Coding Information and Problems of Storage in Dynamical Systems

Transcription

1 FACTA UNIVERSITATIS (NIŠ) SER.: ELEC. ENERG. vol. 7 December Coding Information and Problems of Storage in Dynamical Systems Ilhem Djellit and Rezi Chemlal Abstract: A method of storage of information in one-dimensional piecewise continuous maps as proposed in Rouabhi [] is considered. In this report questions in examining the feasibility of information coding process with recognition and retrieving are discussed and analyzed an approach or an extension is proposed. Keywords: Piecewise continuous maps information coding storage of information. Introduction The last decade saw a growth of interest in memorizing storing and recognizing information. Information theory coding and cryptography are the three load-bearing pillars of modern digital communication systems. All the three topics are vast and there is a vast literature that deals with these topics individually. In this paper an attempt has been made to incorporate all the important concepts of coding process according to the papers [ 3]. Therefore as soon as a new concept is introduced I have tried to provide at least one counterexample. This is intended as a simple paper on the subject. Mathematical tools based on discrete dynamical systems for implementing basic functions of information processing are related to a storage method and information recognition described since 99. The first publications are due to the results of Dmitriev [ ] Andreyev [8 9 0] Chua and Wu in [8]. They aimed at the problem of information process in dynamical systems. This new technology of the use of dynamical systems for storage implies that the information is associated with a dynamical object such as an attractor a piece of a trajectory a Manuscript received February The authors are with Department of Mathematics Faculty of sciences Badji-Mohtar University B.P. Annaba Algeria ( i djellit@hotmail.com). 355

2 356 I. Djellit and R. Chemlal: part of the space phase this information is stored as equilibrium states which play the role of the information carriers. The importance of these applications has served to focus the coding theory community on the complexity of coding techniques and then coding theorists have attempted to construct structured codes. Recognition problems are a real challenge to information processing systems. Even most important problems such as recognition of handwritten text or speech are solved only partially. This fact testifies to inefficiency of the approaches developed by far and prompts to search for new approaches and among new candidates is a use of nonlinear dynamical systems. Mathematical models based on piecewise linear maps are used as a storage method proposed and given in references therein. These maps can be one-dimensional continuous and piecewise linear as in [4]-[8] two-dimensional maps as in [ 9 0]. The information to be stored is represented in the form of discrete sequences composed of the elements from a finite-length alphabet. In the original version [ ] one dimensional maps have the following form: x n f x n where x n is called the image of x n and x n 0. Each information of length to be stored is called information bloc and composed of symbols of an alphabet (of N symbols). Coding an information bloc of length consists in associating a vector X 0 0 to this information bloc. An occurrence of the system trajectory in a part of the unit interval with the system iterates is treated as generation of the corresponding alphabet symbol. In the case of one-dimensional map the components must be all different. If the bloc of information contains coinciding terms the process of storage will be impossible the cycle will overlap and the retrieval of the stored information will not occur. In this contribution we will focus only on the coding process we will discuss some techniques found in literature and their limits First the Section deals with the concept of information coding and its efficient representation. In the section 3 the coding method (filter method) proposed by [] permits according to the author to associate with each information bloc a period cycle which points are all different even if the information to be stored contain coinciding symbols and supposed to overcome simply the problem of repetition of a symbol in information blocs encountered in previous publications we show the limits of this method. The concept of coding is introduced and coding technique is discussed in detail in Section 4. The last section ends with a concluding remar.

