1 Ovrviw In this ppr w onsir two istint, ut intrtwin, thms. Th rst is nighorhoo xpnsion grmmrs whih onstitut onsirl hng rom mor tritionl phrs strutur

Transcription

1 Nighorhoo Expnsion Grmmrs John L. Pltz Dpt. o Computr Sin Univrsity o Virgini Chrlottsvill, VA Mrh 1, 1999 Dpt. o Computr Sin Thnil Rport, TR Astrt Phrs strutur grmmrs, in whih non-trminl symols on th lt si o proution n rwrittn y th string on th right si, togthr with thir Chomsky hirrh lssition, r milir to omputr sintists. But, ths grmmrs r most tiv only to gnrt, n prs, strings. In this rport, w introu nw kin o grmmr in whih th right si o th proution is simply ppn to th intrmit strutur in suh wy tht th lt si oms its "nighorhoo" in th nw strutur. This prmits th grmmtil nition o mny irnt kins o "n-imnsionl" isrt struturs. Svrl xmpls r givn. Morovr, ths grmmrs yil orml thory groun in ntimtroi losur sps. For xmpl, w show tht rstrit nighorhoo xpnsion grmmrs ptur th ssn o nit stt n ontxt r phrs strutur grmmrs. Rsrh support in prt y DOE grnt DE-FG05-95ER

2 1 Ovrviw In this ppr w onsir two istint, ut intrtwin, thms. Th rst is nighorhoo xpnsion grmmrs whih onstitut onsirl hng rom mor tritionl phrs strutur grmmrs in tht intrmit symols r not rwrittn. Inst, ompltly nw pis r simply to th isrt systm ing gnrt. Suh isrt systms w mol y grphs, G k, in this ppr. Th son thm is tht o losur sps n thir ssoit lttis [19]. I th lss o isrt systm ing gnrt is losur sp, w will l to n \goo" grmmrs s thos whih homomorphilly prsrv th inu ltti strutur o th systm with h stp o th gnrtion. Finlly, w will l to vlop hirrhy o grmmrs tht inlus th Chomsky hirrhy, n y whih w n qulittivly msur th omplxity o isrt systms. 1.1 Nighorhoo Expnsion Grmmrs In phrs strutur grmmr, sustring, or mor usully singl non-trminl symol A, is rwrittn y som othr longr string. Whn phrs strutur rwriting is ppli to mor omplx, non-linr systms, sugrph H, or singl no, o G is rwrittn s lrgr sugrph H 0 whih must thn m into G 0 = G? H. Dsriing th ming pross is not lwys sy [20]. Nighorhoo xpnsion grmmrs o not rwrit ny portion o th xisting intrmit systm. Inst, nw portions r simply to th xisting systm G. Som sugrph H o G is signt to om th nighorhoo o this nw strutur. In proution orm, th right si H 0 nots th nw strutur, otn singl point, n th lt si nots th sugrph H o G whih will th nighorhoo o H 0 in G 0. This pross is mor sily visuliz y smll xmpl. A grmmr G horl to gnrt horl grphs n sri y singl proution: K n := p tht is, ny liqu o orr k in G n srv s th nighorhoo o nw point p. Evry point in K n will jnt to p in G 0. 1 Figur 1 illustrts rprsnttiv gnrtion squn. Eh xpn nighorhoo (in this s liqu) hs n m ol; n th xpnsion point irl. Th sh gs init thos whih n th liqu s th nighorhoo o th xpnsion point p. It is not hr to s tht ny grph gnrt in this shion must horl. Bus xtrm points r simpliil (nighorhoo is 1 I w rwrit th proution so tht th right si is oth th nighorhoo n th nwly rt point p within it, th proution oms K n := K n+1 tht is, to ny liqu on n lmnts n nw lmnt to rt liqu on n + 1 lmnts. 2

