A Sardinas-Patterson Characterization Theorem for SE-codes

Transcription

1 A Sadinas-Patteson Chaacteization Theoem fo SE-codes Ionuţ Popa,Bogdan Paşaniuc Facuty of Compute Science, A.I.Cuza Univesity of Iaşi, 6600 Iaşi, Romania May 0, 2004 Abstact The aim of this pape is to pesent a Sadinas-Patteson ike chaacteization theoem fo SE-codes. The eationship between this esut and the simia one fo cassica codes is aso discussed. Intoduction The concept of a Synchonized Etension system (SE-system, fo shot), intoduced in [3], is a powefu geneative mechanism, intoducing a new point of view in foma anguage theoy and combinatoics on wods. SE-systems poved to be vey usefu when deaing especiay with nonstandad geneative devices as conditiona gammas o time-vaying mechanisms [3, 6], to give shote and eegant poofs to esuts concening egua canonica systems [5], o to obtain owe bounds fo pobems concening monadic sting ewiting systems [7]. SE-systems can aso be used to define a new concept of a code, caed in this case an SE-code (Section 5 in [3]). Howeve, not to many esuts ae known about SE-codes. In genea, the SE-code popety is undecidabe (Remak 4. in [3]), and it is decidabe fo the vey paticua cass of non-etuning SE-systems of type (f, f, f). In this pape we want to make a step futhe and give a Sadinas- Patteson ike theoem fo SE-codes. Unfotunatey, this esut can not be used diecty fo obtaining an effective agoithm fo testing the SEcode popety. The pape is stuctued as foows. Section 2 pesents basic concepts and notations egading codes and SE-codes. Ou main esut is deveoped in Section 3. Finay, we concude by some emaks concening z-codes and SE-codes.

2 2 Peiminaies on SE-systems and codes A Synchonized Etension system (SE-system, fo shot) [3] is a 4-tupe G = (V, L, L 2, S), whee V is an aphabet and L, L 2 and S ae anguages ove V. L is caed the initia anguage, L 2 the etending anguage, and S the synchonization set of G. Let G = (V, L, L 2, S) be an SE-system. Define the binay eation G, on V, caed deivation, as foows: u G, v iff ( w L 2)( s S)(, y V )(u = s w = sy v = y). As usua, G, denotes the efeive and tansitive cosue of G,.The anguage geneated by an SE-system G is L(G) = {w V v L such that v G, w} A deivation u v is caed an -deivation of u into v. Two -deivations u u 2 u n and u u 2 u m ae caed distinct if n m o thee is an i such that u i u i. An SEsystem G is caed -ambiguous if thee is a wod v L(G) having at east two distinct -deivations in G. An -deivation u u 2 u n is caed educed if it does not contain cyces, that is, thee ae no i and j such that i j and u i = u j. If an SE-system has the popety that fo any wod v thee is at most a educed -deivation of v, then it is caed weak -nonambiguous o an -code. It is aso geneicay caed an SE-code. Reca now the concept of a code []. A code ove an aphabet V is any set C V + satisfying ( u,..., u m, v,..v n C)(u...u m = v...v n u = v ) Given a set C V + we conside the foowing sets: C = { V + c C, c C}, C i+ = { V + c C, c C i} { V + c C i, c C} fo a i. The Sadinas-Patteson chaacteization theoem states that C is a code iff C C i =, fo a i. Codes ae paticua cases of SE-codes. Indeed, a set C V + is a code iff the SE-system (V, C, C, {λ}) is -nonambiguous. 2

