P Sys ems wi h Ac i e Cells
Da id O ellana-Ma ´ın
Resea ch G oup on Na u al Compu ing
Depa men o Compu e Science and A ificial In elligence
Uni e sidad o Se illa
[email p o ec ed]
Summa y. P sys ems wi h ac i e memb anes is a widely s udied amewo k wi hin he
field o Memb ane Compu ing since he c ea ion o he discipline. The abs ac ion o he
s uc u e and beha io o li ing cells is eflec ed in he ee-like hie a chy and he kinds
o ules ha can be used in hese kinds o sys ems.
Resembling he o ganiza ion and communica ion be ween cells wi hin issues
ha o m o gans, issue-like P sys ems we e defined as hei abs ac ions, using
sympo /an ipo ules, ha is, mo ing and exchanging elemen s om one cell o an-
o he one. All he cells a e loca ed in an en i onmen whe e he e exis an a bi a y
numbe o some elemen s.
La ely, sympo /an ipo ules ha e been used in he amewo k o cell-like mem-
b ane sys ems in o de o s udy hei compu a ional powe . In e es ing esul s ha e been
eached, since hey ac simila ly o hei coun e pa s in he amewo k o issue P sys-
ems.
He e, he use o he o me defined ules ( ha is, e olu ion, communica ion, dis-
solu ion and di ision/sepa a ion ules) is conside ed, bu no wo king wi h a ee-like
s uc u e. Some ema ks abou choosing good seman ics a e gi en.
Key wo ds: Memb ane Compu ing, Ac i e cells, Compu a ional Complexi y, P
e sus NP p oblem.
1 In oduc ion
Memb ane Compu ing is a dis ibu ed pa allel compu ing pa adigm inspi ed by he
way he li ing cells p ocess chemical subs ances, ene gy and in o ma ion. The p o-
cesso uni s in he basic model a e abs ac ions o biological memb anes, selec i ely
pe meable ba ie s which gi e cells hei ou e bounda ies (plasma memb anes)
and hei inne compa men s (o ganelles). They con ol he flow o in o ma ion
be ween cells and he mo emen o subs ances in o and ou o cells and hey a e
also in ol ed in he cap u e and elease o ene gy. Biological memb anes play an
ac i e pa in he li e o he cell. In ac , he passing o a chemical subs ance
176 D. O ellana-Ma ´ın
h ough a biological memb ane is o en implemen ed by an in e ac ion be ween
he memb ane i sel and he p o ein channels p esen in i . Du ing his in e ac-
ion, he chemical subs ance and he memb ane i sel can be modified a leas
locally.
P sys ems wi h ac i e memb anes [7] include ules inspi ed on he beha io o
he p o eins inside he cells. Recalling, e olu ion ules a e he abs ac ion o he
mu a ion o he chemical compounds wi hin singula o ganelles, communica ion
ules gi e us he idea o he anspo o he p o eins h ough he memb anes o
he cells, dissolu ion ules emembe he p ocess o apop osis, which makes he cell
o “kill i sel ” (in his case, we ake he inspi a ion and apply i o memb anes).
A las , di ision and sepa a ion ules a e he ules ha can c ea e an exponen ial
wo kspace in polynomial ime. These a e inspi ed by he asexual and sexual cell
p ocesses, ha gi e bi h o new cells.
All o hose ules can be success ully applied in he amewo k o issue-like P
sys ems. Mo eo e , i would be a mo e na u al way o desc ibe he unc ioning o
hese ules a he cells ha in he memb anes. As an analogy o P sys ems wi h
ac i e memb anes, we a e going o call hem P sys ems wi h ac i e cells
The pape is o ganized as ollows. Nex sec ion b iefly in oduces some p elimi-
na ies needed o make he wo k sel -con ained. Sec ion 3 will be de o ed o p esen
bo h syn ax and seman ics o issue-like P sys ems wi h ac i e cells, le ing Sec-
ion 4 dedica ed o p esen some esul s conce ning he compu a ional complexi y
classes eached by his kind o memb ane sys ems. The pape ends wi h some open
p oblems and concluding ema ks.
