The Theory of Definite Automata

TLDR: A notion of equivalence of definite events is introduced and the uniqueness of the minimal automaton defining an event in an equivalence class is proved.

Abstract: A definite automaton is, roughly speaking, an automaton (sequential circuit) with the property that for some fixed integer k its action depends only on the last k inputs. The notion of a definite event introduced by Kleene, as well as the related concepts of definite automata and tables, are studied here in detail. Basic results relating to the minimum number of states required for synthesizing an automaton of a given degree of definiteness are proved. We give a characterization of all k-definite events definable by k+1 state automata. Various decision problems pertaining to definite automata are effectively solved. We also solve effectively the problem of synthesizing a minimal automaton defining a given definite event. The solutions of decision and synthesis problems given here are practical in the sense that if the problem is presented by n units of information, then the algorithm in question requires about n3 steps of a very elementary nature (rather than requiring about 2n steps as some algorithms for automata do, which puts them beyond the capacity of the largest computers even for relatively small values of n). A notion of equivalence of definite events is introduced and the uniqueness of the minimal automaton defining an event in an equivalence class is proved.