Some Properties of Brzozowski Derivatives of Regular Expressions

Transcription

1 Itertiol Jourl of Computer Treds d Techology (IJCTT) volume 13 umber 1 Jul 014 Some Properties of Brzozoski erivtives of Regulr Expressios NMuruges #1, OVShmug Sudrm * #1 Assistt Professor, ept of Mthemtics, Govermet Arts College (Autoomous), Coimbtore , Tmil Ndu, Idi, * Assistt Professor, ept of Mthemtics, Sri Shkthi Istitute of Egieerig & Techology, Coimbtore 64106, Tmil Ndu, Idi Abstrct :- Brzozoski s derivtives of regulr expressio re developed for costructig determiistic utomt from the give regulr expressio i the lgebric y I this pper, some lemms of the regulr expressios re discussed d the regulr lguges of the derivtives re illustrted Also the geerliztios of the Brzozoski s derivtives re proved s theorems ith help of properties d ko results AMS MSC010 Certifictio: 68Q45, 68Q70 Keyords Regulr expressios, derivtives, d Kleee Closure I INTROUCTION Regulr expressios re declrtive y of defiig regulr lguges recogized by FA or NFA They re equivlet to oe other i the sese tht, for give regulr expressio, it c be costructed fiite stte utomt recogizig the sme lguge described by the regulr expressio, d vice vers All over the yers, vrious ttempts hve bee mde to ccomplish this tsk I the yer 1960, RMcNughto d HYmd [6] provided lgorithm to costruct o determiistic fiite utomto from regulr expressio GBerry d RSethi [1] discussed the theoreticl bckgroud for the RMcNughto d HYmd lgorithm VMGlushkov [4] hs lso give similr lgorithm i the yer 1961 A elegt costructio of determiistic fiite utomt bsed o the derivtives of regulr expressios s proposed by JA Brzozoski [] i the yer 1964 JEHopcroft d JUllm [5] discussed the costructio of - NFA from the give regulr expressio JMChmprud d others [3] described vrit of the step by step costructio hich ssocites stdrd d trim utomt to regulr lguges I this pper, e discuss some bsic set theoretic properties ivolved i Brzozoski y of costructios of utomt hve bee discussed II REGULAR EXPRESSIONS Let be lphbet of symbols A ord over lphbet is fiite sequece of symbols from tht lphbet The set of ll ords over is deoted by The empty ord is deoted by A regulr expressio is defied iductively s (i) is regulr expressio (ii) For y, the symbol is regulr expressio (iii) If E d F re regulr expressios, the,, E F EF E re ll regulr expressios The regulr expressios E F, EF, E re clled respectively uio, coctetio, Kleee closure of the correspodig regulr expressios The lguge of regulr expressio E is deoted s L E, d defied the sme for vrious regulr expressios s follos ( i) L ( ii) L ( iii) L E F L E L F ( iv) L EF L E L F ( v) L E L E The empty set is lso cosidered s lguge of regulr expressio deoted by the symbol itself It is ssumed tht E E E; E E ; E E E The properties of the regulr lguges re discussed i [9] The folloig lemm gives some lgebric type idetities ith respect to regulr expressios 1 Lemm Let E d F re y to regulr expressios The, i E F F E ii EF FE oly he iii E F G E F G E F or b oeof E, F is or iv E E v E F G E F E G ISSN: Pge 9

2 Itertiol Jourl of Computer Treds d Techology (IJCTT) volume 13 umber 1 Jul 014 Not ll lgebric type idetities re hold i the cse of regulr expressios Lemm: For y to regulr expressios E d F, the, i E F E F ii EF E F iii EF FE 3 Lemm: Provided tht E d F re ot equl to or i ( vi) ( ii) ( vii) b b ( iii) ( viii) ( iv) b b b ( ix) ( v) b b b ( x) xi bc b c ( xiv) ( xii) b c c bc ( xv) ( xiii ) b b b Some of the proofs of the equivlet regulr expressios give i the bove lemms re proved i [7] III erivtives of Regulr Expressios 31 efiitio Give lguge L d symbol, the derivtive of L ith respect to symbol is defied s L b b L The derivtives of regulr expressios ith respect to symbol re defied s follos: 1 b O th er ise E F F if L E E F E F o th e r ise 3 E F E F 4 5 E E E if b 6 E E 7 E E The opertor is treted s prefix opertor ith high precedece th +, d * The derivtives ivolvig the opertors itersectio, d complemet re defied by E F E F d E F E F It c be verified tht b b b b 3 Exmples 1 3 Let E b b E The E b Let E b b b E The E b b Let E b The E b b b b b b b b b E b b b b b b b b b b b b b 33 efiitio Let L be regulr lguge We defie if L E if L It c be esily see tht i, fo r y ii, d iii E F E F iv E 34 efiitio Let 1 d E be regulr expressio The, E E 1 1 E E ISSN: Pge 30

