Decision P Sys ems and he P=NP Conjec u e
Ma io J. P´e ez Jim´enez, ´Al a o Rome o Jim´enez,
and Fe nando Sancho Capa ini
Abs ac . We in oduce decision P sys ems, which a e a class o P
sys ems wi h symbol-objec s and ex e nal ou pu . The main esul o
he pape is he ollowing: i he e exis s an NP–comple e p oblem ha
canno be sol ed in polynomial ime, wi h espec o he inpu leng h, by
a de e minis ic decision P sys em cons uc ed in polynomial ime, hen
P=NP. F om Zand on-Fe e i-Mau i’s heo em i ollows ha i P=
NP, hen no NP–comple e p oblem can be sol ed in polynomial ime,
wi h espec o he inpu leng h, by a de e minis ic P sys em wi h ac i e
memb anes bu wi hou memb ane di ision, cons uc ed in polynomial
ime om he inpu . Toge he , hese esul s gi e a cha ac e iza ion o
P=NP in e ms o de e minis ic P sys ems.
1 In oduc ion
In [2] a new model o compu a ion, called P Sys ems, is in oduced wi hin he
amewo k o Na u al Compu ing (bio-inspi ed compu ing). I is based upon
he no ion o memb ane s uc u e ha is u sed o enclose compu ing cells in
o de o make hem independen compu ing uni s. Also, a memb ane se es as a
communica ion channel be ween a gi en cell and o he cells adjacen o i . This
model s a s om he obse a ion ha he p ocesses which ake place in he
complex s uc u e o a li ing cell can be conside ed as compu a ions.
Since hese compu ing de ices we e in odu ced se e al a ian s ha e been
conside ed. A ai ly comple e compendium abou P sys ems can be ound a [8].
In pa icula , P sys ems wi h ex e nal ou pu a e s udied in [4].
The diffe en a ian s o P sys ems ound in he li e a u e a e in gene al gen-
e a ing de ices. Many o hem ha e been p o ed o be compu a ionally comple e:
hey compu e all Tu ing compu able se s o na u al nu mbe s o all ecu si ely
enume able languages, depending on he a ian conside ed.
The model we conside he e wo ks wi h symbol–objec s and i has wo cha ac-
e is ics ha ha e seldom been conside ed be o e: we wo k wi h decision de ices
whose wo k is igge ed by ce ain inpu da a. The aim is o use his kind o P
sys ems o deal wi h decision p oblems.
The main goal o his pape is o show a sufficien condi ion o he ela ion P =
NP o be e ified: i he e exis s an NP–comple e p oblem ha canno be
sol ed in polynomial ime, wi h espec o he inpu leng h, by any amily o
de e minis ic decision P sys ems, cons uc ed in polynomial ime, hen P=NP.
To achie e his, we p o e ha e e y decision p oblem which can be sol ed
by a de e minis ic Tu ing machine in polynomial ime can also be sol ed by a
amily o de e minis ic decision P sys ems in polynomial ime.
The pape is o ganized as ollows: Sec ion 2 b iefly p esen s some basic con-
cep s abou P sys ems wi h ex e nal ou pu ; Sec ion 3 in oduces he new model
(wi h symbol–objec s) o decision P sys ems; Sec ion 4 shows how o simula e de-
e minis ic Tu ing machines by amilies o such P sys ems; Sec ion 5 es ablishes
ou main esul s abou decision P sys ems and he P=NP conjec u e.
2 Mul ise s, Memb ane S uc u es, E olu ion Rules
Amul ise o e a se , A, is a mapping m:A→IN ; m(a) is he numbe o copies
o a∈Ain he mul ise m. The se {a∈A:m(a)>0}is called he suppo o m
and i is deno ed by supp(m). A mul ise , m, is said o be emp y ( esp. fini e) i
i s suppo is emp y ( esp. fini e). I mis a fini e mul ise o e A, we will deno e
i m={{a1,...,a
m}}, whe e he elemen s ai∈supp(m) a e possibly epea ed.
We w i e M(A) o he se o all he mul ise s o e A. Fo wo mul ise s m1,m2
o e Awe define hei union by (m1∪m2)(a)=m1(a)+m2(a), o each a∈A.
