Edi ing Con igu a ions o P Sys ems
E zs´ebe Csuhaj-Va j´u1, An onio Di Nola2, Gheo ghe P˘aun3,4,
Ma io J. P´e ez–Jim´enez4, Gy¨o gy Vaszil1
1Compu e and Au oma ion Ins i u e
Hunga ian Academy o Sciences
Kende u ca 13–17, H-1111 Budapes , Hunga y
E-mail: {csuhaj, aszil}@sz aki.hu
2Depa men o Ma hema ics and Compu e Science
Uni e si y o Sale no
84081 Ba onissi, Sale no, I aly
E-mail: [email p o ec ed]
3Ins i u e o Ma hema ics o he Romanian Academy
PO Box 1-764, 014700 Bucu e¸s i, Romania
E-mail: [email p o ec ed]
4Resea ch G oup on Na u al Compu ing
Depa men o Compu e Science and A i icial In elligence
Uni e si y o Se illa
A da. Reina Me cedes s/n, 41012 Se illa, Spain
E-mail: {gpaun, ma pe }@us.es
Summa y. This pape p oposes and p elimina ily in es iga es he possibili y o ans-
o ming a con igu a ion (memb ane s uc u e and mul ise s o symbol-objec s p esen in
he compa men s o his memb ane s uc u e) o a P sys em in o ano he con igu a ion,
by means o a gi en se o ules ac ing bo h on he memb anes and on he mul ise s o
objec s. Al hough such a ans o ma ion can be ob ained du ing a compu a ion o a P
sys em, we conside i as a goal pe se, as a p e-compu a ion phase, when he sys em i sel
is buil . In his amewo k, se e al impo an opics appea , such as he edi -dis ance be-
ween con igu a ions (wi h espec o a gi en se o edi ing ules; ac ually, his is a weak
me ic, because i is no necessa ily symme ic), no mal o ms, eachabili y, exis ence
o single con igu a ions om which a gi en amily o con igu a ions can be cons uc ed,
e c. We in es iga e he e only a ew o hese ques ions; he pape is mainly de o ed o
o mula ing p oblems in he new amewo k, calling a en ion o he possible ex ensions
and use ulness o he p esen app oach.
1 In oduc ion
Memb ane compu ing aims o abs ac compu ing models om he cell s uc u e
and unc ioning, [14], [15]. Wi h such a goal, he main esea ch opics o he domain
conce n he compu ing powe (compa ing he powe o he models ob ained wi h
132 E. Csuhaj–Va j´u e al.
inspi a ion om he cell biology wi h he powe o Tu ing machines and o hei
es ic ions) and he compu ing e iciency (sol ing compu a ionally ha d p oblems
in a easible ime, by making use o a ime-space ade-o which is made possible
by a ious ways o p oducing an exponen ial wo king space in a linea ime). The
in es iga ions we e a he success ul om hese poin s o iew – see de ails in [15]
and in he web page om h p://psys ems.disco.unimib.i .
Roughly speaking, a cell-like P sys em consis s o a memb ane s uc u e (a
hie a chical a angemen o memb anes), in he compa men s o which one places
mul ise s o symbol-objec s; hese wo elemen s, he memb ane s uc u e and he
mul ise s o objec s p esen in i s compa men s, o m a con igu a ion o a sys em
a a gi en ime. In he compa men s o associa ed wi h he memb anes he e also
a e se s o ules, acco ding o which he objec s and he memb ane s uc u e e ol e;
o he wise s a ed, by using hese ules we ob ain ansi ions among con igu a ions.
A sequence o ansi ions is called a compu a ion. A compu a ion is success ul only
i i hal s, and wi h a hal ing compu a ion we associa e a esul , e.g., in he o m
o he ec o desc ibing he mul iplici y o objec s om a gi en compa men o
he hal ing con igu a ion.
He e we swi ch he ocus om compu a ions o con igu a ions, and we conside
he p oblem o passing om a con igu a ion o ano he con igu a ion wi h he help
o a gi en se o ules. This is simila o looking o ansi ions which link he wo
con igu a ions, bu he in e es is di e en : we do no ca e abou he compu a ion
i sel (hal ing/non-hal ing) o abou i s esul , while he ules we conside a e
mainly de o ed o handling memb ane s uc u es. In some sense, ou app oach is
di ec ed o ounding a “memb ane calculus”, as a emp ed also in [4], in ano he
con ex .
Ac ually, he e a e se e al mo i a ions o his kind o in es iga ion. Fo in-
s ance, as al eady men ioned in [6] ( he p esen pape can also be conside ed as a
con inua ion o [6]), i a good (weak) me ic ela ed o he ime o passing om
a con igu a ion o ano he con igu a ion, wi h espec o he ules o a gi en P
sys em, can be ound, hen i can be use ul in a heu is ic s a egy o sol ing ha d
p oblems, based on he A∗algo i hm om [13]. Then, aking in o accoun ha a P
sys em is no hing else han an ini ial con igu a ion and gi en se s o ules associ-
a ed wi h memb anes, cons uc ing he ini ial con igu a ion is a way o cons uc
he sys em i sel ; o he wise s a ed, we can conside a speci ic se o ules o he
p e-compu ing case, when he compu ing model i sel is build, and o he ules o
he compu a ion. This can ha e in e es ing consequences, o ins ance, in building
a amily o P sys ems associa ed wi h a decidabili y p oblem, in o de o sol e i
( he pa icula sys ems om he amily sol e pa icula ins ances o he p oblem
– see de ails, e.g., in [16]); in he s anda d compu a ional complexi y heo y, his
cons uc ion is done in polynomial ime by a Tu ing machine, bu he p oblem
was o mula ed se e al imes o ha e he P sys ems sol ing a p oblem cons uc ed
by ano he P sys em, so ha he whole p ocedu e is “uni o mly bio-inspi ed”.
Links wi h o he a eas (such as he heo y o abs ac amilies o languages, g aph
heo y, o e olu iona y compu ing) will be men ioned below.
Edi ing Con igu a ions o P Sys ems 133
Howe e , as al eady said, he p esen pape is only a p elimina y explo a ion o
he “memb ane calculus” we p opose, wi h se e al esul s and much mo e esea ch
opics o mula ed.
2 P elimina y De ini ions
We s a by ixing some no a ion and e minology. Fo ela ed/ u he de ails ( om
memb ane compu ing), we e e o [15] and o he pape s a ailable in he web page
men ioned abo e.
