FINITE COMPLETION OF COMMA-FREE CODES. Part II
Title Author(s) Citation Issue Date URL FINITE COMPLETION OF COMMA-FREE CODES (Part II) (Algebraic Systems, Formal Languages and Conventional and Unconventional Computation Theory) Lam, Nguyen Huong 数理解析研究所講究録 (2004), 1366: 129-140 2004-04 http://hdl.handle.net/2433/25370 Right Type Textversion Departmental Bulletin Paper publisher Kyoto University 数理解析研究所講究録 1366 巻 2004 年 129-140 129 FINITE COMPLETION OF COMMA-FREE CODES Part II NGUYEN HUONG LAM* Hanoi Institute of Mathematics P.O.Box 631, Bo Ho, 10000 Hanoi, Vietnam Abstract This paper is a sequel to an earlier paper of the present author, in which it was proved that every finite comma-free code is embedded into a sO-called (finite) canonical comma free code In this paper, it is proved that every (finite) canonical comma-free code is embedded into a finite maximal comma-free code, which thus achieves the conclusion that every finite comma free code has finite completions Keywords Comma-free Code, Completion, Finite Maximal Comma-ffee Code \S Introduction, This paper continues the previous one of the present author [L] Taken as a whole, they represent a solution to the problem of finite completion of comma-free codes $\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}_{0}^{\mathrm{i}\mathrm{n}}\mathrm{t}\mathrm{h}_{\mathrm{e}\mathrm{s}}^{\mathrm{i}\mathrm{s}}\mathrm{c}_{\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{h}\mathrm{e}}^{1\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{i}}\mathrm{s}_{\mathrm{S}}$ $\mathrm{S}\mathrm{o}_{\mathrm{o}\mathrm{m}\mathrm{e}}^{\mathrm{m}\mathrm{e}\mathrm{c}}1\mathrm{a}_{\mathrm{t}\mathrm{e}}^{\mathrm{s}}$ $\mathrm{m}^{0}\mathrm{n}\mathrm{g}\mathrm{p}\mathrm{T}_{\mathrm{y}}^{\mathrm{o}\mathrm{b}}1_{\mathrm{a}}^{\mathrm{e}}\mathrm{m}\mathrm{s}$ in For (finite) prefix codes the problem is easy (positive answer), but for finite codes in general, the answer is negative and the argument is more sophisticated (see Restivo [R] situation is same for finite bifix codes: there exist finite or Berstel and Perrin bifix codes which are not included in any finite maximal bifix code [BP] More on the positive side we can mention finite iffix codes [IJST] and we can also prove that every finite outfix code is included in a finite maximal outfix code (a set $X$ is an outfix code provided $uv$ , $uxv\in X$ implies $x=1$ for any words $u,v$ , ) As for comma-ffee code, in [L] we proved that every finite comma-free code is included in a sO-called (finite) canonical comma-free code and in this paper we shall prove further that every finite canonical code is included in a finite maximal commafree code Thus we add one more class of codes having a positive ansewr to the finite completion problem This paper is organized as follows: In the next two sections we review some background and prove several simple technical statements which are almost folklore and will be used in later constructions After that we prove an instrumental proposition, words If which enable us to make a ramification respective to the set of sO-called this set is finite (in \S 4) the completion is straightforward Else, if infinite, this set contains a “short” -word with rich properties and starting from this word we construct $\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{y}\mathrm{o}\mathrm{f}^{\mathrm{c}\mathrm{o}\mathrm{m}}\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{m}\mathrm{o}\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}_{\mathrm{s}[\mathrm{g}_{\mathrm{p}}\mathrm{j}\mathrm{a}_{\mathrm{h}\mathrm{a}}^{\mathrm{C}}\mathrm{o}\mathrm{d}_{\mathrm{h}\mathrm{a}}^{\mathrm{e}}\mathrm{o}\mathrm{f}}^{\mathrm{p}1\mathrm{e}}\mathrm{n}$ $[\mathrm{B}\mathrm{P}]).