Application of Fuzzy Algebra in Automata theory

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1 Amercan Journal of Engneerng Research (AJER) e-issn: p-issn : Volume-5, Issue-2, pp Research Paper Applcaton of Fuzzy Algebra n Automata theory Kharatt Lal Dept. of Appled Scence Mathematcs Govt. Mllennum polytechnc College Chamba Hmachal Pradesh (INDIA) Open Access ABSTRACT: In our frst applcaton we consder strngs of fuzzy sngletons as nput to a fuzzy fnte state machne. The noton of fuzzy automata was ntroduced n [58]. There has been consderable growth n the area [18]. In ths secton present a theory of free fuzzy monods and apply the results to the area of fuzzy automata. In fuzzy automata, the set of strngs of nput symbols can be consdered to be a free monod. We ntroduced the moton of fuzzy strngs of nput symbols, where the fuzzy strngs from free fuzzy submonods of the free monods of nput strngs. We show that fuzzy automata wth fuzzy nput are equvalent to fuzzy automata wth crsp nput. Hence the result of fuzzy automata theory can be mmedately appled to those of fuzzy automata theory wth fuzzy nput. The result are taken from [7] and [34]. Key Words: Fuzzy strngs, Pattern recognton, Membershp functon, Homomorphsm semgroup, nferred fuzzy automata, I. INTRODUCTION The basc dea s that the class of non-fuzzy system, that are approxmately equvalent to a gven type of system from the pont of vew of ther behavours s a fuzzy class of systems. For nstances the class of system that are approxmately lnear.ths dea of fuzzy classfcaton of system was frst hnted at by Zadeh Sards 1975 appled t to the classfcaton of nonlnear systems accordng to ther nonlneares, pattern recognton methods are frst used to buld crsp classes. Generally, ths approach does not answer the queston of complete dentfcaton of the nonlnearles nvolved wthn one class. To dstngush between the nonlnearles belongng to a sngle class, membershp value n ths class are defned for each nonlnearly one of thee consdered as a references wth a membershp value 1.The membershp value of each nonlnearty s calculated by comparng the coeffcents of ts polynomal seres expanson to that of the reference nonlnearlty. Ths technque of classfcaton s smlar to those used n fuzzy pattern classfcaton.we now menton some other way fuzzy abstract algebra has been appled. The paper [35] deals wth the classfcaton of knowledge when they are endowed wth some fuzzy algebrac structure. By usng the quotent group of symmetrc knowledge as algebrac method s gven n [35] to classfy them also the ant fuzzy sub groups constructon used to classfy knowledge. In the paper [20] fuzzy ponts are regarded as data and fuzzy objects are constructed from the set of gven data on an arbtrary group. Usng the method of least square, optmal fuzzy subgroups are defned for the set of data and t s shown that one of them s obtaned as f fuzzy subgroup by a set of some modfed data.in [55], a decomposton of an valued set gven a famly of characterstc functons whch can be consdered as a bnary block code. Condtons are gven under whch an arbtrary block code corresponds to L-valued fuzzy set. An explct descrpton of the Hammng dstance, as well as of any code dstance s also gven all n lattcetheoretc terms. A necessary and suffcent condtons s gven for a lnear code to correspond to an L-valued fuzzy set. Lemma1: 1 2 n As the asymmetrc between code words on whch fuzzy codes wll be based become large, there s only a small ncrease n the measurable dstance between codewords. For undrectonal errors, the case s that the space of the code wll effect the dstance between the fuzzy code words. These ssues must be taken nto account n desgnng fuzzy codes. Lemma 2: 1 < 2 <... n > 2. Proof: If nstead of usng the Hammng dstance between two fuzzy codes, we used the asymmetrc dstance, so that w w w. a j e r. o r g Page 21

2 D ( A, A ) a u u A ( w ) A ( w ) u u V A ( w ) A ( w ) v u V n w F n w F Then the followng theorem holds. Corollary 1: A fuzzy column vectors h 1 A n s ndependent of a set of fuzzy column vectors h,..., h 1 n If S () = for any {1,, m}. Proof: We gve the algorthm for checkng f a non-null column x k n the Sub semmodule F s lnearly dependent on a set of fuzzy vectors at the end of ths secton. In a set of the column vectors g, = 1,...n, s gven a complete set of ndependent fuzzy vectors f, = 1,..., can be selected such that subsemmodule generated by {f,..., f } contans the g s the procedure s shown n the form of the flow chart. m We now consder a postve sample set R + = 0.8ab, 0.8aa,bb, 0.3ab, 0.2bc, 0.99bbc. The fnte submatrx of the fuzzy Hankel matrx H(r) s shown. Usng the algorthm DEPENDENCE the ndependent columns of the fuzzy Hankel matrx have been ndcated as F 1, F 2, F 5, F 6, F 7. The fnte submatrx of the fuzzy Hankel matrx H (r) 2. S S S S S S S S S (a ) 4 S S S S S S S S S S ( b ) 4 S S S S S S S S S S S (c ) S w w w. a j e r. o r g Page 22

