On Parallel Array P Systems

On Pa allel A ay P Sys ems Linqiang Pan1, Gheo ghe P˘aun2 1Key Labo a o y o Image P ocessing and In elligen Con ol School o Au oma ion Huazhong Uni e si y o Science and Technology Wuhan 430074, Hubei, China [email p o ec ed] 2Ins i u e o Ma hema ics o he Romanian Academy PO Box 1-764, 014700 Bucu e¸s i, Romania [email p o ec ed], [email p o ec ed] Summa y. We u he in es iga e he pa allel a ay P sys ems ecen ly in oduced by K.G. Sub amanian, P. Isawasan, I. Venka , and L. Pan. We i s make explici se e al classes o pa allel a ay P sys ems (wi h one o mo e axioms, wi h o al o maximal pa allelism, wi h ules o a ious ypes). In his con ex , some esul s om he abo e men ioned pape by Sub amanian e al. a e imp o ed. A se ies o open p oblems a e o mula ed. 1 In oduc ion The gene ali y/ e sa ili y o memb ane compu ing is al eady a well known ac , he compu ing amewo k abs ac ed om he cell s uc u e and unc ioning can co e a la ge a ie y o p ocesses, dealing – in pa icula – wi h a la ge a ie y o objec s p ocessed in he compa men s o memb ane s uc u es. The a ays (in gene al, 2D and 3D figu es o a ious ypes) a e one o he ypes o objec s conside ed al eady since 2001, see [3]. A di ec ex ension om s ing objec s o wo-dimensional a ays was in oduced in [1] and hen in es iga ed in a se ies o pape s. A ecen con ibu ion o his esea ch a ea is [6], whe e a na u al coun e pa o he a ay P sys ems om [1] is conside ed: pa allel ew i ing o a ays, ins ead o he sequen ial ew i ing om [1]. Ac ually, he kind o pa allelism in es iga ed in [6] is ha sugges ed by Lindenmaye sys ems: all non e minals o an a ay should be ew i en in each s ep. A possible al e na i e, close o he s yle o memb ane compu ing, is o conside he maximal pa allelism: a mul ise o ules is used which is maximal among he mul ise s o applicable ules in a gi en momen . In he p esen pape , we explici ly conside hese wo kinds o pa allelism, and we p o e ha mos o he esul s om [6] hold ue o bo h kinds o pa allelism, 294 L. Pan, Gh. Paun also imp o ing hose esul s (less memb anes a e used in some o hem, while he powe ul p io i y ela ion is a oided in o he esul s). Se e al ques ions emain open; se e al opics o u he esea ch a e o mu- la ed. 2 De ini ions and No a ions I is use ul o he eade o be amilia wi h basic elemen s o memb ane com- pu ing, e.g., om [4] (wi h up-da ed in o ma ion a ailable a [7]), and o a ay g amma s, bu he used no ions will be ecalled below. Ac ually, in wha con- ce ns he a ays, we will usually use he pic o ial ep esen a ion, hence we need a minimal o malism (o he wise, cumbe some i igo ously o mula ed). The a ays we conside consis o fini ely many symbols om a specified alpha- be Vplaced in he poin s (we call hem pixels) o Z2( he plane); he poin s o he plane which a e no ma ked wi h elemen s o Va e supposed o be ma ked wi h he blank symbol #/∈V. Gi en an a ay Wo e V,supp(W) deno es he se o poin s in Z2ma ked wi h symbols in V. In o de o speci y an a ay, i is usual o speci y he pixels o he suppo , by gi ing hei coo dina es, oge he wi h hei associa ed symbols om V, bu , as we said abo e, we will pic o ially ep esen he a ays, indica ing hei non-blank pixels. These pic u es should be in e p e ed as a ays placed in any posi ion o he plane (cong uen , possible o be supe posed by means o a ansla ion). We deno e by V∗2 he se o all wo-dimensional a ays o fini e suppo o e V, including he emp y a ay, deno ed by λ. Any subse o V∗2is called an a ay language. We