Compression of Palindromes and Regularity.
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1 Compression of Plinromes n Regulrity. Kyoko Shikishim-Tsuji Center for Lierl Arts Eution n Reserh Tenri University 1 Introution In [1], property of likstrem t t view of tse is isusse n it is shown tht pge repetitions our for the mjority s very speifi struture, nmely in the form of neste plinromes. A kin of funtion CFP (Compress First Plinrome) require for n lgorithm whih extrts these strutures in liner time is introue. In this pper, we efine rewriting system R whih overs CFP n onsier reltion etween R n CFP. We opt the nme wrinkle wor inste of neste plinrome n it mens wor whih hs non-trivil plinrome s ftor. The set of ll wrinkle wors is regulr, though the set of ll plinromes is not regulr. We lso give utomt on some lphets whih ept ll wrinkle wors. Preliminries We ssume the reer to e fmilir with si onepts s lphet, wor, lnguge, regulr expression n utomton (for more etils see []). Wors together with the opertion of ontention form free monoi, whih is usully enote y Σ for finite lphet Σ. The length of finite wor w is the numer of not neessrily istint symols it onsists of n is written y w. The empty wor is enote y λ n λ = 0. For wor w = 1! n for 1,,!, n Σ, ftor of w is i! j, where 1 i j n, n the reverse w R of w is n! 1. A wor p Σ is si to e plinrome if p = p R. If plinrome p is not in Σ {λ}, the plinrome p is non-trivil, otherwise p is trivil. If wor w Σ hs t lest one non-trivil plinrome s ftor, w is si to e wrinkle. 1
2 A string rewriting system R on Σ is suset of Σ Σ. We efine reution reltion on Σ tht is inue y R is efine s follows: for every u,v Σ, u v if n only if there exists (l,m) R suh tht for some x, y Σ, u = xly n v = xmy. By, we enote the reflexive trnsitive losure of. If x Σ n there is no y Σ suh tht x y, then x is irreuile; otherwise, x is reuile. The set of ll irreuile wors with respet to is enote y IRR(R). For x, y Σ, if x R y n y is irreuile, then y is norml form for x. If, for ll w, x, y Σ, w R x n w R y imply tht there exists z Σ suh tht x R z n y R z, we sy tht R is onfluent. If for ll w, x, y Σ, w x n w y imply tht there exists z Σ suh tht x R z n y R z, we sy tht R is lolly onfluent. If there is no infinite sequene x 1 x! where x 1, x,! Σ, then R is si to e noetherin. Two string rewriting systems R n S on Σ re lle equivlene if w R z implies w S z n w S z implies w R z n then we enote R S. Compress First Plinrome CFP : Σ Σ is efine s follows (see [1]) : if w Σ is wrinkle wor, for the left most of w where Σ, Σ {λ},, there re u,v Σ suh tht w = uv n we efine CFP(w) = uv n if w Σ is non-wrinkle wor, we efine CFP(w) = w. Sine wrinkle wor w hs only finite non-trivil plinromes s ftor n w > CFP(w), then we n efine CFP (w) = CFP n (w) where n is lrge enough numer suh tht CFP n (w) = CFP n+1 (w) ( = CFP(CFP n (w)) ). 3 Regulrity of the set of ll wrinkle wors. Proposition 1. Let R n S e string rewriting systems R = {p Σ, { { }} S p is plinrome} n S = Σ, Σ λ. Then R n re equivlent. Proof) Sine R S, it is ler tht if w z for some w, z Σ, then we hve w z. On the other sie, let w z for some w, z Σ, then there exist u,v Σ, Σ n plinrome p Σ suh tht w = upv n z = uv. If p Σ {λ},
