Construction of State Diagram of. Regular Expressions Using Derivatives

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1 Applied Mthemticl Sciences, Vol. 6, 2012, no. 24, Construction of Stte Digrm of egulr Expressions Using Derivtives N. Murugesn nd O.V. Shnmug Sundrm # Post Grdute & eserch Deprtment of Mthemtics Government Arts College (Autonomous Coimbtore , Tmil Ndu, Indi nmgson@yhoo.co.in Deprtment of Mthemtics Sri Shkthi Institute of Engineering nd Technology L & T Bye-Pss od, Coimbtore Tmil Ndu, Indi ovs3662@gmil.com Abstrct: In this pper, we discuss the successive derivtive of regulr expressions hving ny number of logicl connectives including Boolen opertors nd lso some of their properties. We lso discuss the successive derivtives of regulr expressions replcing the ech position of the expressions with new symbol. The stte digrm of such modified expressions hs lso been discussed. Mthemtics Subject Clssifiction: 68Q45, 68Q70 Keywords: egulr expressions, derivtives, stte grph utomt, nd Kleene Closure 1. Introduction A lnguge is subset of the set of strings over n lphbet. It cn be recognized by mchine, lso hve representtion interms of genertive device s grmmr, which is clled n cceptor or the finite stte utomton (FSA. There re two importnt types, such s Deterministic Finite Automt (DFA nd Non-Deterministic Finite Automt (NFA. A DFA is 5 tuple M = ( Q,, q0, δ, F where Q is finite set of sttes, is finite set of input symbols, q 0 in Q is the strt stte or initil stte, F Qis the set of finl sttes, ndδ, the trnsition function from Q x to Q. Also δ ( q, = pmens, if the utomton M is in stte q nd reding symbol, then it moves one stte to p t once.

2 1174 N. Murugesn nd O.V. Shnmug Sundrm Fig 1. An exmple for DFA In NFA, on reding symbol t prticulr stte, next stte is not uniquely determined, where the mchine hs choice of moving into one of few sttes [4] Fig 2. An exmple for NFA Finite stte utomt re in generl constructed to recognize the lnguges defined over given lphbet. It is sid tht finite utomton recognize the given lnguge, if it moves from initil stte to finl stte on reding ech nd every word of the lnguge for which the utomton is constructed. A lnguge is sid to be regulr if nd only if there exists finite utomton recognizing the lnguge. egulr expressions re the declrtive wy of defining regulr lnguges. The syntx of regulr expressions over set of input symbols is E:: = 0 / 1 / / E E / E E / E In this pper, we discuss some of derivtives of regulr expressions. 2. Derivtives of egulr Expressions 2..1 Definition Let be regulr expression over n lphbet, nd lso let s be ny finite string over the sme lphbet. Brzozowski [2] defined the derivtive of with respect to s, denoted by Dnd s is defined s Ds = { t st } 1. If = b, then D= b, Db = 2. If = b b, then D Exmples = b ε, Db = ε 3. If = 1 0, then D 1 = ε, D 0 = ε 4. If = , then D = 1 01, D = 0 1

3 Construction of stte digrm Definition: Suppose the regulr expression contins, then for given set of strings of symbols, we define δ ( to be δ ( = ε if ε otherwise if ε 2.3 Note It cn esily seen tht δ( = for ny, δ( P = ε, δ( = nd δ( PQ = δ( P δ( Q. It cn lso be found tht for Boolen function = f (P, Q, the δ is defined s δ ( P Q = δ( P δ( Q follows: δ( PQ = δ( P δ( Q ' ε if δ( P = δ ( P = if δ ( P = ε 2. 4 Theorem If is regulr expression, the derivtive of with respect to symbol is found recursively s follows: D = ε, Db =, if b = ε or b = nd b D( P = D( P P D( P Q δ ( P( D( Q if δ( P = ε D ( PQ = D ( P Q, otherwise D ( f( P, Q = f( D ( P, D ( Q 2.5 Theorem The derivtive of regulr expression with respect to finite set of input symbols s= r is found recursively s follows: D = D ( D, D = D ( D nd D = D ( D r r r 1 For completeness, if s = ε, then Dε = The bove two theorems re due to Brzozowski [2] nd others [1] nd hve been illustrted in the exmple Corollry s =... nd ssume tht = =... = =. Then, we hve Let 1 2 r 1 2 r 1 2 r D = D = D, D = D ( D = D = D nd D... r times = D = D i.e., the derivtive of successive derivtion forms loop t the prticulr stte with sme symbol s lbel. 2.7 Derivtive of regulr expressions with Kleene Closure In this section, we consider the derivtives of regulr expression contining Kleene closure.

4 1176 N. Murugesn nd O.V. Shnmug Sundrm (. i = 0, then D0 = 0, D00 = 0, D = 0 ( ii. = (0 1, then D0 = (0 1, D 1 = (0 1 D = (0 1 ( Loop nd D = (0 1 ( Loop ( iii. = (01 1, then D0 = 1(01 1, D 1 = (0 1 = D = (01 1 = ( Loop ( The bove exmples re illustrted with the following exmples. Fig 3: Stte digrm (i. = 0, (ii. = (0 1, nd (iii. = ( Properties of derivtives of regulr expressions 1. The derivtive D ε of ny regulr expression with respect to ny symbol s is regulr expression [2]. Exmple: D ε =. Here is zero length input nd lso regulr by the definition of regulr expression. 2. Every regulr expression hs finite number d of types of derivtives. Exmple: For the symbol, we hve d = 1. Also for the symbolε, we hve d = 2 nd ny input symbol, other thn the bove two, d = 3 3. At lest one derivtive of ech type must be found mong the derivtives with respect to set of symbols of length not exceeding d Every regulr expression cn be written in the form = δ ( D where the terms in the sum re disjoint. Exmple: If = b, then δ ( =.Hencee the expression become b= b D= b. 5. The reltionship between the d chrcteristic derivtives of cn be represented by unique set of d equtions of the form D s = δ ( D s D u, where D s is chrcteristic derivtive nd Duis chrcteristic derivtive equl to D s. Such equtions will be clled the chrcteristic eqution of. 6. The set of chrcteristic equtions of cn be solved for uniquely (up to equlity., b