The se o memb ane s uc u es,MS, is defined by ecu sion as ollows:
1. [ ] ∈MS;2.I µ1,...,µ
n∈MS, hen [µ1...µ
n]∈MS.
A memb ane s uc u e, µ, can also be seen as a oo ed ee, V(µ),E(µ).
Then, he nodes o his ee a e called memb anes, he oo node he skin mem-
b ane, and he lea es elemen a y memb anes. The deg ee o a memb ane s uc-
u e is he numbe o memb anes in i .
The memb ane s uc u e wi h en i onmen associa ed wi h he memb ane
s uc u e, µ,isµE=[
Eµ]E. I we conside µEas a oo ed ee, hen he oo
node is called he en i onmen o µ.
Gi en an alphabe , Γ, we associa e wi h e e y memb ane o a memb ane
s uc u e a fini e mul ise o elemen s o Γ, which a e called he objec s o he
memb ane.
We also associa e wi h e e y one o hese memb anes a fini e se o e olu ion
ules. An e olu ion ule o e Γis a pai (u, ), usually w i en u→ , whe e u
is a s ing o e Γand = o = δ, whe e is a s ing o e
Γ×({he e, ou }∪{inl:l∈V(µ)})
and δis a special symbol no in Γ. The idea behind a ule is ha he objec s
in u“e ol e” in o he objec s in , mo ing o no o ano he memb anes and
possibly dissol ing he o iginal memb ane.
The leng h o a ule is he numbe o symbols in ol ed in he ule ( o ins ance,
he leng h o u→ is |u|+| |+ 1).
3 Decision P Sys ems
Defini ion 1. A decision P sys em is a cons uc
Π=(Γ,Σ,µΠ,i
Π,M1,...,Mp,(R1,ρ
1),...,(Rp,ρ
p)),
whe e:
–Σis an alphabe , called he inpu alphabe .
–Γis an alphabe such ha Σ⊆Γ; i s elemen s a e called objec s; he e a e
wo dis inguished objec s, Y ES,NO ∈Γ−Σ.
–µΠis a memb ane s uc u e o deg ee p, he memb anes o which we suppose
labeled om 1 o p.
–iΠ∈{1,...,p}is he inpu memb ane o Π.
–Miis a mul ise o e Γ−Σassocia ed wi h he memb ane labeled by i, o
e e y i=1,...,p.
–Riis a fini e se o e olu ion ules o e Γassocia ed wi h he memb ane
labeled by i, and ρiis a s ic pa ial o de o e Ri, o e e y i=1,...,p.
To o malize he seman ics o his model we define fi s wha a configu a ion
o such a P sys em is, and hen he no ion o compu a ion.
Defini ion 2. Le Πbe a decision P sys em wi h ex e nal ou pu .
1. A configu a ion o Πis a pai (µE,M), whe e µis a memb ane s uc u e
such ha V(µ)⊆V(µΠ)and i has he same oo han µΠ, and Mis an
applica ion om V(µE)in o M(Γ). Fo e e y node nd ∈V(µE)we deno e
Mnd =M(nd).
2. The ini ial configu a ion o Π o he mul ise m∈M(Σ)is he pai (µE,M),
whe e µ=µΠ,ME=∅,MiΠ=m∪M
iΠand Mj=Mj, o e e y j=iΠ.
The idea is ha o e e y inpu mul ise m∈M(Σ), we add ha mul ise
o he inpu memb ane, iΠ, o he P sys em and hen s a he wo k o Π.
We can pass, in a non-de e minis ic manne , om one configu a ion o Π
o ano he by applying o i s mul ise s he e olu ion ules associa ed wi h hei
co esponding memb anes. This is done as ollows: gi en a ule u→ o a
memb ane i, he objec s in ua e emo ed om Mi; hen, o e e y (ob, ou )∈
an objec ob is pu in o he mul ise associa ed wi h he pa en memb ane (o
he ex e nal en i onmen i iis he skin memb ane); o e e y (ob, he e)∈ an
objec ob is added o Mi; finally, o e e y (ob, inj)∈ an objec ob is added o
Mj(i jis no a child memb ane o i, hen he ule canno be applied). Finally,
i δ∈ , hen he memb ane iis dissol ed (i iis he skin memb ane, he ule
canno be applied), ha is, i is emo ed om he memb ane s uc u e. The
objec s o a dissol ed memb ane emain in he egion su ounding i , while he
ules a e emo ed. Mo eo e , he p io i y ela ion among he ules o bids he
applica ion o a ule i ano he one o highe p io i y is applied.