handle he a ays by means o ew i ing ules. An a ay ew i ing ule (o e an alphabe V) is w i en as a usual s ing ew i ing ule, in he o m W1→W2, whe e W1, W2a e iso onic a ays o e V:W1and W2co e he same pixels, no ma e whe he hey a e ma ked wi h symbols in Vo wi h #. When g aphically ep esen ing an a ay, usually we igno e he blank pixels, bu , when ep esen ing ew i ing ules, he pixels ma ked wi h # a e also explici ly shown. A ule as abo e is used o ew i e an a ay Win he na u al way: a posi ion in Wis iden ified whe e W1can be supe posed, wi h all pixels ma ching, whe he o no hey a e ma ked wi h symbols in Vo wi h #, and hen hose pixels a e eplaced wi h W2 ( he ac ha W1and W2a e iso onic ensu es he ac ha his eplacemen is possible). I he esul is he a ay W′, we w i e W=⇒W′. The eflexi e and ansi i e closu e o he ela ion =⇒is deno ed by =⇒∗. Simila o s ing ew i ing ules, he a ay p oduc ions can be classified ac- co ding o hei o m. We conside he e only wo ypes o ules, con ex - ee and egula . Remembe ha all ules we wo k wi h a e iso onic ( he shapes o he le hand side and he igh hand side a e iden ical, only he ma king, by blank o non-blank symbols, diffe s). Thus, a con ex - ee ule is an iso onic one wi h only one non-blank pixel in i s le hand side. On Pa allel A ay P Sys ems 295 No e ha we ha e no dis inguished be ween e minal and non e minal sym- bols, like in Chomsky g amma s; o egula ules we need such a dis inc ion. Thus, a egula ule o e he alphabe s Tand N,Nbeing he non e minal one, is a ule o one o he ollowing o ms: A#→a B, #A→B a, # A→B a,A #→a B, A → B, A →a, whe e A, B ∈Nand a∈T. Because in wha ollows we wo k, like in [6], in a Chomsky amewo k, wi h e minal and non e minal symbols, we also impose ha in a con ex - ee ule he single non-blank pixel in he le hand side is ma ked wi h a non e minal symbol. Be o e in oducing he a ay P sys ems, we ecall a no ion use ul below: wo- dimensional igh -linea g amma s. Such a g amma [2] is a cons uc G= (Vh, V , Vi, T, S, Rh, R ), whe e Vh, V , Vi a e he ho izon al, e ical, and in e media e alphabe s o non e minals, Vi⊆V , Tis he e minal alphabe , S∈Vhis he axiom, Rhis he fini e se o ho izon al ules, o he o ms X→AY, X →A, o X, Y ∈Vh, A ∈Vi, and R is he fini e se o e ical ules, o he o ms A→aB, A →a, o A, B ∈V , a ∈T. A de i a ion in Ghas wo phases, an ho izon al one, which uses ules om Rh, and a e ical one, which uses ules om R . The ho izon al de i a ion is as usual in a s ing g amma . In he e ical phase, he ules a e used in pa allel, downwa ds, wi h he es ic ion ha he e minal ules a e used simul aneously o all e ical non e minals. Thus, in he end, a ec angle is ob ained, filled wi h symbols in T. The se o all ec angles gene a ed in his way by Gis deno ed by L(G) and he amily o all languages o his o m is deno ed by 2RLG. Two a ay languages which will be used below a e LR, o all hollow ec angles wi h he edges ma ked wi h a(one elemen o his language is shown in Fig. 1), and LS, o all hollow squa es wi h he edges ma ked wi h a. aaaaaaaaa a a a a a a aaaaaaaaa Fig. 1. A hollow ec angle in LR. 3 Pa allel A ay P Sys ems We pass now o define he pa allel a ay P sys ems. Such a de ice (o deg ee m≥1) is a cons uc Π= (V, T, #, µ, F1, . . . , Fm, R1, . . . , Rm, io), whe e: Vis he o al alphabe , T⊆Vis he e minal alphabe , # is he blank symbol, µis a memb ane s uc u e wi h mmemb anes labeled in a one- o-one way 296 L. Pan, Gh. Paun wi h 1,2, . . . , m,F1, . . . , Fma e fini e se s o a ays o e Vassocia ed wi