3 Figur 1: A squn o nighorhoo xpnsions gnrting horl grphs liqu), n us vry horl grph must hv t lst two xtrm points [8, 6], vry horl grph n so gnrt. I th lt si o th xpnsion rul o G horl is rstrit to only K 1, i.. singl point, thn G gnrts ll, n only, unirt trs. Ths two grmmrs lrly illustrt th ssntil tr-lik strutur o horl grphs. This lttr rsult is wll known, sin vry horl grph is th intrstion grph o th sutrs o tr [7, 14]. 1.2 Nighorhoos W sy is omplt nighorhoo oprtor i or ll X U, X X:, X Y implis X: Y:, (X [ Y ): = X: [ Y:, (X \ Y ): = X: \ Y:. (possily unnssry) Clrly, nighorhoo oprtor xpns th sust. W r otn mor intrst in th lt nighorhoo,, n X: = X:? X; us it rprsnts this inrmntl ition. Lt us onsir rprsnttiv nighorhoo oprtor. I G is grph, w n n lt nighorhoo y X: j = y 62 Xj9x 2 X jnt to y g. Rily, this is th nighorhoo onpt w us in G horl. It pprs in mny grph-thorti lgorithms. 3

4 2 Othr Expnsion Grmmrs W ll G lok grph i vry mximl 2-onnt sugrph is liqu K n [10]. A simpl grmmr G lok onsisting o only th singl rul p := K n n 1 will gnrt th lok grphs. Figur 2 illustrts on suh rivtion squn in G lok. Hr, h p to xpn is moln n its onntion to th nw K n init Figur 2: A squn o nighorhoo xpnsions gnrting lok grphs y sh lins. Rily, th nighorhoo o h nw xpnsion K n is th singl point. Contrsting G horl with G lok is invitl. Th liqu K n on th lt si o th proution in G horl is just xhng with th nw point p on th right si. Both gnrt tr-lik isrt systms. A grmmr to gnrt tringulr plnr ntworks is mor omplx n introus th us o lll gs. Th us o g lls to ontrol grmmrs is not nw,.. [3, 13]. := * := u u u := 4

5 Th ll inits tht th g is xtrior. Whn th rst proution is ppli nw point is rt in th xtrior in suh wy tht th g is its nighorhoo. Sin, th originl g is no longr xtrior, its ll is rmov n th two onnting gs lll s \xtrior". A nw point my insrt into th intrior o ny tringl, s in th son proution. But, insrtion o two suh points stroys plnrity (xpt in th spil s whn two, or mor, gs r xtrior gs.) Consquntly, h o th thr gs onstituting th nighorhoo o th nw point r lll with u (or us). Ths u lls r nvr rmov. Th son proution n ppli only i two o th gs r unll ( nots on't r onition). Th thir proution is it nomolous; it mois nighorhoo inst o ing point. Only th init g is. 2 Th strt strutur or plnr tringultions onsists o singl tringl, ll o whos gs r xtrior. Figur 3 illustrts on suh rivtion squn in G tringl. u u u u u u Figur 3: A squn o nighorhoo xpnsions gnrting plnr tringulr gris I w onsir isrt prtilly orr systms, thr r two istint nighorhoo onpts: X: < = z 62 Xjz < x n z < y x implis y 2 Xg n X: > = z 62 Xjx < z n x y < z implis y 2 Xg. Th nighorhoo oprtor < is ll th ovring oprtor whn th prtil orr is rgr s ltti. 2 This proution ws to hnl th spil ss ssoit with rul 2. It is not iult to writ two spil nighorhoo xpnsion ruls to o this; ut th rul, s givn, is so usul whn xpning tringultion on th \outsi", n oppos to suiviing it on th \insi", tht it ws rtin in its urrnt orm. 5

