Sardinas-Patterson like algorithms in coding theory

Transcription

1 A.I.Cuza University of Iaşi Facuty of Computer Science Sardinas-Patterson ike agorithms in coding theory Author, Bogdan Paşaniuc Supervisor, Prof. dr. Ferucio Laurenţiu Ţipea Iaşi, September 2003

2 Contents 1 Introduction 2 2 Variabe ength codes Definitions Basic characterization resuts Sardinas-Patterson characterization S-P Agorithm for cassica codes S-P Agorithm for z-codes S-P Agorithm for Time-Varying codes S-P Agorithm for SE-codes Concusions 37 Bibiography 38 1

3 Chapter 1 Introduction Nowadays, with the eponentia growth in the amount of data that the human society deas with, data managing becomes a very important probem. Despite of technoogic deveopment there have aways been a probem in transmitting and storing data. Athough there have been important progresses in the quaity of transmissions one can not say that a certain transmission coud happen without errors. The technoogy of data storage is deveoping quicky, but it fais to keep up with the necessities of today s society. The eponentiay increasing amount of data echanged between computers ed to a rapid deveopment in the technoogy of detecting and correcting errors that can occur in it s transmission. We need to encode data in such a manner that we coud detect as many errors that coud occur or, if we may think that we coud encode data in order to decrease it s size in order to reduce the time of the eposure of data to transmission errors. Usuay these two ideas are combined in order to have a faster and more reiabe transmission. The origins of coding theory are in the theory of information deveoped by Shannon in the 1950s. The are severa fieds in which this theory evoved. One is error-correcting codes which is an important appication of agebra. Another one is the theory of entropy that inks codes to probabiity theory. In this paper we present an important part from the theory of codes, the probem of deciding how and when can we use a given set to encode, decode data. The probem of detecting which sets can be used to encode is an important probem in the theory of codes. This probem is soved in the case of cassica 2

4 codes by the Sardinas-Patterson characterization theorem, resut that eads to en effective agorithm for deciding if a given set is a code or not. The aim of this paper is to present how the origina Sardinas-Patterson characterization theorem evoved from cassica codes to new types of codes ike z-codes, TV-codes and SE-codes. In the first part of this paper we present basic properties and definitions of codes. We present the Shutzenberger criterion for codes, the Sardinass- Patterson criterion for codes, and resuts on codes using probabiity distributions. In the second part we present agorithms that decide for a specified set if it is a code or not. We present here how the Sardinas-Patterson characterization theorem eaded to an effective agorithm that decides if a given set is a code or not. We continue by presenting a resut obtained in [2] resut that generaizes the Sardinas-Patterson characterization theorem to z-codes. This resut aso eads to an effective agorithm for z-codes, agorithm based on an upper bound on words that might have two distinct factorizations. Net we present resut for TV-codes, obtained in [9], and the agorithm corresponding to it. The net section is dedicated to Synchronized Etension codes. The resut presented here is a Sardinas-Patterson theorem ike, resut that unfortunatey does n ead to an effective agorithm for deciding the secode property for the entire cass of se-codes. There are subcasses for which this resut has a corresponding effective agorithm. 3

5 Chapter 2 Variabe ength codes In this chapter we define some basic notions on codes. We concentrate our efforts in presenting ony the main notations and ony the most important characterizations resuts that wi be used in the foowing chapters. This basic resuts on codes are presented here ony for giving an idea on codes, especiay on variabe ength codes. For more detais the reader is directed to [1] Some important characterizations resuts presented here are the Schutzenberg criterion for codes and the Sardinas-Patterson theorem for codes. 4

6 2.1 Definitions We sha ca aphabet a finite and nonempty set A. We sha ca the eements of A etters. For every sets A and B, A B denotes the incusion of A into B and A is the cardinaity of the set A; the empty set is denoted by φ. The set of natura numbers is denoted by N and by N we denote the set N \ {0}. Let A = (A i i 1) be a famiy of sets. We denote by A i the set j i A i A j and by A the set i 1 A i. A word w over the aphabet A is a finite sequence of etters: w = (a 1, a 2,...a n ), i a i A. A word w over the aphabet A can be seen as a mapping w : {1,, n} A, where n is a natura number caed the ength of the word. The ength of the word is aso denoted by w. The empty sequence is caed the empty word and it is denoted by λ. Aso we can define λ as the function λ : A. Evidenty the ength of λ is 0. Furthermore we denote the set of a words over an aphabet A as A. We denote A + as the set A \ λ. We define the operation concatenation (or product) : A + A + by: { w(i), if 1 i w (w v)(i) = v(i w ), if w + 1 i w + v This operation is associative and thus we can write w = a 1 a 2 a n instead of: w = (a 1, a 2,, a n ). The concatenation operation can be easiy etended to A if we consider λ as the neutra eement( w λ = λ w = w ). Then (A, ) is a free monoid generated by A. 5