3 ab a abc bc c a b b c b b c a a b b a b b b b ( c ) a a b a b c b c a b b a b b c a a aab a a b b F F F F F F F The algorthm depends also dentfes column f any column vector h ( j) s dependent on the set of generator H (m) = H (m ) {f ˆ,..., f ˆ } of the Hankel matrx as constructed ntable.it also dentfes 1 m the coeffcent S, usng the procedure ARRANGE CS(), N, CARD() and the procedure COMPARESO(k), SO(K 1). Once the ndependent set of the column vectors are extracted, the next steps s to fnd out mathematc, s (x), xv T. In order to determne the matrces (x), x v the expresson x F has to be computed for x = a, b, c and = 1,...,7, the matrces (a), (b) and (c) gven n table. The x s can be computed from the relatonshp = F 1 (),..., F m ()), where the vector corresponds to the entres n the set of ndependent columns F 1,..., F m for the row n H(r) labelled by thus = ( ) The vector = ( ) because the column F 1 s an ndependent column. Once, and (a), (b) and (c) have been determned the fuzzy automaton can be constructed by the method descrbed. The fuzzy automaton that accepts the strngs s shown Fg 2: Inferred Fuzzy Automata PROCEDURE DEPENDENFE Step 1 = 1 from S () such that S () = {v j h j h } Fnd card (S ()) Step 2: If card (S () = 0 go to step 12 Else do w w w. a j e r. o r g Page 23

4 Step 3: If card (S ()) = 1 and S () = {j k }. Step4:If (S()) > 1 h for j = f k, f1 j u for any other j j k, = 0 and j 1 = 1 go to step 5. Else do j 1 = h j for all j = j k u for any other j j k = = 0, = 1 j 1 j u Step 5: = + 1 repeat the procedure untl = m. Step 6: j = 1 fnd v = { } and{ }, k = 1,...,n f v = {j j kl j j k } > { } go to step 12. Else do kl jk u Step 7: Select a j such that / M ax / m n and set R jkl j jku j = Step 8: From R j = R j V ( 1,..., n) such that h = j h j ) Step 9: j = j + 1 f j < n, go to step 6 Step 10: Check f R j converse all {1,...n} f R j = {1,...n} go to step 11. Step 11: Else do go to step 12. ĥ s dependent, pont the value of j. Step 12: ĥ s ndependent. Theorem 1.1: Kleen Schutzenberger for the free monod V T.The set A rec [[V T ]] and A rat [[V T ]] concde. We now defne the Hankel matrces. They can be used to characterze ratonal power seres. Defnton: The Hankel matrx of r A [[V T ]] s a doubly nfnte matrx H(r) whose rows and columns are ndexed by the word sv T and whose elements wth the ndces 4 are equal to (r, x v ) A formal power seres ra [[V T ]] s a functon from, V T to A, we denote the set of all functon from V T to A by AV T. The set AV T also provdes a convenent way to vsualze the column of H(r) as element n AV T we note that wth the column H(r) correspondng to the word v V T. We may assocate the functon f u AV T as follows F u (u) = (r, u v), u V T Defnton 1.1: A code C over the alphabet A s called a prefx (suffx) code f t satsfes CA + C = (A + C C = ). C s called a bprefx code of t s a prefx and a suffx code. A submonod M of any monod N. Satsfyng of proposton.ca + C = s called the left untary n N. M s called rght untary n N f t s satsfes the dual of x A, Mw M mples w M namely A + C C =. Let M be a submonod of a free monod A and c ts base then the followng condtons are equvalent: () w A, Mw M mples w M () CA + C = By the condton () n ths proposton no word of C s a proper left factor of another word of C defne the relaton on A * by x, v A *, x v f v s the left factor of u. Then 1 s a partal orderng of A. In the dagram to follow we dsplay the top part of A partally ordered when A = {a, b} and when A = {a, b, c} A partally ordered (when A = {a, b} w w w. a j e r. o r g Page 24

