Separating OR, SUM, and XOR Circuits

Magnus Find; Mika Göös; Matti Järvisalo; Petteri Kaski; Mikko Koivisto; Janne H. Korhonen

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arXiv · 1304.0513

Separating OR, SUM, and XOR Circuits

Magnus Find·Mika Göös·Matti Järvisalo·Petteri Kaski·Mikko Koivisto·Janne H. Korhonen

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Abstract

Given a boolean n by n matrix A we consider arithmetic circuits for computing the transformation x->Ax over different semirings. Namely, we study three circuit models: monotone OR-circuits, monotone SUM-circuits (addition of non-negative integers), and non-monotone XOR-circuits (addition modulo 2). Our focus is on \emph{separating} these models in terms of their circuit complexities. We give three results towards this goal: (1) We prove a direct sum type theorem on the monotone complexity of tensor product matrices. As a corollary, we obtain matrices that admit OR-circuits of size O(n), but require SUM-circuits of size Ω(n^{3/2}/\log^2n). (2) We construct so-called \emph{k-uniform} matrices that admit XOR-circuits of size O(n), but require OR-circuits of size Ω(n^2/\log^2n). (3) We consider the task of \emph{rewriting} a given OR-circuit as a XOR-circuit and prove that any subquadratic-time algorithm for this task violates the strong exponential time hypothesis.

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Magnus Find, Mika Göös, Matti Järvisalo, Petteri Kaski, Mikko Koivisto, Janne H. Korhonen. 2013-04-23. Separating OR, SUM, and XOR Circuits. https://arxiv.org/abs/1304.0513

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