Spatial Dynamic Structures and Mobility in Computation

Institute of Computer Science Romanian Academ y Spatial Dynamic Structures and Mobilit y in Computation Bogdan Aman Sup ervisor Prof. Dr. Gabriel Cioban u Octob er 2009 Abstract Mem brane com puting is a w ell -establi shed and successful r esearch field which b elong s to t he more g eneral area of mo lecular computi ng. Mem- brane computi ng aims at defining para llel and non-deter minist ic comput- ing mo del s, cal led mem bra ne systems or P Systems, which a bstract from the functi oning and str ucture of the cel l. A membrane system consists of a spa tial str uct ure, a hiera rc hy of m em bra n es w hi c h do not i n tersect , with a di stingui shable membrane cal led skin surrounding all of them. A membrane witho ut any other membranes inside is elemen ta ry , while a no n- elementary membrane is a co mp osi te membrane. The mem bra nes define demarca tions b etw een regio ns; fo r each membrane ther e i s a unique a sso- ciated r eg ion. Since w e hav e a one-to-one corr esp ondence, w e someti mes use membrane inst ea d of regi on, and vice-versa. The spa ce o utside t he skin m embrane i s called t h e en viro nmen t. In this thesi s w e define and in vestigate v ar i ants of systems of m o- bile mem branes as mo dels for molecular com puting and as mo delling paradi gms for bi olog ical systems. On one ha n d, we fol low the st andard approa c h of researc h in mem brane co mputin g : defining a no t ion of com- putati on for systems of mo b i le membranes, and in vestigating the compu- tati onal p o wer of suc h co mputi n g devi ces. Specifica lly , we address i ssues concerning the p ow er of o p erat ions for m o di fying the membrane str uc- ture o f a system of mo b i le membranes b y mobility: endo cytosi s (moving a mem bra ne inside a neigh b our ing mem brane) and endo cytosi s (m o ving a membrane outside the membrane where it is placed). On the ot her ha nd, w e rel a te system s of mobi le membranes to pro cess algebr a (m obile am- bien ts, tim ed m obile am bients, π -calcul us, bra ne ca lculus) b y providing some enco dings and add i ng so m e concepts inspi r ed from pro cess a lgebra in th e fr amework of mo b i le mem bra ne com puting . Ac kno wledgemen ts I thanks to m y sup erviso r Dr. Ga briel Cioba n u fo r providing t hi s o pp or - tunity , and f or his const an t supp or t and a dvice thr o ughout this P hD. I thank s the a non ym o us referees for their comments and suggesti ons which help ed imp r o ve the qua lity of the pap ers in whic h parts of this thesis w ere publ i shed. I w oul d a lso like to t hanks my exam iners: Dr. Gheo rghe P˘ aun, Dr. Dorel Lucan u, Mitic˘ a C raus, Matteo Ca v alier e, for their careful reading of the thesi s and thei r h el pful com men ts. I am gra teful t o a ll my coll eagues from the gro up of F ormal Meth- o ds Lab or ator y , Insti t ute of Comput er Sci ence, Ro m ania Aca dem y , Ia¸ si Branch for t heir suggesti o ns and encoura gements. 1 Con ten ts 1 In tro ducti o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1 Co n text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Mot iv at ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Out l ine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Preli minar ies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.1 Al phab et s, str ings and la nguages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.2 Cho msky Gramm ars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3 L Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Mat rix Gram m ars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.5 Regi ster Mac hines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.6 Pr o cess Alg ebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.6. 1 π -calcul us . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.6. 2 Sa fe Mobile Ambien ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.6. 3 Br ane Cal cu l i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6 2.7 P Syst ems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 7 2.7. 1 T r ansiti on P Sy stems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.7. 