7 Definition Let A be an aphabet(a finite and nonempty set). A code over the set A is a subset C A such that for any n, y 1 y 2...y m, two sequences over C( i C and y j C, i n, j m) if: it foows that: 1. n = m and 2. i = y i, i n. 1 2 n = y 1 y 2 y m In other words, a set C A is a code if it has the unique descifrabiity property; that is any word in C + can be written uniquey as a product of words from C (it has a unique factorization in words from C). It foows naturay that a code C never contains the empty word λ that is why we usuay say that C is a subset of A + = A \ λ. Eampe Let A = {a, b}. C = {a, ab, ba} is not a code over A. It is easy to see that aba has two distinct factorizations over C. aba = (ab)a = a(ba) thus we can concude that C is not a code. For two sets X, Y A we can define a new set XY in the foowing way: XY = {y X, y Y }. We define inductivey C n in the foowing way: C 0 = λ A ; C n+1 = C n C. Now we introduce the is a factor of reation on words. We say that a word w A is a factor of v if there eist, y A such that v = wy. This reation is a partia order on A. Furthermore we say that w is a eft factor of v (or w is a prefi of v) if w is a factor of v and = λ(v = wy). The factor w is caed proper if w v or 1 There are authors[5] that define a code ony as a subset of A. That impies that there eists codes without the unique descifrabiity property. 6

8 in other words y λ. This reation is eft factor is a partia order on A too. We wi write w pref v whenever w is a factor of v and w < v whenever w is a proper factor of v. In a symmetric manner we define is right factor (or is suffi of) and is proper right factor. Athough we sha present some different casses of codes in a foowing section we wi define now some important casses of codes. Definition Let A φ and C A + be a code. Then: 1. C is caed bock code if a words from C have the same ength; 2. C is caed prefi code if there are not two words from C such that one is prefi of the other; 3. C is caed suffi code if there are not two words from C such that one is suffi of the other; 4. C is caed biprefi code if C is in the same time prefi and suffi code; 5. C is caed binary code if the number of eements in A is 2. There are two main fieds in which codes are used: one is for detecting and correcting errors that appear during the transmission of data and the other is data compression. Bock codes are used especiay for detecting and correcting errors. If the transmission is noiseess, that is no errors can occur in the process, we may concentrate our efforts in encoding the data as efficienty as possibe. Thus we must define a method for comparing efficiency of codes. If we consider that at one time a etter can appear with a certain probabiity we obtain a probabiity distribution for an aphabet. In other words a probabiity distribution for A is a function π : A [0, 1] with the property that a Aπ(a) = 1. In this manner if we associate an aphabet with a probabiity distribution we obtain an information source. The coupe (A, π) where A is an aphabet and π is a probabiity distribution for A is caed information source, Usuay it is convenient to etend π at A in the foowing way: π(w) = { 1, if w = λ, Π n i=1π(a i ), if w = a 1 a 2...a n. 7

9 The probabiity of C A is made up by the sum of the probabiities of the words from C. If C = φ then π(c) = 0. We obtain a vaue(π(c)) between 0 and + for the set C, vaue reative to the probabiity distribution π. This vaue is caed the measure of C reative to π. 1 For eampe the probabiity distribution π(c) =, c C is caed Card(C) the uniform distribution. 8

10 2.2 Basic characterization resuts Given a set C over an aphabet A the main question that we try to answer is: Can we determine if C is a code or not? In this section we wi give some theoretica resuts on codes, resuts that wi answer this question. First of a we wi present two simpe resuts on codes. Proposition Let C A + be a code. Then n 1, C n is a code over A. Proof The proof is very simpe. If we consider two different factorizations for the same word over C n then we can rewrite every word in those two factorizations as a product of n words from C(it foows from the definition of C n ). In this manner we obtain two different factorizations over C. This is not possibe(c has the unique descifrabiity property ). Contradiction. We concude that C n is a code. The other resut aows us to view codes as injective morphisms from an aphabet Σ into A. Thus whenever we want to codify an aphabet (more generay a set of eements) over another aphabet we must consider an injective function. Conversey the injective morphisms h : Σ A are caed codifications of Σ in A because h(σ) is a code. Proposition Let A φ be an aphabet and C A + (C φ. C is a code if and ony if h : Σ A an injective morphism such that C = h(σ), with Σ a we chosen set. Proof The proof is very simpe and it can be found in [1] Theorem The Schutzenberger criterion for codes. Let C A +, where A is an aphabet. C is a code if and ony if: 1. C ( n 2 C n ) φ 2. ( w A + )(C w C φ and wc C φ w C ) Unfortunatey this criterion for determining if a given set is a code or not doesn t ead us to an effectivey appyabe agorithm. It is impossibe to determine C n and A + in an acceptabe time. We sha present net a resut that eads to an acceptabe agorithm for deciding if a given set is a code or not. This resut is aso known as the Sardinas-Patterson theorem for codes. 9

11 First of a we must define inductivey the famiy of sets C n, aso caed the sets of reminders, as: C 1 = { A + c C : c C} C i+1 = { A + c C : c C i } { A + c C i : c C} Theorem The Sardinas-Patterson criterion for codes. Let A be an aphabet and C A +. C is a code if and ony if C C i = φ, i 1. Instead of giving a forma proof 2 we wi epain how and why we must compute C i using an eampe. Let C be a subset of {0, 1} +. C = {010, 0001, 0110, 1100, 00011, 00110, 11110, }. We say that the word w = has two distinct factorizations over C. Let us find these two distinct factorizations. 1. We begin from eft to right and we try to find 2 different words from C that are prefi in w. We have found 1 = 0001 and y 1 = We notice that 1 pref y 1. In order to continue our computation we must remember 1 which is the remainder between 1 and y 1 (y 1 = 1 1). We add the word 1 to C 1. More generay in order to determine a computations that might ead to different factorizations we must add to C 1 a such that c is in C with c C. c 2. In order to continue our computation we wi repeat the foowing step: We suppose that we are at the step i. We make the foowing remark: the reminder r from C i is the ony difference between our two distinct factorizations that we are trying to construct, that is: 2 A compete proof can be found in [1] 10