3 3 A Sadinas-Patteson chaacteization theoem fo SE-codes In this section we sha deveop a Sadinas-Patteson chaacteization theoem fo SE-codes. We begin fist by a emak. Given a set C of wods, conside the foowing pocedue: - stat with w = λ and w 2 = λ, and etend both wods to the ight by codewods (w := w c, w 2 := w 2c 2, whee c, c 2 C), such that, at any step w is a pefi in w 2 o vice-vesa. The diffeence between w and w 2 is the wod given by w = w 2 o w 2 = w. If the pocedue above outputs = λ at some step, then C is not a code. Roughy speaking, the Sadinas-Patteson agoithm fo codes computes a the possibe diffeences that can be obtained by the pocedue above. When tying to geneaize this agoithm to SE-codes, two impotant things shoud be taken into consideation: a. we have to aow initiay w and w 2 diffeent than λ but identica because synchonization is used; b. it is not enough to keep ony the diffeence between w and w 2, because these wods can incease and decease thei size in an abitay way, and synchonization is used. The foowing eampe iustates these two emaks. Eampe 3. Conside the SE-system G = (v, L, L 2, S), whee: V = {a, b, c, d, e, f, #, $, &}, L = {a#}, L 2 = {#bc$, $def&, f&ff&, &f&, fff&&} and S = {#, $, &, f&, fff&}, and et a# #bc$ abc$ $def& abcdef& &f& abcdeff& a# #bc$ abc$ $def& abcdef& f&ff& abcdeff& be two deivations. The fist two steps in these deivations ae identic but the thid step is diffeent (as pointed out in (a)). The second deivation, fo instance, can be etended moe: abcdeff& &f& abcefff& fff&& abce& &f& abcef& showing that the wods obtained can incease and decease thei size (as pointed out in (b)). Without oss of geneaity we may assume that fo any SE-system G = (V, L, L 2, S), the anguages L and L 2 do not contain λ. Definition 3. Let G = (V, L, L 2, S) be an SE-system. A configuation of G is a stuctue [(, s ), (, )] (z,s),f whee, V, s,, s S {λ}, z L L 2 {λ}, and f {0, }. 3

4 We denote by Config(G) the set of a configuations of the SE-system G. A configuation [(, s ), (, )] (z,s),f descibes a possibe anaysis step of two distinct deivations of the same wod, as it is pictoiay epesented in the foowing figue: s The fag f is ony when the -deivations ae distinct. The pai (z, s) is used to avoid identic -deivations when f =. The oes of z, s and f ae bette epained by means of an eampe. Eampe 3.2 Conside the SE-system G = (V, L, L 2, S), whee: Let V = {a, b, c, #, $}, L = {#}, L 2 = {#ab$, $c$, c$#, #cab$, $ab$},and S = {#, $, c$}. # #ab$ abc$ c$# ab# #cab$ abcab$ # #ab$ abc$ $ab$ abcab$ be two deivations. We descibe beow possibe stages of these, showing the vaues of (z, s) and f at each step. # (z, s) = (#, #) λ f = 0 2 # (z, s) = (λ, λ) # f = 0 3 # #ab$ ab$ (z, s) = (#ab$, $) # f = 0 4 # #ab$ ab$ (z, s) = (λ, λ) # #ab$ ab$ f = 0 5 # #ab$ abc$ (z, s) = ($c$, $) # #ab$ ab$ f = 0 6 # #ab$ abc$ (z, s) = (λ, λ) # #ab$ abc$ f = 0 7 # #ab$ abc$ c$# ab# (z, s) = (c$#, #) # #ab$ abc$ f = 0 8 # #ab$ abc$ c$# ab# #cab$ abcab$ (z, s) = (c$#, #) # #ab$ abc$ f = 9 # #ab$ abc$ c$# ab# #cab$ abcab$ (z, s) = (λ, λ) # #ab$ abc$ $ab$ abcab$ f = 4

5 When two deivations ae the same, the fag f is 0. In this case, (z, s) = (λ, λ) (as in step 6). In the net step the pai (z,s) indicates that one of the deivation is etended by z and the synchonization wod is s (as in step 7). The vaue of (z, s) emains unchanged in the net step if the same deivation is etended (as in step 8). The fag f emains 0 whie the deivations ae identica up to the cuent o pevious step. The net definition gives the tansition eation between configuations of an SE-system. Due to ou definition of a configuation, tweve cases ae to be consideed. They ae not difficut at a, and ae the foowing. Definition 3.2 The passing fom a configuation to anothe one is given by a binay eation on Config(G). We define this eation by consideing a geneic configuation [(, s ), (, )] (z,s),f and showing a possibe steps fom it:. [(, s ), (λ, s )] (λ,λ),0 s [(, s ), (, s 2)] (s,s 2 ),0, whee = s 2. s s s s s 2 2. [(, s ), (λ, s )] (λ,λ),0 s [(, s ), (, s )] (s,s ),0, whee = and s =. s s s 3. [(, s ), (, )] (z,s),0 [(, s ), (, s 2)] (z,s ),f, whee = s 2 and s s { ((z, s ), f ((λ, λ), 0), if = λ and s = s 2 ) = ((z, s), ), othewise s s s 2 s 4. [(, s ), (, s 2 2)] (z,s),0 [(, s ), (, s )] (z,s ),f, whee =, s = and { ((z, s ), f ((λ, λ), 0), if = λ and s = s ) = ((z, s), ), othewise s s s 5

