We develop a theory characterizing the fundamental capability of deep neural networks to learn, from evolution traces, the logical rules governing the behavior of cellular automata (CA). This is accomplished by first establishing a novel connection between CA and Lukasiewicz propositional logic. While binary CA have been known for decades to essentially perform operations in Boolean logic, no such relationship exists for general CA. We demonstrate that many-valued (MV) logic, specifically Lukasiewicz propositional logic, constitutes a suitable language for characterizing general CA as logical machines. This is done by interpolating CA transition functions to continuous piecewise linear functions, which, by virtue of the McNaughton theorem, yield formulae in MV logic characterizing the CA. Recognizing that deep rectified linear unit (ReLU) networks realize continuous piecewise linear functions, it follows that these formulae are naturally extracted from CA evolution traces by deep ReLU networks. A corresponding algorithm together with a software implementation is provided. Finally, we show that the dynamical behavior of CA can be realized by recurrent neural networks.

翻译：我们发展了一套理论，刻画了深度神经网络从演化轨迹中学习支配细胞自动机（CA）行为的逻辑规则的基本能力。这一理论首先通过建立CA与卢卡西维茨命题逻辑之间新颖的联系得以实现。尽管二值CA数十年来本质上是执行布尔逻辑运算，但一般CA并无此类对应关系。我们证明多值（MV）逻辑，特别是卢卡西维茨命题逻辑，构成了将一般CA刻画为逻辑机器的恰当语言。这是通过将CA转移函数插值为连续分段线性函数实现的，依据麦克诺顿定理，这些函数可导出刻画CA的MV逻辑公式。认识到深度修正线性单元（ReLU）网络实现连续分段线性函数，可推知深度ReLU网络能自然地从CA演化轨迹中提取这些公式。我们提供了相应算法及软件实现。最后，我们展示了CA的动力学行为可由循环神经网络实现。