Dictionary Search and Update by P Systems with String-Objects and Active Membranes

Dic iona y Sea ch and Upda e by P Sys ems wi h S ing-Objec s and Ac i e Memb anes A iom Alhazo 2,1, S e lana Cojoca u1, Ludmila Malaho a1, Yu ii Rogozhin3,1 1Ins i u e o Ma hema ics and Compu e Science Academy o Sciences o Moldo a, Academiei 5, Chi¸sin˘au MD-2028 Moldo a {a iom,s e a,mal, ogozhin}@ma h.md 2IEC, Depa men o In o ma ion Enginee ing, G adua e School o Enginee ing Hi oshima Uni e si y, Higashi-Hi oshima 739-8527 Japan 3Ro i a i Vi gili Uni e si y, Resea ch G oup on Ma hema ical Linguis ics Pl. Impe ial T`a aco 1, Ta agona 43005 Spain Summa y. Memb ane compu ing is a o mal amewo k o dis ibu ed pa allel compu - ing. In his pape we implemen wo king wi h he p e ix ee by P sys ems wi h s ings and ac i e memb anes. 1 In oduc ion Sol ing mos p oblems o na u al language p ocessing is based on using ce ain linguis ic esou ces, ep esen ed by co po a, lexicons, e c. Usually, hese collec ions o da a cons i u e an eno mous olume o in o ma ion, so p ocessing hem equi es much compu a ional esou ces. A easonable app oach o ob aining e icien solu- ion is ha based on applying pa allelism; i has s a ed o be p omo ed al eady in 1970s. Fo ins ance, he possibili ies o applying massi e pa allelism in Machine T ansla ion a e conside ed in [4, 1]. We men ion ha many o he s ages o ex p ocessing ( om okeniza ion, segmen a ion, lema izing o hose dealing wi h na - u al language unde s anding) can be ca ied ou by pa allel me hods. This jus i ies he in e es o applying me hods o e ed by he biologically inspi ed models, and by memb ane compu ing in pa icula . Howe e , he e a e some issues ha by hei na u e do no allow comple e pa alleliza ion, ye exac ly hey a e o en hose “compu a ional p imi i es” ha a e ine i ably used du ing sol ing majo p oblems, like he elemen a y a i hme ic ope a ions a e always p esen in sol ing di icul compu a ional p oblems. Among such “p imi i es” in he compu a ional linguis ics he e a e handling o he dic io- na ies, e.g., dic iona y lookup and dic iona y comple ion. Exac ly hese p oblems cons i u e he subjec o he p esen pape . In ou app oach we speak abou dic- iona y ep esen ed by a p e ix ee. 2 A. Alhazo e al. Memb ane sys ems a e a con enien amewo k o desc ibing compu a ions on ees. Since memb ane sys ems a e an abs ac ion o li ing cells, he memb anes a e a anged hie a chically, yielding a ee s uc u e. 2 De ini ions Memb ane compu ing is a ecen domain o na u al compu ing s a ed by Gh. P˘aun in [2]. The componen s o a memb ane sys em a e a cell-like memb ane s uc u e, in he egions o which one places mul ise s o objec s which e ol e in a synch onous maximally pa allel manne acco ding o gi en e olu ion ules associa ed wi h he memb anes. The necessa y de ini ions a e gi en in he ollowing subsec ion; see also [3] o an o e iew o he domain and [5] o he comp ehensi e bibliog aphy. 2.1 Compu ing by P sys ems Le Obe a ini e se o elemen s called symbols; he se o wo ds o e Ois deno ed by O∗, and he emp y wo d is deno ed by λ. De ini ion 1. A P sys em wi h s ing-objec s and inpu is a uple Π=¡O, Σ, H, E, µ, M1,· · · , Mp, R, i0¢, whe e: •Ois he wo king alphabe o he sys em whose elemen s a e called objec s. •Σis an inpu alphabe . •His an alphabe whose elemen s a e called labels. •Eis he se o pola iza ions. •µis a memb ane s uc u e (a oo ed ee) consis ing o pmemb anes injec i ely labeled by elemen s o H. •Miis an ini ial mul ise o s ings o e Oassocia ed wi h memb ane i,1≤i≤ p. •Ris a ini e se o ules de ining he beha io o objec s om Oand memb anes labeled by elemen s o H. •i0iden i ies he inpu egion. A con igu a ion o a P sys em is i