Distinction Graphs and Graphtropy: A Formalized Phenomenological Layer
  Underlying Classical and Quantum Entropy, Observational Semantics and
  Cognitive Computation
  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."