3 Itertiol Jourl of Computer Treds d Techology (IJCTT) volume 13 umber 1 Jul 014 I geerl, e hve E E 13 E Theorem Let E, F re to regulr expressios d the ord 1 strig over the Kleee closure of lphbet The, E F E F 36 Theorem Let E, F re to regulr expressios d the ord 1 The, EF E F E F E F EF E F E E F I geerl, E F 1 1 E F 1 1 E F E F E F E F P F Theorem Let the ord 1 The, E E E E E 1 1 E E E E 1 E 3 1 E E 3 E 1 E E 3 E E E E E E E E E E E E E 1 3 Similrly, e c geerlize E 1 3 E E E E E E E E E Lemm, here, L,,,,,,,,,,, L Hece the correspodig regulr expressio is 39 Lemm Let E be regulr expressio, the LE LELE, here,,, L E E E E,, L E E E E, E, E E, E L E L E 310 Theorem Let E be y regulr expressio d be y symbol over the lphbet L E L E Cse (i): Let E Also, the L E, d, d L O the other hd, if E, the L E L E L E d E Hece the theorem is true he E Cse (ii): Let E The, L E Hece Also LE, d L E ISSN: Pge 31

4 Itertiol Jourl of Computer Treds d Techology (IJCTT) volume 13 umber 1 Jul 014, the If E b L E L E Cse (iii):let E F G We prove first E F G Suppose, if F F ' E F G d ' ' ' ' L E L F L G d G G ',the Let F d G re to regulr expressios begi ith symbol other th The F G Hece E d L E I the first cse, LF d LG u u Hece LE, u, u, Hece LE LF ' LG ' ie, LE LF ' LG ' I the secod cse, LF b b, d LG bu b, u LE b, bu b,, u Therefore, Hece L E L E L E, Whe E F G Cse (iv): Let E FG Suppose if F d G re to regulr expressios begi ith symbol, the it c be foud s i the cse (iii), tht L E L E L F' L G' Similrly, if F d G re regulr expressios begi ith other th, it c be foud s L E L E Cse (v): E F the E F F F, Let Agi there re to possibilities, sy F F ' or F bf ' he b I the first cse, F F ' d ' I the secod cse, F Therefore F F F E Hece the sttemet trivilly true If F F ' F, the L E L E is ' L F ' L F L E L F L F O the other hd, if E F d F F ' The Hece ' L E L F L F ' L E L F L F L E L E Hece This proves the theorem ' LF 311 Theorem Let 1, d E be regulr expressio over lphbet The E E Suppose if the E E, As illustrtio, let b d E bb the E b b b b b b b b b b b b b Geerlizig the bove illustrtio, the folloig theorems re obtied 31 Theorem Let d E m; m ; the E 313 Theorem If E E, the 314 Theorem If u, u, d E v, v The, E v here Let u 1 The u ISSN: Pge 3

5 Itertiol Jourl of Computer Treds d Techology (IJCTT) volume 13 umber 1 Jul 014 E E E E v u v v v 315 Theorem If u; E bv, here, b d u, v, The E III CONCLUSION Brzozoski derivtives of the regulr expressios re lys helpful tool for costructig FA The geerliztios of the derivtives re useful for trsformig the size of the derivtives of the expressios reserch orieted ork REFERENCES [1] Berry G, Sethi R, From regulr expressios to determiistic utomt, Theoreticl Computer Sciece 48 (1986), [] Brzozoski JA, erivtives of regulr expressios, JACM 11(4): , 1964 [3] Chmprud JM, Poty JL, Zidi, From regulr expressios to fiite utomt, ItJ ComM, 7(4), 1999, [4] Glushkov VM, The Abstrct Theory of Automt, Russi Mthemticl Surveys 16(1961), 1-53 [5] Hopcroft JE d Ullm J,, Itroductio to Automt Theory, Lguges d Computtio, Addiso Wesley, 1979 [6] McNughto R, Ymd H, Regulr expressios d stte grphs for utomt, IRE Trs o Electroic Computers EC-9:1 (1960) [7] Muruges N, Priciples of Automt theory d Computtio, 004, Shithi Publictios [8] Muruges N d Shmugsudrm OV, Costructio of Stte digrm of regulr expressiousig erivtives,m- hikricom/ms/ms-01/ms/sudrmams pdf,Applied Mthemticl Scieces, Vol 6, 01, o 4, [9] Yu S, Regulr Lguges, i: ASlom, eds, Hdbook of Forml Lguges, Vol I, Spriger-Verlg, Berli, 1997, ISSN: Pge 33