\mathrm{T}\mathrm{h}\mathrm{e}$ $x$ $\mathrm{i}\mathrm{l}\mathrm{r}$ $\mathrm{i}\mathrm{l}\mathrm{r}$ finite maximal comma-free codes, more or less explicit, that all contain the original comma-free code (in \S 5) \S Notions and Notation We briefly specify our standard vocabulary and state some prerequisites denotes the set of words on including the Let be a finite alphabet Then denotes the set of non-empty words For subsets of words empty word and as usual we use interchangeably the plus and minus signs to denote the union and difference of them, besides the ordinary notation The set of words is equipped with the concatenation as product with the empty $A$ $A^{*}$ $\mathit{4}^{+}$ $*$ -mail: nhlarn Gthevinh.ncst $\mathrm{E}-$ $\mathrm{a}\mathrm{c}$ $A$ 130 word as the unit For subsets $X$ and $X’$ $XX’=\{xx’ : of $A^{*}$ we denote x\in X, x’\in X’\}$ $X^{0}=\{1\}$ $X^{i+1}=X^{i}X$ $i=0,1,2$ , , $X^{*}= \bigcup_{i>0}X^{i}$ $X^{*}=$ ,$J_{i\geq 0}Xi$ $\ldots$ Our subject-matter is comma-free codes which are defined as follows [S] DEFINITION 2.1 A subset $X\subseteq A^{+}is$ said to be a comma-free code $ifX^{2}\cap A^{+}XA^{+}=\emptyset$ aximal if it is not a proper subset of any other commaAcomma-ffee A comma-ffee code is called maximal is amaximal a comma-free a maximal comma-free code containing it of Acompletion completion A ffee code every always has completions comma-free comma-ffee code lemma, In view of Zorn’s $m$ EXAMPLE 2.2 Every primitive word constitutes a comma-free code This means that for a primitive word , $p^{2}=up^{2}v$ implies $u=1$ or $v=1.$ $p$ * $u\{u,v\}^{*}$ v\}$ and $\{u, v\}$ ’ frequently the following result (Fine and Wilf): If $u\{u, We shall use ffequently $uv=vu$ , then have a common left factor of length at least $|u|+|v|$ , in particular, if $uv=vu,$ and are copowers Comma-ffee codes are closely connected to the notion of overlap We say that two words tt and , not necessarily distinct, overlap if $\{u, v\}^{*}$ $|u|+|\mathrm{t}$ $u$ $|$ $v$ $u$ $v$ $u=tw,$ A^{+}$ , $t\inA^{+}$ and for some non-empty words $s,t\in $s$ $v=ws$ $w\in A^{+}$ , or equivalently, $us=tv$ We call an overlap, and for some non-empty words , such that a right border and a left border of the two overlapping words , We say also that self-Overlaps if and overlap, that is, overlaps itself A right (left) border of a set $X$ is a right (left, resp.) border of any two overlapping words of $X$ We denote the sets of right and left borders of $X$ by $R(X)$ and $L(X)$ , respectively With each conuna-ffee code $X$ we associate the following set, which plays a central role in our treatment $s$ $t$ $u$ $u$ $u$ $w$ $|s|6K\geq 6m.$ , $\mathrm{a}_{\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{a}^{X}\{!\mathrm{o}\mathrm{m}\mathrm{t}\mathrm{h}\mathrm{e}1\mathrm{a}\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{e}.\mathrm{T}\mathrm{h}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{b}\mathrm{o}\mathrm{r}_{\mathrm{d}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{a}\mathrm{r}}^{\mathrm{a}\mathrm{n}\mathrm{d}v}\mathrm{e}}\mathrm{o}\mathrm{W}\mathrm{S}|y|\mathrm{f}\mathrm{o}\mathrm{r}g\mathrm{o}\mathrm{e}\mathrm{s}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{a}\dot{\mathrm{m}}$ 138 (c) Put $p=vg.$ So is both an L- and an -good word and $|p|>3|g|>9K\geq 9m.