3 then w z. If p Σ {λ}, then p is written xx R y Σ {λ}, Σ n x Σ. Sine w = uxx R v uxx R v n so on, we hve w = uxx R v S uv = z. q.e.. The following lemm is well-known (see [3]). Lemm 1. If string rewriting system R is onfluent. R is noethrin n lolly onfluent, then Proposition. Let R n S e string rewriting systems R = {p Σ, { { }} p is plinrome} n S = Σ, Σ λ onfluent.. Then R n S re Proof) By Proposition 1 n Lemm 1, it is enough to prove tht S is lolly onfluent. () If w = v 1 u v where,v 1,u,v, Σ n v 1,v for some, Σ, then w u v, w v 1 u n u v u, v 1 u u. () If w = v n v {,,, } where, Σ,, Σ. Sine S, S, S, S, we hve w S v where v = when v {,, } n v = when v =. q.e.. Proposition 3. Let R' e one of string rewriting systems of Proposition. If, for w, z Σ n nturl numer n, CFP n (w) = z, then w ' z. On the other hn, if w ' z for w, z Σ, then there exist nturl numer n n z' Σ suh tht CFP n (w) = z' n w ' z'. Proof) If CFP(w) = z, it is ovious tht w ' z. If w hs no non-trivil plinrome s ftor, then we hve w = z n CFP(w) = w. We my ssume tht w is wrinkle. There exists finite sequene: w = w 0, CFP(w) = w 1,!,CFP n (w) = w n suh tht w n 1 is wrinkle n w n is not wrinkle. Then we hve sequene w w 1! w n. Sine w n is not wrinkle, hs no non-trivil plinrome s ftor n w n IRR(R'). By w n Proposition, the rewriting system R' is onfluent n then we hve w R' w n. The following orollry is ler y Proposition 3. 3
4 Corollry 1. Let R e the string rewriting system p is plinrome}. Then we hve CFP (Σ ) = IRR(R). R = {p Σ, By Corollry 1, we hve CFP (Σ ) = {the set of ll non-wrinkle wors} The following lemm is well-known (see [3]). Lemm. A string rewriting system R is finite, then IRR(R) is regulr set. By Proposition 3 n Lemm, we hve the following theorem. Theorem 1. The lnguge NW = {the set of ll non-wrinkle wors} n the lnguge W = {the set of ll wrinkle wors} re oth regulr. Proof) The lnguge NW = CFP (Σ ) is regulr n then W =(NW) is regulr. q.e.. The set of ll plinromes re ontest-free ut not regulr. W = {the set of ll wrinkle wors}= { w w hs t lest one non-trivil plinrome s ftor} is regulr. 4 Exmples. on Σ. For Σ 4, we give n utomton on Σ whih epts ll wrinkle wors Exmple 1. If Σ = {}, then NW = { w Σ w is non-wrinkle wor} is the empty set. Exmple. If Σ = {,}, then NW is the set {, } Exmple 3. If Σ = {,,}, then NW is the set { u Σ u is ftor of ()* ()* suh tht u > 1}. By the following utomton A = (Q, Σ, δ, i, F) (Figure 1), NW is epte, where Q = {i,1,,!, 9}, i Q is the initil stte n F = Q is set of finl sttes. 4
5 Figure 1 i Exmple 4. If Σ = {,,,}, then NW is the lnguge whih is epte y the following utomton A = (Q, Σ, δ, i, F) (Figure ), where Q = {i,1,,!,1}, i Q is the initil stte n F = Q is set of finl sttes. In Figure, sttes i,13,14,1,1 n the following trnsition funtions whih strt from these sttes re omitte for simpliity: :i 13, :i 14, :i 14, :i 1, :13, :13 8, :13 10, :14 11, :14 7, :14 1, :1, :1, :1 9, :1 3, :1 1, :1 4 (see Figure 3). Figure
6 Figure i Referenes. [1] Mihel Speiser, Ginlu Antonini, Aerrhim Li, Julin Sutnto: On neste plinromes in likstrem t. The 18th ACM SIGKDD Interntionl Conferene on Knowlege Disovery n Dt Mining, KDD '1, Beijing, Chin, August 1-1, 01. ACM 01: [] Peter Linz, An Introution to Forml Lnguge n Automt, Jones n Brtlett Pulishers, In., USA 00, ISBN, [3] Ronl V. Book, Frierih Otto: String-Rewriting Systems. Texts n Monogrphs in Computer Siene, Springer 1993, ISBN Center for Lierl Arts Eution n Reserh Tenri University 100 Somnouhi, Tenri, Nr 3-810, Jpn E-mil ress: tsuji st.tenri-u..jp
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