s “snapsho ”, i.e., he cu en memb ane s uc u e and he mul ise s o s ings o objec s p esen in egions o he sys em. While ini ial con igu a ion is C0= (µ, M1,···, Mp), each subsequen con igu a ion C0is ob ained om he p e ious con igu a ion Cby maximally pa allel applica ion o ules o objec s and memb anes, deno ed by C⇒C0(no u he ules a e applicable oge he wi h he ules ha ans o m Cin o C0). A compu a ion is hus a sequence o con igu a ions s a ing om C0, espec ing ela ion ⇒and ending in a hal ing con igu a ion (i.e., such one ha no ules a e applicable). I Mis a mul ise o s ings o e he inpu alphabe Σ⊆O, hen he ini ial con igu a ion o a P sys em Πwi h an inpu Mo e alphabe Σand inpu egion i0is (µ, M1,· · · , Mi0−1, Mi0∪M, Mi0+1,· · · , Mp). Dic iona y Sea ch by P Sys ems wi h S ing-Objec s 3 2.2 P sys ems wi h ac i e memb anes To speak abou P sys ems wi h ac i e memb anes, we need o speci y he ules, i.e., he elemen s o he se Rin he desc ip ion o a P sys em. Due o he na u e o he p oblem o his pape , he s anda d model was gen- e alized in he ollowing: •Coope a i e ules: a ule can conside consecu i e symbols in a s ing (o he - wise, he ime complexi y would be much highe ). •S ing eplica ion ( o e u n he esul wi hou emo ing i om he dic io- na y). •Memb ane c ea ion ( o add wo ds o he dic iona y). Hence, he ules can be o he ollowing o ms: (a∗) [ a→b]e h, o h∈H, e ∈E, a, b ∈O∗ (e olu ion ules, associa ed wi h memb anes and depending on he label and he pola iza ion o he memb anes, bu no di ec ly in ol ing he memb anes, in he sense ha he memb anes a e nei he aking pa in he applica ion o hese ules no a e hey modi ied by hem); (a∗ ) [ a→b||c]e h, o h∈H, e ∈E, a, b, c ∈O∗ (like he p e ious case, bu wi h s ing eplica ion); (b∗)a[ ]e1 h→[b]e2 h, o h∈H, e1, e2∈E, a, b ∈O∗ (communica ion ules; an objec is in oduced in o he memb ane; he objec can be modi ied du ing his p ocess, as well as he pola iza ion o he memb ane can be modi ied, bu no i s label); (c∗) [ a]e1 h→[ ]e2 hb, o h∈H, e1, e2∈E, a, b ∈O∗ (communica ion ules; an objec is sen ou o he memb ane; he objec can be modi ied du ing his p ocess; also he pola iza ion o he memb ane can be modi ied, bu no i s label); (d∗) [ a]e h→b, o h∈H, e ∈E, a, b ∈O (dissol ing ules; in eac ion wi h an objec , a memb ane can be dissol ed, while he objec speci ied in he ule can be modi ied); (g∗) [ a→[b]e2 g]e1 h, o g, h ∈H, e1, e2∈E, a, b ∈O∗ (memb ane c ea ion ules; an objec is mo ed in o a newly c ea ed memb ane and possibly modi ied). Addi ionally, we will w i e ∅in place o some s ings on he igh -hand side o he ules, meaning ha he en i e s ing is dele ed. The ules o ypes (a∗),(a∗ ) and (g∗) a e conside ed o only in ol e objec s, while all o he ules a e assumed o in ol e objec s and memb anes men ioned in 4 A. Alhazo e al. hei le -hand side. An applica ion o a ule consis s in eplacing a subs ing de- sc ibed in he le -hand side o a s ing in he co esponding egion (i.e., associa ed o a memb ane wi h label hand pola iza ion e o ules o ypes (a∗),(a∗ ) and (d∗), o associa ed o a memb ane wi h label hand pola iza ion e1 o ules o ype (c∗), o immedia ely ou e o such a memb ane o ules o ype (b∗) ), by a s ing desc ibed in he igh -hand side o he ule, mo ing he s ing o he co esponding egion ( ha can be he same as he sou ce egion immedia ely inne o immedi- a ely ou e , depending on he ule ype), and upda ing he memb ane s uc u e acco dingly i needed (changing memb ane pola iza ion, c ea ing o dissol ing a memb ane). The ules can only be applied simul aneously i hey in ol e di e en objec s and memb anes (we epea ha ules o ype (a) a e no conside ed o in ol e a memb ane), and such pa allelism is maximal i no