2 P elimina ies
An alphabe Γis a non-emp y se and hei elemen s a e called symbols. A s ing u
o e Γis an o de ed fini e sequence o symbols, ha is, a mapping om a na u al
numbe o n∈Non o Γ. The numbe nis called he leng h o he s ing uand i
is deno ed by |u|. The emp y s ing (wi h leng h 0) is deno ed by λ. The se o all
s ings o e an alphabe Γis deno ed by Γ∗. A language o e Γis a subse o Γ∗.
A mul ise o e an alphabe Γis an o de ed pai (Γ, ) whe e is a mapping
om Γon o he se o na u al numbe s N. The suppo o a mul ise m= (Γ, )
is defined as supp(m) = {x∈Γ| (x)>0}. A mul ise is fini e ( espec i ely,
emp y) i i s suppo is a fini e ( espec i ely, emp y) se . We deno e by ∅ he
emp y mul ise . We deno e by M (Γ) he se o all fini e mul ise s o e Γ. The
ca dinal o a fini e mul ise mis defined as ∑
x∈Γ
m(x).
Le m1= (Γ, 1), m2= (Γ, 2) be mul ise s o e Γ, hen he union o m1and
m2, deno ed by m1+m2, is he mul ise (Γ, g), whe e g(x) = 1(x) + 2(x) o
each x∈Γ. We say ha m1is con ained in m2and we deno e i by m1⊆m2, i
1(x)≤ 2(x) o each x∈Γ. The ela i e complemen o m2in m1, deno ed by
m1 m2, is he mul ise (Γ, g), whe e g(x) = 1(x)− 2(x) i 1(x)≥ 2(x), and
g(x) = 0 o he wise.
P Sys ems wi h Ac i e Cells 177
A oo ed ee is a connec ed, acyclic, undi ec ed g aph in which one o he
e ices (called he oo o he ee) is dis inguished om he o he s. Gi en a node
x(diffe en om he oo ) in a oo ed ee, i he las edge on he (unique) pa h
om he oo o he node xis {x, y}(so x=y), hen yis he pa en o node x
and xis achild o node y. We deno e i by y=p(x) and x∈ch(y). The oo is
he only node in he ee wi h no pa en . A node wi h no child en is called a lea
(see [2] o de ails).
Le us ecall ha he pai unc ion ⟨n, m⟩= ((n+m)(n+m+ 1)/2) + nis
a polynomial- ime compu able unc ion which is also a p imi i e ecu si e and
bijec i e unc ion om N×N o N.
Adecision p oblem Xis one whose solu ion is ei he “yes” o “no”. This can
be o mally defined by an o de ed pai (IX, θX), whe e IXis a language o e a
fini e alphabe and θXis a o al boolean unc ion o e IX. The elemen s o IXa e
called ins ances o he p oblem X.
2.1 Recognize memb ane sys ems
In his sec ion, a memb ane sys em designa es any a ian o P sys em. Recognize
memb ane sys ems we e in oduced in [4] and hey p o ide a na u al amewo k
o sol e decision p oblems by means o de ices in Memb ane Compu ing.
De ini ion 1. A memb ane sys em Πis a ecognize memb ane sys em i he ol-
lowing holds:
1. The wo king alphabe Γo Πhas wo dis inguished objec s yes and no.
2. The e exis s an (inpu ) alphabe Σs ic ly con ained in Γ.
3. The ini ial mul ise s M1, . . . , Mqo Πa e mul ise s o e Γ Σ.
4. The e exis s a dis inguished memb ane called he inpu memb ane.
5. The ou pu egion iou is he en i onmen .
6. All compu a ions hal .
7. I Cis a compu a ion o Π, hen ei he objec yes o objec no (bu no bo h)
mus ha e been eleased in o he en i onmen , and only a he las s ep o he
compu a ion.
In ecognize memb ane sys ems any compu a ion is ei he an accep ing com-
pu a ion (when objec yes is eleased in o he en i onmen a he las s ep).
