Uria-Albizuri Jone, Carmantini Giovanni Sirio, Beim Graben Peter, Rodrigues Serafim
Department of Mathematics, University of the Basque Country, Leioa, Spain.
foldAI, Munich, Germany.
Cogn Neurodyn. 2024 Dec;18(6):3291-3307. doi: 10.1007/s11571-023-09977-5. Epub 2023 May 31.
Computational modeling of neurodynamical systems often deploys neural networks and symbolic dynamics. One particular way for combining these approaches within a framework called leads to neural automata. Specifically, neural automata result from the assignment of symbols and symbol strings to numbers, known as Gödel encoding. Under this assignment, symbolic computation becomes represented by trajectories of state vectors in a real phase space, that allows for statistical correlation analyses with real-world measurements and experimental data. However, these assignments are usually completely arbitrary. Hence, it makes sense to address the problem which aspects of the dynamics observed under a Gödel representation is intrinsic to the dynamics and which are not. In this study, we develop a formally rigorous mathematical framework for the investigation of symmetries and invariants of neural automata under different encodings. As a central concept we define for such systems. We consider different macroscopic observables, such as the mean activation level of the neural network, and ask for their invariance properties. Our main result shows that only step functions that are defined over those patterns of equality are invariant under symbolic recodings, while the mean activation, e.g., is not. Our work could be of substantial importance for related regression studies of real-world measurements with neurosymbolic processors for avoiding confounding results that are dependant on a particular encoding and not intrinsic to the dynamics.
神经动力学系统的计算建模通常采用神经网络和符号动力学。在一个名为神经自动机的框架内，将这些方法结合起来的一种特定方式会产生神经自动机。具体而言，神经自动机是通过将符号和符号串赋值给数字（即哥德尔编码）而产生的。在这种赋值下，符号计算由实相空间中状态向量的轨迹来表示，这使得能够对真实世界的测量值和实验数据进行统计相关性分析。然而，这些赋值通常是完全任意的。因此，探讨在哥德尔表示下观察到的动力学的哪些方面是动力学固有的，哪些不是，是有意义的。在本研究中，我们为研究不同编码下神经自动机的对称性和不变量建立了一个形式上严格的数学框架。作为一个核心概念，我们为这类系统定义了[此处原文缺失相关内容]。我们考虑不同的宏观可观测量，比如神经网络的平均激活水平，并研究它们的不变性属性。我们的主要结果表明，只有在那些相等模式上定义的阶跃函数在符号重新编码下是不变的，而平均激活水平等则不是。我们的工作对于使用神经符号处理器对真实世界测量进行相关回归研究可能具有重要意义，有助于避免因依赖特定编码而非动力学固有特性而产生的混淆结果。