6 Consquntly, w hv t lst thr irnt grmmrs to gnrt prtil orrs, pning on whthr on uss <, >, or <> (whih omins thm oth) s th nighorhoo onpt. For xmpl, th grmmr G right tr with th singl proution p := q lrly gnrts th lss o root trs whos orinttion pns on whthr p is to q: < or q: >. I w lt th nighorhoo not y th lt hn o this proution n ritrry sugrph N o G, s in th proution N := q whr q: < is still th nighorhoo oprtor, w gt G lt root gnrting ll lt root yli grphs. Figur 4 illustrts on suh gnrtion squn. I inst o th singlton Figur 4: A squn o nighorhoo xpnsions gnrting lt root yli grphs point o Figur 4, w llow n ritrry ntihin, A n, to th strt strutur, thn rily G lt root oms G yli whih gnrts ll yli grphs or prtil orrs. I inst, w us th nighorhoo oprtor >, th sm proution yils ll right root yli grphs, whih with th ntihin strt strutur gnrts yli grphs s or. Som intrsting sumilis o th yli grphs n vlop i on uss th two-si nighorhoo oprtor, X: <> = X: < [ X: >. Th mjor prolm with ths two-si nighorhoos is tht or n ritrry sust o H o G, it my not vint whih lmnts long to whih si. Suh grmmr w woul ll miguous. An unmiguous grmmr or two trminl, prlll sris ntworks n spi y rquiring th lt hn nighorhoo to singl g. For xmpl, G ttspn hs th two rwrit ruls, p 1 p 2 := q p 1 p 2 := q 1 q 2 A rprsnttiv xmpl o squn o struturs gnrting y G ttspn, givn singl g 6

7 s th strt strutur is shown in Figur 5. Agin, nw points o th xpnsion r not Figur 5: A squn o two trminl, prlll sris ntworks gnrt y, G ttspn y irls with sh lins initing th onntion to thir nighorhoos. Purists my osrv tht nighorhoo xpnsion grmmrs r xpt to only nlrg th intrmit struturs. G ttspn lts th g on th lt si tr ing th right hn lmnts or whih it is to th nighorhoo. Isn't this rlly phrs strutur grmmr in whih gs r rwrittn inst o symols? Although on oul opt this intrprttion, it is mor urt to osrv tht i w rtin th lt hn g tr th xpnsion, th rsulting strutur is simply th trnsitiv losur o TTSPN, so thir rsur is not rlly hnging nything. 3 Closur Sps Whn phs strutur proution is ppli to string it prsrvs th strutur o th string. Evry intrmit orm is string. W liv tht ll grmmrs shoul prsrv th ssntil strutur o thir unrlying ojts. But wht is th ssntil strutur o mily o isrt ojts? W liv uniquly gnrt losur oprtors rprsnt si wy o sriing th strutur o isrt systm. Nighorhoo xpnsions hv n n so s to homomorphilly prsrv this onpt o strutur. Any losur oprtor ' must stsiy th xioms: X X:', X Y implis X:' Y:', n X:':' = X:', whr X; Y U r ritrry susts in univrs U o intrst [12]. I in ition, X:' = Y:' implis (X \Y ):' = X:', w sy tht ' is uniquly gnrt. For xmpl, monophoni losur [6], in whih X:' nots ll points lying on horlss pths twn istint points p; q 2 X is uniquly gnrt ovr horl grphs. A vtor sp, or mtroi M, is th losur (or spn ) o st o sis vtors [21]. M must stisy th xhng xiom: p; q 62 X:, n q 2 (X [ p): imply tht 7