An alphabe is a ini e and non-emp y se o abs ac symbols. Fo an alphabe
Awe deno e by A∗ he se o all s ings o symbols om A, including he emp y
s ing, deno ed by λ; he se A∗− {λ}, o non-emp y s ings o e A, is deno ed by
A+. The leng h o a s ing w∈A∗is deno ed by |w|.
Amul ise o e an alphabe Ais a mapping om A o N, he se o na u al
numbe s. We ep esen he mul ise s by s ings om A∗; he numbe o occu ences
o a symbol a∈Ain a s ing w, deno ed by |w|a, ep esen s he mul iplici y o a
in he mul ise ep esen ed by w(hence all s ings ob ained by pe mu ing symbols
in a s ing w ep esen he same mul ise ). Because o his di ec co espondence
be ween mul ise s o e Aand s ings o e A, we will use he e ms “mul ise ” and
“s ing” in e changeably ( o ins ance, we speak abou he “leng h” o a mul ise ,
wi h he ob ious meaning ha his is he o al mul iplici y o elemen s in he
mul ise , equal wi h he leng h o he s ing which ep esen s he mul ise ).
In one o he ollowing sec ions we need he no ion o a (non–de e minis ic)
egis e machine. Such a de ice consis s o a gi en numbe o egis e s each o
which can hold an a bi a ily la ge non-nega i e in ege numbe , and a se o
labelled ins uc ions which speci y how he numbe s s o ed in egis e s can change
and which ins uc ion should ollow a e any used ins uc ion.
De ini ion 1. A(non–de e minis ic) egis e machine is a cons uc M=
(m, Lab, l0, lh, P ), whe e m≥1is he numbe o egis e s, Lab is a nonemp y
ini e se (whose elemen s a e called ins uc ion labels),l0∈Lab is he s a label,
lh∈Lab − {l0}is he hal label (assigned o ins uc ion HALT), and Pis a ini e
se o (labelled) ins uc ions o one o he ollowing h ee o ms:
•l1: (ADD( ), l2, l3), o l1, l2, l3∈Lab, 1≤ ≤m,
•l1: (SUB( ), l2, l3), o l1, l2, l3∈Lab, 1≤ ≤m,
•lh:HALT ( he hal ins uc ion).
Each label om Lab labels a mos one ins uc ion o P, and l0labels one ins uc-
ion o P.
As a e o a egis e machine M= (m, Lab, l0, lh, P ) is an m- uple (a1, . . . , am)
o na u al numbe s. We in e p e he numbe aias he con en o egis e i. A
snapsho o ins an aneous desc ip ion o a egis e machine M= (m, Lab, l0, lh, P )
134 E. Csuhaj–Va j´u e al.
is a pai (l, s), whe e l∈Lab is a label, and sis a s a e o M. In ui i ely, he label
lindica es wha is he ins uc ion which is abou o be execu ed.
A snapsho σ= (l, s) o a egis e machine Mis called ini ial i l=l0and
s= (0,0, . . . , 0), and i is called hal ing i l=lh.
De ini ion 2. Le M= (m, Lab, l0, lh, P )be a egis e machine, and σ= (l, s)
wi h s= (a1, . . . , am)a non–hal ing snapsho o M. A successo o σis a snapsho
σ0= (l0, s0)de ined as ollows:
1. I llabels an ins uc ion (ADD( ), l2, l3), hen s0= (a1, . . . , a +1, . . . , am), and
l0=l2o l0=l3.
2. I llabels an ins uc ion (SUB( ), l2, l3), hen s0= (a1, . . . , a −1, . . . , am)and
l0=l2i a >0, and s0=sand l0=l3i a = 0.
Tha is, he execu ion o an ins uc ion (ADD( ), l2, l3) adds 1 o egis e and hen
go o one o he ins uc ions wi h labels l2and l3, non-de e minis ically chosen.
In he execu ion o an ins uc ion (SUB( ), l2, l3), i egis e is non-emp y, hen
we sub ac 1 om i and con inue wi h he ins uc ion labelled wi h l2, o he wise
we con inue wi h he ins uc ion labelled wi h l3.
Acompu a ion o a egis e machine M= (m, Lab, l0, lh, P ) is a ( ini e o
in ini e) sequence σ0, σ1, . . . , σko snapsho s o Msuch ha : (a) σ0is he ini ial
snapsho , (b) σi+1 is a successo snapsho o σi, o i= 0, . . . , k −1, and (c) i
k∈N hen σkis hal ing (in his case, we say ha he compu a ion is hal ing).
The se N(M) o na u al numbe s compu ed by a egis e machine Mis he
se o numbe s x∈Nsuch ha he e exis s a hal ing compu a ion such ha i
σ= (lh,(a1, . . . , am)) is i s hal ing snapsho , hen a1=x.
Tha is, a egis e machine Mcompu es a se N(M) o numbe s in he ollowing
way: we s a wi h all egis e s emp y (hence s o ing he numbe ze o) wi h he
ins uc ion wi h label l0and we p oceed o apply ins uc ions as indica ed by
he labels (and made possible by he con en s o egis e s); i we each he hal
ins uc ion, hen he numbe s o ed a ha ime in egis e 1 is said o be compu ed
by Mand hence i is in oduced in N(M) (because o he non-de e minism in
choosing he con inua ion o he compu a ion in he case o ADD ins uc ions,
N(M) can be an in ini e se .)
I is known (see, e.g., [12]) ha in his way we can compu e all se s o numbe s
which a e Tu ing compu able (e en wi h machines wi h a small numbe o egis e s,
bu his de ail is no o in e es he e).
Theo em 1. I Qis a Tu ing compu able se , hen he e exis s a egis e machine
M, such ha N(M) = Q.
Wi hou loss o gene ali y, we may assume ha when hal ing, all egis e s a e
emp y, wi h he excep ion o egis e 1, which con ains he gene a ed/compu ed
numbe .
Edi ing Con igu a ions o P Sys ems 135
3 Con igu a ions
In o mally speaking, a memb ane s uc u e is a 3D a angemen o esicles, placed
in a unique ex e nal memb ane, called he skin memb ane, wi hou o he ela ions
aken in o conside a ion han he inclusion (child-pa en ) ela ion. Ma hema ically,
his co esponds o a oo ed ee s uc u e, wi h he oo associa ed wi h he skin.