6 5. [(, s ), (, )] (z,s),0 s [(, ), (, s 2)] (z,s ),f, whee = s 2, = and { ((z, s ), f ((λ, λ), 0), if = λ and = s 2 ) = ((z, s), ), othewise s s s 2 s 6. [(, s ), (, )] (z,s),0 [(, s ), (, )] (z,s ),f, whee = s, = = and { ((z, s ), f ((λ, λ), 0), if = λ and s = ) = ((z, s), ), othewise s s s 7. [(, s ), (, )] (z,s),0 s [(, s ), (, )] (z,s),, whee =,s =, =. s s s. 8. [(, s ), (, )] (z,s), s [(, ), (, s 2)] (λ,λ),, whee(z, s) (s, s 2), = and = s 2. s s s 2 9. [(, s ), (, )] (z,s), s [(, ), (, s 2)] (λ,λ),, whee (z, s) (s, s 2), = and = s 2. This case is pictoiay epesented in the foowing figue: s s s 2 0. [(, s ), (, )] (z,s), s [(, s ), (, )] (λ,λ),, whee (z, s) (s, s ), =, = and s =. 6

7 s s s... [(, s ), (, )] (z,s), [(, s ), (, s 2)] (λ,λ),, whee (z, s) (, s 2), = s 2. s s s 2 2. [(, s ), (, )] (z,s), [(, s ), (, )] (λ,λ),, whee (z, s) (, s ), = and s =. s. s. s Definition 3.3 Let G = (V, L, L 2, S) be an SE-system. We define the sets Q i Config(G), fo a i 0, as foows: Q 0 = {[(λ, λ), (, )] (,s2 ),0 L, S}, Q i+ = {[(, s ), (, s 2)] (z,s ),f [(, s), (, s2)] (z,s),f Q i [(, s ), (, )] (z,s),f [(, s ), (, s 2)] (z,s ),f }. Lemma 3. Let G = (V, L, L 2, S) be an SE-system and u = u s u 2 = u 2 u m = u m s m be an -deivation in G. Fo any i, i m, the foowing popety hods: if u i = u iu i s i and u i is the maima pefi of u i such that it is a pefi in u j fo a i + j m, then thee is k, i k < m, such that u k = u is k. Poof If we assume, by contadiction, that thee is i such that the popety in Lemma does not hod, the pefi u i cannot be maima. Indeed, the popety in Lemma is obvious fo i = m. Suppose now, that the popety hods fo i 2 and we show that it hods fo i. Let u i = u i u i s i and u i = u iu i s i as in the Lemma s hypothesis. Since u i u i s i u iu i s i, we have that thee is an wod such that s i L 2 and u i u i = u iu i s i. By the induction hypothesis fo u i = u iu i s i thee is k such that u k = u is k. We have to conside fou cases: Case : if u i = λ then i satisfies the Lemma fo u i. Case 2: u i is a pope pefi in u i. This case does not hod since u i is not maima. 7

8 Case 3: u i = u i. Then the same k satisfies the Lemma fo u i. Case 4: u i is a pope pefi in u i. We show that this does not hod. In ode to do that we conside anothe two sub-cases: Case 4.: if u i u i u i then u i is not maima because u i u i is a pefi in a u j, i j m; Case 4.2: if u i < u i u i then u i is not maima because u i is a pefi in a u j, i j m. Theefoe, this case does not hod. Any step between two configuations of G, () [(, s ), (, )] (z,s),f [(, s ), (, s 2)] (z,s ),f, eads to at east an -deivation in a step in G, u v. Indeed, u = s and v = s o v = s 2 veify the popety. We sha say that the step u v is associated to (). As a esut, a computation of ength n between configuations eads to at east an -deivation of ength k, fo any k n. We sha say that each such an -deivation is associated to that computation. Now, we ae in a position to pove ou main esut. Theoem 3. Let G = (V, L, L 2, S) be an SE-system. G is an code if and ony if fo any i 0, Q i does not contain configuations [(, s ), (λ, s )] (z,s),. Poof Let us suppose that G is an -code. We pove the statement in Theoem by contadiction. Assume that thee is an i such that Q i contains a configuation [(, s ), (λ, s )] (z,s),. Theefoe, thee is a computation [( 0, s 0 ), ( 0, s 0 2)] (z 0,s 0 ),f 0 [(, s ), (, s 2)] (z,s ),f [( i, s i ), ( i, s i 2)] (z i,s i ),f i = [(, s ), (λ, s )] (z,s), whee [( j, s j ), (j, s j 2 )] (z j,s j ),f j Q j fo a j, j i. Let k = min{j j i f j = }. Fom definition 3.2 it foows that k < i. Let ( ) v 0 v... v k = k s k ( ) u 0 u... u k u k = k k s k 2 be two -deivations associated to the computation above, one of ength k and the othe of ength k and such that v p = u p fo a 0 p k. We etend these two -deivations accoding with the foowing cases: Case. If the tansition [( k, s k ), ( k, s k 2)] (z k,s k ), [( k+, s k+ ), ( k+, s k+ 2 )] (z k+,s k+ ), is of type 8-0 as in definition 3.2, then the deivation (*) can be etended in accodance with the definition of the tansition eation as foows: 8