5 Constructionn of stte digrm Construction of Stte digrms First we find the derivtions of egulr expression with respect to input sequences of length 0, 1, 2... till there re no new derivtives for lll sequences of some length. We ssocite n internl stte for ech chrcteristic derivtive. If chrcteristic eqution ssocited with ny stte continsε, then tht prticulr stte is mrked s n ccepting stte. The forml lgorithm of construction of deterministic utomton D is s follows. This lgorithm is due to G..Berry nd.sethi [1] Algorithm Step 1: Sttes of D re the distinct derivtives D for ll strings w. Step 2: Construct trnsition under from stte p to stte q if nd only if p is for derivtive D w for some w, nd q is for D. w w Step 3: The stte for is the strt stte. A stte is n ccepting if nd only if it is for derivtive D, w for some w, nd δ ( Dw = ε ; thtt is, the empty string is in L ( Dw 3..2 Exmple Let = ( b( D ε =, D = ( b( = D b = b D = ( b ( = D = D = b ε D = b ε D = ε..... bbb.... b bb b Fig 4: Stte grph for = ( b( 3..3 egulr Expressions with Distinct Symbols Automt will be constructed by explicitly computing derivtives, then derivtives of derivtives, nd so on s neededd [1].

6 1178 N. Murugesn nd O.V. Shnmug Sundrm If two regulr expressions E 1 nd E 2 which reduce to the sme expression using ssocitivity, commuttivity, nd idempotence of re clled similr or equivlent [2]. Without bove given condition tht, duplicte sub-expressions would cuse successive derivtives of E = ( by, to be distinct. These properties llow sum of expressions to be treted s set of expressions, thereby removing duplictes [2]. Following McNughton nd Ymd [5], we mrk ll input symbols in regulr expression to mke them distinct. The mrks re written s subscripts; mrked version of = ( b ( is ( 1 b2 b3( b, wh here 1 to b 5 re treted s different symbols. G.Berry nd vi Sethi [1] proved the theorem tht llows ech symbol in mrked expression to be viewed s stte utomton. Fig 5 is motivting exmple. Ech stte of the utomton in fig 5 is lbeled with symbol representing derivtivee of the expression ( 1 b2 b3( 4 b 5. Furthermore, s in utomt constructed by the given lgorithm 3.1, the stte for C hs trnsition under mrked symbol to stte for the derivtive of C by thtt symbol. By construction, ll the trnsitions entering stte re lbeled with the sme symbol; for exmple, the trnsitions of the lbel b 3 into the stte C 3 3. Let E be regulr expression with distinct symbols. For ny symbol, the continution of in E is ny expression D we for some w. By structurl induction on E, ll such expressions re equivlent. Hencee the continution of in E to refer to some expression in the equivlent clss. The sttes of the utomton in fig. 5 re lbeled with continutions of mrked input symbols in the expression ( 1 b 2 b3( 4 b 5. For ll i, 1 i 5, C i is the continution of the symbol mrked i, nd C 0 represents the entire expression. C0 = ( 1 b2 b3( 4 b5 C1 = ( 1 b2 b3( 4 b5 C2 = ( 1 b2 b3( 4 b5 C3 = ( 4 b5 C4 = ε nd C5 = ε Here C 0 nd C 2 re equivlent expressions to the bove fct. The modified construction of fig 4 is given in fig 5 s below Fig 5: Grph for ( 1 b2 b3( 4 b5

7 Constructionn of stte digrm 1179 Exmple If the regulr expression = ( , then the mrked input symbols of the given regulr expression becomes ( For ll i, 1 i 5, C i is the continution of the symbol mrked i, nd C 0 represents the entire expression. C0 = ( C1 = 1( C2 = ( C3 = ( C = 0 nd C = ε 5 The construction of the stte digrm is given below in the figure 6. Fig 6: Grph for ( Conclusion The derivtives of regulr expression hve been n interesting re of reserch in the field of Theory of Computtion, since its introduction by J.A.Brzozoswki [2] nd others [1] [3]. ecently mny reserchers follow prtil derivtives of regulr expressions. Also similr ttempts cn be mde for Kleene closures with regulr expressions. eferences [1]. G.Berry,.Sethi, From regulr expressions to deterministic utomt, Theoreticl Computer Sciencee 48 (1986, [2]. J.A.Brzozowski, Derivtives of regulr expressions, J.ACM 11(4: , [3]..McNughton, H.Ymd, egulr expressions nd stte grphs for utomt, IE Trns. on Electronic Computers EC-9:1 ( [4]. Kml Krithivsn, m, Introduction to Forml lnguges, Automt theory nd Computtion, 2009, Person publishers. [5]. N.Murugesn, Principles of Automt theory nd Computtion, 2004, Shithi Publictions. eceived: August, 2011