2 P Sy stems wi th Active Membranes . . . . . . . . . . . . . . . . . . . 18 3 Mobile Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.1 Si m ple Mobile Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1. 1 Defini t ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.1. 2 C o mputa t iona l Po w er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Enha nced Mobile Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2. 1 Mot iv at ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.2. 2 Defini t ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2. 3 Mo delli ng p ow er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 3.2. 4 C o mputa t iona l Po w er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.3 Mutua l Mo bile Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3. 1 Mot iv at ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.3. 2 Defini t ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.3. 3 C o mputa t iona l Po w er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.4 Co nclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5 2 4 Mobi le Mem bra nes with Ob jects on Sur face . . . . . . . . . . . . . . . . . . . 46 4.1 Mutual Membranes with Ob ject s o n Su r face . . . . . . . . . . . . . 47 4.1. 1 Definiti on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1. 2 Com putati onal Po w er . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.1. 4 Relat ionship of PEP Calculus and Mut ua l Mem branes w i th Ob jects on Surf ace . . . . . . . . . . . . . . . . . . . . 53 4.2 Concl u si ons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 8 5 Membrane Systems and Pro cess Alg ebra . . . . . . . . . . . . . . . . . . . . . . . 59 5.1 Mobile Membranes and Mobi le A m bients . . . . . . . . . . . . . . . . 60 5.1. 2 Relat ionship of Saf e Mobile Ambien ts w ith Mutual Mobi le Membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.1. 3 Decidabi lity Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.2 Tim e i n Mobile Membranes a nd Mobi l e Am bients . . . . . . . 89 5.2. 1 Tim e in Membrane Co m puting . . . . . . . . . . . . . . . . . . . 89 5.2. 2 Mobil e Mem branes wi th Tim ers . . . . . . . . . . . . . . . . . . . 90 5.2. 3 Mobil e Mem branes Wit h and Wit hout Ti m ers . . . . 92 5.2. 4 Mobil e Mem branes wi th Tim ers a n d Timed Mobil e Ambien ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 5 5.3 Typed Mem brane Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.3. 1 Typed Mem brane Systems . . . . . . . . . . . . . . . . . . . . . . . . 99 5.3. 2 Typed π -calcul us . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 0 8 5.4 Concl u si ons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6 Co nclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.1 Contributio n s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 6.2 Other Co n tri butio ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 5 3 Prelimi naries This chapter con tai ns so m e basic notio n s of forma l la nguage theo r y , as w ell as automat a. F or furt her informat ion ab out these topi cs t he reader is referred to t he monogra phs [9, 12, 13, 24] and to sp eci fic pap ers cited in th e nex t sections. 4 Mobile Mem branes In this chapter we define the syst ems o f si mple, enhanced an d mutual mobil e membranes a nd study their mo d el ling and com putati onal p o wer. The syste ms of simple mobile mem br anes a re a v a riant of P systems with acti v e mem bra nes having none of t he features like p olar izati ons, la - b el change, division of non-elemen ta ry m em bra n es, pri orit ies, or co op er- ative r ules. The additio n a l feature considered inst ea d ar e the op erat i ons of endo cy t o sis and ex o cytosis : m o vi n g a membrane inside a neig h b our ing membrane, or outsi de the mem bran e where i t is placed. How ev er, these op era tions are sligh tly d i fferen t in the pa p ers intro ducing t hem: in [8] one ob ject is sp ecified i n each mem brane in volv ed in the op era tion, while in [18] one ob ject is men tioned only in the mo vin g membrane. Ano t her v a ri- an t of P system