Fo each fini e mul ise mo e he inpu alphabe Σ, he compu a ion o he
sys em Πwi h inpu ms a s om he configu a ion ob ained by adding he inpu
mul ise m o he con en s o he inpu memb ane, in he ini ial configu a ion o
Π. The e o e, in his kind o sys ems we ha e an ini ial configu a ion associa ed
wi h each inpu mil ise m(o e he inpu alphabe Σ). We deno e Π+m he
memb ane sys em Πwi h inpu mul ise m.
178 D. O ellana-Ma ´ın
2.2 Polynomial complexi y classes o ecognize memb ane sys ems
Nex , le us ecall he concep o efficien sol abili y by means o a amily o
ecognize memb ane sys ems (see [4] o mo e de ails).
De ini ion 2. A decision p oblem X= (IX, θX)is sol able in polynomial ime by
a amily Π={Π(n)|n∈N}o ecognize memb ane sys ems om a class R, in a
uni o m way, deno ed by X∈PMCR, i he ollowing s a emen s hold:
• he amily Πis polynomially uni o m by Tu ing machines, ha is, he e exis s
a de e minis ic Tu ing machine wo king in polynomial ime which cons uc s
he sys em Π(n) om n∈N;
• he e exis s a pai (cod, s)o polynomial- ime compu able unc ions o e he
se IXsuch ha :
– o each ins ance u∈IX,s(u)is a na u al numbe and cod(u)is he inpu
mul ise o he sys em Π(s(u));
– o each n∈N,s−1(n)is a ini e se ;
– he amily Πis polynomially bounded wi h ega d o (X, cod, s), ha is,
he e exis s a polynomial unc ion p, such ha o each u∈IXe e y
compu a ion o Π(s(u)) + cod(u)is hal ing and i pe o ms a mos p(|u|);
– he amily Πis sound wi h ega d o (X, cod, s), ha is, o each u∈IX, i
he e exis s an accep ing compu a ion o Π(s(u))+cod(u), hen θX(u) = 1;
– he amily Πis comple e wi h ega d o (X, cod, s), ha is, o each u∈IX,
i θX(u) = 1, hen e e y compu a ion o Π(s(u)) + cod(u)is an accep ing
one.
The polynomial complexi y class PMCRis closed unde polynomial- ime educ-
ion and unde complemen [5].
3 Tissue-like P Sys ems wi h Ac i e Cells
This new kind o P sys ems keeps he inspi a ion keeps he ounda ions o classical
issue P sys ems, ha is, he exchange o elemen s be ween he cells. He e, ins ead
o he use o sympo /an ipo ules, we a e going o in oduce he applica ion o
he ules ypically used in cell-like P sys ems wi h ac i e memb anes.
3.1 Syn ax
De ini ion 3. A issue-like P sys em wi h ac i e memb anes and sepa a ion ules
o deg ee q≥1is a uple (Γ, Γ0, Γ1, H, H0, H1, µ, M1, . . . , Mq,R, iou ), whe e:
•Γis a ini e alphabe and H={1, . . . , q};
• {Γ0, Γ1}is a pa i ion o Γand {H0, H1}is a pa i ion o H;
• M1, . . . , Mqa e ini e mul ise s o e Γ;
• R is a ini e se o ules o e Γo he ollowing o ms:
P Sys ems wi h Ac i e Cells 179
(a) [a→u]α
h o h∈H, α ∈ {+,−,0}, a ∈Γ, u ∈M (Γ)(objec e olu ion
ules).
(b) a[ ]α1
h→[b]α2
h o h∈H, α1, α2∈ {+,−,0}, a, b ∈Γ(send-in commu-
nica ion ules).
(c) [a]α1
h→b[ ]α2
h o h∈H, α1, α2∈ {+,−,0}, a, b ∈Γ(send-ou com-
munica ion ules).
(d) [a]α
h→b o h∈H, α ∈ {+,−,0}, a, b ∈Γ(dissolu ion ules).