8 p 2 (X [ q):. 3 It n shown [19] tht uniquly gnrt losur sps must stisy th nti-xhng xiom: p; q 62 X:', n q 2 (X [ p):' imply tht p 62 (X [ q):'. From this oms th jtiv ntimtroi losur sp. Closur sps orm kin o ul to vtor sps. I th sts o losur sp (U; ') r prtilly orr y X ' Y i n only i Y \ X:' X Y:' w otin losur ltti [19], in whih th sultti o los susts is lowr smimoulr. 4 W hv ssrt tht this losur ltti L (U;') sris th strutur o th unrlying isrt systm. Figur 6() illustrts th losur ltti o th sixth horl grph gnrt in Figur 1. This is omplx igrm; ut thr r signint rgulritis () Ø () Figur 6: A horl grph n its losur ltti tht mk it mnl to nlysis. For xmpl, th sultti o los susts (thos whih r onnt y soli lins) n shown to lowr smi-moulr ltti [15]. Consir th los sust g whih ovrs th los susts g; g; g. Thy, in turn, ovr thir pirwis in's, g; g; g, whih in turn ovr thir ommon in, g, to orm istriutiv sultti. This is gnrl proprty o lowr smi-moulrity, s shown in [4]. Th los sust g is gnrt y th sust g, whih is ov, to th lt, n onnt y sh lin. Suh gnrtors, not X:, r th uniqu miniml susts with givn losur. 5 Th susts ontin twn ny los st X:' 3 This prmits th milir hng o sis o vtor sp. 4 Th ltti o vtor susps prtilly orr y inlusion is uppr smi-moulr. 5 Mny trms r oun in th litrtur or ths gnrting sts pning on ons pproh. With onvx losur in isrt gomtry on spks o xtrm points [5]. With rspt to trnsitiv losurs in 8

9 n its gnrtor X: ompris ooln lgr, [X:'; X:], whih w skth in with sh imon. A w o th orring rltionships twn non-los susts hv n init y ott lins. Although ths losur lttis grow xponntilly, or xmpl, th nl horl grph gnrt in Figur 1 on 9 points hs 146 los sts, on n still rson out thm. W will s this prsntly. But, thy om too lrg to illustrt sily. Although thr r similritis, nighorhoo oprtor is not in gnrl losur oprtor us (1) nighorhoo oprtor n not impotnt, X: X::, n (2) losur oprtor n not union prsrving, X:' [ Y:' (X [ Y ):'. Thr n mny irnt losur oprtors, '. Similrly, thr n mny nighorhoo oprtors,. W sy tht n ' r omptil i X los implis X: is los. It is not hr to show tht i G is horl thn j n ' monophoni r omptil on horl grphs. Similrly, j n ' gosi r omptil on lok grphs. W know o no uniquly gnrt, losur oprtor on tringult plnr gris. But on my xist. In [17] it is shown tht t lst n n istint oprtors n n ovr ny n lmnt systm with n 10. Howvr, suh losur oprtor nnot omptil with th nighorhoo oprtor. Figur 7 inits why. Assum tht vry point is los, so p: j is los or Ø Figur 7: Clos sts o non-horl grph ll p. Consquntly, th our quriltrls ; ;, n r los. Bus los sts r los unr intrstion, th tringls ; ;, n must los s wll. This trmins th uppr portion o th ltti 6 whih nnot losur ltti. rltionl lgrs on lls thm th irruil krnl [1]. In [19], w ll thm gnrtors, ut not thm with th symol, suggstiv o sis th gnrtors o mtroi sp. 6 In Figur 7, ll gs r lso los. Bus los sts r los unr intrstion, th gs ; ;, n must los. On oul ssum tht th rmining 4 gs r not los, in whih s oms th gnrtor o, t. But, this will not hng th ssn o this ountr xmpl. 9