12 1 2 r = y 1 y 2 y k or 1 2 = y 1 y 2 y k r or We continue our computation by: a) finding words in C that begin with the reminders from C i ; that is continuing our computation with an k ; b) finding words in C that are prefi in the reminders from C i ; that is continuing our computation with an y. If we reach λ as a reminder it means that at that step we have two distinct factorizations ( we have started with different words from C) for the same word(based on the remark made above the ony difference between those two factorizations is λ). In our case we have reached: w = (00011)(010)(11110)(00110) = (0001)(101011)(1100)(0110) This theorem offers a test for codes that consists in a systematic organizations of computations required to verify if a subset of A + is a code or not. Basicay we try to find distinct derivations for the same word by starting from eft to right and trying to match words to the set of reminders. We compute a the reminders in a the attempts to find distinct factorizations. We discover distinct factorizations by finding the empty word as one of the reminders and keeping in mind that the computations have different starting words. Now we can say that for any subset C of A it is effectivey decidabe if C is a code or not. The agorithm suggested by the resut above is caed The Sardinas- Patterson agorithm for codes. We wi describe more of this agorithm in a foowing section. We have previousy defined the probabiity distribution for an aphabet A and we have etended it to words over A. We have defined the measure of sets as the sum of the probabiities for the words in that set. We sha present a resut that inks the measure of a set with the code property for that set. Theorem Let C A + and π a probabiity distribution for the aphabet A. 3 The compete proof can be found in [1] 11

13 1. If C is a code then π(c n ) = π(c n ), n If π is positive (π(a) > 0, a A), if π(c) is finite and if then C is a code. π(c n ) = π(c n ), n 1, Theorem Let A be an aphabet. Then C A + and C a code impies that π(c) 1. We sha discus the case when π is a particuar case of distribution the uniform distribution. The uniform distribution for an aphabet is a function: π : A [0, 1], π(a) = 1, a A. A In the case of A finite and the distribution uniform then we obtain that any code C over the aphabet of k impies that the foowing equation hods: k c 1. c C This theorem has its imitations in determining if a given set is a code or not as shown in the foowing eampe[1] Eampe Let A = a, b and C = {ab, aba, aab}. It is easy to verify that this set is not a code since: (aba)(ab) = (ab)(aab). If we consider any probabiity distribution π over A we obtain π(c) < 1. Indeed, et π(a) = p, π(b) = q, with p + q = 1. Then π(c) = pq + 2p 2 q. It is easy to verify that p + q = 1 impies that we aways have pq 1 4 and p 2 q Then π(c) < 1. 12

14 This eampe shows that the converse of the Theorem2.2.4 is fase. Theorem Let A be an aphabet and 1 d 1 d 2 d n be a set of natura numbers. There eists a prefi code C consisting of n words with the engths (d 1, d 2,, d n ) if and ony if 1 A d A d A d 1. n This resut is aso known as the Kraft Inequaity and gives an important too for verifying if there eists codes with predetermined word engths. In 1956 McMian etends this inequaity to finite codes not just prefi. McMian Inequaity: Let A be an aphabet. Then C = c 1, c 2,, c n A + code the foowing inequaity hods: 1 A d A d dn A 13

15 Chapter 3 Sardinas-Patterson characterization Usuay it is not easy to verify if a given subset of A has the unique descifrabiity property and thus being a code. The usua test in order to determine if a set of words is a code or not consists in finding two different decompositions for the same w C and thus determining that the given set is not a code. Intuitivey the process of finding different decompositions is usuay easy if the sets C and A have few eements. If we ook at a set C = {a, ab, b} subset of {a, b} + it is very simpe to see that ab has two distinct factorizations. Usuay sets that might be codes have many eements and it is very hard to see two distinct factorizations or to prove that there doesn t eist different factorizations for the same word. If we ook at the set C = {aa, ab, aab, bba} it is not easy to determine two different factorizations or to prove that there aren t such. These simpe eampes show us that it is necessary to find agorithm that effectivey decide if a given set is a code or not. The Sardinas-Patterson agorithm (SP-agorithm for short) suggested by Theorem2.2.2 consists merey in computing a the reminders in a attempts at a doube factorization. It discovers a doube factorization by the fact that the empty word is one of the reminders. 14

16 3.1 S-P Agorithm for cassica codes Theorem2.2.2, aso known as the Sardinas-Patterson theorem for codes, eads us to an effective agorithm that soves the probem if a given set is a code or not. The theorem states that C is a code iff C C i =, for a i i. In other words C is a code iff C i does not contain the empty word for any i 1. It is very easy to see that these two forms of the Sardinas-Patterson theorem are equivaent. Indeed, if C C i = then from the definition of C i+1 it foows that λ C i+1. The converse hods true; that is if λ C i then there C C i. We sha present the agorithm as a proposition. Proposition Let A be an aphabet and C A +, C-finite and different from emptyset.it is effectivey decidabe if C is a code or not. Proof C is a code C C i = φ, i 1. We show that because C is finite then the foowing sequence is cycic: C 1, C 2,, C n Indeed if C is finite then we can effectivey determine a natura number n 1 such that: ( c C)( c n) We wi show net that: ( i 1)( c C i )( c n) We wi prove the statement by induction over i. For i = 1 C 1 = { A + c C : c C} = ( C 1 )( c C)( < c n) For i 1 if we suppose that ( c C i )( c n) then if we consider how C i is constructed: C i+1 = { A + c C : c C i } { A + c C i : c C} it is easy to see that hods. ( c C i+1 )( c n) 15