(e) [a]α1
h→[b]α2
h[c]α3
h o h∈H, α1, α2, α3∈ {+,−,0}, a, b, c ∈Γ(di ision
ules o elemen a y memb anes).
(e) [a]α1
h→[Γ0]α2
h[Γ1]α3
h o h∈H, α1, α2, α3∈ {+,−,0}, a ∈Γ(sepa a-
ion ules o elemen a y memb anes).
( ) [[ ]α1
h0[ ]α2
h1]α
h→[[ ]α3
h0]α5
h[[ ]α4
h1]α6
h o h, h0, h1∈H,
α, α1, α2, α3, α4, α5, α6∈ {+,−,0}(di ision ules o non-elemen a y
memb anes).
( ) [[ ]α1
h0[ ]α2
h1]α
h→[Γ0[ ]α3
h0]α5
h[Γ1[ ]α4
h1]α6
h o h∈H, h0∈H0, h1∈H1,
α, α1, α2, α3, α4, α5, α6∈ {+,−,0}(sepa a ion ules o
non-elemen a y memb anes).
•iou ∈H∪ {en }, whe e en ∈ Γ∪H.
A issue-like P sys em wi h ac i e cells o deg e q≥1 can be iewed as a se
o qcells, labelled by elemen s o H, a anged in a di ec ed s uc u e µgi en by a
di ec ed g aph ( he cell s uc u e) whose nodes h ha ha e ou deg ee(h) = 0 a e
called elemen a y cells, such ha : (a) M1, . . . , Mq ep esen he fini e mul ise s
o objec s (symbols o he wo king alphabe Γ) ini ially placed in he qcells o he
sys em; (b) Ris a fini e se o ules o e Γassocia ed wi h he sys em; and (c)
iou ∈H∪ {en }indica es he ou pu egion. We use he e m egion i o e e
o cell iin he case i∈Hand o e e o he “en i onmen ” o he sys em in he
case i=en . I he memb ane sys em makes no use o sepa a ion ules o non-
elemen a y cells, hen se s H0and H1will be omi ed. I sepa a ion ules ei he o
elemen a y and non-elemen a y cells a e no used, hen we can omi ei he he se s
H0and H1and Γ0and Γ1. The leng h o a ule is he numbe o objec s in ol ed
in i ( o ins ance, he leng h o he objec e olu ion ule [ a→u]α
his 1 + |u|. Le
us no ice ha in his amewo k we can change (classical) objec e olu ion ules
by coope a i e e olu ion ules (see [11] o mo e de ails).
Fo each cell hdiffe en o cells hwi h indeg ee(h)= 0, we deno e p(h)
he label o he pa en o hin µ. By con en ion, he “pa en ” o cells hwi h
indeg ee(h) = 0 is he en i onmen o he sys em
3.2 Seman ics
An ins an aneous desc ip ion o a con igu a ion C a an ins an o a P sys em
wi h ac i e cells is desc ibed by he cell s uc u e a ins an and all mul ise s
o objec s o e Γassocia ed wi h all he memb anes p esen in he sys em a he
momen .

180 D. O ellana-Ma ´ın
An objec e olu ion ule [a→u]α
his applicable o a configu a ion C a an
ins an , i he e exis s a cell labelled by hwi h pola iza ion αin C which con ains
objec a. When applying such a ule, objec ais consumed and all objec s om
mul ise ua e p oduced in ha memb ane.
A send-in communica ion ule a[ ]α1
h→[b]α2
his applicable o a configu a ion
C a an ins an , i he e exis s a cell labelled by hwi h pola iza ion α1in C such
ha indeg ee(h)>0 and i s pa en one o i s pa en cells con ain objec a. When
applying such a ule, objec ais consumed om he selec ed pa en cell and objec
bis p oduced in he co esponding cell h, and he pola iza ion o cell hchanges o
α2.
A send-ou communica ion ule [ a]α1
h→b[ ]α2
his applicable o a configu a ion
C a an ins an , i he e exis s a cell labelled by hwi h pola iza ion α1in C
such ha i con ains objec a. When applying such a ule, objec ais consumed
om such cell and objec bis p oduced in he one o i s pa en cells chosen in a
non-de e minis ic way, and he pola iza ion o cell hchanges o α2.
