Fix point internal hierarchy specification for context free grammars

Transcription

1 Proceedgs of the th WEA Itertol Coferece o COMPUER, Agos olos, Crete Isld, Greece, July 26-28, Fx pot terl herrchy specfcto for cotext free grrs AILE CRĂCIUEA, RALF FABIA, DAIEL HUYADI, EMIL MARI POPA Deprtet for Coputer ceces d Ecooc Ifortcs Luc Blg Uversty of bu Io Rţu treet, o 5-7, bu ROMAIA crcue@slro, rlffb@ulbsburo, dyhuyd@yhooco, elpop@ulbsburo Abstrct: - I the prtculr cse whe cotext free grr s used s odel for coputto syste, ech oterl wll be turlly ssocted to eg, e type, d every geerto rule represets operto the coputtos syste hus, the dervto tree (sytx tree) of ths grr yelds herrchc structure of types s well s herrchc structure of syste opertos Cosequetly, o ech herrchc level there exsts set of types d set of opertos, ely lguge Recursve specfctos re useful tool for represetg d studyg ths d of lguge herrches As we show ths pper, the recursve specfctos re d of costructor for these lguges Key-Words: - coputto syste, fx pot, recursve specfcto, terl herrchy, type herrches Itroducto Wth ths pper we ddress the ssue of geerto d specfcto ethods for syste desg bsed o recursve specfctos, cosderg cotext free grrs s the bscs lestoe curret oder progrg lguges herefore we l the propertes of Kleee s fx pot theory to terl herrchy specfctos of cotext free grrs hs leds to the possblty of specfy bstrcto levels for pplcto geerto tht rely o prevous (older) levels I other words, there re ulted possbltes to exted the exstg specfctos We ssue flrty wth bsc deftos d results of forl lguge theory Frst we stress out soe portt results fro set theory, eeded the forl odelg of recursve systes Further we preset lgorth tht deteres the types d lgorth for the proper herrchy of types At the ed we fsh wth sple exple showg the edte results of the preseted theory 2 Prelry cocepts d otos I ths prgrph we recll soe bsc cocepts d otos relted to our further dscussos Fro the set theory we ow tht posets (prtl ordered set: s reflexve, tsyetrc d trstve) hve the propertes y s upper boud of subset Z of poset ( P, ) ff y P d, for ll z Z, z y y s the lest eleet ( ) of Z ff y Z d, for ll z Z, y z y s xl eleet of Z ff y Z d there s o z Z such tht z y d y z he oto of lower boud, gretest eleet, d l eleet receve dul deftos (e deftos obted by replcg by ) y s the supreu, Z, of Z ff y s upper boud of Z d y s the lest of the upper bouds of Z y s the fu, Z, of Z ff y s lower boud of Z d y s the gretest of the lower bouds of Z Defto 2 A prtl ordered set ( P, ) s clled do f t hs oe lest eleet d f y scedg sequece over P hs upper boud P Defto 22 Let ( D, ),,( D, ), > 0 be dos he the product do of dos s do ( D, ), where: D= D D d ( x, x2,, x) ( y, y2,, y) ff x y, =,, ( x, x2,, x),( y, y2,, y) D D Defto 2 Let D be do A recursve specfcto over D s totl fucto : D D 2 such tht ( ) ( ) ( ) I ths 2 codtos the sequece ( ) ( ) ( ) s clled Kleee sequece for [],[0] Defto 2 A eleet f D, defed by f = = ( ), () s clled Kleee setc of []

