Descriptive Complexity of $\#\textrm{AC}^0$ Functions

Arnaud Durand; Anselm Haak; Juha Kontinen; Heribert Vollmer

doi:10.1016/j.jcss.2020.04.002

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arXiv · 1604.06617

Descriptive Complexity of $\#\textrm{AC}^0$ Functions

Arnaud Durand·Anselm Haak·Juha Kontinen·Heribert Vollmer

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Abstract

We introduce a new framework for a descriptive complexity approach to arithmetic computations. We define a hierarchy of classes based on the idea of counting assignments to free function variables in first-order formulae. We completely determine the inclusion structure and show that #P and #AC^0 appear as classes of this hierarchy. In this way, we unconditionally place #AC^0 properly in a strict hierarchy of arithmetic classes within #P. We compare our classes with a hierarchy within #P defined in a model-theoretic way by Saluja et al. We argue that our approach is better suited to study arithmetic circuit classes such as #AC^0 which can be descriptively characterized as a class in our framework.

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Arnaud Durand, Anselm Haak, Juha Kontinen, Heribert Vollmer. 2016-04-22. Descriptive Complexity of $\#\textrm{AC}^0$ Functions. https://doi.org/10.1016/j.jcss.2020.04.002

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