s w ith mobil e membranes is mo bile P systems [23] having rules inspi r ed from mobil e ambien ts [4]. T uring com pleteness is obtai ned b y using nine membranes t ogether with the op erat ions of endocy t osis and exo cyt osis [18]. Using a l so some con text ual ev olutio n rul es (togeth er with endo cyt osis and exo cytosis) , in [14] i t is prov en tha t fo ur mobil e mem - branes are eno ugh t o get the p o wer of a T uri ng mac hine, while in [BA5] w e decr ease the n um b er of membranes t o three. In o rder to simplify the presen ta t ion, we use systems of simple mobile membr anes instead of P systems wit h mobil e membranes. The s y stems of enhanc e d mob ile membr anes a re a v ar iant of m embrane systems whi c h we pr op o sed in [BA10] fo r describing som e biol ogica l mec h- anisms of the immune syst em. The op erat ions gov erning the mobil ity of the systems of enhanced mo bile membranes ar e endo cytosis (endo), exo- cytosis (exo), forced endo cyt osis ( fendo) and for ced exo cy t osis (fexo ). The computa tiona l p ow er of the systems of enhanced mo bile membranes usi ng these f our op er atio ns was studied in [17] where i t is prov ed tha t tw elv e membranes can provide the comput atio nal universality , whi le in [ B A5] w e improv ed t he resul t by reducing the n um b er of membranes t o ni ne. It is w ort h to n o te that unlike the previ ous results, the rewr itin g of ob ject by means of con text-f r ee rules is n o t used in an y of the results (pr o of s). F ollowing our appr oach from [BA 14] we define systems of mutual mo - bile membr anes representing a v ar iant of systems o f sim ple mo bile mem- branes in whic h the endo cyto sis and t he exo cytosi s work whenever t he in- 5 v olved membranes “ agree” on the m o vemen t; thi s agr eemen t is descr i b ed b y using dual ob jects a and a in the inv olv ed m em bra nes. The op erat ions gov erning the mobil ity of t he system s of mutual mobil e membranes are m utua l endo cytosi s (mutual endo ), and m utual ex o cyt o sis (mutual exo). It i s eno ugh t o consider the biol ogica lly inspired o p era t ions of m utua l en- do cyt osis and m utual exo cytosi s a nd t h r ee membranes (compa rtments) to get t he full computatio nal p ow er of a T uring ma c hine [BA 8]. Three rep- resen ts the minimum n umber of mem bra nes i n order to discuss prop erly ab out the mov emen t pr ovided by endo cytosis and exo cy tosis: w e w ork with tw o m em bra nes insi d e a skin membrane. 6 Mobile Mem branes with Ob je cts on Surface Mem brane systems [2 1, 2 2] and bra ne cal culus [ 3] hav e b een i nspired from the structur e and the function i ng of t he l i ving cell . Al t hough these m o dels start from t h e same observ atio n, th ey are bu i ld ha ving in mind differen t goal s: membrane systems in v estig ate formall y the computati onal nature and p ow er of v ario us features of membranes, while the brane calcul us is capable to g ive a fai thful a nd intuitive r epresen ta tion of the biolo gical re- ality . In [5] the i nitia tors of these t w o forma l isms describ e the goal s they had in mind : “While membrane computi ng is a br anc h o f natur al comput - ing which tries to abstra ct com puting mo del s, in the T uring sense, fro m the structu r e and the funct ioning o f the cel l, maki n g use esp eci ally of a u- toma ta, l anguag e, a nd com plexity theoreti c to ol s, brane calcul i pa y mor e attention to t h e fidel ity to the bio logi cal real ity , ha v e a s a pri mary t arget systems biol ogy , and use esp ecial ly t he fra mework of pr o cess a lgebra .” In [BA1 8] we define a new cla ss of systems of mobil e m em bra nes, namely the systems of mutual me mbr ane s with obje ct s on surfac e . The inspira tion to a dd ob ject s on membrane and t o use t he bio logi cally in- spired r u l es pino/ exo/pha go co mes fro m [2, 5, 6, 15, 16]. The nov elty comes fr om the fact t hat we use ob jects and co-ob jects in phago and exo rules i n o rder to il l ustrat e the fact that b ot h inv olv ed membranes agree on the mo vemen t. W e inv estigate in [BA 6] the com putati onal p o