17 We have proved that the words in C i have engths ess or equa to the words in C. It is easy to see that the sequence: C 1, C 2,, C n is not infinite. Then there eists k, such that: (1 k )(C k = C ) The foowing agorithm decides if C is a code or not: begin i = 1 Compute C 1 whie(c i does not contain empty word) begin i := i + 1 Compute C i if( (eists j < i : C j = C i ) (C i = φ)) return C is a code endwhie return C is not a code end This Agorithm aways ends because the sequence C 1, C 2,, C n is cycic, that is, there eists 1 i j such that C i = C j. This agorithm is aso known as the S-P agorithm for codes. If we consider the same eampe as in Theorem2.2.2 : C is a subset of {0, 1} +. C = {010, 0001, 0110, 1100, 00011, 00110, 11110, } the resuts obtained by appying the Sardinas-Patterson agorithm are shown in Figure 1. If we want to obtain the shortest word w that has two different factorizations over C we must ook in the resut tabe from right to eft. In our case λ is a reminder from 0110 C 7 and 0110 C then w ends in 0110 (w = w ) is obtained as a reminder between 0 C 6 and C. Thus w 1 = w 2 0. Continuing in the same manner we obtain w =

18 010 C C C C C C C C λ... F igure 1 17

19 3.2 S-P Agorithm for z-codes In this section we wi present an agorithm for testing if a given set is a z-code or not. This agorithm that soves the probem in the case that the given set is finite was given in [2] and it is based on a suitabe etension of the Sardinas-Patterson criterion for cassica codes. First of a we sha reca the concept of a z-code and we sha present some basic resuts on z-codes. There are many cases where we need to etend words not ony to the right as in cassica codes but to the eft too by taking in consideration the fina etters of the word. This specia case of a code that generates a word that doesn t necessary increase in size but can decrease the size of the constructed word is caed a zigzag-code or for short a z-code. A factorization over a z- code can continue with a word to the right as in cassica codes but can aso continue with a word on the eft and thus deeting that codeword from the computation and thus decreasing it s size. A more forma definition is given net. Let A be an aphabet and A be the free monoid generated by A. Let X A. We define a reation on A A generated by the set T = {((u, v), (u, v)) u, v A, X}. We define the step reation between the pairs (u, v) and (u, v ) if and ony if ((u, v), (u, v )) T or ((u, v ), (u, v)) T. We denote a step between two pairs by: (u, v) (u, v ). We say that a step is to the right on or to the eft on if (u, v) (u, v), respectivey (u, v) (u, v). A path is defined as a sequence of steps. We define by X the set that contains a words w A such that eists at east one finite path between the pairs (λ, w) and (w, λ) (eists and it is finite (λ, w) + (w, λ)). A z-factorization f of w over X of ength m, where X is a subset of A and w X is a path (u i, v i ) (u i+1, v i+1 ) for i = 0, 1,, m, which verifies the foowing conditions: (1) u 0 = v m+1 = λ, (2) v 0 = u m+1 = w, (3) (u j, v j ) (u k, v k ) for j k. 18

20 Condition (3) is necessary in order to ecude the presence of cyces in the factorization. Definition A set X A is a z-code if and ony if any word w A has at most one z-factorization over X. Remark (1) Any z-code is a code; the converse is fase. There are codes that are not z-codes. Let us consider X = {b, aab, bba, aabba}. X is a code. Indeed, if we appy the Sardinas-Paterson agorithm discussed in the precedent section we obtain the foowing resut: C 1 = {ba}; C 2 = {a}; C 3 = {ab, abba}; C 4 = φ. and thus we concude that X is a code. X is not a z-code. Indeed, the word w = aabba over the aphabet A = {a, b} + has two distinct z-factorizations over X. Let s number the eements of X as foows: 1 = b 2 = aab 3 = bba 4 = aabba. Two distinct factorizations for w are: a) (λ, aabba) right 1 (aabba, λ) b) (λ, aabba) right 2 (aab, ba) eft 1 (aa, bba) right 3 (aabba, λ) We concude that X is not a z-code. (2) Any prefi(suffi) code is a z-code. There are z-codes that are neither prefi or suffi. The foowing agorithm computes a the factorizations in a attempts at a doube factorization for a singe word. First of a we must define the concept of a configuration. If in Sardinas-Patterson agorithm for codes there are produced a sequence of sets of words unti one of this sets contains λ or is 19

21 equa to empty set, this agorithm produces a sequence of sets with eements from A A {1, 2,, n} 3. In the origina S-P agorithm it is sufficient to take in account ony the different reminders from the right side because the factorization ony continues to the right. For z-codes it is not sufficient because an z-factorization can continue with a step to the eft, and thus decreasing the ength of the word to have doube factorization; in order to continue a z-factorization to the eft we must know the eft side of a computation (usuay denoted by ), not ony the different reminders. We define a configuration for a set X any tupe A A {1, 2,, n} 3. A configuration is any tupe q = (, r, i, j, k), where is the eft part in the attempt at a doube configuration and r is the reminder at that step. At that moment we have constructed two different z-factorizations: one for, and one for r. It is obvious that if we reach a configuration q = (, λ, i, j, k ) we have found two distinct z-factorizations for the same word. The agorithm proposed in [2] constructs sets of configurations unti reaches a set that contains configuration q = (, λ, i, j, k ) or unti the ength of a possibe doube z-factorization reaches the maimum vaue K. It is shown that for a given set X it is possibe to determine the ength K of the shortest words that might have a doube factorization. The agorithm finds two distinct z-factorizations with the property that these different z-factorizations are the shortest that we can find. First of a we must define how we can obtain the sets of configurations. This is an iterative process beginning with the set: Q 1 = {( h, r, h, 0, k) h, h X : k = k r} and continuing with the set Q n which is the set of a configurations produced from configurations from Q n 1. The configurations are obtained according to the actua step in the derivation. The configuration q = (, r, i, j, k) produces a configuration q = (, r, i, j, k ) according to the foowing three cases: 1. we continue our derivation to the right with an h which is an prefi of the reminder r: 20