2 Proceedgs of the th WEA Itertol Coferece o COMPUER, Agos olos, Crete Isld, Greece, July 26-28, Rer: Fro the do defto, ths supreu exsts Le 2 If ( D, ) s do d :( D, ) ( D, ) s ootoe the s recursve specfcto (Proof: see []) Defto 25 Let ( P, ) be prtl ordered set d : P P totl fucto A fxed pot of s eleet f P tht verfes ( f ) = f he lest fx pot of (f t exsts) s the lest eleet fro the set of fx pots [],[8] heore 2 Let ( P, ) be prtl ordered set d :( P, ) ( P, ) ootoe fucto If there exsts f = { h h ( h), the t s fx pot of (Proof: see []) Defto 26 Let ( D, ) d ( E, ') be dos A ootoe ppg : ( D, ) ( E, ') s cotuous f t preserves the leses upper bouds of the cresg sequeces, ely ( f) = ( ( f)) [] A portt result, o whch ths pper reles o, s the followg theore, ow s Kleee s fx pot theore heore 22 (Kleee s fx pot theore) Let (D, ) be do d : ( D, ) ( D, ) cotuous fucto he Klee s setc f = ( ) s the lest fx pot of ([],[8]) = Exple 2 We deote Pf( X, Y ) the set of ll prtl fuctos f : X Y he ( Pf( X, Y ), ) s do, where the relto s defed s follows: f g DD( f) DD( g) d g( x) = f( x), ( ) x DD( f ) hus, f f f2 f we defe f s DD( f ) = DD( f ) = 0 ( f)( x) = f( x), ( ) such tht x DD( f ) he fucto f s well defed becuse f x DD( f ) DD( f), the f ( x) = f( x) I ths crcustces t c esly bee see tht the 2 Kleee sequece ( ) ( ) ( ) s deed cresg sequece Pf( X, Y ), d Kleee s setc DD( f ) = DD( ( )), = f ( x) = ( )( x), ( ), x DD ( ( )) stsfes the relto = ( ) f Rer: he relto f g shows us tht g offers t lest s uch forto th f dose Fx pot d forl lguges I ths prt we focus o cotext free grrs (type 2 grrs Chosy s clssfcto) Fst, we show o exple, tht the lguge geerted by cotext free grr (CFG) c be obted fro the sllest fx pot of well chose recursve specfcto Exple uppose the followg sple CFG: G = (,,, P) where P= { A B A Ab A b B bb B b = {, A, B, = {, b d s the grr xo (strtg sybol) Frst we rewrte the producto rules the followg wy A+B A Ab+b B bb+b where + deotes the uo he we defe the recursve specfcto: :(2 ) (2 ) (, A, B)=(A+B, Ab+b, bb+b) Applyg Kleee s theore to detere the leses fx pot for the bove chose (the fct tht s cotues wll be show lter) d coputg the Kleee sequece we hve: 0 (,, ) = (,, ) (,, ) = (,, ) = {, b, b) (,, ) = (, b, b) = { b + b, b + b, 2 2 b + b It c be esly see tht ducto over proofs tht ( ) = ({ b { b, { b,{ b ) hus, the sequece ( ) s truly cresg sequece d we hve ( ) = ( LG ( ),{ b,{ b ) 0 (here L(G) es the lguge geerted by the grr G) However, L(G) s the frst copoet of the lest fx pot of More geerlly speg, for

3 Proceedgs of the th WEA Itertol Coferece o COMPUER, Agos olos, Crete Isld, Greece, July 26-28, =,2,, the -th copoet of ( ) s 0 { w w X d v w, where v {, A, B A closer loo o the bove reltos wll revel soe other forto to Let deote the power of the relto, e w w' es tht w derves (geertes) w ' steps f there exsts sequece w, w2,, w such tht w= w w2 w w = w' w w' es tht w= w' Hece, the -th eleet of ( ) s { w w X d v w', for (2) he represetto { w w X d v w, s trcy, s the followg exple shows Exple 2 Cosderg grr wth sgle vrble d the productos ++bb, the, X X ths ples :2 2 d ( )=++bb hus, 0 ( ) = ( ) = { 2 ( ) = {, bb { But 2 ( ) s ot cceptble sce the shortest dervto for the word bb s v v bbv bbv bb, whch eeds steps ( sted of les the 2) Aywy, loog t the dervto tree, depct fgure, for the word bb, we see tht he s of heght 2 b Fgure Dervto tree for