hankala, Miika Hannula, Minna Hirvonen, Vivian Holzapfel, Juha Kontinen, Arne Meier, Laura Strieker. 2025-02-18. Logic and Computation through the Lens of Semirings. https://arxiv.org/abs/2502.12939

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