As usual in memb ane compu ing, we ep esen he memb ane s uc u es by s ings
o labelled pa en heses ( he labels a e associa ed wi h he memb anes, hence also
wi h he nodes o he unde lying ee) o , g aphically, as Eule -Venn diag ams (o a
pa icula ype: no in e sec ion is allowed and he e is a unique ex e nal memb ane,
he skin). Figu e 1 con ains bo h he g aphical and he ee ep esen a ion o he
ollowing memb ane s uc u e
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Fig. 1. Example o a memb ane s uc u e.
The numbe o memb anes om a memb ane s uc u e µis called he deg ee
o µ, and he numbe o le els o he associa ed ee (wi h he oo in le el 1) is
called he dep h o µ. Some imes, we deno e he deg ee wi h deg(µ) and he dep h
wi h dep(µ). In he case o he memb ane s uc u e om Figu e 1, we ha e he
deg ee equal o 10 and he dep h equal o 5.
In he example abo e, he labels a e associa ed in a one- o-one way
o memb anes. In he case o memb ane s uc u es which e ol e du ing a
(p e)compu a ion, e.g., by memb ane di ision, we allow mul iple memb anes o
be labelled wi h he same label. In such a case, he iden i ica ion o memb anes
h ough labels will be no longe possible, ha is why we use a double labelling o
memb anes, by means o pai s (i, j), whe e iis he “ eal” label, and jiden i ies

136 E. Csuhaj–Va j´u e al.
he copy o memb ane iin he memb ane s uc u e. Thus, i we use labels om
a se H, hen he ac ual labels a e om H∪(H×N), wi h elemen s o Hused
alone when hey label unique memb anes and in couples when hey label se e al
memb anes a he same ime. This ex ended se o labels, H∪(H×N), is deno ed
by He. The se o labels om Hewhich ac ually appea in a memb ane s uc u e
µis deno ed by He(µ); clea ly, his se is ini e, as we wo k he e only wi h ini e
memb ane s uc u es.
A sys ema ic, ecu si e p ocedu e o cons uc ing he ex ended labels can
be based on an encoding hi1, j2io numbe s j1, j2, o ins ance, using Can o
pai unc ion: we s a wi h labels hin e p e ed as (h, 0); when di iding such
a memb ane, we label he esul ing memb anes by (h, h0,1i) and (h, h0,2i); he
memb anes ob ained by di iding he i s o hese memb anes a e labelled by
(h, hh0,1i,1i) and (h, hh0,1i,2i), while he memb anes ob ained by di iding he
second memb ane a e labelled by (h, hh0,2i,1i) and (h, hh0,2i,2i), and so on and
so o h. Because also his a numbe , we can encode also h(i can be uniquely
eco e ed om a code), bu we p e e o keep i “ isible”, in pai -labels as abo e.
Howe e , he ollowing con en ion is made: we use na u al numbe s as labels o
memb anes, he skin memb ane is always labelled wi h 1, and no o he memb ane
has his label (hence he skin memb ane does no need a double labelling).
The unique di ec ly uppe memb ane o each memb ane i, excep he case o
he skin (i= 1), which has no memb ane abo e i , is called he pa en o iin µand
deno ed by pa µ(i); con e sely, he di ec ly inne memb anes placed in memb ane i
a e called he child en o iand hei se is deno ed by chdµ(i). O cou se, he labels
men ioned he e a e ei he om Ho om He, depending on he ci cums ances.
Fo an elemen a y memb ane hwe ha e chdµ(h) = ∅.
I in each compa men (we also say egion) io a memb ane s uc u e µ
we place a mul ise wi, o objec s om a gi en alphabe O, hen we ob ain a
con igu a ion. No e ha a mul ise can be emp y (and hen i is ep esen ed by
λ).
De ini ion 3. A con igu a ion Co e a se o objec s Oand wi h memb anes
labelled wi h elemen s o some se His a pai C= (µ, M), whe e µis a memb ane
s uc u e wi h labels om Heand M:He(µ)−→ O∗is a mapping which associa es
mul ise s o e Owi h he egions o µ.
4 Rules o P ocessing Con igu a ions
Because a con igu a ion means bo h a memb ane s uc u e and he associa ed
mul ise s, we need ules o p ocessing memb anes and mul ise s o objec s.
The ypes o ules we conside he e a e indica ed in Table 1 (in all cases,
a, b, c, d a e objec s om an alphabe O(in ules o ype (10), cplays he ole o
a ca alys ), h, h0, h00 a e labels om a se H, and u, a e mul ise s o objec s o e
he alphabe O).
Edi ing Con igu a ions o P Sys ems 137
Table 1. Types o ules o con igu a ion edi ing
N . Iden i ica ion Fo m o he ule Ac ion
1 di [ha]h→[h0b]h0[h00 c]h00 di ide
2 me [ha]h[h0b]h0→[h00 c]h00 me ge
3 endo [ha]h[h0b]h0→[h[h0d]h0c]hendocy osis
4 exo [ha[h0b]h0]h→[hc]h[h0d]d0exocy osis
5 c e a→[hb]hc ea e
6 dis [ha]h→bdissol e
7 in a[h]h→[hb]hmo e in
8 ou [ha]h→[h]hbmo e ou
9 ncoo [ha→ ]hnon-coope a i e objec s e olu ion
10 ca [hca →c ]hca aly ic objec s e olu ion
11 coo [hu→ ]hcoope a i e objec s e olu ion
No e ha he i s ou pai s o ules a e one he in e se o he o he : he
ope a ion o me ging wo memb anes is he in e se o he ope a ion o di iding
a memb ane, endocy osis is he in e se o exocy osis, c ea ing a memb ane is
he in e se o dissol ing a memb ane, mo ing an objec inside a memb ane is
he in e se o mo ing i ou side a memb ane. The coope a i e ules o objec s
e olu ion can be conside ed as hei own in e se.
When applying a ule [ha]h→[h0b]h0[h00 c]h00 , he con en s o memb ane h,
objec s and memb anes alike, a e eplica ed and in oduced in bo h memb anes h0
and h00, wi h he excep ion o objec a, which is eplaced by bin he i s memb ane
and by cin he second memb ane. Con e sely, when me ging he memb anes h0, h00,
by using he ule [h0b]h0[h00 c]h00 →[ha]h, hen he con en s o bo h memb anes is
accumula ed (in he mul ise sense) in memb ane h, wi h he objec s b, c eplaced
by a. Simila ly, in he exocy osis/endocy osis ules, he whole con en s o he
mo ed memb ane is mo ed oge he wi h he memb ane, wi h he objec s a, b
eplaced by c, d, espec i ely.