exple 2 I other words, f we dt prllel substtutos (ll the vrbles of sequece c be replced durg sgle step), the our word s deed dervble two steps: bb bb hs s odel of the setc of totl cll by, due the fct tht ll vrbles re replced by ech step however ths odel s odeterstc becuse y producto fro the set of productos c be chose to replce every occurrece of vrble b 0 Iterl herrchy of cotext free lguges he geertg grr for cotext free lguge offers terl lguge herrchy By ths es, we cosder the lguge setcs equl to the clculus syste obted the followg wy: every oterl sybol of the grr represets the e of set of eleets clled type every producto represets heterogeeous operto, ely, f A α P d α = s0as A2 s A s +, the the heterogeeous operto ssocted to ths producto s deoted by s0s s s + d opertes o the sets A, A2,, A d produces result of type A, e ( s0s ss+ ) : A A2 A A If p = A α P the t( p ) deotes the type of p, t( p) = A d( p ) deotes the do of p, d( p) = { A, A2,, A s( p ) deotes the stte word (or sybol) of p, s( p) = s0s s s +, d p ( ) the rty of p, p ( ) = Gve cotext free lguge G = (,,, P), we cosder herrchy of oterl sybols buld s follows: 0 = { A A α P, α () + = { A A s0as A2 sas+ P () d A, A2,, A he ext lgorth deteres the types herrchy for cotext free grr G= (,,, P) Algorth Iput: A CFG G = (,,, P) Output: he set of types Method: cossts buldg sequece of sets { tht coply wth the property 0 = = hus, wll cot ll the oterl sybols tht c geerte words over Obvously, f the lguge s equl to the epty set 00 AR 0 : = = { A A α P, α 0 DO UIL ( = ) + = { A 0 A s As A s A s P 0 2 +, 2,, d A A A

4 Proceedgs of the th WEA Itertol Coferece o COMPUER, Agos olos, Crete Isld, Greece, July 26-28, : = + 06 EDDO 07 OP Property For y gve cotext free grr G = (,,, P), there exsts fte uber of herrchc type levels (he coplete proof c be foud []) Exple We cosder ow s exple, clssc CFG, ely the grr tht geertes rthetc expressos: G = (,,, P), where ={E,, F ={,,+,(,) =E P={F, F, F, E, E E+, F (E) We obt the followg er herrchy of types: 0 ={F ={F, 2 ={F,, E ={F,, E Buldg er herrchy of productos leds us to: 0 P = { p P d( p) = d t( p), + P = P { p P d( p) d t( p) For gve cotext free grr G= (,,, P), the followg lgorth deteres the correspodg herrchy of types Algorth 2 Iput: A cotext free grr G = (,,, P) Output: he set of productos P Method: cossts of buldg sequece of sets { P wth the property 2 P P P = P = he set P wll cot ll useful productos of the grr G 00 AR 0 : = P = { p P d( p) = d t( p) 0 DO UIL ( P = P ) + P = P { p P d( p) 0 d t( p) 05 : = + 06 EDDO 07 OP Property 2 For y gve cotext free grr G= (,,, P) there exst fte uber,, of producto levels (For the coplete proof see []) I buldg the type herrchy we use pproprte recursve specfcto for every herrchc level Every = { v, v2,, vr hs the recursve specfcto: r r :(2 ) (2 ), ( v, v2,, vr) = ([ v ],[ v2],,[ vr]), where [ v ] = { α = s0as A2 s A s+ p= A α Pd d( p) = v Exple 2 For the cotext free grr of rthetc expressos, for exple, we hve the followg producto herrchy: P ={F P 2 ={F, F P ={F, F, F, E P =P d the correspodg recursve specfctos re: :(2 ) (2 ), (F)=() :(2 ) (2 ), 2 (F, )=(, F) :(2 ) (2 ), (F,, E)=(, F F, ) :(2 ) (2 ), (F,, E)=( (E), F F, E+) he fx pots of ths specfctos yeld the pproprte types: ( ) = ( ) 2 ( ) = ( ) ( ) = ( ) hus f = ( ), e v = { 2 (, ) = (, ) 2 2( ) = 2(, ) = (, ) 2 ( ) = (, ) hus f (, ) v =, e 2 = {, v2 = { (,, ) = (,, ) 2 (,, ) = (,, ) = (,, ) (,, ) = (,, ) = (,{,, ) (,, ) = (,{,, ) = (,{,,, {, ) ( ) = (,{,,,,, 2 tes tes {,,,, )