A dissolu ion ule [ a]α
h→bis applicable o a configu a ion C a an ins an
, i he e exis s a cell labelled by hwi h pola iza ion αin C , diffe en om he
ou pu egion, such ha i con ains objec a. When applying such a ule, objec a
is consumed, cell his dissol ed and i s objec s a e sen o one o he pa en s cells,
chosen non-de e minis ically (o ances o s ha ha e no been dissol ed). Fo all
h′such ha (h′) = hand h′′ such ha (h) = h′′, when his dissol ed, hen new
edges om all h′′ o all h′a e c ea ed, and edges om h′′ o hand om h o h′
a e emo ed.
A di ision ule [ a]α1
h→[b]α2
h[c]α3
his applicable o a configu a ion C a an
ins an , i he e exis s a cell labelled by hwi h pola iza ion α1in C , diffe en
om he ou pu egion, such ha i is an elemen a y cell and con ains objec a.
When applying such a ule, he cell is di ided in o wo cells wi h he same label,
one wi h pola iza ion α1and he o he one wi h pola iza ion α2; a he same ime,
objec ais consumed and objec bappea s in he fi s cell, and cin he second
one, and he emaining objec s ge duplica ed in he wo c ea ed cells. Fo all h′
such ha (h′) = hand h′′ such ha (h) = h′′, when his dissol ed, hen edges
om all h′′ o hand om h o h′a e duplica ed.
A sepa a ion ule [ a]α1
h→[Γ0]α2
h[Γ1]α3
his applicable o a configu a ion C a
an ins an , i he e exis s a cell labelled by hwi h pola iza ion α1in C , diffe en
om he ou pu egion, such ha i is an elemen a y cell and con ains objec
a. When applying such a ule, he cell is sepa a ed in o wo cells wi h he same
label, one wi h pola iza ion α1and he o he one wi h pola iza ion α2; a he same
ime, objec ais consumed and he mul ise o objec s con ained in memb ane h
ge dis ibu ed: he objec s om Γ0a e placed in one cell, hose om Γ1a e placed
in he second one. Fo all h′such ha (h′) = hand h′′ such ha (h) = h′′,
when his dissol ed, hen edges om all h′′ o hand om h o h′a e duplica ed.
A di ision ule [[ ]α1
h0[ ]α2
h1]α
h→[[ ]α3
h0]α5
h[[ ]α4
h1]α6
his applicable o a configu-
a ion C a an ins an , i he e exis s a cell labelled by hwi h pola iza ion αin
C , diffe en om he ou pu egion, such ha i is he pa en o a cell labelled by
P Sys ems wi h Ac i e Cells 181
h0wi h pola iza ion α1and o ano he cell labelled by h1wi h pola iza ion α2.
When applying such a di ision ule o a cell labelled by hin a configu a ion C ,
ha cell is di ided in o wo cells wi h he same label wi h pola iza ions α5and
α6, in such a way ha he con en s (mul ise o objec s) and ela ions (child en
and pa en cells) a e duplica ed in o he wo new cells, excep om cells labelled
by h0, ha becomes a child cell o he fi s one, wi h pola iza ion α3, and h1, ha
becomes a child cell o he second one, wi h pola iza ion α4. Fo all h′such ha
(h′) = hand h′′ such ha (h) = h′′, when his dissol ed, hen edges om all h′′
o hand om h o h′a e duplica ed (excep o edges om h o h0and h1, which
ones emains one o each new c ea ed cell).
A sepa a ion ule [[ ]α1
h0[ ]α2
h1]α
h→[Γ0[ ]α3
h0]α5
h[Γ1[ ]α4
h1]α6
his applicable o a
configu a ion C a an ins an , i he e exis s a cell labelled by hwi h pola iza ion
αin C , diffe en om he ou pu egion, such ha i is he pa en o a cell labelled
by h0wi h pola iza ion α1and o ano he cell labelled by h1wi h pola iza ion α2.
