On the error correcting capacity of finite automata

Abstract

MICHAEL A. HARRISON Department of Electrical Engineering and Electronics Research Laboratory, University of California, Berkeley, California INTRODUCTION One interesting application of sequential machines is to consider them us encoders and decoders for an information channel. In a sequence of papers, Neumann (1962, 1962, and 1964) has exploited the properties of automata to study such codings. Schiitzenberger (1964) has studied the synchronizing properties of such codes. In Neumann's work, the possibility of noise in the channel was considered. Winograd (1964) generalized the model to the case where the decoder was a finite au- tomaton which corrected a set of errors prescribed by an error relation; the idea being that the noise in the channel is describable by a relation. A related study of errors on regular sets has been made by Hartmanis and Stearns (1963). In this paper, we obtain results concerning the structure of error correcting automata. Another reason for the study of these automata is that the regular sets which they recognize are particularly simple. In fact, these events are the natural generalization of the definite events (Perles, Rabin, and Shamir, 1963; Brzozowski, 1963. NOTATIONAL CONVENTIONS The notation used in this paper is derived from Rabin and Scott (1959) and Biichi (1962). Let Z be a fixed finite nonempty set called the alphabet. Z* is the free monoid generated by Z with identity element A and the operation of concatenation. If x E Z*, then the length of x is the number of ele- ments of Z in x; of course lg (A) = 0. * This study was supported by the Air Force Office of Scientific Research Grant AF-AFOSR-639-64. Some of the material in this paper was presented at an Inter- national Colloquium on Automata Theory and Algebraic Linguistics, Jerusalem, Israel, August 24-25, 1964. 43O