5 Proceedgs of the th WEA Itertol Coferece o COMPUER, Agos olos, Crete Isld, Greece, July 26-28, herefore v { f = (,{,{ ) d =, v2 = {, v = {, where = tes (,, ) = (,, ) 2 (,, ) = (,, ) = (,, ) (,, ) = (,, ) = (,{,, ) (,, ) = (,{,, ) = ({,( ), {,,, {,, +, + ) 5 (,, ) = ({, (),{,,, {,, +, +)= ({, (), (), (+), (+), {, (),,,, (), (), (), {,,, +, +, ++, ++, +, +, ++, ++, +, +, ++, ++) Geerlly, f bulds the set of bse prry types of lguge L(G) d represets the geertg level or the lexco of lguge L(G), other words, the lexcl level of lguge L(G) 5 Coclusos d future wors I geerl, cocrete coputtol structures re coposed of fly of obect sets, uber of (prtl detered opertos) o these obects d seres of propertes of these opertos Hece, coputtol structure s lgebrc structure ce coplex pplctos wll requre the possblty to reso bout reltol lgebrs buld fro other reltol lgebrs v cert costructo prcples, t becoes ecessrly to llow resog ot oly wth sgle relto lgebr but lso bout severl structures d the coectos betwee the Recursve specfctos ephsze lguge costructo fro sple levels to coplex oes Eve f we poted out the terl herrchy of gve grr, ths tur c be exteded ultedly, by ddg, to ew types d opertos buld upo the lredy defed levels he here preseted des re prt of our reserch o forl ethods for progrg lguge d o of the reserch drectos of our locl Reserch Cetre rous coottos d prctcl spects wll be prt of further scetfc ppers As purpose, we re curretly tedg to pply ths ethod systes for pplcto geerto bsed o ult lyer specfcto We wll otor wth gret terest the evoluto of the syste, over the te, to see f the ulted extesblty s relble log ru soluto Refereces: [] Erest G Mes, Mchel A Arbb, Algebrc Approches to progr setcs prger erlg, ew Yor, Berl, Hedelberg, Lodo, Prs, oyo, 986 [2] Alfred Aho, Rv eth, Jeffrey D Ull, Coplers: Prcples, echques, d ools, Addso Wesley, 200 [] eodor Rus, Mecse forle petru specfcre lbelor, Ed Acdee Roâe, Bucureşt, 98 [] Crăcue, rsltore ş copltore, Ed Al Mter, bu, 2002 [5] El M Pop, Modele forle coputtole, Edtur Al Mter, bu, 2000 [6] El M Pop, Lbe forle Fudetele lbelor de progrre, Edtur Al Mter, bu, 200 [7] El M Pop, Progrre geetc s evolutv, Edtur Al Mter, bu, 200 [8] El M Pop, Forl ytx d etcs of Progrg Lguge, Edtur Al Mter, bu, 200 [9] Rlf Fb, Lbe forle, eore, Exeple, Problee, Edtur Uverstăţ Luc Blg bu, 2006 [0] Cregă, C Rescher, D ovc, Itroducere lgebrcă î fortcă Lbe forle, Ed Jue, Iş, 97 [] Cregă, C Rescher, D ovc, Itroducere lgebrcă î fortcă eore utotelor, Ed Jue, Iş, 97 [2] oder Juc, Lbe forle ş utote, Ed MtrxRo, 999 [] Gbrel Or, Lbe forle ş cceptor, Ed Albstră, Clu-poc, 2002 [] Luc D Şerbăţ, Lbe de progrre ş copltore, Ed Acdee Roâe, Bucureşt, 987 [5] Jürge Dssow, Gheorghe Pău, Regulted Rewrtg Forl Lguge heory, Adee erlg Berl, 989 [6] Gh Pău, Grtc cotextule, Bucureşt, 982 [7] Gh Pău, Mecse geertve le proceselor ecooce, Ed ehcă, Bucureşt, 988