Codes, Simplifying Words, and Open Set Condition

In formal language theory the concept of simplifying words is considered in connection with the deciphering delay of codes (cf. BP, Section II.8], LP]). In this paper we show that transferring a deenition from the theory of iterated function systems (IFS) sheds some new light on simplifying words.: Formal languages may be considered as IFS in the Cantor space of innnite strings over a nite alphabet X (cf. Fe], S2]). If those IFS satisfy the so-called Open Set Condition (OSC) then the underlying language is a code having simplifying words. Moreover, the simplifying words of a code C are naturally subdivided into nonsubwords (of the set of code messages C) and into simplifying preexes. It can be shown that the existence of simplifying preexes for a code C is equivalent to the fullllment of a strong version of the OSC for the related IFS. 1 Preliminaries Given an alphabet X, we consider the free monoid of ((nite) words X generated by X with the concatenation as operation and with the empty word e as identity element. As is well-known, codes over X are deened as the bases of free submonoids of X , that is, C X is a code if and only if every word w in the submonoid C of X generated by C has exactly one factorization over C.