[论文解读] Distinction Graphs and Graphtropy: A Formalized Phenomenological Layer Underlying Classical and Quantum Entropy, Observational Semantics and Cognitive Computation

Ben Goertzel|arXiv (Cornell University)|Feb 2, 2019

Computability, Logic, AI Algorithms参考文献 14被引用 3

一句话总结

本文引入了‘区分图’——一种表征观察者对实体之间感知区别的图结构——并定义了‘图熵’（graphtropy）为节点间连接概率的平均值，从而为信息提供了现象学基础。研究表明，图熵可统一逻辑熵与香农熵，与热力学和量子熵相关联，并通过动态区分图建模认知、意识与量子动力学，构建了一个以观察者为基础的统一框架。

ABSTRACT

A new conceptual foundation for the notion of "information" is proposed, based on the concept of a "distinction graph": a graph in which two nodes are connected iff they cannot be distinguished by a particular observer. The "graphtropy" of a distinction graph is defined as the average connection probability of two nodes; in the case where the distinction graph is a composed of disconnected components that are fully connected subgraphs, this is equivalent to Ellerman's logical entropy, which has straightforward relationships to Shannon entropy. Probabilistic distinction graphs and probabilistic graphtropy are also considered, as well as connections between graphtropy and thermodynamic and quantum entropy. The semantics of the Second Law of Thermodynamics and the Maximum Entropy Production Principle are unfolded in a novel way, via analysis of the cognitive processes underlying the making of distinction graphs This evokes an interpretation in which complex intelligence is seen to correspond to states of consciousness with intermediate graphtropy, which are associated with memory imperfections that violate the assumptions leading to derivation of the Second Law. In the case where nodes of a distinction graph are labeled by computable entities, graphtropy is shown to be monotonically related to the average algorithmic information of the nodes (relative to to the algorithmic information of the observer). A quantum-mechanical version of distinction graphs is considered, in which distinctions can exist in a superposed state; this yields to graphtropy as a measure of the impurity of a mixed state, and to a concept of "quangraphtropy." Finally, a novel computational model called Dynamic Distinction Graphs (DDGs) is formulated, via enhancing distinction graphs with additional links expressing causal implications, enabling a distinction-based model of "observers."

研究动机与目标

• 通过将信息建立在特定观察者所作的区分之上，而非抽象的熵度量，来建立信息的新概念基础。
• 形式化观察者依赖的区分与既有的熵概念（包括香农熵、逻辑熵、热力学熵与量子熵）之间的关系。
• 将观察者建模为动态的、因果关联的区分网络，从而为认知、意识与信息处理提供现象学方法。
• 探讨图熵在以观察者为中心的框架下对热力学第二定律与最大熵产生原理的启示。
• 通过量子区分图与量子图熵（quangraphtropy）将该框架扩展至量子系统，关联至量子演化与纠缠。

提出的方法

• 将区分图定义为：若观察者无法区分节点 a 与 b，则在二者之间建立边。
• 引入‘图熵’（graphtropy）作为节点间连接概率的平均值，其在特定结构假设下可退化为埃勒曼（Ellerman）的逻辑熵。
• 提出概率图熵，并将其与热力学熵的类比形式联系起来，包括图上的最大熵分布。
• 将观察者建模为动态区分图（DDGs），通过在区分之间引入因果蕴含关系，以模拟认知与物理动力学。
• 推导区分图的量子版本，其中区分可处于叠加态，从而引出‘量子图熵’（quangraphtropy）作为混合态纯度的度量。
• 通过证明图熵与观察者相对节点的平均算法信息之间呈单调关系，将图熵与算法信息理论联系起来。

实验结果

研究问题

• RQ1 如何将信息正式地建立在特定观察者所作的区分之上，而非抽象的信息论度量？
• RQ2 图熵与经典熵度量（如香农熵与逻辑熵）之间存在何种关系？
• RQ3 在何种意义上，图熵可重新诠释热力学第二定律及其与观察者记忆结构的关系？
• RQ4 图熵如何与意识状态及认知智能相关联，特别是通过记忆缺陷？
• RQ5 如何通过量子图熵与量子动态区分图（quantum DDGs）将区分图扩展至量子系统，以建模叠加与纠缠？

主要发现

• 图熵推广了逻辑熵，并在区分图由完全连通分量构成时退化为逻辑熵，从而与香农熵建立明确联系。
• 在认知结构一致且无记忆损失的假设下，热力学第二定律被重述为：‘观察者记忆图的图熵永不减少’。
• 具有中等图熵的状态（既非最大也非最小）与复杂智能相关，其源于违反第二定律假设的记忆缺陷。
• 在算法信息理论中，图熵与观察者相对节点的平均算法信息呈单调关系，从而将其与计算复杂性联系起来。
• 量子区分图可建模混合量子态，‘量子图熵’（quangraphtropy）作为状态纯度的度量，且量子DDG中的因果链接等价于通过复矩阵乘法实现的幺正演化。
• 动态区分图（DDGs）为观察者提供了一种演化区分与因果关系网络的计算模型，从而可模拟认知与物理信息动力学。