The notion of an F code is introduced as a generalization of the notion of an L code. All interrelations between ordinary codes of bounded delay, L codes of bounded delay and F codes are established. Attention is also focused on unary morphisms. Many of them are F codes

Michalik Marek

English

University of Szeged

Acta Cybernetica 14 (2000) 485-489. F codes Marek Michalik * Abstract The notion of an F code is introduced as a generalization of the notion of an L code. All interrelations between ordinary codes of bounded delay, L codes of bounded delay and F codes are established. Attention is also focused on unary morphisms. Many of them are F codes. 1 Introduction Consider a nonerasing morphism h : A* —• A*, where A is a finite alphabet. We emphasise that all morphisms discussed in this paper are nonerasing, that is, h(a) 1 (the empty word) for every a 6 A. A morphism h is a code if h is injective. We will denote by C the class of all codes. A morphism h is an L code if the function h given by h{a\a2...an) = /i(ai)/i2(a2).../in(an) (aj 6 A and hl(ai) is the i-th iterate of the morphism h) is injective. For a positive integer k and a word w, we denote by prefi.(it>) the prefix (initial subword) of w of length k. If a word w is shorter then k, then preffc(w) = w. The first letter of the word w, we denote by first(w). A morphism h is of bounded delay k if, for all words u and w, the equation prefk(h(u)) = prefk(h(w)) implies the equation first(u) = first(w). A morphism h is of bounded delay if it is of bounded delay k, for some k. A morphism h is of weakly bounded delay k if, for all words u and w, the equation pref k(h{u)) = preik(h(w)) implies the equation first(u) = first(w). If for all i > 0, the equation pref fc(^/i(w)) =pref k{ti-h{w)) implies the equation first(u) = first(w), then h is of strongly bounded delay k. In general, h is of weakly or strongly bounded delay if it is so for some k. A morphism 'Faculty of Applied Mathematics, University of Mining and Metallurgy, Al. Mickiwicza 30, 30-059 Cracow, Poland, e-mail: grmichal@kinga.cyf-kr.edu.pl 485 486 Marek Michalik h is of medium bounded delay if. for some recursive function / and all i >0, u and w, the equation pref f { i )(h*h(u)) = Pref/(i) [hlh(iu)) implies the equation first(u) = first(?u). Morphism h is a prefix code if for every dif-ferent words u, w h(u) is not a prefix of h(w). We will denote by L, B, W, S, M, P the corresponding classes of the morphisms. The next diagram due to [1] shows all inclusion between the classes introduced. The arrow stands for strict inclusion. 2 F codes From now on, / , g denote functions N —¥ N. We say that f < g if there exists n0 £ N such that f(n) = g(n) for n < no and / (no ) < g{no)-We will use the symbol hj to denote function hj : A* A* given by /¿/(moa-on) = h^l\a1)hi^{a2)...h^n\an) (at £ A,h^l](ai) is the / ( i ) -th iterate of the morphism h) We call the morphism h an F code if there exists a minimal function / : N —» N such that hf is injective. We will denote by F the class of all F codes. It is easy to see that every L code is an F code. If function h is injective then there exists a minimal function / such that h := hf is injective. Thus we conclude that every L code is an F code. We show that F-L^D. . Lemma 2.1 Let A = { a i , . . . , an},h : A* —• A*, mink:= min{||/ifc(ai)| - |/ifc(aj)||J|/i'!(ai)| : i ^ j\i,3 e {1 , . . . , n } } , maxfc := max{|/ifc(oj)| : i £ { 1 , . . . , n } } . If for each n € N there exists k, such that minfc > n then we can define the function f as follows / ( 1 ) = min{fc : mm*, > 0}, := maxy^j, Vi £N f(i + 1) := min{fc : min^ > di}, dj+i := m a + d i The function hf is injective. Proof. It suffices to prove that different words have different length. The proof of this is by induction on word length. By the definition of / (1 ) we have ||/i/(aj)| — F codes 487 ¡Ma, )|| > 0 for all a¿,aj G A,i ^ j. Let w = ai...ak,u = a[ ... a'ka'k+1. It is clear that \hf(u)\ > |h/(*+1)(a'fc+1)| > dk > 1^/(^)1- The proof is completed by showing that for all u, w G A, |w| = |w;| = k +1, it holds that ||/i/(u)| — |/i/(w)|| > 0. Consider w = a\ ... ak+i,u = a[... a'k+1 and a¿ + i ^ a'k+i- We see at once that ll/i^+^K+OI - \h«k+1Ha'k+l)\\ > dk,\hf(ai...ak)\ < dk,\hf(a'1:..a'k)\ < dk] thus \\hf(ai .. .a f c + 1 )| - \hf(a[.. ,a'fc+1)|| > 0 and finally ||/i/(u)| - \hf(w)\\ > 0. • Lemma 2.2 Let A = { a i , . . . ,an},h : A* A*, min f c := min{||/i'!