10 A unmntl proprty o uniquly gnrt losur lttis n oun in [19] Lmm 3.1 I ' is uniquly gnrt, n i Z 6= is los, p 2 Z: i n only i Z? pg is los, in whih s Z:? pg (Z? pg):. Using this proprty on n u tht g: = g, g: = g, g: = g, n g: = g. But, this is impossil. Th st g nnot gnrt two irnt los sts! Nithr n g. Bus Figur 7 is th simplst non-horl grph, w hv Thorm 3.2 Lt th iniviul lmnts o n unirt strutur G los. A losur oprtor ' n omptil with th jny nighorhoo oprtor, j i n only i G is horl. Proo: I G is horl, thn th monophoni losur oprtor ' monophoni stiss th thorm. Only i ollows or th xmpl ov. 2 Conjtur 3.3 Lt F mily o isrt unirt systms G. I thr xists nighorhoo oprtor n omptil losur oprtor ' in whih vry singlton is los, thn G is ssntilly tr-lik. For \tr-lik", w xpt nition s on intrstion grphs [14]. In [19], thr irnt losur oprtors r n ovr prtilly orr systms. Thy r Y:' L = xj9y 2 Y; x yg, Y:' R = zj9y 2 Y; y zg, Y:' C = xj9y 1 ; y 2 2 Y; y 1 x y 2 g. On prtilly orr systms, ' L, n ' R r il losur oprtors, ' C is n intrvl, or onvx, losur oprtor. 7 It is not iult to show tht < is omptil with oth ' L n ' R, s is >. Th inu losur ltti rsulting rom th losur oprtor ' L on th 7 point post o Figur 4 is shown in Figur 8. As in Figur 6, th sultti o los susts is not y soli lins; n th ooln intrvls oun y th los susts X:' n thir gnrting sts X: r not y sh lins whih hv n orint towr th uppr lt. This xmpl is lss luttr, so th hrtristi losur ltti strutur is mor vint. For instn, th sust g is th uniqu gnrtor or th los st g. All 2 4 susts in th ooln ltti 7 Prtil orrs my strutur with rspt to t lst ths thr irnt losur oprtors, ' L ; ' R or ' C. Th ility to intrprt isrt systms with rspt to irnt losur onpts n quit powrul. S [18]. 10

11 g g g g g g g g () () Ø Figur 8: A losur ltti L G;' L thy limit (inluing g), hv g s thir losur. Th orring twn ths ooln gnrtor su-lttis, whih hv n suggst y w ott lins, mirror th orring o th los sts low thm. 4 Strutur Prsrvtion Why shoul on opt pross tht gnrts lmily o isrt struturs y ing som ompltly nw portion within nighorhoo rthr thn rwriting n xisting lmnt, whih o ours must hv nighorhoo? A prtil nswr ws givn in stions 1 n 2. Th nighorhoo xpnsion pproh n us to gnrt svrl milis o intrsting isrt systms. In Stion 5, w will monstrt tht th lsss o rgulr n ontxt r lngugs in th Chomsky hirrhy o th phrs strutur grmmrs n susum y nighorhoo xpnsion lsss. In this stion w xmin mor sutl rgumnt s on th homomorphi prsrvtion o isrt systm strutur. Homomorphi (strutur prsrving) gnrtion is most sily n in trms o its invrs, homomorphi prsing. In this s prsing onsists o simply lting th point 11

12 n ll its inint gs. Figur 9() illustrts th ltion o th lmnt rom th horl grph o Figur 2(). W lt, whih is suggstiv o \striking out", not th ltion oprtor. Th susript nots th lmnt, or st, ing lt. In th χ Ø Ø Figur 9: A \ltion" trnsormtion, sour ltti L G, pr-img quivln sts r init y ott lins. Th lmnt is mpp onto in th trgt ltti, n mny sts ontining, suh s g, r mpp onto th st otin y rmoving. But, s w n s rom th mortly omplx pr-img prtition in L G o Figur 9, ltion is not just simpl st xtnsion o point untion. 8 For xmpl, th st g mps onto th st g. This is nssry i th th trnsormtion is to prsrv th strutur o ll th susts, tht is, i th ltti o G:, not L G:, is to smi-homomorphi ontrtion o L G. For isrt systm with n lmnts, its losur ltti will hv 2 n lmnts. Consquntly, ruing th siz o systm grtly rus its stuturl omplxity, s is lr in Figur 9. 8 Dltion o lmnts mk littl sns within th usul nition o isrt untions us lt lmnt hs no img in th oomin. Howvr, in th ontxt o ltti ovr ll susts o isrt sp, lt lmnt or susp n simply mpp onto th mpty st,, in th oomin. 12