$ It may self-Overlap only with borders longer than $6m$ If for almost all 1-words $w\in E_{l}(X)$ , either $wp$ contains a factor in $X$ or contains has only a finite number of good words then we are done, the comma-free code (of course, the hypotheses can be effectively tested) , we can complete it at least by trial Otherwise we can choose (again, effectively) an 1-word $q\in$ Ei(X) with ($7|\geq 2|p|$ such that $qp$ avoids $X+p$ and does not contain any occurrence of other than the last $\mathrm{R}$ $p$ $w$ $X\mathit{1}$ $p$ $p$ $|$ $p$ $q$ one By Lemma 3.6 integer satisfying $qp^{i}$ is primitive for all positive integers $i$ We choose a positive $n$ $(n-1)|p|>|q|+6N.$ Note that $n>2.$ We have first REMARK 5.1 It is routine to check that Let , for every $G_{i}$ $i=0,1$ , $\ldots$ $qp^{n+1}$ is a good word for $X$ , $n-$ l, be the set consisting of words of the form $up^{i}qp^{n}$ satisfying the following conditions (i) $|u|\geq|p|$ (i) is an 1-word and up avoids $X:u\in (iii) is not a right or left factor of is primitive $up^{i}qp^{n}\in Q.$ (iiii) $u$ E_{l}(X)$ , $up\not\in A^{*}XA^{*}$ $u$ $p$ $up^{i}qp^{n}$ We have a few preliminary remarks REMARK 5.2 Since $|p|>9m>m$ and is primitive, are not m-sesquipowers , all words of $p$ $|q|\geq 2|p|$ and $p$ is not a factor of $G_{i}$ $\mathrm{g}$ REMARK 5.3 All words of REMARK 5.4 All words of is an -good word $G_{i}$ avoid $G_{i}$ are $X$ $\mathrm{i}\mathrm{l}\mathrm{r}$ and are not factors of words, $G_{i}\subseteq E(X)$ $X^{2}$ , because $u$ is an 1-word and $p$ $\mathrm{R}$ has another occurrence of , apart from the last one, then REMARK 5.5 If it must occur in up if $i>0$ and in $uq$ if $i=0.$ This is because $|q|\geq 2|p|$ , does not contain , $n>2$ and is primitive $up^{i}qp^{n}$ $p^{n}$ $q$ $p$ $p$ These remarks give rise to the following assertion PROPOSITION 5.6 (g) Every word of are not factors of ords of (gg) All words $G_{i}$ $G_{i}$ $w$ ord for $X$ is a good word $\mathrm{w}$ $p^{n}qp^{n}$ Next, we define the set $H$ as follows: $H$ consists of the words of the form satisfying satisffing 2|p|>|p|)$ $|v|\geq|q|(\geq 2|p|>|p|)$ (j) $|v|\geq|(\mathrm{j}|(\geq $vp\in E_{l}(X)$ Ei(X) 1-word: $vp\in$ (jj) is 1-word and $vp$ avoids $X$ , in other words, $vp$ is l-wOrd: le ft factor of , is not a right factor of (jjj) is not a right or left is primitive: $vp^{n}\in Q.$ (jjjj) $vp^{n}$ $v$ $v$ $p$ $q$ $v$ $vp^{n}$ It is routine to verify that the counterparts of Remarks 5.2 –5.4 and Proposition al valid for $H$ (instead of ) Also, by the similar reasons, we have 5.6 are also $0$ $G_{i}$ 137 REMARK 5.7 If must be one in $vp$ $vp^{n}$ has another occurrence of different bom the last one, then it $p^{n}$ $\mathrm{b}^{1}\mathrm{e}\mathrm{t}$ $\overline{G}_{i}=G_{i}-A^{+}G_{i}$ $\overline{H}=H-A^{+}H$ and $H$ The following proposition says that the as the sets of “minimal” words of are finite and “minimal” words are of bounded length, hence $G_{i}$ $\overline{H}$ $\overline{G}_{i}$ and if is is a -word with $n>i\geq 0$ , PROPOSITION 5.8 (i) If , in hence in factor has a right not a right factor of then and if both , are not right factors of is an -word with (ii) If $H$ has a right factors in , hence in then $lr$ $wp^{i}qp^{n}$ $wp^{i}qp^{n}$ $w$ $lr$ $wp^{n}$ $|\mathrm{t}\mathrm{p}|\geq 6N+|p|$ $G_{i}$ $|\mathrm{r}\mathrm{p}|\geq 6N+|q|$ Proof, (i) Since $|\mathrm{t}\mathrm{P}|\geq 6N+|p|$ and $\mathit{4}l\mathit{1}$ $w’\in A^{*}$ $w$ $q$ $\overline{H}$ $wp^{n}$ where $p$ $p$ $\overline{G}_{i}$ , $|w_{0}|=|p|$ , $X$ is $N$ -canonical, we can write $=w’w_{6}w_{5}w_{4}w_{3}w_{2}w_{1}w_{0}$ $|w_{j}|$ $\leq N$ $|w_{j}|\leq and $w_{j}\ldots w_{1}w0p^{i}qp^{n}$ $UJ_{j}$ $\mathrm{f}\mathrm{f}_{1^{\mathrm{j}\mathrm{j}7}}\mathrm{o}p^{i}qp^{n}$ a -word) for $j=1$ , ,’ In view of Proposition 3.2, there exist 1-word (hence alr-word) is an 1-word(hence integers two different $1\mathrm{r}$ $\ldots$ $1\leq s\leq 3