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arXiv · 2608.29094

Defining the Crick-Franklin-Watson genes of any robot (equals a finite state machine) and defining the Shannon genetic code attached to each gene. We also define the Krohn-Rhodes complexity of any regular maximal prefix code

John Rhodes
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Abstract

This paper will discuss the relationships among three pillars of mathematics: important research in codes and automata by Marcel-Paul Schutzenberger and others; random walks on finite semigroups by Persi Diaconis and others; and advanced techniques from finite semigroup theory used in proving Krohn-Rhodes complexity c is decidable by Stuart Margolis, Anne Schilling, and myself. Very surprising connections exist between the Fundamental Lemma of Complexity c (epimorphisms between finite semigroups that are one-to-one on subgroups preserve c) and coupling from the past in Markov chains, Diaconis' strong stationary time, and the Crick-Franklin-Watson genes of a finite automaton (which will be defined in the paper).

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John Rhodes. 2026-08-29. Defining the Crick-Franklin-Watson genes of any robot (equals a finite state machine) and defining the Shannon genetic code attached to each gene. We also define the Krohn-Rhodes complexity of any regular maximal prefix code. https://arxiv.org/abs/2608.29094

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