9 if the tansition step is of type (8): ( ) v 0 v... v k = k s k 2 k+ k+ s k+ 2 ( ) u 0 u... u k u k = k k s k 2 = k+ s k+ if the tansition step is type (9): ( ) v 0 v... v k = k s k 2 k+ s k+ ( ) u 0 u... u k u k = k k s k 2 = k+ k+ s k+ 2 if the tansition step is type (0): ( ) v 0 v... v k = k s k 2 k+ s k+ ( ) u 0 u... u k u k = k k s k 2 = k+ k+ s k+ 2 Case 2. If the tansition [( k, s k ), ( k, s k 2)] (z k,s k ), [( k+, s k+ ), ( k+, s k+ 2 )] (z k+,s k+ ), is of type -2 as in definition 3.2, then the deivation (**) can be etended in accodance with the definition of the tansition eation as foows: if the tansition step is type () : ( ) v 0 v... v k = k s k 2 = k+ s k+ ( ) u 0 u... u k u k = k k s k 2 k+ k+ s k+ 2 if the tansition step is type (2) : ( ) v 0 v... v k = k s k 2 = k+ k+ s k+ 2 ( ) u 0 u... u k u k = k k s k 2 k+ s k+ By epeating the pocedue descibed above (cases and 2) with the net tansition [( k+, s k+ ), ( k+, s k+ 2 )] (z k+,s k+ ), [( k+2, s k+2 ), ( k+2, s k+2 2 )] (z k+2,s k+2 ), and so on, unti the ast tansition is pocessed, we get finay two wods, i s i and i i s i. Moeove, = λ. Theefoe, we have obtained two distinct -deivations of the same wod i s i, contadicting the fact that G is an -code. Convesey, we assume that, fo any i 0, Q i does not contain configuations [(, s ), (λ, s )] (z,s),, but G is not an -code. Then, thee ae two distinct -deivations ( ) ( ) u u 2... u n v v 2... v m fo the same wod u n = v m. Let u i = u iu i s i and v j = v jv j s j as in Lemma 3., fo a i n and j m (fo instance, u i is the maima pefi in u i such that it is a pefi in u q fo a i q n). 9

10 We sha pove that the deivations (*) and (**) can be simuated by computations between configuations. In this way, we sha obtain a natua numbe p such that [(u n, s n), (λ, s n)] (z,s), Q p, contadicting the hypothesis. We have to discuss two cases. Case. u v o, if u = v then the fist step in the deivations above uses diffeent synchonization wods. Without oss of geneaity we can assume that u v. Consideing the foowing pocedue: P0. initiaization cuent configuation [(, s ), (, )] (z,s),f := [(λ, λ), (u u, s )] (u u,s ),0 ; deivation D : u u 2... u n and deivation D2 : v v 2... v m P. Fo the deivations D of the fom, D : 2... e and fo the deivation D2 of the fom, D2 : y y 2... y f compute k < e and < f such that u k = u s k and v = v s Lemma 3.); P2. If k+ then find a computation fom cuent configuation cuent configuation [(, s ), (, s k)] (,s ), ony by tansitions of type -2, etend it by a tansition of type 8-0 [(, s ), (, s k)] (,s ), [(, s k), (αy, s )] (z,s), (as in whee α = y, then etend it futhe by tansitions of type -2 [(, s k), (αy, s )] (z,s), [(, s k), (α, s )] (z,s), and finay, etend it by a tansition of type 8-0 [(, s k), (α, s )] (z,s), [( α, s ), (β k+, s k+)] (z,s),, whee y β = u k+; update cuent configuation [(, s ), (, )] (z,s),f := [(α, s ), ( k+ k+, s k+)] (z,s), ; if u k+ v 2 then D became D : k+... e and D2 became D2 : y 2... y f. othewise, D became D : y 2... y f and D2 became D2 : k+... e. goto P; 0