h he m egions o µ,R1, . . . , Rma e fini e se s o a ay ew i ing ules o e Vassocia ed wi h he m egions o µ; he ules ha e a ached a ge s he e, ou , in (in gene al, he e is omi ed), hence hey a e o he o m W1→W2( a ); finally, iois he label o a memb ane o µspeci ying he ou pu egion. In wha ollows, we only conside a ay P sys ems wi h egula (REG) and con ex - ee (CF) ules – wi h he symbols in V−Tconside ed as non e minals. A compu a ion in an a ay P sys em is defined in he same way as in a symbol objec P sys em, wi h he ollowing de ails. Each a ay om a compa men o he sys em mus be ew i en by he ules in ha compa men . The ew i ing is pa allel, wi h wo ypes o pa allelism: (1) he o al one, indica ed by allP, which means ha all non e minal symbols om he a ay a e ew i en, and (2) he maximal one, indica ed by maxP, which means ha a mul ise o ules is applied which is maximal, no u he ule can be added o i . Fo any wo ules used simul aneously, no pixel o hei le hand sides may o e lap (i.e., co e he same pixel o he ew i en a ay). An impo an poin appea s he e in wha conce ns he a ge indica ions o he ules: in each compa men , in a s ep we apply a mul ise o ules wi h he same a ge indica ion. This is a e y s ong es ic ion, because i e e s o all a ays om he compa men . In his pape , we wo k unde his es ic ion. A weake and somewha mo e na u al condi ion, which emains o be in es iga ed (e.g., a e he esul s p o ed below alid also in his case?), is o impose he es ic ion o use ules wi h he same a ge sepa a ely o each ew i en a ay ( hus, sepa a e a ays may be ew i en by ules wi h diffe en a ge s). O cou se, he wo a ian s coincide o sys ems wi h only one axiom in he ini ial configu a ion. The a ays ob ained by an allP o a maxP ew i ing a e placed in he egion indica ed by he a ge associa ed wi h he used ules, in he usual way in mem- b ane compu ing. I is impo an o s ess he ac ha all a ays om a gi en compa men a el oge he du ing a compu a ion. A compu a ion is success ul only i i hal s; ha is, i eaches a configu a ion whe e no ule can be applied o he exis ing a ays. The esul o a hal ing compu- a ion consis s o he a ays composed only o symbols om Tplaced in he egion wi h label ioin he hal ing configu a ion. The se o all such a ays compu ed (we also say gene a ed) by a sys em Πis deno ed by A(Π). No e ha a compu a ion which p oduces a e minal a ay (hence no ule can be applied o i ), bu s ill can ew i e ano he a ay, is no hal ing; i he ew i ing o one a ay con inues o e e , no ma e how many e minal a ays we e p oduced, hen no esul is ob ained. We deno e by PAPm(axk, α, β) he amily o all a ay languages A(Π) gene - a ed by sys ems Πas abo e, wi h a mos mmemb anes, a mos kini ial a ays in i s compa men s (Σm i=1ca d(Fi)≤k), wi h ules o ype α∈ {REG, CF }, wo king in he β∈ {allP, maxP}mode. When mo kis no bounded, hen i is eplaced wi h ∗. On Pa allel A ay P Sys ems 297 The ollowing esul s we e p o ed in [6] (p i indica es he use o a p io i y ela ion on he ules): Lemma 1 (Lemma 3 in [6]). 2RLG ⊆PAP3(ax1, CF, allP ). Lemma 2 (Lemma 4 in [6]). P AP3(ax1, CF, allP )−2RLG =∅. Lemma 3 (Theo em 3 in [6]). LR∈P AP2(ax1, REG, allP, p i). Lemma 4 (Theo em 4 in [6]). LS∈P AP3(ax1, REG, allP, p i). In wha ollows, we will imp o e all hese esul s in e ms o he numbe o memb anes in he fi s wo lemmas and a oiding he p io i y ela ion in he las wo lemmas ( hese wo esul s a e ob ained a he p ice o using mo e han one axioms o using he maxP way o applying he ules). 