Computing Maximal Error-detecting Capabilities and Distances of Regular Languages

A (combinatorial) channel consists of pairs of words representing all possible input-output channel situations. In a past paper, we formalized the intuitive concept of “largest amount of errors” detectable by a given language L, by defining the maximal error-detecting capabilities of L with respect to a given class of channels, and we showed how to compute all maximal error-detecting capabilities (channels) of a given regular language with respect to the class of rational channels and a class of channels involving only the substitution-error type. In this paper we resolve the problem for channels involving any combination of the basic error types: substitution, insertion, deletion. Moreover, we consider the problem of finding the inverses of these channels, in view of the fact that L is error-detecting for γ if and only if it is error-detecting for the inverse of γ. We also discuss a natural method of reducing the problem of computing (inner) distances of a given regular language L to the problem of computing maximal error-detecting capabilities of L.

[1]  S. Gubser.  Time warps , 2008, 0812.5107.

[2]  J. Berstel,et al.  Theory of codes , 1985 .

[3]  Shirley Dex,et al.  JR 旅客販売総合システム（マルス）における運用及び管理について , 1991 .

[4]  Christian Choffrut,et al.  Distances between languages and reflexivity of relations , 2002, Theor. Comput. Sci..

[5]  Grzegorz Rozenberg,et al.  Handbook of formal languages, vol. 1: word, language, grammar , 1997 .

[6]  Paul N. Schofield,et al.  State Complexity of Additive Weighted Finite Automata , 2007, Int. J. Found. Comput. Sci..

[7]  Christopher L. McAloney.  Error-Correction and Finite Transductions , 2004 .

[8]  David Sankoff,et al.  Time Warps, String Edits, and Macromolecules: The Theory and Practice of Sequence Comparison , 1983 .

[9]  Lila Kari,et al.  Descriptional Complexity of Error/Edit Systems , 2002, DCFS.

[10]  Pedro V. Silva,et al.  Maximal Error-Detecting Capabilities of Formal Languages , 2008, J. Autom. Lang. Comb..

[11]  Stavros Konstantinidis.  Transducers and the Properties of Error-Detection, Error-Correction, and Finite-Delay Decodability 1 , 2002 .

[12]  Cristian S. Calude,et al.  Additive Distances and Quasi-Distances Between Words , 2002, J. Univers. Comput. Sci..

[13]  Vladimir I. Levenshtein,et al.  Binary codes capable of correcting deletions, insertions, and reversals , 1965 .

[14]  Stavros Konstantinidis,et al.  Computing the edit distance of a regular language , 2007, Inf. Comput..

[15]  Stavros Konstantinidis.  Error-Correction, and Finite-Delay Decodability , 2002, J. Univers. Comput. Sci..

[16]  J. Howard Johnson.  Rational Equivalence Relations , 1986, Theor. Comput. Sci..

[17]  Joseph B. Kruskal,et al.  Time Warps, String Edits, and Macromolecules , 1999 .

[18]  F. Lemmermeyer.  Error-correcting Codes , 2005 .