13 W now hv mor orml wy o ning wht homomorphi nighorhoo xpnsion, N, rlly is. Lt G (n)?! p G (n?1) ny ltion in isrt systm with n points. By n xpnsion N with rspt to th nighorhoo N w mn n invrs oprtor to p, whr N is th img in G (n?1) o th nighorhoo N o p not p:n in G (n). Thus, w lwys hv G (n)?! p G (n?1)?! N G (n) An, sin ltions lwys inu homomorphi mppings twn th orrsponing losur lttis, nighorhoo xpnsion n rgr s homomorphi s wll. Lmm 4.1 I thr xists losur oprtor, ' tht is omptil with th nighorhoo oprtor,, us y nighorhoo xpnsion grmmr G, thn ny squn o ltions or xpnsions homomorphilly prsrvs th intrnl strutur o h intrmit systm. Th sns o this lmm is illustrt y Figur 10. As w hv sn with G tringl, not ll εn ε ε N N G G G G χ χ χ (1) (2) (3) (4) N ϕ p p p p ε χ... ϕ ϕ ϕ ϕ LG LG LG LG (1) (2) (3) (4)... Figur 10: A squn o ltions/xpnsions xpnsion grmmrs hv omptil losur oprtors. But, i on xists it nors kin o rgulrity in th grmmr's rivtions. 5 Expnsion Grmmrs n th Chomsky Hirrhy How irnt r nighorhoo xpnsion grmmrs rom mor ustomry phrs strutur grmmrs? W will lim thy r vry similr, xpt tht nighorhoo xpnsion sms mor xil. Thorm 5.1 Lt L ny rgulr lngug. Thr xists nighorhoo xpnsion grmmr G in whih th lt si (nighorhoo) o vry proution is los singlton lmnt, suh tht L(G) = L. Proo: Lt G rgulr ny rgulr, or nit stt, n st into ithr lt, or right rursiv orm. W ssum G rgulr is right rursiv, tht is ll proutions r o th orm V := V 0 j, so only th rightmost lmnt o th string is vril. W us < n ' R s our nighorhoo n losur oprtors rsptivly. Th rightmost lmnt 13

14 is los, s r ll rightmost intrvls. This rightmost lmnt provis th nighorhoo, <, or th xpnsion. I < D is ll with V, it is ithr rll with n th nw xpnsion point ll with V 0, or ls < D is just rll with th trminl symol, thry ning th gnrtion. I G rgulr is lt rursiv, w woul us > n ' L s th nisghorhoo n losur oprtors to otin th sm rsult. 2 Th only istintion is tht xpnsion grmmrs with ths proprtis n lso gnrt lngugs o root trs s wll. Thorm 5.2 Lt L ny ontxt r lngug. Thr xists nighorhoo xpnsion grmmr G in whih th lt si o vry proution is los, suh tht L(G) = L. Proo: Bus ontxt r grmmr G prmits non-trminl symols to rwrittn nywhr in th string, w rquir onvx losur ' C whr X:' C = qjp q r; p; r 2 Xg, to mult it. W lso ssum Chomsky norml orm [2, 9], in whih vry proution is o th orm V := V 1 V 2 or V :=. Agin, it is pprnt tht th nighorhoo ll with V n rll with V 1 n th xpnsion lmnt ll V 2. 2 By thir vry ntur, w woul ssum tht nighorhoo xpnsion grmmrs n mult ny ontxt-snsitiv grmmr, whih must non-rsing. Nighorhoo xpnsion grmmrs too must non-rsing, n th nighorhoo pnnt proutions woul sm to xtly ptur th ontxt snsitiv qulity. W liv this to tru. But, s yt, w hv oun no onvining proo. It is possil to tlog nighorhoo xpnsion grmmrs oring to whthr th lt si (nighorhoo) o vry proution is: simpl los st, los st, singlton lmnt, or hs som othr wll n proprty, whih w ll nighorhoo ritri; n whthr th right si (nighorhoo) o vry proution is: singlton lmnt, or hs som othr wll n proprty, whih w ll rplmnt ritri. W osrv tht th lt si o th proution o G horl is los nighorhoo in th intrmit strutur, s in G lok s wll. But, in th lttr grmmr vry lt si is lso singlton lmnt. Th lngug L lok is sust o L horl [11]. I this proution is urthr rstrit to hv only singlton lmnts on th right si, s in G tr, it lngug is L tr whih is sust o L lok. Th grmmr G lt root with hrtristi gnrtion s illustrt in Figur 4, stiss non o th nighorhoo ritri sri ov. I in ition to th nighorhoo 14