3 Coding Information and Problems of Storage in Dynamical Systems 357 Coding Information For Storing One-dimensional maps are simple models with complicated nonlinear dynamics. They are used in studying attractors in investigating of chaotic dynamics and in analyzing information processing. Dynamics appears in iteration of the maps and the information carriers here are dynamical attractors: stable or unstable limit cycles or strange attractors. In these systems the recognition problem can be solved in the sense of the search for the reference most close to the presented one []. Many techniques are proposed in order to deal with the problem of repeated symbols the q-level storage consists in associating a cycle to a sequence of q symbols which permits to deal with information blocs that do not contain a series of q symbols. The orthogonalization of data is used as a complement to the previous methods; this consists in considering a repeated sequence as a new symbol. Another solution is the use of two-dimensional maps to lighten the constraints on repetition and for high information capacity. Indeed since their appearance (techniques) in 99 the methods were developed their capabilities were essentially extended by means of using multi-dimensional maps as the information storehouses.. Coding scheme A coding scheme was proposed in []-[3] as coding an information of length which consists in associating a vector X 0 to this information. In the case of one-dimensional maps all the components of X must be all different which maes the memorization possible. In piecewise-linear maps with stored information the dynamical system dimension is fixed (e.g. equal to ). The information bloc defines a vector X x T the components of which belong to the interval l 0 ; each point x j is associated with the symbol c j. We define a new vector X :.X A X where A A i j is a square matrix of ran. It is chosen in the following form: A ii a A j j a A a i j where the coding parameter a A i j represents the component of the i th line and j th column of the matrix A.. Storing information in piecewise-linear maps Let us consider the process of storing. A sequence of symbols the symbols must be different (important condition to apply the coding process) so this sequence

4 358 I. Djellit and R. Chemlal: can be represented by a -cycle of one-dimensional piecewise linear (or multidimensional in presence of q repeated symbols) dynamical system. This dynamical system is designed as follows: the interval is divided into N equal subintervals and mapped onto itself by the map. Each point of the j th subinterval is associated with the j th symbol of alphabet and each point of the -cycle related to one of the symbols is associated with the middle point of the corresponding subinterval (see [] for more details). The sequence of symbols corresponds to straight lines and then the associated map is defined by these segments. Multi-dimensional map permits to store information blocs where there is no more than q repeated symbols. For example a level storage of the bloc x 0 x n will create the cycle for the points x 0 x n x n x n x 0 This method is limited by the precision of computer for an alphabet of M=56 symbols and a 6-level storage the length of sub-intervals defining the cycle will be which is less than the precision of computer in single precision it is so impossible to distinguish between two points..3 The orthogonalization of data This technique is used to perform the last one; a method close to the Lempel Ziv Welch algorithm of compression (see []) is applied to delete the repetition of q symbols. This consists in replacing a sequence of q symbols by a new symbol and extend the alphabet by this end. The following example gives an idea about the method and its limits. Let be the alphabet A 0 9. We want to memorize the information bloc at the level q. We notice that the sequences 3 and 35 are repeated twice in this bloc to eliminate this difficulty we define two new symbols a 3 and b 35. The precedent bloc is then written abab where no sequence of two symbols is repeated. This information bloc can therefore be memorized by the alphabet A 0 9 a b in a level storage However this method maes the comparison between two information blocs very complex for example the two information blocs a5 and b are apparently different only the taing into consideration of the association a 3 and b 35 permits to conclude that these two information blocs are identical and correspond to the sequence 35.

5 Coding Information and Problems of Storage in Dynamical Systems 359 The research of an information bloc is then very fastidious; we must effectively loo for the different forms that it can tae 3 Filter Method (according to []) This method consists in defining the coded vector in the form where A is given by X B X A X () A ii a A j j a A a i j All the other elements of A being equal to zero. The authors in []-[3]stated that this technique permits to deal with all information bloc regardless of symbol repetition. In [] a choice criterion for a is given according to information blocs to code. In the next point we will discuss the limits of this method. 3. Limits of this method Let us consider the family of vectors of the form: )+ X " $ # # # %'& # ( () All these vectors possess a same sequence of q symbols repeated n - q times. If we try to use the alphabet. 0 9 / we will find a new vector of the form where X # # It is obvious that the new vector contain a sequence of identically symbols. Applying the q-level storage will not give better result as we will find the following vector: 0 x 3 # x 3 # x 3 #

6 " " " 360 I. Djellit and R. Chemlal: It is easy to verify that each element of the vector is equal to the element at distance q that means v i v i 3 q : 4 i Then the q level storage does not permit to deal with the problem we must also remember that this method presents limitations in relation with computers capacity. 4 Analysis of the Coding Process In this method for storing and retrieving information based on piecewise linear maps proposed by S. Rouabhi [] and where examples demonstrating the method were given. We discuss the first step (which seems problematical) of the proposed method in detail and analyze it in various ways. The matrix B defined in is given by A B This matrix is organized as follows B a 0 0 a a a a a a a a a a a a a a a a a a a a a a a a a a 3 a a a a a a a a a a a 3 a a 4 a 3 a a a a a a a a a a a a a 3 a a a a a a a a a a If we note B B B respectively the elements of the first line the matrix B can be written as B B B B B B B B 676 B 4 B 5 B B 4 B B )+ )+ )+

