[Paper Review] 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 Algorithms14 references3 citations

TL;DR

This paper introduces 'distinction graphs'—graphs representing an observer's perceptual distinctions between entities—and defines 'graphtropy' as the average connection probability between nodes, formalizing a phenomenological foundation for information. It shows graphtropy generalizes logical and Shannon entropy, links to thermodynamic and quantum entropy, and models cognition, consciousness, and quantum dynamics via Dynamic Distinction Graphs, offering a unified framework grounded in observer-based distinctions.

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."

Motivation & Objective

• To establish a new conceptual foundation for information by grounding it in the distinctions made by a specific observer, rather than abstract entropy measures.
• To formalize the relationship between observer-dependent distinctions and established entropy concepts, including Shannon, logical, thermodynamic, and quantum entropy.
• To model observers as dynamic, causal networks of distinctions, enabling a phenomenological approach to cognition, consciousness, and information processing.
• To explore the implications of graphtropy for the Second Law of Thermodynamics and the Maximum Entropy Production Principle in observer-centered frameworks.
• To extend the framework to quantum systems via quantum distinction graphs and quangraphtropy, linking to quantum evolution and entanglement.

Proposed method

• Define a distinction graph as a graph where an edge between nodes a and b indicates the observer cannot distinguish a from b.
• Introduce 'graphtropy' as the average connection probability between nodes, which reduces to Ellerman’s logical entropy under specific structural assumptions.
• Formulate probabilistic graphtropy and connect it to thermodynamic entropy analogues, including maximum entropy distributions on graphs.
• Model observers as Dynamic Distinction Graphs (DDGs), incorporating causal implication links between distinctions to simulate cognitive and physical dynamics.
• Derive a quantum version of distinction graphs where distinctions can be in superposition, leading to 'quangraphtropy' as a measure of mixed state impurity.
• Link graphtropy to algorithmic information theory by showing it is monotonically related to the average algorithmic information of nodes relative to the observer.

Experimental results

Research questions

• RQ1 How can information be formally grounded in the distinctions made by a particular observer, rather than in abstract information-theoretic measures?
• RQ2 What is the relationship between graphtropy and classical entropy measures such as Shannon and logical entropy?
• RQ3 How can the Second Law of Thermodynamics be reinterpreted in terms of graphtropy and observer memory structures?
• RQ4 In what way does graphtropy relate to states of consciousness and cognitive intelligence, particularly through memory imperfections?
• RQ5 How can distinction graphs be extended to model quantum systems, including superposition and entanglement, via quangraphtropy and quantum DDGs?

Key findings

• Graphtropy generalizes logical entropy and reduces to it when distinction graphs are composed of fully connected components, establishing a clear link to Shannon entropy.
• The Second Law of Thermodynamics is rephrased as: 'The graphtropy of an observer’s memory graph will never decrease' under assumptions of consistent cognitive structure and no memory loss.
• States of consciousness with intermediate graphtropy—neither maximal nor minimal—are associated with complex intelligence, arising from memory imperfections that violate the assumptions of the Second Law.
• In algorithmic information theory, graphtropy is monotonically related to the average algorithmic information of nodes relative to the observer, linking it to computational complexity.
• Quantum distinction graphs model mixed quantum states, and 'quangraphtropy' emerges as a measure of state impurity, with causal links in quantum DDGs equivalent to unitary evolution via complex matrix multiplication.
• Dynamic Distinction Graphs (DDGs) provide a computational model of observers as evolving networks of distinctions and causal relations, enabling simulation of cognitive and physical information dynamics.