22 h r r Case 1 2. we continue our derivation to the right with an h such that the reminder r is a prefi of h : r h r Case we continue our derivation to the eft with an h : h r r Case 3. The configuration q = (, r, i, j, k ) is defined formay as foowing: Definition We say that a configuration q = (, r, i, j, k) produces on X a configuration q = (, r, i, j, k ) if there eists h X such that: or h j, r = h r, = h, i = h, j = 0 k = k; h j, h = rr, = r, 21

23 or i = k, j = 0 k = h; h i, r = h r, = 1 h, i = 0, j = h k = k. After defining how a set of configurations produces another set of configurations we present an agorithm given in [2] that decides if a given set X is a z-code or not. The vaue K that occurs is the upper bound on the ength of the shortest words in A that might have two distinct factorizations. We define two famiies of sets in order to hep us see better the computations that we are processing. These famiies of sets are obtained from the sets of configurations and have no computationa vaue. and C n = {r A (, r, i, j, k) Q n } W n = {r A (, r, i, j, k) Q n } W n is aso caed the set of remainders of n th eve. The Agorithm begin n = 1; Compute Q 1, C 1, W 1 whie (λ is not in W n ) and ( Q n φ ) and( r < K, for any r C n ) begin n := n + 1 Compute Q n, C n, W n endwhie if (λ is in W i ) then X is not a z-code ese X is a z-code end 22

24 Remark The agorithm aways ends because of the imit imposed on the shortest word that might have two distinct factorizations. In [2] it is shown that K can be computed before the eecution of the agorithm. If we consider X a subset of A that is not a z-code (there are words with distinct factorizations), and w a word with two distinct factorizations with the property that the ength of w is minimum then: K = a A p a i=1 C X a 2i, where for any w A, w a denotes the number of occurrences of the etter a in w and for any X = { 1, 2,, n } and a A we denote by: n X a = i a. i=1 23

25 3.3 S-P Agorithm for Time-Varying codes Time-Varying codes(tv-codes for short) have been introduced in [9] as a proper etension of L-codes[3]. We sha reca these two concepts and present the etension of the Sardinas-Patterson agorithm to TV-codes. L-codes are introduced in [3] as functions g : Σ Σ such that g : Σ Σ given by: g(λ) = λ, and g(σ 0 σ 1 σ n 1 ) = g 1 (σ 0 )g 2 (σ 1 ) g n (σ n 1 ) for a σ 1, σ 2,, σ n Σ +, is injective. g i denotes the i th iteration of the unique homomorphic etension of g, for a i 1 (g 1 = g and g i+1 = g i g for a i 1, where is the function composition). A TV-code over an aphabet A is any function h : Σ N A +, where Σ is an aphabet, such that the function h : Σ A given by: h(λ) = λ, and h(σ 1 σ 2 σ n ) = h(σ 1, 1)h(σ 2, 2) h(σ n, n) for a σ 1, σ 2,, σ n Σ +, is injective. The main idea behind a TV-code is to encode a etter by considering it s position in the word to be coded. Because each symbo is coded by taking in consideration it s position in the word to be encoded we can view any TV-code as a matri with Σ rows and an infinite number of coumns: Σ \ N σ 1 h(σ 1, 1) h(σ 1, 2) h(σ 1, 3) σ 2 h(σ 2, 1) h(σ 2, 2) h(σ 2, 3) σ Σ h(σ Σ, 1) h(σ Σ, 2) h(σ Σ, 3) 24

26 For any TV-code h : Σ N Σ + we define the sets H i (the i th coumn in the matri above) in the foowing manner: H i = {h(σ, i) σ Σ} and we denote by H the famiy of sets (H i i 1). The set H i is caed the i th section of h. In [9] there have been presented some resuts concerning reations eisting between TV-codes, cassica codes and L-codes. We wi present briefy some of these resuts. Any code g : Σ : A + is a TV-code. Indeed we can define h : Σ N A + as h(σ, i) = g(σ) for a σ Σ and i N. It is obvious that g = h. Any L-code g : Σ Σ + is a TV-code. Indeed, we can define h : Σ N Σ + by h(σ, i) = g i+1 (σ), for a σ Σ and i N. It is obvious that g = h. A major resut on TV-codes is that there are TV-codes that are not L-codes. This assures us that TV-codes are a proper etension of L-codes. The Schutzenberger criterion for codes has been etended to TV-codes in the foowing manner: Theorem A function h : Σ N A + is a TV-code if and ony if h is reguar, H is catenativey independent, and the foowing property hods true: ( w A + )(w not in H = ( i, j 1)(H 1 H i w H 1 H j = φ wh i+1 wh j+1 = φ)) We are going to present a Sardinas-Patterson characterization theorem for TV-codes 2. First of a we are going to anayze the cases where we can get different factorizations. As seen in the Sardinas-Patterson theorem for cassica codes we begin by growing two strings by different code words. First of a we begin by constructing a set of reminders such that any word member of this set is a reminder from two different codewords, in fact codewords from H 1. H 1 1,1 = H 2 1,1 = { A + σ σ Σ : h(σ, 1) = h(σ, 1)} 1 More detais and the compete proof can be found in [9] 2 The compete proof can be found in [9] 25