wer of systems of mutual membranes with ob ject s on surf ace co ntrolled by pairs of r u l es: pino/exo or pha go/ex o , proving that t hey ar e universal with a small num b er of membranes. Si mil a r rules are used by another form alism called brane cal culus [3]. W e co mpare in [BA1 8] the systems of mutual membranes with ob jects on surface with brane calculus, a nd enco de a frag - men t of brane cal culus in to the newl y defined cla ss of systemm s of mobil e membranes. Even brane calculus ha v e a n in terleaving semantic and mem- brane systems ha v e a par allel one, b y perf o rming this translat ion we sho w that t he di fference b et ween the tw o mo dels is no t signi fi ca n t. 7 Mem brane Sy stems a nd Pro cess Al gebra The m em bra ne systems [21, 22] and t he m obile ambien ts [ 4] hav e simi- lar structures and common concepts. Bo th ha v e a hierarc hica l str ucture representing lo cati ons, and ar e used to m o del v ari ous asp ects of biologica l systems. The mobil e a m bients are suit a ble to represent t h e mo vemen t of am bien ts through a m bients and the co mmunication which takes place inside the bounda ries of ambien ts. Mem brane systems are suita ble t o r ep- resen t the mov emen t of ob jects and mem branes t hrough membranes. W e consider these new com puting mo dels used in describ i ng v arious bi o log- ical pheno mena [ 3, 7] , and enco de the ambien ts in to m em bra ne systems [BA11]. W e pr esen t suc h a n enco ding, a nd use i t to descri b e t he so d i um- p ota ssium exc hange pump [BA4]. W e provide an op er atio n a l corr esp on- dence b etw een the saf e ambien ts and their en co ding s, as well as v ario us relat ed pro p ert i es of the membrane systems [BA4]. In [ BA3] we inv estigate the pr oblem of reaching a co nfigurat ion from another configuratio n in a sp ecial class of sy st ems of mobile membranes. W e pro v e t hat the reachability can b e decided by reduci n g it to the reach- abili t y problem of a v ersion of pure and publi c ambien t ca lculus wit hout the capa bili ty op en . A fea ture o f cu r rent membrane system s is t he fa ct that ob jects and membranes a r e p ersist ent. Ho wev er, this is not qu i te tr ue in the real w orl d. In fact, cell s and i ntracellu l ar pro teins hav e a w ell-defin ed life- time. Inspired fr om these bio logi cal fa cts, w e define in [BA16] a mo del of mobil e m em bra nes i n which eac h membrane a nd eac h ob ject has a ti mer atta c hed representing thei r li fetim e. Thi s new feature is i nspired from bi- olog y where cel ls and intracellul a r proteins hav e a well-defined l ifeti m e. In order to simulate the pa ssage of tim e w e use rules of th e form a ∆ t → a ∆( t − 1) and [ ] ∆ t i → [ ] ∆( t − 1) i for t h e ob jects and membranes which ar e not in v olved in other rul es. If the ti mer of an ob jects reaches 0 then th i s ob ject i s consumed by applyi ng a rule of the for m a ∆0 → λ , while if t he ti mer of a membrane i r ea c hes 0 then the membrane is dissolved b y a pplying a rul e of the form [ ] ∆0 i → [ δ ] ∆0 i . After dissolv ing a membrane, all ob jects a nd mem- 8 branes previo usly presen t i n it beco me elements of the immedia tely upp er membrane, while the r ules o f the dissolved mem brane ar e remov ed. So me results show tha t mutual mobil e membranes with a nd wi thout t imers hav e the sam e computa tiona l p ow er. Since w e ha v e defined a n extensi on with time fo r mo bile ambien ts in [BA 1, BA2, BA7], and one for mobile mem- brane in [BA16], we study t he relat ionship b etw een these t wo extensions: timed safe mobi le ambien ts are enco ded into mutual mobil e membranes with ti mers. Mem brane systems [21, 2 2] a re kno wn to b e T ur ing complet e and ar e often used to m o del biolog ical syst ems and their evolution. In o rder to increase the m o dell ing p ow er of t his for mali sm, we define i n [BA9] a t yp ed v ersio n, w hic h l ea ds in turn t o a decrease o f com putati onal p ow er. W e en- rich t he symp ort /antip ort membrane systems with a typ e discipline whi ch allow t o guarantee t h e so u n d ness of reduct ion