Gi en wo configu a ions, Cand C,o Π, we say ha Cis ob ained om C
in one ansi ion s ep, and we w i e C⇒C, i we can pass om he fi s o he
second one by using he e olu ion ules appea ing in he memb ane s uc u e o
Cin a pa allel and maximal way in each memb ane, and o all he memb anes
a he same ime.
Defini ion 3. Le Πbe a decision P sys em. A compu a ion, C,o Πwi h inpu
m∈M(Σ)is a sequence, possibly infini e, o configu a ions o Π,C0⇒C1⇒
...⇒Cq,q≥0, such ha
–C0is he ini ial configu a ion o Π, wi h he mul ise mplaced in memb a-
ne iπ.
–Each Ci(1 ≤i≤q)is ob ained om he p e ious configu a ion by one
ansi ion s ep.
We say ha Cis a hal ing compu a ion o Πi he e is no ule applicable o
he objec s p esen in i s las configu a ion. In his case, we say ha Cqis he
hal ing configu a ion o C.
We say ha Πis de e minis ic i o each m∈M(Σ) he e exis s an unique
compu a ion wi h inpu m.
The philosophy o he P sys ems wi h ex e nal ou pu is ha we canno know
wha is happening inside he memb ane s uc u e, bu we can only collec he
in o ma ion sen ou om i o he en i onmen . Thus, i is na u al ha he
hal ing compu a ions o hese P sys ems epo o he en i onmen when hey
ha e eached hei final configu a ions (accep ing o ejec ing). Fu he mo e, he
idea behind he decision P sys ems is o use hem as languages decision de ices.
These conside a ions lead us o he ollowing no ions.
Defini ion 4. A de e minis ic decision P sys em, Π, is said o be alid when
he ollowing is e ified:
–All compu a ions o Πhal .
–Fo each compu a ion o Πonly one ule o he o m u→ (ob, ou ), whe e
ob =Y ES o ob =NO, may be applied in he skin memb ane o µΠ, and
only in he las s ep o he compu a ion.
Defini ion 5. Le Πbe a de e minis ic alid decision P sys em. We say ha
a configu a ion (µE,M)o Πis an accep ing ( esp., ejec ing) configu a ion i
Y ES ∈ME( esp., NO ∈ME).
We say ha Cis an accep ing ( esp., ejec ing) compu a ion o Πi i s as-
socia ed hal ing configu a ion is an accep ing ( esp., ejec ing) configu a ion.
Defini ion 6. A de e minis ic alid decision P sys em, Π, accep s ( espec i ely,
ejec s) a mul ise m∈M(Σ)i he compu a ion o Πwi h inpu mis an
accep ing ( esp. ejec ing) compu a ion.
We deno e by D he class o all de e minis ic alid decision P sys ems.
4 Simula ing Tu ing Machines by Decision P Sys ems
In wha ollows we a e going o define wha we mean by simula ing a Tu ing
machine (as a languages gene a ing de ice) h ough a amily o de e minis ic
decision P sys ems. This has o be done in such a way ha e e y solu ion o
a decision p oblem gi en by a Tu ing machine p o ides a solu ion o he same
p oblem by a decision P sys em. Mo eo e , he addi ional cos s o he educ ion
om one solu ion o ano he mus be polynomial in e ms o he inpu size.
We ake as a model he concep o complexi y classes in memb ane sys ems
in oduced by G. P˘aun in [1].
Defini ion 7. We say ha a de e minis ic Tu ing machine, TM, is simula ed
in polynomial ime by a amily o de e minis ic alid decision P sys ems ΠTM =
(ΠTM(1),Π
TM(2),...,Π
TM(k),...)i :
1. The amily ