The p-adic Theory of Automata Functions

Abstract

This is mostly a survey paper on the p-adic (and wider, an ulrametric) theory of automata functions, though sketch proofs are given in a few cases. In the paper, by the automaton we mostly mean a transducer, i.e., a sequential machine which maps symbols of a finite input alphabet to symbols of a finite output alphabet so that any output symbol depends on corresponding input symbol and on current state of the machine, whereas any input symbol changes current state of the machine. Therefore an automaton whose input and output alphabets consist of p symbols, p a prime, produces a mapping from input infinite words to output infinite words over a p-letter alphabet. As the words can naturally be associated with p-adic integers, the mapping produced by the automaton is a function whose domain and range are p-adic integers. It can be shown that the function is necessarily 1-Lipschitz w.r.t. p-adic metric; and moreover, any 1-Lipschitz mapping from p-adic integers to p-adic integers is a mapping associated with some automaton. Therefore one can study behaviour of automata by studying dynamics of corresponding 1-Lipschitz mappings, the automata functions. The paper focuses on dynamical (especially, ergodic) and other properties of automata functions as functions from p-adic integers to p-adic integers. Settings of problems considered in the paper are mostly motivated by (or related to) applications in computer science, cryptography, pseudorandom numbers, digital economy (smart contracts modelling), physics, quantitative biology, etc. Basically the paper is a demonstration of what can be called an ultrametric approach to the phenomenon of causality; that is why we also briefly touch the problem of automata over continuous time although automata over discrete time constitute main body of the paper.