rules w ith resp ect to some relev a n t pr op er t ies of the biol ogica l systems. The k ey tec hnica l t o ol s w e use are type inference a nd pr incipa l typing [2 5], i.e. we asso ciate to each reductio n rule the minim al set o f cond i tion s tha t must b e sa tisfied in o rder to assure that applyi ng this rul e to a correct membrane system , t hen we get a correct membrane system as w ell. The typ e system for membrane systems with symp o rt/a n tip ort r ules is (up to o ur k no wl edg e) the first attem pt to con tr ol the ev olu t ion of mem bra ne systems using typing rules. The pr esen tat i on of the typ ed so dium-p otassi u m pump is a n exam ple ho w to intro duce and use types i n mem bra ne systems. The π -calcul u s typed pump i s presen ted i n or der t o see what is the desired mo del ling p ow er we w ant to ha v e in m em bra ne systems b y in tro ducing a type system. Typ e descripti ons of bi olog ical inspired forma lisms, along w ith the typ e infer - ence algori thm can also be found in [BA12, 10]. Other stat ic t ec hniques ha ve b een appl i ed t o biolog ical systems, suc h as Co n tro l Flow Analysi s [1, 19, 2 0] and Abst ract In terp r etati on [1 1]. 9 Conclusions Con tributions Mem brane com puting is a w ell-est a blished a n d successful resear ch field which b elong s to the mor e g eneral a r ea of m olecula r comput ing. Mem- brane computi ng aims at defining para llel and non-deter minist ic comput- ing mo del s, cal led mem bra ne systems or P Systems, which a bstract from the funct i oning a nd struct ure o f t he cell. Since th e in tr o ducti on of thi s mo del , m a n y v ar iants hav e b een prop osed a nd t he li terat ure on the sub- ject is now rapidly growing. Ther e are tw o st andard wa ys of in v estig ating membrane syst em s: consideri ng their comput atio nal p ow er in compari son with the classical not ion of T ur ing computa bili ty , or consi d er ing their ef- ficiency in solvi n g algor i thmi ca lly hard problems, lik e NP-pr oblems, in a p oly nomia l tim e. In this resp ect , we defin ed new classes o f mem bra ne systems which are p ow erful, mo stly equiv al en t t o T uring machines, and fo r whi c h w e establi shed links wit h pr o cess a lgebra . The syste ms of simple mobile mem br anes a re a v a riant of P systems with act ive mem bra nes ha ving none o f the fea tures like p o l ariza tions, la b el c hange, divisi o n of non-elemen ta r y membranes, pri orit ies, or co op erati v e rules. The additi o nal featur e considered instead are th e op erati ons of endo cytosis and exo cy t o sis : moving a mem bra ne i nside a neig h b o ur ing membrane, or outside t he m embrane where it is placed. In [BA5] w e prov ed t h a t t hree mobi le membranes are enough t o g et t h e p ow er of a T uring machine. The systems of enhanc e d mobile membr anes a r e a v a riant of syst ems of simple mobile mem branes tha t we pr op osed in [BA10] fo r descr ibing some biolo gical mec hanisms of the im m une system. The op erati ons go v erning the m obili t y of the systems of enhanced mobi le membranes are endo cy to- sis (endo), exo cyto sis ( exo), enhanced endo cyto sis (f en d o ) and enhanced exo cyt osis ( f exo). In [ BA5] we st udied the comput atio nal p ow er of the systems of nine enhanced m o bile membranes usi ng these fo ur op er atio ns. F ollowing our appr oach fr om [BA1 4] w e defined systems of mutual mo- bile membr anes representing a v ar iant of systems o f sim ple mo bile mem- 10 branes in whi ch the endo cyt osis a nd exo cyt osis work whenever the inv olved membranes “agree” on the mov emen t; t his agreement is describ ed b y using dual ob jects a and a in the inv olv ed membranes. The op er atio ns gov erning the mobil ity of the systems of m utua l mobil e membranes are mutual endo- cytosis (mutual endo), and mutual exo cyto sis (mutual exo) . It is eno ugh to consider the bi olog icall y inspi r ed op erat ions of m utual endo cyto si s and m utua l exo cyt osis and three mem bra nes ( compar tments) to get the full computa tiona l p ow er of a T uring ma c hine [BA8]. In [BA 18] w e defined