5 A partally ordered by When A = {a, b, c} A necessary and suffcent condton for a subset C of A to be prefx code s that6 for every c C, w A, w n, c and w c mples w C. Ths to obtan example of prefx codes, t suffces to select subsets C of A that wll be end ponts for. For Example: The trees dsplayed below gve the prefx codes. C 1 = {a 2,aba, ab 2, b} over {a, b} and c 2 = {a 2, ab, ac, ba, b 2, cb,c 2 } over {a, b, c}. The set B = {a n b n } s an example fallng tree wth end ponts a n b, n Prefx codes over {a, b} Prefx over code {a, b, c} Ths s no smple characterzaton general codes analogous to condton. The proposton s to for prefx codes. Proposton 1.1: Let A be a free monod and C be a subset of A. Defne the subset D of A recursvely by D 0 = C and D = {w A D 1 w c 0 or C w D 1 }, = 1, 2,... Then c s a code over A f and only f C D = for = 1, 2,... Suppose e s a fnte then the length of the word n C. Hence there s only a fnte number of dstnct D and ths proposton gves an algorthm for decdng whether or c s a code. Proposton 1.2: Let S be a semgroup C a column system for S and ga homomorphsm of B + nto S. Then the functon A : B C defned by x B + A ( x ) g ( x ) b a c - sub-semgroup of B +. If x, y B + f xy B + () x y x, xy, y () x y x, xy () x x n n w w w. a j e r. o r g Page 25

6 (v) (v) (v) x y x, y x xy, y x y x, y > xy, y then A s free, pure, every pure, left untary, rght untary, or untary, respectvely. Moreover, 1 A X g (X ) for every X C. Example 1.1: () For e = {c = a, a 3 b, aba } We have D 0 = C, D 1 = {a 2 b, ba} D 2 = {ab} and D 3 = {a, b} Snce C AD 3, C s not a code () C = {a, a 2 b, bab, b 2 } we have D 0 = C, D 1 = {ab}, D 2 = {b} and D = {ab, b} For = 3, 4,... Snce C D = For = 1, 2,... C s a code. Therefore we now consder the constructon of codes usng fuzzy subsemgroups. Let L s the partally ordered set L s called a - sem lattce f x, y L x: x and y have a greatest lower bound least upper bound, say x y (xy. A - semlattce s called complete f for every subset of L has a greatest lower bound n L. Let A A : B L s a sem lattce whose elements are L-subsets of the free semgroup B +. An L-subset A of B + s an L-subsemgroup of B + f for t t L, the level set A x B A ( x ) t y B + A ( x y ) A ( x ) A ( y ) s a subsemgroup of B +. Then A s a L-sub-semgroup of B + f x, The search for sutable codes for communcaton theory s known. It was proposed by Garla-that-Lsem-group theory be used. To ths end free, pure, very pure, left untary, rght untary, untary such L- submsemgroup there s a famly of codes assocated wth t. An L-subsemgroup of a free semgroup f free, one are, pure, very pure, left untary rght untary respectvely.thus any method used to construct an L- =subsemgroup of a free semgroup of one of these types yelds a famly of semgroups of the same types. Namely the level sets of the L - subsemgroup. V. CONCLUSION We have assumed that error n the transmtted of words across a nosy channel were symmetrc n nature.e., the probablty of 1 0 and 0 1 cross over falures were equally lkely. However error n VSLI crcut and many computer memores are on a undrectonal nature [8] A undrectonal error model assumes that both 1 0 1cross overs can occur, but only are type of error occurs n a partcular data word. Ths has provded the bass for a new drecton n codng theory and fault tolerance computng. Also the falure of the memory cells of some of the LSI transstor cell memores and NMOS memores are most lkely caused by leakage of charge. If we represent the presence of charge n a cell by 1 and the absence of charge by 0, then the errors n those type of memores can be modeled as 1 0 type symmetrc errors, [8].The result n the remander of ths secton are from [15]. Once agan F denotes the feld of ntegers module 2 and F n the vector space of n - tuples over F, we let p denotes the transmtted 1 wll be receved as a1 and a transmtted 0 wll be receved as a 0, Let q = 1p. Then q s the probablty that there s an error n transmsson n an arbtrary bt. REFERENCE [1] Malk D.S. Mordeson J.N. and Sen M.K. Free fuzzy submonods and fuzzy automata. Bull. Cal. Math. Soc. 88: , [2] Mordeson J.N. Fuzzy algebrac varetes II advance n fuzzy theory and technology Vol. 1: 9-21 edted by Paul Wang [3] Gerla, G, Code theory and fuzzy subrngs-semgroup. J. Math. Anal, and Appl. 128: , [4] Kandel A, Fuzzy Mathematcal technology wth applcaton. Addson-Wasely Pub. Co [5] Fuzzy Automata and Decesson Process M.M. Gupta, G.N. Sards, and B.R. Ganes, Eds, pp-77-88, North Holand Publ. Amsterdam. [6] Bourbak, Elements of mathematcs commutatve algebra, Sprnger Verlag, New York, [7] Gerla, G, ʽʽ Code theory and fuzzy subrngs- semgroup ʼʼ. J. Math. Anal, and Appl. 128: , w w w. a j e r. o r g Page 26