Information Processing Letters Information Processing Letters 58 ( 1996) 297-301 Codes, simplifying words, and open set condition Ludwig Staiger ’ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQ Martin-Luther-Universit~t Halle- zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONML Wittenberg, Institut fiir Informatik, Wcinbwgweg 17, D-06120 Halle, Germany Received 12 February 1996 Communicated by T. Lengauer Keywords: Formal languages; Codes; Deciphering delay; Fractal geometry; Iterated function systems zyxwvutsrqponmlkjihgfedcbaZYXWVU 1. Introduction In formal language theory the concept of simpli- fying words is considered in connection with the deciphering delay of codes (cf. [ 3, Section II.81 and [ 71). In this paper we show that transferring a definition from the theory of iterated function sys- tems (IFS) sheds some new light on simplifying words. Formal languages may be considered as IFS in the Cantor space of infinite strings over a finite alphabet X (cf. [ 5,101) . 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 prefixes. It can be shown that the existence of simplifying zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA prefixes for a code C is equivalent to the fulfillment of a strong version of the OSC for the related IFS. 1 Email: staiger@informatik.uni-halle.de. The results presented in this paper have been obtained while the author was with Lehrstuhl fiir Informatik II, Rhein.-Westf. Techn. Hochschule Aachen, Germany. 2. Preliminaries Given an alphabet X, we consider the free monoid of (finite) words X* generated by X with the concate- nation . as operation and with the empty word e as identity element. As is well known, codes over X are defined as the bases of free submonoids of X*, that is, C G 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. An iterated function system (IFS) is an (12 + l)- tuplcZ= (M,q I,..., p,,) where (M, p) is a met- ric space and ~1,. . . , pPnarc contracting mappings on M, that is mappings pi : M ---f M satisfying zyxwvutsr p((oi(X), Pi(Y)) < c. p(n,Y> for SOmeC < 1 (cf. [ 2,4] ) . Then there is a unique compact set 8 Z kZ c M which is the solution of the equation K = i)f+Di(K). i=l K: is called the attractor of 1. An IFS is referred to as satisfying the Open Set Condition (OSC) provided there is a nonempty open subset 0 C M such that spi( 0) c c3 and pi( 0) zyxwvutsrq n cpi(O) =@fori,j E {l,...,n}andi # j,anditisre- 0020-0190/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved PI1 SOO20-0 190( 96) 00074-9