Logical characterizations of algebraic circuit classes over integral domains

IF 0.4 4区 计算机科学 Q4 COMPUTER SCIENCE, THEORY & METHODS Mathematical Structures in Computer Science Pub Date : 2024-05-13 DOI:10.1017/s0960129524000136
Timon Barlag, Florian Chudigiewitsch, Sabrina A. Gaube
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Abstract

We present an adapted construction of algebraic circuits over the reals introduced by Cucker and Meer to arbitrary infinite integral domains and generalize the $\mathrm{AC}_{\mathbb{R}}$ and $\mathrm{NC}_{\mathbb{R}}^{}$ classes for this setting. We give a theorem in the style of Immerman’s theorem which shows that for these adapted formalisms, sets decided by circuits of constant depth and polynomial size are the same as sets definable by a suitable adaptation of first-order logic. Additionally, we discuss a generalization of the guarded predicative logic by Durand, Haak and Vollmer, and we show characterizations for the $\mathrm{AC}_{R}$ and $\mathrm{NC}_R^{}$ hierarchy. Those generalizations apply to the Boolean $\mathrm{AC}$ and $\mathrm{NC}$ hierarchies as well. Furthermore, we introduce a formalism to be able to compare some of the aforementioned complexity classes with different underlying integral domains.
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积分域上代数电路类的逻辑特征
我们介绍了卡克和米尔引入的对任意无限积分域的实数代数回路的改编构造,并针对这种情形推广了 $\mathrm{AC}_{\mathbb{R}}$ 和 $\mathrm{NC}_{\mathbb{R}}^{}$ 类。我们给出了一个类似于伊默曼定理的定理,它表明对于这些经过调整的形式主义,由恒定深度和多项式大小的电路决定的集合与由一阶逻辑的适当调整定义的集合是相同的。此外,我们还讨论了杜兰、哈克和沃尔默对有保护谓词逻辑的广义化,并展示了 $\mathrm{AC}_{R}$ 和 $\mathrm{NC}_R^{}$ 层次的特征。这些概括也适用于布尔 $\mathrm{AC}$ 和 $\mathrm{NC}$ 层次。此外,我们还引入了一种形式主义,以便能够比较上述一些具有不同底层积分域的复杂性等级。
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来源期刊
Mathematical Structures in Computer Science
Mathematical Structures in Computer Science 工程技术-计算机:理论方法
CiteScore
1.50
自引率
0.00%
发文量
30
审稿时长
12 months
期刊介绍: Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.
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