15 oprtor <, w hoos ' L (s illustrt in Figur 8) to its omptil losur oprtor, n w rquir h lt si nighorhoo to los, w otin rivtions suh s Figur 11. Rily, rquiring th lt si nighorhoo to los gnrts lngug Figur 11: Anothr squn o nighorhoo xpnsions gnrting lt root yli grphs o trnsitivly los yli grphs. Simply swithing th omptil losur oprtor to ' R inlus th st o ll si rprsnttions o th prtil orr, tht is i x < y n y < z thn th g (x; z) is not in G [16]. I urthr, w rquir th lt si nighorhoo to los ntihin, G lt root gnrts only ths si rprsnttions o prtil orr. In ny s, prtilly orr systms r mor omplx thn irt, or unirt, trs with rspt to th nighorhoo xpnsion hirrhy. This ors with our lgorithmi intuition. Morovr, th st o ll trs is omprl to rgulr lngug whil th st o ll prtil orrs is omprl to ontxt r lngug. Agin, this sms intuitivly orrt. Thus w gin to s linguisti hirrhy mrging y whih w n qulittivly ompr th omplxity o irt n unirt isrt systms. 6 Nighorhoo Oprtors So r w hv only onsir two kins o nighorhoo oprtor simpl no jny, j, n th omprtiv nighorhoos, < ; > ; <> in prtil orrs. Ths r milir, sy to work with, n mk goo illustrtiv xmpls. Howvr, thr xist mny mor wys o ning nighorhoo oprtors n thir mhnisms or ining nw lmnts to n xpning ongurtion. Th xmpls w vlop is this stion r suggstiv o th wy toms my on to orm lrgr moluls, th wy smll prours with som 15

16 wll-n intrs my inorport into lrgr tsk, or how iniviul prts my omin to rt lrgr ssmlis. Our xmpl is surly simpl. Th lmnts r quiltrl tringls with gs ll (in lokwis orr) ; ; or ; ; s show in Figur 12. W hv suprimpos Figur 12: Two sptil lmnts n rrow to mk th orinttion vint in our gurs; ut thy r not tully prt o th grmmr or lngug. W suppos tht ths lmnts r sujt to th n trnsormtions, trnsltion n rottion, in th pln. Now suppos tht nighorhoo oprtor 1 spis tht n ut, n ut, n n ut. W lt th grmmr G 1 onsist o th singl rul p := q Now, pning on th initil orinttion o th initil lmnt rivtion must yil uniorm pttrn suh s Figur 13. Figur 13: A uniorm l in th lngug L(G 1 ) Th rigiity o th nighorhoo oprtor llows no xiility in G 1. Suppos w join th possiility tht n lso ut to th oprtor 1 to otin 2 n grmmr G 2. Rily L(G 1 ) L(G 2 ), ut L(G 2 ) hs mny itionl ongurtions. 16

17 I, using th nw nighorhoo rul, ntrl hxgonl \rottor" s shown in th ntr o Figur 14 is initilly onstrut, n i this is xpn in \rth-rst" shion, thn rigily rotting pttrn must gnrt. This ollows us th only g lls tht Figur 14: A \rotting" l in th lngug L(G 2 ) otin y xpning th ntrl hxgonl ongurtion n ppr on ny onvx xtrior ounry r ; ;, n whih uniquly trmin th orrsponing orinttion o th xpnsion lmnts s thy xpn out rom th ntrl or. O ours, thr xist mor irrgulr ongurtions in th lngug L(G 2 ) whih n gnrt y ltrntly ltting n gs nighors o g. Figur 15 illustrts on suh ongurtion. Not tht nithr o th tringls not y x n y x y Figur 15: A irrgulr rivtion in th lngug L(G 2 ) n oupi in nl ongurtion. It woul ppr tht L(G 2 ) n ritrrily irrgulr; ut this is not tru. W osrv tht, o th 6 possil orinttions o th two lmnts, only 2 ppr in Figur 15, n only 1 mor oul ppr. Similrly, w osrv tht only 3 orinttions ppr in Figur 14; n thy r not thos o Figur 15. Th nighorhoo xpnsion grmmr onpt hs ptur notion o symmtri prity in this vry simpl hxgonl sp. W r xploring wht othr proprtis it n sri. 17