We s ess he impo an ac ha he labels appea ing in hese ules a e om
H, no om He−H; ha is, a ule associa ed wi h a memb ane wi h label his
applied o all copies o his memb ane. Then, because Oand Ha e ini e se s,
he numbe o ules o ypes (1) – (8) is ini e; imposing a bound on he o al
mul iplici y o mul ise s uand (on he leng h o s ings uand ) om ules o
ypes (9), (10), (11), we can also ensu e ha he numbe o ules o hese ypes is
ini e.
Rules o memb ane di ision, dissolu ion and c ea ion, o sending objec s in-
side o ou side a memb ane a e cu en ly used in P sys ems wi h ac i e memb anes;
non-coope a i e and ca aly ic ules o objec s e olu ion a e also cus oma y. Rules
o me ging memb anes and o mo ing a whole memb ane ( oge he wi h i s con-
138 E. Csuhaj–Va j´u e al.
en s) inside o ou side ano he memb ane we e also occasionally used – e.g., in
[3], [1], [11]. In many cases, a ian s o ules o hese ypes a e conside ed, o
ins ance, in wha conce ns he labels o he in ol ed memb anes. In he case o
di ision ules, he basic a ian does no allow he change o he labels (which is
allowed o di ision ules in Table 1 abo e); hen, we can also allow he change o
he label o in/ou ules, and his ex ension was used, e.g., in [2].
These ules a e used as s anda d in memb ane compu ing, choosing he mem-
b anes and he objec s in a non-de e minis ic manne , in such a way ha he
applica ion o ules is maximally pa allel. Each objec and each memb ane can be
in ol ed in he applica ion o only one ule, wi h he men ioning ha a ule o
he ypes (9), (10), (11) is no conside ed as using he memb ane h, bu only he
objec s a, c and hose om he mul ise u, espec i ely. Thus, any memb ane can
be in ol ed in only one ule o ypes (1) – (8), bu he numbe o ules o ypes
(9), (10), (11) which e e o a memb ane hdoes no ma e . In all ules o ypes
(1) – (8) he e a e in ol ed objec s ( he ules hemsel es indica e how he mem-
b ane s uc u e is changed, unde he in luence o objec s); hese objec s canno
be used a he same ime also by ules o ypes (9), (10), (11). The use o ules o
any ype in a memb ane s uc u e is done in a bo om-up manne . Fo ins ance,
i an elemen a y memb ane his di ided (by a ule o ype (1)), o mo ed in o
ano he memb ane (by a ule o ype (3)), hen i s all i s objec s di e en om
he objec in ol ed in he di ision o in he mo e should e ol e h ough objec
e olu ion ules, and, when di ided o mo ed, he memb ane will eplica e o mo e
he objec s ob ained by he maximally pa allel use o ules o objec e olu ion.
Simila ly, i a memb ane con ains inside o he memb anes, hen i s all lowe le el
memb anes e ol e and hen he uppe le el memb anes e ol e, in he same s ep.
I is impo an o no e ha we ha e no es ic ed he e he use o ules o any
ype o elemen a y memb anes, al hough his migh be an in e es ing es ic ion
o in es iga e.
The p e ious lis con ains hose ules which we conside as basic, bo h bio-
logically mo i a ed and ma hema ically elegan , bu , o cou se, o he ules can
be conside ed. This esea ch opic, ela ed o he se o ules o use, is le o he
eade , and his is an a ac i e di ec ion o in es iga ion. He e we ha e men ioned
hese ele en ypes, bu we will examine in some de ails only some combina ions
o ules, as an illus a ion o he ype o p oblems which we wan o aise in his
pape .
We close his sec ion by men ioning ha by c ea ing memb anes and by en-
docy osis we can inc ease he dep h o a memb ane s uc u e, while by means o
he con e se ope a ions (by dissol ing memb anes and exocy osis) we can dec ease
he dep h o a memb ane s uc u e. In u n, he memb ane di ision inc eases he
“wid h” o a memb ane s uc u e (hence i can inc ease he ou -deg ee o he ee
desc ibing i ). Rules o ypes (7) – (11) do no modi y he memb ane s uc u e,
hey only mo e objec s ac oss memb anes o handle objec s inside compa men s.
Edi ing Con igu a ions o P Sys ems 139
5 The Edi -Dis ance Among Con igu a ions
We now in oduce one o he basic no ions o ou app oach, he edi -dis ance
among wo con igu a ions wi h espec o a gi en se o ules.
Le us ix an alphabe o objec s, O, and a se o labels, H. Le kbe a na u al
numbe . The amily o all con igu a ions C= (µ, M), wi h he memb ane s uc u e
o deg ee a mos nand dep h a mos d, wi h labels in Heand mul ise s o objec s
o e O, such ha each mul ise wp esen in a egion o µhas |w| ≤ k, is deno ed
by CF Gn,d(O, H, k).
Because bo h he deg ee o he memb ane s uc u e and he o al mul iplici y
o elemen s in he mul ise s a e bounded, he amily CFGn,d(O, H, k) is ini e o
all n, d, k ≥1.
Then, we de ine
CF G(O, H, k) = [
n≥1,d≥1
CF Gn,d(O, H, k).
This is he amily o all con igu a ions wi h memb anes labelled wi h elemen s
o Heand mul ise s (each o hem o o al mul iplici y a mos k) o e O. Because
we allow memb anes wi h he same label om H, his amily is in ini e.
Le us now conside a se Ro ules, o ins ance, o some o he ypes (1) –
(11). This can be a se o speci ic ules, which a e lis ed explici ly, o can be a
se o ypes o ules ( ha is, he se o all ules o he gi en ypes). I we use all
possible ules o gi en ypes, hen we w i e he lis o ule iden i ica ions a e R;
o ins ance, R(di , in, cook) ( esp., R(di , in, ca k)) is he se o all ules o e a
gi en alphabe and a gi en se o labels, o ypes (di ), (in), (coo) ( esp., (di ),
(in), (ca )), wi h he le -hand and he igh -hand membe o coope a ing ules
o leng h a mos k( esp., using a mos kca alys s); he alphabe and he se
o labels ollow om he con ex whe e he ules a e used. I he size o objec
p ocessing ules is no ele an (o no known), hen we omi men ioning i , and
we w i e coo ins ead o cook. Some imes we also w i e ncook, indica ing he ac
ha we use non-coope a ing ules a→uwi h |u| ≤ k.