27 It is obvious that if H 2 H 1 1,1 φ then we can find two distinct factorizations thus h not being a TV-code. After constructing H 1,1 we continue by adding a new codeword to the first(second) string. We encounter two cases. Case 1: The codeword from H 2 is shorter than the reminder w H 1,1 (h(σ, 2)w = w, where w is the new reminder). In this manner we define two the sets H 1 2,1 by taking in consideration if we catenate the new codeword to the first string: H 1 2,1 = { A + σ Σ : h(σ, 2) H 1 1,1}, and H 2 1,2 if we catenate to the second string: H 2 1,2 = { A + σ Σ : h(σ, 2) H 2 1,1}. Case 2: The codeword from H 2 is onger than the reminder w H 1,1 (h(σ, 2) = ww, where w is the new reminder). In this manner we define two the sets H2,1 2 by taking in consideration if we catenate the new codeword to the first string: H2,1 2 = { A + y H1,1 1 : y H 2 }, and H 1 1,2 if we catenate to the second string: H 1 1,2 = { A + y H 2 1,1 : y H 2 }. In this manner we define inductivey, for any k 1, the famiy of sets H k = {H 1 i,j, H 2 i,j i, j k}: H k,k = H 1 k,k = H 2 k,k = { A + σ σ Σ : h(σ, k) = h(σ, k)}; H 1 i,j = { A + (i > k H i H 1 i 1,j φ) (j > k H 2 i,j 1 H j φ)}, for a i, j k with either i > k or j > k; H 2 i,j = { A + (j > k H j H 2 i,j 1 φ) (i > k H 1 i 1,j H i φ)}, for a i, j k with either i > k or j > k. 26

28 Let us see en eampe[9]. Let h be the function given in the tabe beow. Σ \ N σ 1 a baa ab ab ab σ 2 baa b aab aab aab σ 3 aba bab a a a We compute the foowing sets members of the famiy H 1 : H 1,1 = {a, b}, H 1 2,1 = {a} = H 2 1,2, H 2 2,1 = {a, b} = H 1 1,2; H 1 3,1 = φ = H 2 1,3, H 2 3,1 = {b, ab} = H 1 1,3, H 1 2,2 = {aa, ab} = H 2 2,2; H 1 4,1 = H 2 4,1 = H 1 1,4 = φ, H 1 3,2 = {a, b} = H 2 2,3, H 2 3,2 = {b} = H 1 2,3. Theorem Sardinas-Patterson criterion for TV-codes 3 A function h : Σ N A + is a TV-code if and ony if h is reguar and the foowing property hods true: ( k 1)( i, j k)(h 1 i,j H i+1 = φ = H 2 i,j H j+1 ) This resut eads to en effective agorithm for deciding if a given reguar function h is a code or not. 3 A compete proof can be found [9] 27

29 The Agorithm begin i = 1; j = 1 whie(true) begin Compute H 1 i,j Compute H 2 i,j, if(h 1 i,j H i+1 φ H 2 i,j H j+1 φ) return C is a not code endwhie end 28

30 3.4 S-P Agorithm for SE-codes Synchronized Etension systems (SE-systems for short) are powerfu generative toos, a new and eegant approach in forma anguage theory and combinatorics on word. Introduced in [6], those are very usefu when deaing with non-standard generative devices as time-varying mechanisms, conditiona grammars or monadic string rewriting systems [6, 9, 8, 10]. Those can aso be used to define a new concept of a code(section 5 in [6],[4]). A Synchronized Etension system (SE-system, for short) [6] is a 4-tupe G = (V, L 1, L 2, S), where V is an aphabet and L 1, L 2 and S are anguages over V. L 1 is caed the initia anguage, L 2 the etending anguage, and S the synchronization set of G. Let G = (V, L 1, L 2, S) be an SE-system. We define the binary reation G,r on V as foows: u G,r v iff ( w L 2 )( s S)(, y V )(u = s w = sy v = y). s y s As seen in the picture the word s S acts as a synchronizing word between u and w. The reation G,r is defined in such a manner that the synchronizing word from S is negected in the fina resut. The anguage generated by an SE-system G and G,r is defined as: L r (G) = {w v L 1 such that v G,r w} If we want the synchronizing word to appear in the fina resut we must define a new reation G,r + in the foowing manner: u G,r + v iff ( w L 2 )( s S)(, y V )(u = s w = sy v = sy). s y s 29