4 Resul s The fi s wo lemmas abo e can be easily imp o ed. P oposi ion 1. 2RLG ⊆PAP2(ax1, CF, β), β ∈ {allP, maxP}. P oo . Le G= (Vh, V , Vi, T, S, Th, R ) be a wo-dimensional igh -linea g am- ma . We cons uc he a ay P sys em Πindica ed in Figu e 2. The ho izon al de i a ion is done in memb ane 2. When he ho izon al phase is comple ed, he a ay is mo ed o he skin memb ane, whe e he e ical phase is pe o med. A - e he use o e minal ules, he a ay is mo ed back in o he inne memb ane. I i is no e minal, hen he compu a ion con inues o e e , by means o he ules A→A, A ∈Vi. These ules a e also in oduced in o de o make he maxP compu a ions o be allP compu a ions. The equali y L(G) = A(Π) is clea . ⊓⊔ P oposi ion 2. PAP2(ax1, CF, β)−2RLG =∅, β ∈ {allP, maxP }. P oo . Le us conside he a ay P sys em om Figu e 3. F om he axiom AB, we gene a e a s ing XnYn(Agoes o he le , simul aneously wi h Bgoing o he igh ), hen, like in a wo-dimensional igh -linea g amma , in he skin egion we go e ically, ma king he pixels wi h ain he columns o Xand wi h bin he columns o Y. We ob ain (mo ed in he cen al memb ane) a ec angle wi h he same numbe o columns ma ked wi h aand wi h b, which is no in he amily 2RLG ( he same example was used also in [6]). The sys em wo ks iden ically in he allP and maxP modes. ⊓⊔ Remo ing he p io i y om he o he wo esul s om [6] can be done, bu making use o he possibili y o ha ing wo axioms in he ini ial configu a ion o he sys ems. One o hem will gene a e he desi ed a ays, he o he one will gene a e “ win” a ays, which con ol he compu a ion o he o me a ays. 298 L. Pan, Gh. Paun ' & $ % ' & $ % 1 2 S X#→AY , o X→AY ∈Rh A→A, o A∈Vi X→A(ou ), o X→A∈Rh A→A(ou ), o A∈Vi A #→a B, o A→aB ∈R A→a(in), o A→a∈R Fig. 2. The a ay P sys em om he p oo o P oposi ion 1 ' & $ % ' & $ % 1 2 AB #A→AX B#→Y B X→X Y→Y A→X(ou ) B→Y(ou ) X #→ a X Y #→b Y X→a(in) Y→b(in) Fig. 3. The a ay P sys em om he p oo o P oposi ion 2 P oposi ion 3. LR∈PAP2(ax2, REG, β), β ∈ {allP, maxP}. P oo . We conside he a ay P sys em om Figu e 4. The axioms and he ules a e w i en on wo columns, in he le one hose which lead o he desi ed a ays, in he igh one hose handling he “ win” (con ol) a ays. The ules a e simila , wi h he igh column ones dealing wi h p imed non e minal symbols. On Pa allel A ay P Sys ems 299 The axiom in he le column is placed in he skin egion, he one in he igh column is placed in he inne memb ane. In he fi s s ep, we mo e he inne axiom o he skin memb ane, while emo ing he subsc ip 0 om all symbols A, B, A′, B′. Now we s a he gene a ion o he hollow ec angles. The bo om ho izon al line o he a ays is gene a ed in he skin memb ane, by mo ing Band B′ o he igh . A a gi en s ep, we swi ch o mo ing e ically, wi h he a ays mo ed o memb ane 2. We go upwa ds, synch onously, hen he a ays a e again mo ed o he skin memb ane, wi h Dand D′being he cu en non e minals. Bo h Dand D′go o le and, a some momen , Dis eplaced wi h a(hence he “le ” a ay is e minal (bu we do no know whe he he ec angle is comple ed). Mo ed again in memb ane 2, he le a ay emains idle, while he igh one can be ew i en – and, because o he pa allelism, his mus be done – i (and only i ) D′has a non-ma ked pixel in i s le . This happens i and only i he e minal a ay is no a comple e hollow ec angle. The symbol D′′ will go up indefini ely, hence he compu a ion ne e hal s. Thus, only he hal ing compu a ions p oduce elemen s in he language LR.