I o each ule o ype (1), (3), (5), (7) we also use he e e se ule o ypes
(2), (4), (6), (8), espec i ely, and o each objec e olu ion ule (o ypes (9), (10),
(11)) [hu→ ]hwe also ha e he ule [h →u]hin he se R, hen he se Ris
said o be e e sible. I o a se Ro ules we add all ules e e se o he ules om
R, hen we ob ain a e e sible se o ules, deno ed by cR and called he e e sible
comple ion o R. (Clea ly, i Rcon ains ules o a ype (1), (3), (5), (7), hen cR
also con ains ules o ypes (2), (4), (6), (8), espec i ely, and, i Rcon ain ules
o any ype (ncoo), (ca ), hen cR may con ain ules o ype (coo). Fo ins ance,
cR(di , in, ca ) = R(di , me , in, ou , coo), and cR(c e, ncoo) = R(c e, dis, coo).)
Fo wo con igu a ions C1, C2∈CFG(O, H, k) we w i e C1=⇒RC2i we can
pass om C1 o C2by using ules om R( his is a ansi ion, in he cus oma y
meaning in memb ane compu ing, wi h he ules used in he maximally pa allel
manne ). We deno e by =⇒∗
R he e lexi e and ansi i e closu e o he ela ion
=⇒R.
146 E. Csuhaj–Va j´u e al.
a e in oducing a mul ise M(g) in a memb ane g, he ules associa ed wi h g
can s a wo king, al hough he memb anes om lowe le els a e no ye p esen .
This leads o he de-synch oniza ion o he sys em, which is no co ec .
S ill, his di icul y can be o e come, by in oducing he mul ise s only in he
las momen o he cons uc ion. Because his is a a he impo an aspec , we
gi e he e he ull de ails, modi ying he cons uc ion om he p oo o Theo em 5
o his case.
Theo em 7. Fo e e y con igu a ion C= (µ, M)∈CF Gn,d(O, H, k), he e is a
ini e se Ro ules o ypes (c e)and (ncoo), o e an alphabe O0⊃O, such
ha edi R(C0, C)≤2d−1and he mul ise s o Ca e a ailable only in he las
s ep o he passage om C0 o C. Tha is, he e exis con igu a ions C1, . . . , Cp∈
CF Gn,d(O, H, k)such ha p≤2d−1,C0=⇒RC1=⇒RC2=⇒R. . . =⇒RCp=
C, and o all i∈He(µ),M(i)∈C−(C0∪C1∪. . . ∪Cp−1).
P oo . This ime, o each label g∈He(µ) o a memb ane in µwe conside he
objec s hgi,hgi0indexed wi h he s ep when such symbols a e supposed o e ol e.
We p oceed in he same way wi h he mul ise s M(i). Speci ically, we ake
O0=O∪ {hgis,hgi0
s|g∈He(µ),2≤s≤2d−1}
∪ {hM(g)is|g∈He(µ),2≤s≤2d−1},
and he ollowing ules in R:
1. [1a→ hM(1)i2hj1i2hj2i2...hj 1i2]1, whe e {j1, j2, . . . , j 1}=chdµ(1).
2. hgi2s→[ghgi0
2s+1]g,
[ghgi0
2s+1 → hM(g)i2s+2hh1i2s+2hh2i2s+2 ...hh gi2s+2]g, whe e
{h1, h2, . . . , h g}=chdµ(g) and g∈He(µ) labels a memb ane in a le el so
µwi h s < d.
3. hgi2d−2→[ghgi0
2d−1]g,
[ghgi0
2d−1→M(g)]g, whe e gis a memb ane in le el do µ.
4. [ghM(g)ij→ hM(g)ij+1]g, o all g∈He(µ) and 1 ≤j≤2d−2,
[ghM(g)i2d−1→M(g)]g, o all g∈He(µ).
This ime, each mul ise M(g) is in oduced in he “p o ec ed” o m hM(g)is,
wi h sindica ing he s ep o he compu a ion (p e-compu a ion, i we ake in o
accoun he ac ha we wan o cons uc he ini ial con igu a ion o a P sys em);
only in he las s ep he symbols hM(g)i2d−1a e eplaced wi h he ac ual mul ise s
M(g), all o hem a he same ime, hence om now on he compu a ion can s a ,
synch onously, in all compa men s o µ.2
The ema k made a e Theo em 5, abou he possibili y o bounding he mul-
ise om ules [ga→ ]g, is alid also o he cons uc ion om he p oo o
Theo em 7, bu his ime we ha e o be ca e ul wi h he synch oniza ion, hence e -
e y symbol om a mul ise should be “p o ec ed”, such ha i becomes ac i e only
in he las s ep o he p e-compu a ion. This can be again achie ed by indexing

Edi ing Con igu a ions o P Sys ems 147
each objec om Owi h coun e s which inc ease wi h he s eps o he compu a-
ion. Because he leng h o he p e-compu a ion depends on he ou -deg ee o he
ee o µ, we ha e o coun in he indices o symbols hM(i)is o each le el exac ly
+ 1 s eps, whe e is he maximal ou -deg ee o µ, e en i he ou -deg ee o he
node whe e we wo k migh be smalle han . Thus, a e ( + 1)d−1 s eps, we
pass o p oducing he symbols o mul ise s M(i), one by one, du ing k−1 s eps,
whe e kis he maximal leng h o a mul ise . Tha is, we un-p o ec he symbols,
passing om objec s o ype hais o aonly a e ( + 1)d−1+(k−1) s eps. The
echnical de ails, al hough clea om he p e ious discussion, a e cumbe some,
hence we omi hem.
We illus a e he cons uc ion om he p oo o Theo em 7 (wi hou es ic ions
on he size o ules o ype (ncoo)) wi h he case o he con igu a ion Cwi h he
memb ane s uc u e as in Figu e 1 and mul ise s M(i),1≤i≤10, which we do
no speci y. The se Rcon ains he ules gi en in Table 2 (we p esen he ules in
he o de hey a e used in he 9 s eps o he passage om C0 o C). The eade is
asked o ollow he wo k o hese ules, on he way om C0 o C.
Table 2. Rules o ypes (c e), (ncoo) o ob aining he con igu a ion om Figu e 1
S ep Rules
1 [1a→ hM(1)i2h2i2h3i2]1
2h2i2→[2h2i0
3]2,h3i2→[3h3i0
3]3, [1hM(1)i2→ hM(1)i3]1
3 [2h2i0
3→ hM(2)i4h4i4]2, [3h3i0
3→ hM(3)i4h5i4h6i4]3, [1hM(1)i3→ hM(1)i4]1
4h4i4→[4h4i0
5]4,h5i4→[5h5i0
5]5,h6i4→[6h6i0
5]6,
[1hM(1)i4→ hM(1)i5]1, [2hM(2)i4→ hM(2)i5]2, [3hM(3)i4→ hM(3)i5]3
5 [4h4i0
5→ hM(4)i6h7i6h8i6h9i6]4, [5h5i0
5→ hM(5)i6]5, [6h6i0
5→ hM(6)i6]6,
[1hM(1)i5→ hM(1)i6]1, [2hM(2)i5→ hM(2)i6]2, [3hM(3)i5→ hM(3)i6]3
6h7i6→[7h7i0
7]7,h8i6→[8h8i0
7]8,h9i6→[9h9i0
7]9,
[ihM(i)i6→ hM(i)i7]i, o i= 1,2,3,4,5,6
7 [7h7i0
7→ hM(7)i8]7, [8h8i0
7→ hM(8)i8h10i8]8, [9h9i0
7→ hM(9)i8]9,
[ihM(i)i7→ hM(i)i8]i, o i= 1,2,3,4,5,6
8h10i8→[10h10i0
9]10,[ihM(i)i8→ hM(i)i9]i, o i= 1,2,3,4,5,6,7,8,9
9 [10h10i0
9→M(10)]10,[ihM(i)i9→M(i)]i, o i= 1,2,3,4,5,6,7,8,9
Theo em 6 has a se ies o in e es ing consequences.
Fo an alphabe Oand a se Ho labels, le OHbe he alphabe de ined in
he p oo o Theo em 5. Le R(c e, ncoo2) be he se o all ules o ypes (c e) and
(ncoo) wi h objec s in O∪OHand labels in H. Clea ly, his is a ini e se .
Co olla y 1. I C1∈CF Gn1,d1(O, H, k1)and C2∈CF Gn2,d2(O, H, k2), hen
edi cR(c e,ncoo2)(C1, C2)< d1·(n1+k1) + d2·(n2+k2)−2.
148 E. Csuhaj–Va j´u e al.
P oo . We jus w i e he iangle inequali y o C1, C2, C0and we use he e-
sul om Theo em 6. Because we wo k wi h he e e sible comple ion o he se
R(c e, ncoo2), we can pass bo h om C0 o C1,and C2, and con e sely, hus ha ing
a pa h om C1 o C2.2
This means ha any wo con igu a ions om CF G(O, H, k) can be eached
om each o he , hey a e a a ini e dis ance wi h espec o he se cR(c e, ncoo2)
o ules.
The p e ious co olla y gi es only an uppe bound on he dis ance among he
conside ed con igu a ions. Can his es ima ion be imp o ed? This seems o be
always possible, a leas wi h a (small) cons an : when passing di ec ly om C1
o C2we can “sa e” ha pa o he memb ane s uc u e o C1which can be
ound also in C2( o ins ance, he skin memb ane). The ques ion which is he
la ges imp o emen in gene al, o o wo con igu a ions wi h ce ain “simila i ies”
emains as a esea ch opic.
7 Gene a o s o Families o Con igu a ions
Ano he in e es ing consequence o Theo em 6 is he ac ha he con igu a ion C0
can be seen as a gene a o o he whole amily CFG(O, H, k), wi h espec o ules
om R(c e, ncoo2), o , in e ms o P sys ems, as a no mal o m o all sys ems
which can use ules o memb ane c ea ion ( he “minimali y” o he con igu a ion
C0is somewha su p ising and, o cou se, pleasan : any ecu si ely enume able se
o numbe s can be gene a ed by a sys em wi h he ini ial con igu a ion equal o
C0). We place his idea in a mo e gene al amewo k, esembling he one o AFL
(abs ac amily o languages) heo y, see [8].
De ini ion 5. Le C ⊆ CF G(O, H, k)be a amily o con igu a ions and Ra se o
ules. We de ine
genR(C) = {C∈CF G(O, H, k)|edi R(C0, C)<∞ o some C0∈ C}.
Tha is, genR(C) is he amily o con igu a ions which can be eached, in a ini e
numbe o s eps, when s a ing om con igu a ions om C, by using ules om R.
De ini ion 6. Fo C1,C2⊆CF G(O, H, k)and a se Ro ules, we say ha :
1. C1is an R-gene a o o C2i genR(C1) = C2;
2. C1is an R-co e o C2i C2⊆genR(C1).
O cou se, any se o con igu a ions is an R-co e o i sel wi h espec o any
se o ules, hence o in e es a e minimal gene a o s and co e s, in pa icula ,
single on gene a o s and co e s.
De ini ion 7. A amily C ⊆ CF G(O, H, k)o con igu a ions is said o be p incipal
wi h espec o a se Ro ules i he e is a con igu a ion C∈CF G(O, H, k)such
ha genR({C}) = C. In such a case we say ha Cis an R-gene a o o C.
Edi ing Con igu a ions o P Sys ems 149
In his amewo k, Theo em 6 di ec ly implies he ollowing in e es ing esul :
Theo em 8. The amily CF G(O, H, k)is p incipal wi h espec o he se o ules
R(c e, ncoo2), and C0is an R(c e, ncoo2)-gene a o o CF G(O, H, k).
Because in a p incipal amily each con igu a ion can be eached om any gi en
gene a o , i he espec i e se o ules is e e sible, hen i also ollows ha any
con igu a ion can be eached om any o he con igu a ion (passing h ough he
gi en gene a o ), hence we ha e:
Co olla y 2. In he amily CF G(O, H, k), any con igu a ion is a gene a o wi h
espec o cR(c e, ncoo2) = R(c e, dis, coo2).