31 Aso we can etend word to the eft by defining in a simiar manner the reations G, and G, +. The reation between these reation is that by etending to the eft we obtain the mirror image of what we obtain by etending to the right: L (G) = L r ( G), where L (G) is the anguage generated by an SE-system G etending to the eft and L r ( G) is the anguage generated by the SE-system G etending to the right. A derivation can be considered as word π over the aphabet L 2 S (a sting containing pairs ike:the word used in the current derivation step, the synchronization used). If L 2 and S are indeed sets, a derivation can aso be considered as a sting of pairs of indees. In this case we say that π in an r -derivation of u into v, and we write u π r v. A derivation u r v is caed an r -derivation of u into v. Two r - derivations u 1 r u 2 r r u n and u 1 r u 2 r r u m are caed distinct if n m or there is an i such that u i u i. An SE-system G is caed r -ambiguous if there is a word v having at east two distinct r -derivations in G. An r -derivation u 1 r u 2 r r u n is caed reduced if it does not contain cyces, that is, there are no i and j such that i j and u i = u j. If an SE-system has the property that for any word v there is at most a reduced r -derivation of v, then it is caed r -nonambiguous or an r -code. It is aso genericay caed an SE-code. Furthermore on we sha use ony r derivations because any resut obtained for this kind of derivations can be easiy transated to the other three types of derivations. Definition A SE-system G = (V, L 1, L 2, S) is caed k-bounded directed if S = V k and the derivation reation is defined as foow: u G,r v iff ( w L 2 )( s S)(, y V )(u = s w = sy v = y), such that s is the ongest word in S, suffi in u and prefi in w. 30

32 Remark Codes are particuar cases of SE-codes. Indeed, a set C A + is a code if and ony if the SE-system is r -nonambiguous. (A, C, C, {λ}) Z-codes are not particuar cases of SE-codes. If the SE-system: G = (A {#}, X{#}, {#}X{#} X{##}, {#} X{#}) associated to the set of words X is r -nonambiguous then X is a z-code, but not conversey 5. This happens because in the case of a z-code the words removed by eft derivations must be kept because they are used for right derivations. In the case of SE-codes these words are not kept and the right derivations do not depend on the eft derivations made before. The Sardinas-Patterson characterization theorem for codes can be etended to SE-codes but it doesn t ead to an effective agorithm for testing the SEcode property. It has been pointed in [6] (Remark 4.1) that this property is undecidabe in genera. In order to deveop a Sardinas-Patterson theorem ike we must compute a the reminders in a attempts at a doube factorization. In order to do that we must define a structure that describes a step from a computation. It is not sufficient to describe ony the reminders as in the origina theorem because an derivations can decrease or increase the ength of the computation. Definition A configuration of G is a structure [(, s 1 ), (r, s 2 )] (z,s),f where, r V, s 1, s 2, s S {λ}, z L 1 L 2 {λ}, and f {0, 1}. We denote by Config(G) the set of a configurations of the SE-system G. A configuration [(, s 1 ), (r, s 2 )] (z,s),f describes a possibe anaysis step of two distinct r derivations of the same word, as it is pictoriay represented in the foowing figure: 4 For more detais see [6] 5 For more detais the reader is directed to [4] 31

33 s 1 r s 2 The fag f is 0 if both r -derivations are etended in the same way, and 1 otherwise. The pair (z, s) is used to avoid identic r -derivations when f = 1. We have defined a possibe anaysis step of two distinct r derivations. We must define a way for a configuration to produce another configurations stricty depending on how can we continue these two distinct derivations. This passing from a configuration to another is given by a binary reation on Config(G). Definition The passing from a configuration to another one is given by a binary reation on Config(G). We define this reation by considering a generic configuration [(, s 1 ), (r, s 2 )] (z,s),f and showing a possibe steps from it: 1. [(, s 1 ), (λ, s 1 )] (λ,λ),0 s 1 [(, s 1 ), (r, s 2)] (,s 2 ),0, where = r s 2. This case is pictoriay represented in the foowing figure: s 1 s 1 r s 2 s 1 s 1 2. [(, s 1 ), (λ, s 1 )] (λ,λ),0 s 1 [(, s 1), (r, s 1 )] (,s 2 ),0, where = r and s 1 = r. This case is pictoriay represented in the foowing figure: s 1 s 1 r s 1 s 1 s 1 3. [(, s 1 ), (r, s 2 )] (z,s),0 s 2 [(, s 1 ), (r, s 2)] (z,s ),f, where r = r s 2 and { ((λ, λ), 0), if r ((z, s ), f ) = = λ and s 1 = s 2 ((z, s), 1), otherwise This case is pictoriay represented in the foowing figure: 32

34 s 1 s 1 r s 2 r s 1 r s 2 s 4. [(, s 1 ), (r, s 2 )] 2 (z,s),0 [(, s 1), (r, s 1 )] (z,s ),f, where = r, s 1 = r r and { ((λ, λ), 0), if r ((z, s ), f ) = = λ and s 1 = s 1 ((z, s), 1), otherwise This case is pictoriay represented in the foowing figure: s 1 r s 2 r s 1 r s 2 s 1 5. [(, s 1 ), (r, s 2 )] (z,s),0 s 1 [(, s 2 ), (r, s 2)] (z,s ),f, where = rr s 2, = r and { ((λ, λ), 0), if r ((z, s ), f ) = = λ and s 2 = s 2 ((z, s), 1), otherwise This case is pictoriay represented in the foowing figure: s 1 s 1 r s 2 r r r s 2 s 1 s 6. [(, s 1 ), (r, s 2 )] 1 (z,s),0 [(, s 1), (r, s 2 )] (z,s ),f, where = s 1, r = r = and { ((λ, λ), 0), if r ((z, s ), f ) = = λ and s 1 = s 2 ((z, s), 1), otherwise This case is pictoriay represented in the foowing figure: s 1 s 1 s 1 r s 2 r r s 2 7. [(, s 1 ), (r, s 2 )] (z,s),0 s 1 [(, s 1), (r, s 2 )] (z,s),1, where =,s 1 =, r = r. This case is pictoriay represented in the foowing figure: 33