11 P3. If k+ < y then find the east q > k + such that k+ y (thee is such a q i ); k became k := q (such that k + = q), goto P2. Accoding to the pocedue descibed above, the deivations (*) and (**) can be simuated by computations between configuations. In this way, a configuation [(u n, s n), (v m, s m)] (z,s),, whee u ns n = u n = v m = u nv ms m, wi be eached. Since s n = v ms m, the ast step in the computation eading to [(u n, s n), (v m, s m)] (z,s), can be changed by consideing the synchonization wod v ms m instead of s m. Theefoe, a configuation [(u n, s n), (λ, s n)] (z,s), can be eached as we. As a concusion, thee is p > 0 such that [(u n, s n), (λ, s n)] (z,s), Q p, which is a contadiction. Case 2. u = v,..., u q = v q, whee q < n and q < m. The deivations u u q and v v q ae simuated ike in the fist case with the ony diffeence that f = 0. This simuation geneates a configuation [(u q, s q), (α, s q)] (z,s),0 that wi change f = 0 into f = in the net step, accoding to the definition. Case can now be appied. Let C i = { [(, s ), (, )] (z,s),f Q i}, i 0. It is easy to see that the foowing Cooay stands. Cooay 3. Let C V +. The SE-system G = (V, C, C, {λ}) is - nonambiguous if and ony if λ / C i, i 0. As mentioned in Section 2, C is a code if and ony if the SE-system G = (V, C, C, {λ}) is -nonambiguous. In the view of cooay 3., we obtain that C is a code if and ony if λ / C i, i 0. This is an equivaent efomuation of Sadinas Patteson chaacteization theoem fo codes, mentioned in Section 2. Concusions -deivations in a SE-system G etend wods to the ight. Simiay, one can define -deivations which etend wods to the eft [3]. In fact, -deivations can be obtained by -deivations by evesing the wods. Theefoe, the esut we obtained in this pape can be easiy efomuated fo the case of -codes. In [2], a Sadinas-Patteson chaacteization theoem fo z-codes has been poposed. These codes ae not paticua cases of SE-codes. In fact, Poposition 5.2 in [3] woks ony in one diection: if the SE-system G associated to a set of wods X is weak -nonambiguous then X is a z-code, but not convesey. This can be epained as foows: eft and ight etensions fo z-codes by a wod w X ae guided by the synchonization wod w (fo eft etension) o by λ (fo ight etension); the wods emoved by eft synchonizations must be kept because they ae used fo ight synchonization (which is not the case fo SE-codes).

12 Theefoe, ou chaacteization Theoem 3. is not an etension of the chaacteization Theoem 3.6 in [2]. Theoem 3. does not ead to an effective agoithm fo testing the SE-code popety. In fact, as it has been pointed out in [3] (Remak 4.), this popety is undecidabe in genea. Finding paticua subcasses of SE-systems fo which the SE-code popety is decidabe, emains an inteesting subject fo futhe eseach. Acknowedgement The authos ae geaty indebted to F.L. Ţipea fo suggesting the subject of this pape. His numeous emaks and advices heped us to impove the caity and the eadabiity of the pape. Refeences [] J. Beste, D. Pein. Theoy of Codes, Academic Pess, 985. [2] M. Madonia,S. Saemi T. Spotei. A Geneaisation of Sadinas and Pattesons agoithm to z-codes, Theoetica Compute Science 08, 992, [3] F.L. Ţipea, E. Mäkinen, C. Apachiţe. Synchonized Etensions Systems, Acta Infomatica 37, 200, [4] F.L. Ţipea, E. Mäkinen. A Note on Synchonized Etension Systems, Infomation Pocessing Lettes 79, 200, 7-9. [5] F.L. Ţipea, E. Mäkinen. Note on SE-systems and Regua Canonica Systems, Fundamenta Infomaticae 46, 200, [6] F.L. Ţipea, E. Mäkinen, C. Enea. SE-Systems, Timing Mechanisms, and Time-Vaying Codes, Intenationa Jouna of Compute Mathematics 79, [7] F.L. Ţipea, E. Mäkinen. On the Compeity of a Pobem on Monadic Sting Rewiting Systems, Jouna of Automata, Languages and Combinatoics 7(4),