8 Using Rules o Types (di ), (endo), and (ncoo)
The ules o ype (c e) and (ncoo) a e no he only ones o which we can ob ain
esul s as abo e. The use o ules o ype (ncoo) (o (ca ), (coo)) canno be a oided,
because hese ules a e he only ones (in he basic se conside ed he e) which
can inc ease he numbe o objec s in he con igu a ions, bu memb anes can be
c ea ed also by di ision. Howe e , as we ha e no iced al eady, by di ision we canno
inc ease he dep h o he memb ane s uc u e. This can be done by endocy osis
ules, hence a combina ion o ules o ype (di ) and (endo) can simula e ules o
ype (c e).
Fo ins ance, i we wan o c ea e a new memb ane, wi h label g, inside an
exis ing memb ane h, we can i s pass om [ha]h o [hb]h[gc]g( he objec ais
supposed o exis in memb ane h), by using a di ision ule, hen o [h[gc0]gb0]h,
by an endocy osis ule.
S ill, one mo e di icul y a ises: we canno di ide he skin memb ane, hence we
canno s a om he con igu a ion C0used in he p e ious sec ions, we need a
leas wo memb anes in he beginning.
On he o he hand, by using endocy osis/exocy osis ules (as well as ules o
ype (coo)), any con igu a ion wi h nmemb anes can be ans o med in o any
con igu a ion wi h he same numbe o memb anes. Then, i exocy osis is allowed
also wi h espec o he skin memb ane, hen we can expel memb anes om he
con igu a ions, hus dec easing he deg ee o he memb ane s uc u es. O cou se,
by using memb ane dissol ing ules (bu no memb ane c ea ion ules) we can
again dec ease he deg ee o con igu a ions.
The e o e, se e al combina ions o ules can be conside ed which leads o esul s
like hose om Sec ions 6 and 7 (o cou se, wi h edi -dis ances among con igu a-
ions di e en om hose in sec ions abo e).
In wha ollows we conside only he case when we use ules o ypes
(di ), (endo), and (ncoo), again wi h a bound on he leng h o ules o
ype (ncoo), wi hou pe mi ing he exocy osis o he skin memb ane (con-
sequen ly, he numbe o memb anes canno be dec eased). Tha is, we use
150 E. Csuhaj–Va j´u e al.
he se R(di , endo, ncoo2) o ules. Conside also he con igu a ions Cm
0=
([1[2. . . [m]m. . . ]2]1, a1, a2, . . . , am), o all m≥1, whe e a1, a2, . . . , ama e new
symbols, no in he alphabe O, wi h each objec aip esen in egion i, 1 ≤i≤m.
We men ion he e, wi hou a p oo , only he esul co esponding o Theo em
8. Fo m≥1, le us deno e
CF G≥m(O, H, k) = [
n≥d≥m
CF Gn,d(O, H, k),
hence his is he amily o all con igu a ions C= (µ, M) wi h memb ane s uc u es
o deg ee a leas m.
Theo em 9. The amily CF G≥m(O, H, k)is p incipal wi h espec o he se o
ules R(di , endo, ncoo2), and Cm
0is an R(di , endo, ncoo2)-gene a o o his am-
ily.
I is also wo h no ing he esul co esponding o Theo em 7, which p o ides
a u he no mal o m o sys ems which a e allowed o use ules o he di ision o
memb anes and endocy osis, and non-coope a i e ules o objec e olu ion: any
sys em o his ype o deg ee a leas mis equi alen wi h a sys em ha ing he
ini ial con igu a ion equal o Cm
0.
I we conside a se o ules con aining ules o ypes (di ), (endo), (exo), (ncoo),
wi hou es ic ing he exocy osis, o i we allow dissol ing ules, hen we can again
gene a e CF G(O, H, k) s a ing om C0, hence we ha e a gene al no mal o m
heo em, o P sys ems o any deg ee.
In o de o ha e an idea abou he way he pa h om Cm
0 o an a bi a y
con igu a ion C∈CF G≥m(O, H, k) p oceeds, we illus a e he cons uc ion o a
esul like ha om Theo em 5 – hence wi hou he complica ions which appea
i we wan o p o ec he mul ise s o objec s, in iew o a p oo o a esul like
Theo em 7, and wi hou limi ing he size o objec p ocessing ules.
We conside again a con igu a ion Cwi h he memb ane s uc u e µ om
Figu e 1 and mul ise s M(i) placed in he compa men s i= 1,2, . . . , 10 o µ. The
con igu a ion Ccan be gene a ed om C2
0, e en i we use no exocy osis ule, and
we conside he e his case.
The se o ules used is gi en in Table 3, which, like in case o Table 2, p esen s
he ules in he o de o he s eps when hey a e used. The symbols b, c, d, e, used
in he ules a e chosen in an ad hoc manne ; in o de o gene alize he cons uc ion
o an a bi a y con igu a ion, hese symbols should be ela ed in a sys ema ic
manne o he memb anes whose e olu ion hey con ol and o he s ep o he
cons uc ion (which is also ela ed o he le el o he ee whe e we wo k). Such
echnicali ies a e le o he eade .
9 A Ple ho a o Resea ch Topics
Se e al esea ch opics we e al eady men ioned abo e, and many o he s can be
conside ed. We o mula e he e only some o hem, wi hou any in en ion o be
Edi ing Con igu a ions o P Sys ems 151
Table 3. Rules o ypes (di ), (endo), (ncoo) o ob aining he con igu a ion om Fig-
u e 1
S ep Rules
1 [1a1→M(1)]1,[2a2]2→[2b1]2[3b2]3
2 [2b1]2→[2c1]2[4c2]4,[3b2]3→[3c1]3[5c2]5
3 [2c1]2[4c2]4→[2[4c0
2]4c0
1]2,[3c1]3[5c2]5→[3[5c0
2]5c0
1]3
4 [2c0
1→M(2)]2,[3c0
1→M(3)]3,
[4c0
2]4→[4d1]4[7d2]7,[5c0
2]5→[5d1]5[6d2]6
5 [4d1]4[7d2]7→[4[7d0
2]7d0
1]4,[5d1→M(5)]5,[6d2→M(6)]6
6 [4d0
1→M(4)]4,[7d0
2]7→[7e1]7[8e2]8
7 [7e1→M(7)]7,[8e2]8→[8e0
2]8[9e3]9
8 [9e3→M(9)]9,[8e0
2]8→[8 1]8[10 2]10
9 [8 1]8[10 2]10 →[8[10 0
2]10 0
1]8
10 [8 0
1→M(8)]8,[10 0
2→M(10)]10
exhaus i e (and wi hou knowing which o hese opics a e di icul and which a e
s aigh o wa d).
In he p e ious sec ions we ha e in es iga ed only some e y pa icula cases:
he amily CFG(O, H, k), he se o ules R(c e, ncoo2), he gene a o C0, as well
as he amily CFG≥2(O, H, k), he se o ules R(di , endo, ncoo2), and he gen-
e a o Cm
0. Conside o he amilies o con igu a ions. Fo ins ance, wha abou
CF Gn,d(O, H, k)? Wha abou amilies using pa icula ypes o memb ane s uc-
u es (desc ibed by linea ees, bina y ees, ees o dep h 2, e c)? Then, he e
a e amilies o P sys ems – hence o con igu a ions – which appea in a na u al
way in a ious con ex s; such a con ex is ha o sol ing NP-comple e p oblems
in polynomial ime, using a amily o P sys ems cons uc ed in an uni o m man-
ne , s a ing om he p oblem o sol e (see de ails in [16] and in a se ies o o he
pape s). Can such a amily be gene a ed om a gi en unique con igu a ion, using
a sui able se o ules?
Many p oblems a e sugges ed by he pa allelism wi h he AFL heo y. De-
ine a so o “abs ac amily o con igu a ions”, wi h “nice” closu e p ope ies
(unde ope a ions de ined by e olu ion ules, bu also unde se - heo e ical ope -
a ions, such as union and in e sec ion). Conside se s o ules Rand amilies o
con igu a ions Csuch ha genR(C)6=CF G(O, H, k); which p ope ies has such a
amily?
A la ge panoply o p oblems is ela ed o he se o ules aken in o accoun .
Any se o ules o ypes belonging o a subse o he se {(1),. . . , (11)}can be
conside ed, a bi a y o e e sible, wi h speci ic es ic ions on he size o mul ise s
used in ules o objec e olu ion. Also, o he ypes o ules can be conside ed.

152 E. Csuhaj–Va j´u e al.
Find amilies o con igu a ions and se s o ules o which he minimal gen-
e a o s a e no single ons ( ha is, ind amilies o con igu a ions which a e no
p incipal wi h espec o a gi en se o ules). Is he p ope y o being a gene a o
decidable o a gi en amily o con igu a ions, a gi en se o ules, and a gi en
candida e gene a o ? (The answe should be nega i e in gene al, because o he
Tu ing comple eness o mos classes o P sys ems.) P o iding ha we know ha
a amily is p incipal, how can we ind a gene a o ? Is his possible in an e ec i e,
algo i hmic way? Which is he complexi y o such an ope a ion?
The decidabili y and complexi y issues we e al eady men ioned o he ques ion
whe he a con igu a ion can be eached om ano he con igu a ion by means o
ules in a gi en se , and his is o a c ucial impo ance (e.g., o he possible
applica ions o ou app oach). Again, he answe depends in an essen ial manne
on he se o ules – bu how exac ly his happens i emains o be disco e ed.
Connec ions wi h p e ious in es iga ions o eachabili y issues will p obably be
use ul, especially o cases whe e he eachabili y can be decided – see, e.g., [9],
[10].
A se ies o in e es ing ques ions conce ns he possible ela ions o he in es-
iga ions abo e wi h o he a eas o compu e science, o ma hema ics in gene al.
P ocessing con igu a ions means o p ocess he unde lying ees, oo. Can any con-
nec ion wi h ee/g aph g amma s be es ablished? Then, we can na u ally de ine
a g aph on he se CF G(O, H, k), wi h he con igu a ions as nodes and he a cs
indica ing he ansi ions among con igu a ions wi h espec o a gi en se o ules.
How his link can be used? Looking o ( he numbe o ) connec ed componen s
in his g aph seems o be di ec ly ela ed o he ca dinali y o gene a o se s.
Fu he mo e: wha abou he opological p ope ies o he me ic space de ined
by a dis ance edi R? Which is he ele ance o such connec ions o memb ane
compu ing?
On he same line, in e es ing connec ions seems o be possible wi h e olu ion-
a y compu ing. Assume ha we ha e a c i e ion acco ding o which some con ig-
u a ions can be conside ed “be e ” han o he s (and ha his c i e ion can be
checked in an e icien manne ). Then we can y o “imp o e” he con igu a ions
as usual in e olu iona y compu ing: s a om an ini ial popula ion o con igu a-
ions, apply a gi en numbe o ules, andomly, o hese con igu a ions, so ha
a new popula ion is ob ained; selec om his popula ion only pa o con igu a-
ions, in he dec easing o de o hei quali y; epea hese s eps un il a sa is ac o y
con igu a ion is ob ained (o a gi en numbe o s eps ha e been pe o med). Any
applica ion o his s a egy?
We close his sec ion wi h an “exo ic” ques ion, ha o sel - ep oduc ion: by
sending memb anes ou o he skin memb ane (by exocy osis – ules o ype (a)),
we can send a whole con igu a ion/sys em in o he en i onmen . How can his be
done in such a way o s a om a gi en con igu a ion (any gi en con igu a ion?)
and o send ou a copy o i ? This is in e es ing also as an in insic p oblem, and
i can be use ul bo h o c ea ing exponen ial space o sol ing ha d p oblems and
Edi ing Con igu a ions o P Sys ems 153
o c ea ing u he and u he agen s capable o “explo ing he ex a e es ial
space” in he specula i e amewo k ske ched in [5].
10 Final Rema ks
We conclude hese no es wi h he belie ha he in es iga ions s a ed he e dese e
u he e o s, bo h because o he ma hema ical appeal o he many ques ions
aised in his amewo k, bu also because o he in e es o memb ane compu ing
in gene al, and o possible applica ions in pa icula (e.g., in inding heu is ic
algo i hms o add essing ce ain decidabili y o op imiza ion p oblems).
We hope o e u n o his opic in a o hcoming pape .
Acknowledgemen
The wo k o Gh. P˘aun was pa ially done du ing a s ay as a isi ing scien is a
MTA SZTAKI EU Cen e o Excellence, Budapes .
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