An Algebra of Discrete Channels That Involve Combinations of Three Basic Error Types

Abstract

Recently, the author introduced a nonprobabilistic mathematical model of discrete channels, the BEE channels, that involve the error-types substitution, insertion, and deletion. This paper defines an important class of BEE channels, the SID channels, which include channels that permit a bounded number of scattered errors and, possibly at the same time, a bounded burst of errors in any segment of predefined length of a message. A formal syntax is defined for generating channel expressions, and appropriate semantics is provided for interpreting a given channel expression as a communication channel (SID channel) that permits combinations of substitutions, insertions, and deletions of symbols. Our framework permits one to generalize notions such as error correction and unique decodability, and express statements of the form "The code K can correct all errors of type ξ " and "it is decidable whether the code K is uniquely decodable for the channel described by ξ ," where ξ is any SID channel expression.

Error-detecting properties of languages

“…We now deÿne a certain class of SID-channels-see [10] for the full class of SIDchannels. Although the formal deÿnition is not required for the proofs in the present paper, it is included here so that the reader can clearly understand how these channels a ect messages.…”

“…To keep the basic deÿnitions general, we use the framework of P-channels (see [9]) restricted to the case of ÿnite words. This channel model is very general and includes the case of SID-channels which were presented in [8] and further extended in [10]. SID-channels are discrete channels represented by formal expressions that describe the type of errors permitted and the frequency of those errors.…”

Dependences related to strict binary relations

“…Then the minimal spanning sets are less related to independent sets. Secondly, Konstantinidis [9] provided a general framework of discrete channels such that one can express various error situations in a systematic manner. The notion of bounded error effects plays an important role for modelling communication channels.…”