When applying such a sepa a ion ule o a cell labelled by hin a configu a ion C ,
ha cell is sepa a ed in o wo cells wi h he same label wi h pola iza ions α5and
α6, in such a way ha he con en s (mul ise s o objec s) and ela ions (child en
cells) a e dis ibu ed as ollows: The fi s cell ecei es he mul ise o objec s om
Γ0, and all child cells whose label belongs o H0; and he second cell ecei es he
mul ise o objec s om Γ1, and all child cells whose label belongs o H1. Fo all h′
such ha (h′) = hand h′′ such ha (h) = h′′, when his dissol ed, hen edges
om all h′′ o ha e duplica ed, and edges om h o h′a e dis ibu ed depending
on whe he hey belong o H0o H1.
In issue-like P sys ems wi h ac i e cells, he ules a e applied acco ding o he
ollowing p inciples:
•The ules associa ed wi h memb anes labelled wi h ha e used o all copies o
his memb ane.
•A one ansi ion s ep, one objec can be used by only one ule (chosen in a
non-de e minis ic way).
•A one ansi ion s ep, a cell can be subjec o only one ule o ypes (b)–( ),
and hen i is applied a mos once.
•Objec e olu ion ules can be simul aneously applied o a cell wi h one ule
o ypes (b)–( ). Objec e olu ion ules a e applied in a maximally pa allel
manne .
•I a he same ime a memb ane labelled wi h his di ided/sepa a ed by a ule
o ype (e) o ( ) and he e a e objec s in his cells which e ol e by means o
ules o ype (a), hen we suppose ha fi s he e olu ion ules o ype (a) a e
used, changing he objec s, and hen he sepa a ion is p oduced. O cou se,
his p ocess akes only one ansi ion s ep.
•Ou pu cell can ne e ge di ided, sepa a ed, no dissol ed.
Le us conside a issue-like P sys ems wi h ac i e cells ΠWe say ha configu-
a ion C yields configu a ion C +1 in one ansi ion s ep, deno ed by C ⇒ΠC +1,
i we can pass om C o C +1 by applying he ules om he sys em ollowing he
182 D. O ellana-Ma ´ın
p e ious ema ks. A compu a ion o Πis a (fini e o infini e) sequence o configu a-
ions such ha : (a) he fi s e m is he ini ial configu a ion o he sys em; (b) o
each n≥1, he n- h configu a ion o he sequence is ob ained om he p e ious
configu a ion in one ansi ion s ep; and (c) i he sequence is fini e (called hal ing
compu a ion) hen he las e m is a hal ing con igu a ion (a configu a ion whe e
no ule o he sys em is applicable o i ).
All compu a ions s a om an ini ial configu a ion and p oceed as s a ed
abo e; only hal ing compu a ions gi e a esul , which is encoded by he objec s
p esen in he ou pu egion iou associa ed wi h he hal ing configu a ion. I
C={C } < +1 o Π( ∈N) is a hal ing compu a ion, hen he leng h o C, deno ed
by |C|, is , ha is, |C| is he numbe o non-ini ial configu a ions which appea in
he fini e sequence C. Fo each i(1 ≤i≤q) we deno e by C (i) he fini e mul ise
o objec s o e Γcon ained in all cells labelled by i(by applying di ision o sep-
a a ion ules diffe en cells wi h he same label can be c ea ed) a configu a ion C .
3.3 Families o issue-like P sys ems wi h ac i e cells
We use he ollowing no a ions:
• NAC(α, β, δ), whe e α∈ {+e, −e},β∈ {+c, −c}and δ∈ {+d, −d}, is he
class o all ecognize P sys ems wi h ac i e cells wi hou using di ision no
sepa a ion ules.
• DAC(α, β, δ, γ), whe e α∈ {+e, −e},β∈ {+c, −c},δ∈ {+d, −d}and α∈
{+n, −n},is he class o all ecognize P sys ems wi h ac i e cells and di ision
ules.