a new class of syst ems of mobile membranes, namely the systems of mutual me mbr ane s with obje ct s on surfac e . The rules of th i s cl ass are biologi call y inspired, namel y pino/ex o/phag o rules. The nov elt y comes from the fact tha t w e used ob jects and co-ob jects in phago and exo rules in order to ill ustrat e t he fact t hat b oth i n volv ed membranes agr ee on the mo v ement. W e inv estigat ed in [BA 6] the co mpu- tati onal p o wer for sy st ems of mutual membranes with ob jects on surfa ce con tr olled by pairs of rul es: pino/exo or phago/ ex o, pro ving that t hey are universal with a sma l l n umber of m embranes. Simil ar rules ar e used by another for mali sm call ed bra n e calcul us [ 3]. In [BA18] w e defined an op- erati onal sem an ti c for systems of mem bra nes wi th ob jects on surface and compar e the systems o f m utual membranes wit h ob ject s on surface with brane ca lculus, and enco ded a fra gment of brane calculus in to the newly defined cla ss of system s o f mob i le mem bra nes. In [BA11] we enco ded the mobile a m bients in to the mem brane systems and provided an op er atio nal semantic f o r membrane systems. W e pr e- sen ted suc h a n enco ding , and used i t to describ e th e so dium-p o tassium exc hange pump [BA4]. W e provided a n o p erat ional cor resp ondence b e- t ween the safe ambien ts a nd th ei r enco dings, as well as v ario us relat ed prop er ties of the m embrane systems [BA4]. In [BA3] we in v estig ated the probl em of reac hing a configura tion from another configur atio n for a sub cla ss of systems of mobil e membranes, and prov ed t hat the reachability ca n b e decid ed by r educing it to the r eac h- abili t y problem of a v ersion of pure and publi c ambien t ca lculus wit hout the capa bili ty op en . In [BA15, BA1 6] new cl a sses of membranes a r e defined: ti mers are assigned to eac h membrane and each ob ject. This new feat ure is inspir ed from bio logy where cells an d in tr acellul ar protei n s hav e a well-defined lifeti me. In order to simulate the passa g e of ti me w e use rul es o f the form a ∆ t → a ∆( t − 1) and [ ] ∆ t i → [ ] ∆( t − 1) i for the ob jects and mem bra nes which are no t i nv olved in o ther r ules. If the timer of a n o b jects reaches 0 then this ob ject i s consumed by a pplying a rule of the fo rm a ∆0 → λ , whi le if the tim er of a membrane i rea c hes 0 then the mem bra n e is dissolved b y applyi ng a rule of the form [ ] ∆0 i → [ δ ] ∆0 i . After disso l ving a mem brane, 11 all ob jects and membranes previo usly present in it b ecome elem en ts o f the membrane containing it, whil e t he r ules of the dissolved membrane are remov ed. By addi ng t imers to ob jects and mem bran es i n to a system of m utua l mobil e membranes, w e do not o b t ain a more p ow erful formal ism. Accor d- ing to [BA1 6] w e h av e that systems of mutual m obile m em bra nes wi th timer s and systems of mutual mobile membranes without tim er s ha v e the same p ow er. Si nce a n ex t ension w ith tim e for m obile ambien ts al ready exists [BA1, BA2, BA7], and one fo r systems of mob i le m embranes is pre- sen ted in [BA 1 6], the rel a tion shi p b etw een these t wo extensions is studied: timed sa fe mo bile ambien ts are enco ded i n to sy st ems o f m utual mobil e membranes with t imers [BA1 6]. In [ BA9] we enriched the symp ort/ an ti p ort mem bra ne systems with a typ e discipline which allows t o gua r antee the soundness o f reducti on rules with resp ect to som e relev an t pr op er t ies of t he biol ogica l systems. The k ey t echnical to ol s we used a re typ e in f erence and principa l t ypi ng, i.e. w e asso ci ate to each reductio n rul e t he m i nima l set of condit ions w hic h m ust b e satisfied i n order t o assur e tha t applying t his rul e to a cor r ect membrane system, then w e get a correct membrane system as w ell . T h e t yp e sy st em for membrane system s with symp o rt/a n tip ort rules is (up t o our knowledge) th e fir st a ttempt to control t he ev olut ion o f membrane sys- tems using typing rules. The presentation of the typed so dium -p ota ssium pump is an ex ample how t o intro duce and use typ es in membrane systems. The π -calculus typed p u m p is presen ted i n o rder to see what i s the desired mo del ling p o wer w e wan t t o hav e i n membrane systems b y in tro ducing a t yp e sy st em. Other Con tributions W e also fo cused on ot her form a lisms whi c h ar e characteri zed by spat ial dynami c structures and mo bili ty , namel y mo b i le ambien ts and the calculus of lo opi n g sequences. In wha t foll o ws we present the w ork do ne in t his directi on. In [BA1] w e extended mobil e am bients wi th timers and pr o xi miti es, in order to g et a clea r not ion of lo cati on and mobil i ty . Timer s define tim eo uts for v ar i ous resources, ma king them av a i labl e only for a deter mined p eri o d of time; w e add t imers to ambien ts and cap a bili t ies. W e presented an exampl e ho w the new mo del is working . The co ordi n a tion of the am bients in t ime an d space is given b y assig ning sp ecific v al u es to t imers, and by a set of co ordi natio n rules. In [ BA2] w e added timers t o communication channels, capabilit ies and ambien ts, and used a typing system for comm unicat ion. The passage of 12 time i s given b y a discrete glo bal ti me prog ress functio n. W e pr o ved that structur al co n g ruence and passag e o f time do not in ter f ere wi t h t he typing system. M or eo ver, o n ce w ell -t yp ed, an ambien t r em ains well-t yp ed. A timed extension of t he cab proto col il lustra tes ho w t he new for mali sm is w ork ing. In [ BA7] we added timers to ca pabili ties and a mbien ts, a nd pr o vi d ed an op erat i onal sem an ti cs of the new ca lculus. C ertai n r esults are r elated to the passag e of time, and so m e new b ehavioural equi v alences ov er timed mobil e am bients are defined. Ti meout for net work communication (TTL) can b e nat u r ally mo delled by the time constrai n ts ov er capabi liti es and ambien ts. The new forma lism ca n b e used to describ e netw ork pr oto cols; Simple Netw ork Manag em en t Pro to col (SNMP) may impl em en t its own strat eg y fo r t imeout and retra n sm ission i n TCP/IP . In [ BA12] we enriched the ca lculus o f lo oping sequences, a formali sm for descri bing ev olut ion of biolo gical sy st ems b y mea n s o f t erm rewri ting rules, wi th typ e discipli nes t o guarantee the soundness of reducti on rules with resp ect to i n terest i ng biolog ical pro p erti es. 13 Bibliograph y [1] C. Bo dei, A. Braccia li, D . Chi a rugi . 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A guide to the the ory of NP-c ompletenes s . W.H . F r eeman and Co., 1979 . [13] J.E . H op cro ft, J.D. Ul lman . Intr o duction to auto mata t h e ory, lan- guages and c o m putation . Addi son-W esl ey , 1979 . [14] S. N . Kr ishna. T he P ow er of Mobi lity: F o ur Membranes S uffi ce. L e c- tur e Note s i n Computer Scienc e vol.352 6, 242–25 1, Spr inger, 2005 . [15] S. N . Kri shna. U n i v ersal ity results for P system s ba sed on brane cal cul i op era tions. Theo retica l Com puter Sci ence, vol.371 , 83-105 , 200 7. [16] S. N . Kri shna. Mem brane comput ing w ith tr a nsp or t and embedd ed protei ns. Theoreti cal Co mputer Science, vol.41 0 , 355-3 75, 2 009. [17] S. N . Krishna, G. Cio ban u. On t he Computa tiona l P ow er of Enhanced Mobile Membranes. L e ctur e Notes in Computer Sc i e n c e , vol.502 8, 326– 3 35, 20 08. [18] S. N . Krishna, Gh. P˘ aun. P Systems with Mobile Membranes. Natur al Computing vol.4(3) , 255 –274 , 200 5. [19] F. Ni elson, H. Riis-Nielso n, D. Sc huc h-Da-Rosa, C. Priam i. Sta tic analysi s for systems bio logy . ACM International C onfer en c e Pr o c e e d- ing Series , vol. 5 8, 1–6, 2 004. [20] F. Niel son, H. Ri is-Niel so n, C. Priam i , D. Sc h uch-Da-Rosa. C on- trol Flo w Ana lysis for BioAmbien ts. Ele c tr onic Notes in The or eti c al Computer Scienc e , vol. 18 0 (3), Elsevi er , 6 5–79 , 200 7. [21] Gh. P ˘ aun. Co mputi n g wi th Mem branes. Journal of Computer and System Scienc es , vol.61(1 ), 108–14 3, 20 00. [22] Gh. P˘ aun. Membr ane Com puting. An Intr o duction. Springer-V erla ng, Berlin, 20 02. [23] I. Petre, L. Petre. Mobil e A mbien ts a nd P System s. Journal of Universal Computer Scien c e , vol.5(9) , 588-59 8, 199 9 . [24] G. Rozen b erg , A Salomaa . Ha n db o ok of F ormal L angua ges . Spri nger- V er lag, Heidelb erg, 199 7. [25] J. W ells. The E ssence of Principal T ypi ngs. L e ctur e Notes in Com- puter Scienc e , vol. 23 80, 913 – 925, Spr inger, 20 02. 15 My Publications ISI Publicati ons [BA1] B. Am an, G.Cio ban u. Ti mers and Proximit i es for Mobile Ambien ts. L e ctur e Notes in C omputer Scienc e , vol.4649, 3 3 –43, 2007 . [BA2] B. Aman, G.Ci obanu. Mobile Am bients with Ti mers and Typ es. L e ctur e Notes in C omputer Scienc e , vol.4711, 5 0 –63, 2007 . [BA3] B. Aman, G.Ciobanu. On the Reachability P roblem in P Sys- tems wit h Mobile Membranes. L e ctur e Notes in Computer Scienc e , v ol. 4860 , 113–12 3 , 20 0 7. [BA4] B. Ama n , G. Ciobanu. On the Relati o nship Bet w een Mem branes and Ambien ts. Biosystems , vol.91(3 ), 515–5 30, 200 8. [BA5] B. Aman, G.Ci obanu. Sim ple, Enha nced and Mutua l Mobi l e Mem - branes. T r ansac tions on C o mputationa l Systems Biolo gy , accept ed, to app ear in 20 09. [BA6] B. Aman, G.C iobanu. M embrane Sy stems wi th Sur f ace Ob jects. Natur al C omputing , 2009 . ( submitt ed) Other Publ ications [BA7] B. Ama n, G.Cio banu. Timed Mobile Ambien ts for Net w ork Proto- cols. L e ctur e Notes in Computer Scienc e , vol.504 8 , 234– 250, 20 08. [BA8] B. Ama n, G.Ci obanu. T uring Complet eness Usi ng Three Mobi le Mem branes. L e ctur e Notes in Computer S cienc e , vol.571 5, 41– 54, 2009 . [BA9] B. Ama n , G.Ci obanu. Typ ed Mem brane Systems. 10th Workshop on Membr ane C omputing (WMC09) , accepted, to app ea r in 200 9. [BA10] B. Aman, G.Cio banu. Descri bing the Imm une Sy stem Using E n- hanced Mobile Mem branes. Ele ctr onic N otes in The or etic al Compu ter Scienc e , vol.194(3 ), 5–18, 2008 . [BA11] B. Am a n, G. Cioba n u. T ransl ating Mobile Ambien ts in to P Sys- 16 tems. Ele ct r onic Notes in The or etic a l Computer Scienc e , v ol.171 (2), 11–2 3 , 20 0 7. [BA12] B. Aman, M. Dezani-Cia ncagl i ni, A. T ro ina. T yp e Di sciplines for Analysi ng Bio logi cally Relev ant Pro p ert i es. Ele ctr onic Notes in The o r etic al Comp uter Scienc e , vol.227, 97–1 11, 200 9. [BA13] B. Aman, G.Cio banu. Structu r al Prop erties a nd Observ ability in Mem brane Systems. Pr o c e e dings of SYNASC07: 9th Inte rna tional Symp osium on Symb olic and Numeric A lgorithms for Scient i fi c Com- puting , IEEE Co mputing So ci et y , 74 –81, 2007 . [BA14] B. Ama n, G.Cioba n u. Reso ur ce Comp etiti o n and Synchroniza- tion in Membranes. P r o c e e dings of SYNASC 08: 10t h International Symp osium on Symb olic and Numeric A lgorithms for Scient i fi c Com- puting , IEEE Co mputing So ci et y , 14 5-151 , 2 009. [BA15] B. Am a n, G.Cio b a n u. P Systems with Tim ers. IEEE Com puting So ciety , 200 9. ( subm itt ed) [BA16] B. Ama n, G.Ciobanu. Mobile Mem bra nes wit h Tim ers. Ele c tr onic Pr o c e e dings in The or e tic al Compu t er Scienc e , 2009 . (subm i tted) [BA17] B. Am an, G. Cioba n u. Decidab i lity Resul ts for Mobi le Mem branes Derived from Mobile Am bients. Pr o c e e dings of CiE 2 008 (4th Con- fer enc e on Computabilit y in Eur op e) , 15– 25, 2 008. [BA18] B. Am an, G . Cioba n u. Mem brane System s wi th Surfa ce Ob- jects. Pr o c e e dings of the Internatio nal Workshop on Comput i n g w ith Biomole cules (CBM 200 8 ) , 17– 2 9, 2 0 08. [BA19] B. Am a n, G.Ci obanu. Mobil e Am bients and Mobi le Membranes. Pr o c e e dings of CiE 2 007 (3th Confer enc e on Computability in Eu- r op e) , 1 6–27 , 200 7. 17 Bogdan Aman , b o rn at 26 June 19 8 2, Boto- sani, has graduat ed “Al. I. Cuza” Uni v ersity of Ia ¸ si , F aculty o f Mat h em atics, in 2007. He is a Ph.D. student under the sup ervisio n of Dr. Gabriel Ciob a n u at the Ro mania n Academy (Ia ¸ si Branch), Institut e o f C omputer Science. His main r esea rc h fields ar e membrane com - puting, com putat i onal mo dell ing for syst ems biolo gy , pr o cess algebr a, a nd other t heoreti cal asp ects of computer science. 18