18 Rrns [1] Chris Brink, Wolrm Khl, n Gunthr Shmit. Rltionl Mthos in Computr Sin. Springr Vrlg, Win, [2] Nom Chomsky. Forml Proprtis o Grmmrs. In R.D. Lu, R.R. Bush, n E. Glntr, itors, Hnook o Mthmtil Psyhology. Wily, Nw York, [3] M.J.M. Bor n A. Linnmyr. Mp 0L-systms with g ll ontrol, Comprison o mrkr n yli systm. In Hrtmut Ehrig, Mnr Ngl, Grzgorz Roznrg, n A. Rosnl, itors, Grph-Grmmrs n Thir Applition to Computr Sin, Ltur Nots in Computr Sin 291, pgs 378{392. Springr-Vrlg, D [4] Pul H. Elmn. Mt-istriutiv lttis n th nti-xhng losur. Algr Univrslis, 10(3):290{299, [5] Pul H. Elmn n Rort E. Jmison. Th Thory o Convx Gomtris. Gomtri Dit, 19(3):247{270, D [6] Mrtin Frr n Rort E. Jmison. Convxity in Grphs n Hyprgrphs. SIAM J. Algr n Disrt Mthos, 7(3):433{444, July [7] Fni Gvril. Th Intrstion Grphs o Sutrs in Trs r xtly th Chorl Grphs. J. Comintoril Thory B, 16:47{56, [8] Mrtin Chrls Golumi. Algorithmi Grph Thory n Prt Grphs. Ami Prss, Nw York, [9] John E. Hoprot n Jry D. Ullmn. Introution to Automt Thory, Lngugs, n Computtion. Aison-Wsly, Ring, MA, [10] Rort E. Jmison-Wlnr. Convxity n Blok Grphs. Congrssus Numrntium, 33:129{ 142, D [11] Rort E. Jmison-Wlnr. Prtition Numrs or Trs n Orr Sts. Pi J. o Mth., 96(1):115{140, Spt [12] Kzimirz Kurtowski. Introution to St Thory n Topology. Prgmon Prss, [13] J. Luk n H.B. Luk. From 0L n 1L mp systms to intrmint n trmint growth in plnt morphognsis. In Hrtmut Ehrig, Mnr Ngl, Grzgorz Roznrg, n A. Rosnl, itors, Grph-Grmmrs n Thir Applition to Computr Sin, Ltur Nots in Computr Sin 291, pgs 392{410. Springr-Vrlg, D [14] Trry A. MK n Fr R. MMorris. Topis in Intrstion Grph Thory. SIAM Monogrphs on Disrt Mthmtis n Applitions. Soity or Inustril n Appli Mth., Phillphi, PA, [15] B. Monjrt. A us or rquntly risovring onpt. Orr, 1:415{416, [16] John L. Pltz. Computr Dt Struturs. MGrw-Hill, F

19 [17] John L. Pltz. Evluting th inry prtition untion whn N = 2 n. Congrss Numrntium, 109:3{12, [18] John L. Pltz. Prtition oints o yli grphs. In Mnr Ngl, itor, Pro. WG 95, Grph-Thorti Conpts in Computr Sin, Ltur Nots in Comp. Si., #1017, pgs 318{332. Springr-Vrlg, [19] John L. Pltz. Closur Lttis. Disrt Mthmtis, 154:217{236, [20] John L. Pltz n Azril Rosnl. W Grmmrs. In Pro. Intn'l Joint Con on AI, pgs 609{619, Wshington, DC, My [21] W. T. Tutt. Introution to th Thory o Mtrois. Amr. Elsvir,