(o¿)| - 1 ^ ( ^ ) 1 1 , 1 ^ ( ^ ) 1 : i ¿ j\i,j G { l , . . . , n } } . If the sequence min¿ is not bounded then morphism h is an F code. Proof. Let / be defined as in lemma 2.1. There exists a minimal function g such that hg is injective, thus h G F. • Theorem 2.3 The class L is strictly included in the class F. Proof. Every L code is an F code. Let h:{a, b}* —> {a, 6}* be given by h(a) = a 2 , h(b) = a6. Morphism h is not an L code (h(aa) = h(b)). We have min^ = 2k, hence by lemma 2.2, h G F. • Remark 2.4 For the morphism h:{a, &}* —> {a, b}* given by h(a) = a2 , h(b) = a6 the function f(i) = 2i — lisa minimal function such that hf is injective. Proof. To prove this, we observe that for every function p(i) = nt such that hp is an injective function we have V¿ ^ j (n¿ ^ n¿) A (|n¿ — nj\ ^ 1) ( as a consequence of hn(ab) = hn(ba) and hn(a)hn+1(a) = hn{b)), hence p is minimal. • It is easy to check that if h : A* —> A* is an F code then h\A : A —> A* is injective. The reverse implication is not true. Let A = {a, b, c, d} and h be given by h(a) = b, h(b) = a, h(c) = bd: h(d) = d. Function h\A is injective. For every / : TV TV, hf(ad) = / i / ( c ) , hence h i F. Similarly to L codes we have Remark 2.5 The composition of two F codes is not necessarily an F code. Proof. Consider the morphisms g and h defined by g(a) = ab,g(b) = ba,h(a) — a2,h(b) = a. Clearly g is a code and, hence, an F code, h is an L code. However, the composition h o g = a3 is not an F code. • Theorem 2.6 Classes F and W are incomparable. Proof. It suffices to show that W £ F. Consider a morphism h given by h(a) = edb, h(b) = b2,h(c) = deb,h(d) = a,h(e) = a3, then h G W (see [1]). For all i > 2, hl(a) = hl(c) and h(ddd) = h(e), hence h # F. From this we conclude that W and F are incomparable. • 488 Marek Michalik Now we can redraw the diagram as follows: C L « - F If we restrict our considerations to the class of morphisms h : {a, 6}* —¥ {a, 6}* we obtain Theorem 2.7 B = C = M = W and L -Prooj. Let h(a) = ba, h(b) =b2. he P\S (see [1]). Morphism h(a) = a, h(b) = ab is a code but not a prefix code. Morphism h(a) = a2, h(b) = a is an element of L — C. From theorem 2.3 we obtain F \ L 0. To complete the proof it suffices to show that C C B and W C C. (C C B) Every prefix code is of bounded delay. Let h(a) — x, h(b) = xy, x,y £ A+,x^y. Morphism h is of bounded delay k = 2\x\ + |j/|. (W C C) Let h i C, then there exist n,m £ N such that h(a) = xm, h(b) = xn thus h $ W . • 3 F codes and the unary morphism Theorem 3.1 Let A = { a i , . . . , an}, a £ A, h : A* {a}* , minfc := min{||/ifc(ai)| - 1^(^)11,\hk{ai)\ : i ± j-i,j £ { l , . . . , n } } . The unary morphism h is an F code if and only if the sequence min^ is not bounded. Proof. If sequence min^ is not bounded then from lemma 2.1 h e F. Consider h£ F. Suppose that there exists M £ N such that for all A: 6 N min^ < M. The morphism h is a nonerasing morphism, thus for all ai £ A and n £ N we have jhn(ai)| < |/in+1(ai)l- I f minfc < M then 3n0 £ NVi > n0Vi £ {1 , . , . , n } 1^(^)1 = |/ii+1(ai)|. We have a? = hl{ai) = ht+l{ai) = / j ^ K ) ) = Hap) for some p £ N. This implies h(a) = a and finally h(ai) = h(a\h^) which contradicts h £ F. • Remark 3.2 The last theorem is not true for arbitrary morphism. Proof. Let h(a) = a, h(b) = ab, h(c) = b. For every w, ui, u2, u3 £ {a, b, c}* h(wbui) ^ h(wcu2), h(wbui) h{wauz), h(wcu2) ^ h(wauz). To show this we observe that Vu3,v,w £ A* Vk £ N pref f c {ahH+ 1 ( / i ( i i3)) } £ {aM + 1 6u, a^bv}. Thus h is an L code, but mim = 0 and Vfc > 1 minfc = 1. • F codes 489 Corollary 3.3 It is decidable whether the unary morphism is an F code. The morphism h is an almost L code if and only if h is not an L code and 3t G NVffi,u G A*, first(w) ^ first(u) (h(w) = h\u) (\w\ = t V |u| = t)) Remark 3.4 F \ almost L ^ 0. Proof. Let h(a) = a3, h{b) = a6, h{c) = a2, then hn(a) = a 3" , hn{b) = a6 '3""1 , hn{c) = a2 '3" , minn = 3 n _ 1 . From lemma 2.2 we obtain h G F. For every w G A* h(aaw) = h(bcw), thus h $ almost!/ . • Let U be the class of all unary morphisms h : {a, b} —• {a}* such that h(a) = an , h{b) = ar, n ^ r, n > 2. Theorem 3.5 F D U = (L U almost L) D U Proof. It is clear that if n = r or n = 1 then h is neither an F code nor an almost L code. If n ^ r and n > 2 then the sequence min* = min{|nfc— r-nk~x\, nk, r -n f c _ 1 } is not bounded. This implies h G F. Every unary morphism such that h(a) = an, h(b) = ar, n ^ r, n > 2 is either an L code or an almost L code (see [3]). • References [1] H.A. Maurer, A. Salomaa, D. Wood, Bounded delay L codes (Theoretical Computer Science 84 (1991) 265-279) [2] J. Honkala and A. Salomaa, Characterization results about L codes (Infor-matique theorique et Applications vol. 26, n°3, (1992) 287-301) [3] H.A. Maurer, A. Salomaa, D. Wood, L codes and number systems (Theo-retical Computer Science 22 (1983) 331-346) Received January, 1999 

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