7 8 " " " Coding Information and Problems of Storage in Dynamical Systems 36 We note that the line i is the result of moving bac the last i elements of the first line to the beginning of the line. The product of this matrix by a vector of the form gives: B X B X Comparing the first and the q 3 B B B B B B B 666 B 4 B 5 B B 4 B B )+ B 3 3 x B 3 B B 3 B 3 x B 3 B 3 B 4 3 B 5 x B B 4 x B 3 B th element of the vector we have v B 3 3:#3 B q 3 B q 3 B q 393 B v q B q 3 B q 39 B 3 B 3 3:#3 B q Reorganization of terms shows that the first and the q 3 th element of the vector are identical what is in contradiction with the affirmation that it is always possible to choose the coding parameter a so that all the elements will be different. Let us now consider the inverse problem; we will show that the only the class of vectors on which the method fails are those of the form. Without loss of generality we can assume that the first and the q3 th element of the vector result are equal and show that these vectors are of the form. The first vector element is given by v B 3 3:#3 B q 3 B q 3 B q 393 B As the q 3 th line in matrix B is the result of moving bac the last q elements of the first line to the beginning of the line the q 3 th element of the vector is equal to: v q B q 3 B q 393 B q By identification we have: B 3 ; < 3 ; x 3 < 39#3 B x )+ )+ 0

8 36 I. Djellit and R. Chemlal: Using basic properties of polynomials for a and with some algebraic operations on relation between and q we will find conditions that must be satisfied on x i ; < # = x These conditions lead to the construction of vectors of the form (). 5 Conclusion This problem can be treated an idea consists in the use of basic permutation on elements of vectors described above will broe down the periodicity and then the Filter method will be applicable this leads to multiply the matrix B by a Perlis basic operator this is sufficient to deal with the problem. Acnowledgments This wor is supported by a grant from the ANDRU under No CU References [] S. Rouabhi Mémorisation d information dans des récurrences unidimensionnelles et multidimensionnelles Ph.D. dissertation Institut National De Sciences Appliques de Toulous- INSA Laboratoire d Etudes des Syètmes Automatiques Informatiques Toulouse 000. [] Storage of information in one dimensional piecewise continuous maps International Journal of Bifurcation and Chaos vol. 0 no. 5 pp [3] G. C. Mira. and S. Rouabhi Two applications of noninvertible maps in communication and information storage European Journal of Operational Research pp [4] A. S. Dmitriev Storing and recognition of information in -d dynamic systems Radiotehnia i Eletronia vol. 36 no. pp [5] Chaos and information processing in dynamical systems Radiotechnia i Eletrononia vol. 38 no. pp [6] A. S. Dmitriev A. I. Panas and S. O. Starov Storing and recognition information based on stable cycles of one-dimensional maps Phys. Lett. A vol. 55 pp [7] A. S. Dmitriev D. A. Kuminov V. V. Pavlov and A. I. Panas Storing and processing texts in -d dynamic systems Institute of Radioengineering and Electronics of RAS Moscow no. 3(585) 993 preprint.

9 Coding Information and Problems of Storage in Dynamical Systems 363 [8] Y. V. Andreyev A. S. Dmitriev L. O. Chua and C. W. Wu Associative and random access memory using -d maps Int. Journal of Bifurcation and Chaos vol. 3 no. pp [9] Y. V. Andreyev Y. V. Belij and A. S. Dmitriev Storage of information using stable periodic points of two-dimensional and multi-dimensional maps Radiotechnia i Eletrononia vol. 39 no. pp [0] Y. V. Andreyev A. S. Dmitriev D. A. Kuminov and V. V. Pavlov Information processing in -d and -d maps: recurrent and cellular neural networs implementation in Proc. of CNNA Fourth International Worshop on Cellular Neural Networs and their Applications 996 pp