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Maximal Prefix Codes and Subclasses of the Context-Free Languages Class. P. 121–129

Версия для печати

Section: Physics. Mathematics. Informatics

UDC

519.713

Authors

Korabel’shchikova Svetlana Yur’evna
Institute of Mathematics, Information and Space Technologies, Northern (Arctic) Federal University named after M.V. Lomonosov (Arkhangelsk, Russia)
e-mail: kmv@atnet.ru
Mel’nikov Boris Feliksovich
Togliatti Branch of Samara State University (Togliatti, Russia)
e-mail: bormel@rambler.ru

Abstract

This paper analyzes the relation between maximal prefix codes, formal language theory and alphabetic coding. Commutation conditions in the global supermonoid of the free monoid, equivalence criterion of a pair of finite languages and a range of other results, connected with infinite language iterations, are expressed in terms of maximal prefix codes. Many of these results are related to the algorithmic problems of monomial algebra (i.e. associative algebras defined by the so-called languages of obstructions). Prefix codes are mostly used in alphabetic coding, as the prefix property guarantees unambiguous decodability. Maximal prefix codes have a range of other properties: the MacMillan’s inequality becomes equality for them and all the knots of a code tree are saturated. We used the relation between the maximal prefix codes and the code trees and calculated the number of maximal prefix codes of the given power r in the q-alphabetic. In this paper we deduce the general formula and give its typical applications. The maximal prefix codes of power r over the q-alphabetic do not exist if the remainder on dividing r by q-1 isn’t equal to 1. The quotient k of r and q-1 can be explanated as the maximal quantity of tiers in the code tree and as the quantity of beams from q edges of the tree. The setup (n1, n2, n3, …, ns) represents the distribution of these beams over the tiers of the code tree. A number of unsolved problems and our conjectures of necessary conditions of commutation, requiring further verification, are given in the conclusion.

Keywords

context-free language, prefix code, code tree, maximal prefix code
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