35 s 1 s 1 s 1 r s 2. r s 2 r 8. [(, s 1 ), (r, s 2 )] (z,s),1 s 1 [(, s 2 ), (r, s 2)] (λ,λ),1, where(z, s) (, s 2), = r and = rr s 2. This case is pictoriay represented in the foowing figure: s 1 s 1 r r r s 2 r s 2 s 2 9. [(, s 1 ), (r, s 2 )] (z,s),1 s 1 [(, s 1), (r, s 2 )] (λ,λ),1, where (z, s) (, s 1), = r and = rr s 2. This case is pictoriay represented in the foowing figure: s 1 s 1 r s 2 r r s 2 s [(, s 1 ), (r, s 2 )] (z,s),1 s 1 [(, s 1), (r, s 2 )] (λ,λ),1, where (z, s) (, s 1), =, r = r and s 1 =. This case is pictoriay represented in the foowing figure: s 1 s 1 r s 1. s 2. r r s [(, s 1 ), (r, s 2 )] (z,s),1 s 2 [(, s 1 ), (r, s 2)] (λ,λ),1, where (z, s) (, s 2), r = r s 2. This case is pictoriay represented in the foowing figure: s 1 r s 2 s 1 r s 2 r s 2 34

36 12. [(, s 1 ), (r, s 2 )] (z,s),1 s 2 [(, s 1), (r, s 2 )] (λ,λ),1, where (z, s) (, s 1), = r and s 1 = r r. This case is pictoriay represented in the foowing figure: s 1 r s 2 s 2 r. s 1. r s 1 After describing how a configuration can produce another configurations we define the sets Q i Config(G),for a i 0, for any SE-system G = (V, L 1, L 2, S), as foows: Q 0 = {[(λ, λ), (r, s 2 )] (r,s2 ),0 rs 2 L 1, s 2 S}, Q i+1 = {[(, s 1), (r, s 2)] (z,s ),f [(, s 1), (r, s 2 )] (z,s),f Q i [(, s 1 ), (r, s 2 )] (z,s),f [(, s 1), (r, s 2)] (z,s ),f }. Theorem Let G = (V, L 1, L 2, S) be an SE-system. G is an r code if and ony if for any i 0, Q i does not contain configurations [(, s 1 ), (λ, s 1 )] (z,s),1. This theorem eads us to an agorithm for verifying that a given set is a SE-code or not. Unfortunatey we can t give a hating condition for this agorithm. If it stops the answer is that the given set is not a SE-code. Thus the SE-code property is semi-decidabe. 6 The proof can be found in [4] 35

37 The Agorithm begin i = 1; Compute Q 1 whie ([(, s 1 ), (λ, s 1 )] (z,s),1 not in Q i ) and ( Q i φ ) begin i := i + 1 if ([(, s 1 ), (λ, s 1 )] (z,s),1 in Q i ) then X is not SE-code stop; Compute Q i endwhie end The SE-code property is semi-decidabe in genera but there are particuar subcasses of SE-system for which this property is decidabe. Indeed for the k-bounded directed SE-systems 7 this property is decidabe. 7 See Definition

38 Chapter 4 Concusions This paper is dedicated to the study of a very important aspect of the theory of codes: how to determine if a given set is a code or not. This probem is soved, in the case of cassica codes, by the Sardinas-Patterson characterization theorem. This resut eads to an effective agorithm for detecting the code property for a given set. When introducing various casses of codes, the authors have to sove a very important probem: the way of determining how a given set can act as a code or not. This probem is usuay soved etending the Sardinas- Patterson resut for that specific code. This is the case of z-codes, TV-codes and SE-codes. In the case of SE-codes we have deveoped a Sardinas-Patterson characterization theorem but unfortunatey it doesn t ead us to an effective agorithm. We concentrate our efforts in determining specific subcasses of SE-systems for which the SE-code property is decidabe. This remains an interesting open probem. 37

39 Bibiography [1] J. Berste, D. Perrin. Theory of Codes, Academic Press, [2] M. Madonia,S. Saemi T. Sportei. A Generaisation of Sardinas and Pattersons agorithm to z-codes, Theoretica Computer Science 108, 1992, [3] H.A. Maurer, A.Saomaa, D.Wood. L-codes and Number Systems, Theoretica computer Science 22, 1983, [4] I. Popa, B. Paşaniuc. A Sardinas-Patterson Characterization Theorem for SE-codes, 10th Internationa Conference on Automata and Forma Languages - Debrecen Hungary, (Information Processing Letters, submitted). [5] S. Roman. Coding and Information Theory, Springer-Verag, [6] F.L. Ţipea, E. Mäkinen, C. Apachiţe. Synchronized Etensions Systems, Acta Informatica 37, 2001, [7] F.L. Ţipea, E. Mäkinen. A Note on Synchronized Etension Systems, Information Processing Letters 79, 2001, 7-9. [8] F.L. Ţipea, E. Mäkinen. Note on SE-systems and Reguar Canonica Systems, Fundamenta Informaticae 46, 2001, [9] F.L. Ţipea, E. Mäkinen, C. Enea. SE-Systems, Timing Mechanisms, and Time-Varying Codes, Internationa Journa of Computer Mathematics 79,

40 [10] F.L. Ţipea, E. Mäkinen. On the Compeity of a Probem on Monadic String Rewriting Systems, Journa of Automata, Languages and Combinatorics (to appear). 39