Definability by constant-depth polynomial-size circuits

Q4 Mathematics
Larry Denenberg , Yuri Gurevich , Saharon Shelah
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引用次数: 65

Abstract

A function of boolean arguments is symmetric if its value depends solely on the number of 1's among its arguments. In the first part of this paper we partially characterize those symmetric functions that can be computed by constant-depth polynomial-size sequences of boolean circuits, and discuss the complete characterization. (We treat both uniform and non-uniform sequences of circuits.) Our results imply that these circuits can compute functions that are not definable in first-order logic. In the second part of the paper we generalize from circuits computing symmetric functions to circuits recognizing first-order structures. By imposing fairly natural restrictions we develop a circuit model with precisely the power of first-order logic: a class of structures is first-order definable if and only if it can be recognized by a constant-depth polynomial-time sequence of such circuits.

定深多项式电路的可定义性
布尔参数函数是对称的,如果它的值仅取决于参数中1的数量。本文第一部分部分刻画了布尔电路的等深度多项式数列所能计算的对称函数,并讨论了它们的完全刻画。(我们同时处理均匀和非均匀电路序列。)我们的结果表明,这些电路可以计算在一阶逻辑中不可定义的函数。在论文的第二部分,我们将计算对称函数的电路推广到识别一阶结构的电路。通过施加相当自然的限制,我们发展了一个具有一阶逻辑能力的电路模型:一类结构是一阶可定义的,当且仅当它可以被这种电路的等深度多项式时间序列识别。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
信息与控制
信息与控制 Mathematics-Control and Optimization
CiteScore
1.50
自引率
0.00%
发文量
4623
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