u he ules a e applicable o objec s and memb anes ha we e no in ol ed. 3 Dic iona y Dic iona y sea ch ep esen s compu ing a s ing- alued unc ion {ui−→ i|1≤i≤d} de ined on a ini e se o s ings. We ep esen such a dic iona y by he skin memb ane con aining he memb ane s uc u e co esponding o he p e ix ee o {ui|1≤i≤d}, wi h s ings $ i$0in egions co esponding o he nodes associa ed o ui. Due o echnical easons, we assume ha o e e y l∈A1, he skin con ains a memb ane wi h label l. We also suppose ha he sou ce wo ds a e non-emp y. Fo ins ance, he dic iona y {ba −→ lying,bi −→ s o ed}is ep esen ed by [ [ ]0 a[[[$ lying$0]0 ]0 a[[$s o ed$0]0 ]0 i]b[ ]0 c· · · [ ]0 z]0 0 Le A1, A2be he alphabe s o he sou ce and a ge languages, espec i ely. Conside a P sys em co esponding o he gi en dic iona y. Π=¡O, Σ, H, E, µ, M1,· · · , Mp, R, i0¢, O=A1∪A2∪ {?,?0,$,$0,$1,$2, ail} ∪ {?i|1≤i≤11}∪{!i|1≤i≤4}, Σ=A1∪A2∪ {?,?0,!,$,$0}, H=A1∪ {0}, E ={0,+,−}, µand se s Mi,1≤i≤p, a e de ined as desc ibed abo e, i0= 1, so only he ules and inpu seman ics s ill ha e o be de ined. Dic iona y Sea ch by P Sys ems wi h S ing-Objec s 5 3.1 Dic iona y sea ch To ansla e a wo d u, inpu he s ing ?u?0in egion 1. Conside he ollowing ules. S1 ?l[ ]0 l→[?]0 l,l∈A1 P opaga ion o he inpu in o he memb ane s uc u e, eaching he loca ion co - esponding o he inpu wo d. S2 [ ??0]0 l→[ ]− l∅,l∈A1 Ma king he egion co esponding o he sou ce wo d. S3 [ $ →$1||$2]− l,l∈A1 Replica ing he ansla ion. S4 [ $2]e l→[ ]0 l$2,l∈H,e∈ {−,0} Sending one copy o he ansla ion o he en i onmen . S5 [ $1→$ ]0 l,l∈A1 Keeping he o he copy in he dic iona y. The sys em will send he ansla ion o uin he en i onmen . This is a simple example illus a ing sea ch. I he sou ce wo d is no in he dic iona y, he sys em will be blocked wi hou gi ing an answe . The ollowing subsec ion shows a solu ion o his p oblem. 3.2 Sea ch wi h ail The se o ules below is conside ably mo e in ol ed han he p e ious one. How- e e , i handles 3 cases: a) he a ge wo d is ound, b) he a ge wo d is missing in he a ge loca ion, c) he a ge loca ion is un eachable. F1 [ ? →?1||?2]0 0 Replica e he inpu . F2 [ ?2→?3]0 0 Delay he second copy o he inpu o one s ep. F3 ?1l[ ]0 l→[ ?1]+ l,l∈A1 P opaga ion o he i s copy owa ds he a ge loca ion, changing he pola iza ion o he en e ed memb ane o +. F4 ?3l[ ]+ l→[ ?3]0 l,l∈A1 P opaga ion o he second copy owa ds he a ge loca ion, es o ing he pola - iza ion o he en e ed memb ane. 6 A. Alhazo e al. F5 [ ?1l→[ ?4]− l]0 k,l, k ∈A1 I a memb ane co esponding o some symbol o he sou ce wo d is missing, hen he i s copy o he inpu emains in he same memb ane, while he second copy o he inpu es o es i s pola iza ion. C ea ing a memb ane o handle he ailu e. F6 [ ?1?0→?7]0 l,l∈A1 Ta ge loca ion ound, ma king he i s inpu copy. F7 [ ?7]0 l→[ ]− l∅,l∈A1 Ma king he a ge loca ion. In ei he case, some memb ane has pola iza ion −. I emains o send he answe ou , o ail i i is absen . The memb ane should be dele ed in he ail case. F8 [ $ →$1||$2]− l,l∈A1 Replica ing he ansla ion. F9 [ $2]e l→[ ]0 l$2,l∈H,e∈ {0,−} Sending one copy o he ansla ion ou . F10 [ $1→$ ]0 l,l∈A1 Keeping he o he copy in he dic iona y. F11 [ ?3→?5]− l,l∈A1 The second copy o inpu will check i he ansla ion is a ailable in he cu en egion. F12 ?3l[ ]− l→[ ?5]− l,l∈A1 The second copy o inpu en e s he auxilia y memb ane wi h pola iza ion −. By now he second copy o he inpu is in he egion co esponding o ei he he sea ch wo d, o o i s maximal p e ix plus one le e (auxilia y one). F13 [ ?5→?6]− l,l∈A1 I wai s o one s ep. F14 [ ?6→ ∅ ]0 l,l∈A1 I he a ge wo d has been ound, he second copy o he inpu is e ased. F15 [ ?6]− l→[ ]0 l?8,l∈A1 I no , he sea ch ails. F16 [ ?8]0 l→[ ]0 l?8,l∈A1 Sending he ail no i ica ion o he skin. F17 [ ?8l→?8]0 0 Dic iona y Sea ch by P Sys ems wi h S ing-Objec s 7 E asing he emaining pa o he sou ce wo d. F18 [ ?8?0]0 l→[ ]0 l ail Answe ing ail. F19 [ ?4→?9]− l,l∈A1 F20 [ ?9→?10 ]− l,l∈A1 F21 [ ?10 →?11 ]− l,l∈A1 I he a ge loca ion was no ound, he i s inpu copy wai s o 3 s eps while he memb ane wi h pola iza ion −handles he second inpu copy. F22 [ ?11 ]0 l→ ∅,l∈A1 E asing he auxilia y memb ane. 3.3 Dic iona y upda e To add a pai o wo ds u−→ o he dic iona y, inpu he s ing !u$ $0in egion 1. Conside he ollowing ules. U1 [ ! →!1||!2]0 0 Replica e he inpu . U2 [ !2→!3]0 0 Delay he second copy o he inpu o one s ep. U3 !1l[ ]0 l→[ !1]+ l,l∈A1 P opaga ion o he i s copy owa ds he a ge loca ion, changing he pola iza ion o he en e ed memb ane o +. U4 !3l[ ]+ l→[ !3]0 l,l∈A1 P opaga ion o he second copy owa ds he a ge loca ion, es o ing he pola - iza ion o he en e ed memb ane. U5 [ !1→!4]0 l,l∈A1 I a memb ane co esponding o some symbol o he sou ce wo d is missing, hen he i s copy o he inpu emains in he same memb ane, while he second copy o he inpu es o es i s pola iza ion. Ma king he is copy o he inpu o c ea ion o missing memb anes. U6 [ !4l→[ !4]+ l]0 k,l, k ∈A1 C ea ing missing memb anes. U7 [ !4$→$ ]0 l,l∈A1 8 A. Alhazo e al. Releasing he a ge wo d in he co esponding loca ion. U8 [ !3$→ ∅ ]0 l,l∈A1 E asing he second copy o he inpu . We unde line ha he cons uc ions p esen ed abo e also hold in a mo e gen- e al case, i.e., when he dic iona y is a mul i- alued unc ion. Indeed, mul iple ansla ions can be added o he dic iona y as mul iple s ings in he egion as- socia ed o he inpu wo d. The sea ch o a wo d wi h mul iple ansla ions will lead o all ansla ions sen o he en i onmen . The p ice o pay is ha he con- s uc ion is no longe de e minis ic, since he o de o applica ion o ules S4 o F9 o di e en ansla ions is a bi a y. Ne e heless, he cons uc ions emain “de e minis ic modulo he o de in which he ansla ions a e sen ou ”. 4 Discussion In his pape we p esen ed he algo i hms o sea ching in a dic iona y and com- ple ing i implemen ed as memb ane sys ems. We unde line ha he sys ems a e cons uc ed as eusable modules, so hey a e sui able o using as sub-algo i hms o sol ing mo e complica ed p oblems. The scope o handling dic iona ies is no limi ed o he dic iona ies in he clas- sical sense. Unde s anding a dic iona y as in oduced in Sec ion 3, i.e., a s ing- alued unc ion de ined on a ini e se o s ings, leads o di ec applicabili y o he p oposed me hods o handle alphabe s, lexicons, hesau a, dic iona ies o ex- cep ions, and e en da abases. Acknowledgmen s All au ho s g a e ully acknowledge he suppo by he Science and Technology Cen e in Uk aine, p ojec 4032. A iom Alhazo g a e ully acknowledges he sup- po o he Japan Socie y o he P omo ion o Science and he G an -in-Aid o Scien i ic Resea ch, p ojec 20·08364. Yu ii Rogozhin g a e ully acknowledges he suppo o he Eu opean Commission, p ojec MolCIP, MIF1-CT-2006-021666. Re e ences 1. H. Ki ano: Challenges o massi e pa allelism. P oceedings o he 13 h In e na ional Join Con e ence on A i icial In elligence, Chambe y, F ance, 1993, ol. 1, 813–834. 2. Gh. P˘aun: Compu ing wi h memb anes. Jou nal o Compu e and Sys em Sciences, 61, 1 (2000), 108–143. 3. Gh. P˘aun: Memb ane Compu ing. An In oduc ion. Sp inge -Ve lag, 2002. 4. E. Sumi a, K. Oi, O. Fu use, H. Iida, T. Higuchi, N. Takahashi, H. Ki ano: Example- based machine ansla ion on massi ely pa allel p ocesso s. P oceedings o he 13 h In e na ional Join Con e ence on A i icial In elligence, Chambe y, F ance, 1993, ol. 2, 1283–1289. 5. P sys ems webpage. h p://ppage.psys ems.eu/.