Descriptive complexity for distributed computing with circuits

Descriptive complexity for minimal time of cellular automata

Descriptive complexity may be useful to design programs in a natural declarative way. This is important for parallel computation models such as cellular automata, because designing parallel programs is considered difficult. Our paper establishes logical characterizations of the three classical complexity classes that model minimal time, called real-time, of one-dimensional cellular automata according to their canonical variants. Our logics are natural restrictions of the existential second-order Horn logic. They correspond to the three ways of deciding a language on a square grid circuit of si…

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

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 arith…

图神经网络的描述复杂性

The Descriptive Complexity of Graph Neural Networks

该研究从电路与描述复杂性角度分析GNN能力，证明不同配置的GNN分别对应非一致/一致$\text{TC}^0$类及GFO+C逻辑可定义的查询，刻画了GNN的计算表达能力边界。

布尔电路的可拓一致性

Extensional Uniformity for Boolean Circuits

在非均匀电路复杂性类C上施加扩展均匀性条件意味着简单地将C与均匀类L相交。相反，通常的内涵一致性条件要求资源有限的机器能够展示定义C的电路族中的电路。我们说（C，L）具有“一致性对偶性”，如果通过在描述定义C的电路族的连接语言的一阶描述复杂性装置中添加所谓的“L-数值谓词”，可以内涵地捕获外延一致类C \cap L。本文给出了一致性对偶性的正例和反例。

描述复杂性理论中的随机化与去随机化

Randomisation and Derandomisation in Descriptive Complexity Theory

我们从逻辑的角度研究概率复杂性类和去随机化问题。对于每一个逻辑L，我们引入一个新的逻辑BPL，有界错误概率L，这是从L中定义的复杂性类BPP，有界错误概率多项式时间，是从PTIME中定义的类似的方式。我们的主要重点在于问题的去随机化，我们证明，有一个查询是可定义的BPFO，概率版本的一阶逻辑，但不是在Cinf，有限变量无穷逻辑计数。这意味着有限模型理论的许多标准逻辑，如传递闭包逻辑和不动点逻辑，无论有无计数，都不能去随机化。同样，我们提出了一个查询的有序结构，这是在BPFO定义的，但不是在一元二阶逻辑，和一个查询的加法结构，这是在BPFO定义的，但不是在FO。这些查询中的后一个表明，AC0（有界深度多项式大小的电路）的某些统一变体不能被去随机化。这些结果与大多数标准复杂性类可以去随机化的普遍看法相反。最后，我们注意到，BPIFP+C，概率版本的定点逻辑计数，捕获的复杂性类BPP，即使在无序结构。

通过布尔网络研究神经网络的描述复杂性

Descriptive complexity for neural networks via Boolean networks

本文从描述复杂性角度用布尔网络刻画神经网络表达能力，建立二者间的多项式规模转换，并实现任意激活函数间的等价转换。