First Order Extensions of Residue Classes and Uniform Circuit

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First Order Extensions of Residue Classes and Uniform Circuit

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ortiz@arcadia.edu. Abstract. The first order logic Ring(0, +,â,

First Order Extensions of Residue Classes and Uniform Circuit Complexity Argimiro Arratia1, and Carlos E. Ortiz2 1

LSI, Universitat Politècnica de Catalunya, Barcelona, Spain [email protected] 2 Arcadia University, Glenside, PA 19038-3295, U.S.A. [email protected]

Abstract. The first order logic Ring(0, +, ∗, 0, we denote by Zm the finite residue class ring of m elements. Zm , as an algebraic structure, consists of a set of elements {0, 1, . . . , m − 1}, and two binary functions + and ∗ which corresponds to addition and multiplication modulo m, respectively. Definition 1. By Ring(0, +, ∗) we denote the logic of finite residue class rings. This is the collection of first order sentences over the set of built-in predicates {0, +, ∗}, where 0 is a constant symbol, and + and ∗ are binary function symbols. The models of Ring are the finite residue class rings Zm , and in each Zm , the 0

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A. Arratia and C.E. Ortiz

is always interpreted as the 0-th residue class (mod m), and + and ∗ are addition and multiplication modulo m. By Ring(0, +, ∗,