⊓⊔ A simila esul can be ob ained o he language o hollow squa es. P oposi ion 4. LS∈PAP2(ax2, REG, β), β ∈ {allP, maxP}. P oo . We conside he a ay P sys em om Figu e 5, again wi h he axioms and he ules w i en on wo columns, wi h he same significance as abo e. In he fi s s ep, we mo e he axiom om he skin egion o he inne memb ane, while also emo ing he subsc ip 0. In he inne memb ane we s a cons uc ing he squa ea, om he le bo om co ne , g owing he e he le and he bo om edges. A some momen , he wo a ays a e mo ed o he skin memb ane, whe e he uppe and he igh edges a e cons uc ed. The wo k on he “le ” a ay is e mina ed a he momen when he a ays a e mo ed o he inne memb ane. We check he e whe he o no he squa e is comple ed. I is no comple ed i and only i he non e minal B′′ has a non-ma ked pixel abo e i . I his is he case, he compu a ion will con inue o e e , wi h B′′′ going o he igh , hence he compu a ion ne e hal s. Checking all de ails emains as an exe cise o he eade . ⊓⊔ The maxP mode o using he ules is, in ui i ely speaking, able o “appea ance checking”, which is known o be a powe ul ea u e o egula ed g amma s. This is confi med also in ou amewo k: we can gene a e he languages LR, LSby means o a ay P sys ems wi h only one axiom, a he p ice o using con ex - ee ules – ac ually, in he cons uc ion below we ha e only h ee non- egula ules, o a he simple o ms – used in he maxP mode. P oposi ion 5. LR∈PAP2(ax1, CF, maxP). P oo . We conside he a ay P sys em om Figu e 6. We s a in he skin mem- b ane, g owing he bo om edge. A some momen , we pass o g owing he e ical edges, and he a ay is mo ed o he inne memb ane. 300 L. Pan, Gh. Paun ' & $ % ' & $ % 1 2 A0aB0 A0→A B0→B A′ 0aB′ 0 A→A B#→aB # A→A a(in) # B→B a(in) #D→Da D→a(in) A′→A′ B′#→aB′ # A′→A′ a(in) # B′→B′ a(in) #D′→D′a D′→D′(in) # A→A a # B→B a A→a(ou ) B→D(ou ) # A′→A′ a # B′→B′ a A′→a(ou ) B′→D′(ou ) #D′→D′′ a # D′′ →D′′ a A′ 0→A′(ou ) B′ 0→B′(ou ) Fig. 4. The a ay P sys em om he p oo o P oposi ion 3 A e he a ay is mo ed o he inne memb ane, he symbol Bin oduces wo non e minals, Xand B′. The la e one will g ow he igh hand edge, he symbol Xwai s unchanged un il he a ay is mo ed back o he skin memb ane. He e we bo h g ow he uppe edge, by means o he non e minal C, and we change X o Y, which is mo ed o he le , simul aneously wi h he symbol C. A any momen , Cis eplaced wi h Dand Ywi h Y′and he a ay is sen o memb ane 2. He e we check whe he he ec angle is comple ed. Dis eplaced wi h a. The symbol Y′can be ew i en i and only i i has a non-ma ked pixel On Pa allel A ay P Sys ems 301 ' & $ % ' & $ % 1 2 A0 a B0 A0→A B0→B # A→A a B#→aB A→a(ou ) B→B(ou ) # A′→A′ a B′#→aB′ A′→A′(ou ) B′→B′(ou ) # B′′ →B′′′ a B′′′ #→aB′′′ A#→aA # B→B a A#→a¯ A B→¯ B ¯ A→a(in) ¯ B→a(in) A′ 0 a B′ 0 A′ 0→A′(in) B′ 0→B′(in) A′#→aA′ # B′→B′ a A′#→aA′′ B′→B′′ A′′ →a(in) B′′ →B′′ (in) Fig. 5. The a ay P sys em om he p oo o P oposi ion 4 in i s le hand place. I his is he case, hence he ec angle is no comple e, he symbol Zis in oduced; i no , he ule #Y′→Z#(ou ) canno be applied ( he maxP mode o using he ules allows ew i ing only some non e minals). Anyway, he a ay is mo ed back o he skin egion (a leas he ule D→a(ou ) is used). I Y′is s ill p esen , hen i is simply e ased by he ule Y′→#. Symbol Zcanno be emo ed, hence he compu a ion leads o a e minal a ay i and only i he ec angle is comple ed. ⊓⊔ I emains as an exe cise o he eade o use he same idea in o de o p o e ha LS∈PAP2(ax1, CF, maxP}. On he o he hand, we see no way o eplace maxP wi h allP in hese wo esul s: LR∈PAP2(ax1, CF, maxP) and LS∈ PAP2(ax1, CF, maxP).