• SAC(α, β, δ, γ), whe e α∈ {+e, −e},β∈ {+c, −c},δ∈ {+d, −d}and
α∈ {+n, −n},is he class o all ecognize P sys ems wi h ac i e cells and
sepa a ion ules.
The meaning o pa ame e s is he ollowing:
•i α= +e( esp., −e) hen e olu ion ules a e pe mi ed ( esp., o bidden).
•i α= +c( esp., −c) hen communica ion ules a e pe mi ed ( esp., o bid-
den).
•i α= +d( esp., −d) hen dissolu ion ules a e pe mi ed ( esp., o bidden).
•i α= +n( esp., −n) hen di ision/sepa a ion ules o elemen a y and non-
elemen a y cells a e pe mi ed ( esp., only di ision/sepa a ion ules o ele-
men a y cells a e pe mi ed).
3.4 Ano he (no so ele an ) app oach
One ques ion discussed when his amewo k was being c ea ed was:
In issue-like memb ane sys ems, he na u al de ini ion would be he one whe e
when we do a communica ion ule, he cell in e ac s he en i onmen (objec s go
P Sys ems wi h Ac i e Cells 183
o he en i onmen in send-ou communica ion ules and comes om i in
send-in communica ion ules. The same goes o dissolu ion ules, ha is, when a
cell dissol es, i s con en s go o he en i onmen .
This defini ion seems he bes in o de o cap u e he beha io o issue P
sys ems. Bu because o he simple s uc u e c ea ed, i has li le in e es ega ding
he compu a ional complexi y o hese sys ems.
I his kind o sys ems is defined, we can suppose ha he e a e qcells disposed
in he en i onmen , and hey can in e ac wi h i h ough communica ion and
dissolu ion ules. Bu we can simula e his beha io wi h P sys ems wi h ac i e
memb anes wi h q+1 memb anes, whe e qmemb anes a e si ua ed wi hin one ha
ac s as he en i onmen in he p e ious sys em. So complexi y classes whe e hese
amilies o P sys ems we e in ol ed in would be weake han classical P sys ems
wi h ac i e memb anes, he e o e i will no be conside ed.
4 Some Resul s Abou Compu a ional Complexi y
Fi s o all, i is easy o see ha e e y P sys em wi h ac i e cells is a leas as pow-
e ul as i s ac i e memb anes coun e pa . I can be p o ed because e e y P sys em
wi h ac i e memb anes s uc u e is defined by a oo ed ee µ. A ee is a pa icula
case o a g aph, whe e cycles a e no allowed. Fo e e y P sys em wi h ac i e mem-
b anes, we can define a P sys em wi h ac i e cells ha simula es i s beha io . Le
Π= (Γ, Γ0, Γ1, H, H0, H1, µ, M1, . . . , Mq,R, iou ) a P sys em wi h ac i e mem-
b anes. We can c ea e (in polynomial ime) a P sys em wi h ac i e memb ane ha
simula es i s beha io . Le Π′= (Γ, Γ0, Γ1, H, H0, H1, µ′,M1, . . . , Mq,R, iou ) be
he P sys em wi h ac i e cells ha simula es i s beha io . µ′is cons uc ed as
ollows:
•Le µ′be a single node h, whe e his he label o he skin memb ane o Π.
•Fo e e y memb ane h′si ua ed wi hin ano he memb ane hin Π, we c ea e
a node h′in µ′and add an edge om h o h′.
The di ec ed g aph ob ained has he shape o a di ec ed oo ed ee, and as i has
he same se o ules, seman ics o he sys em makes Π′simula e he beha io o
Π′. We can conclude wi h:
Theo em 1. PMCAM(α,β,δ,γ)⊆PMCAC(α,β,δ,γ),
no ma e which kinds o ules we a e using.
4.1 Some complexi y classes
As i happened wi h P sys ems wi h ac i e memb anes, we can use he Milano
Theo em [14] o s a e ha no compu a ionally ha d p oblems can be sol ed in
polynomial ime wi hou using ules allowing he gene a ion o an exponen ial
numbe o memb anes/cells in polynomial ime. Then: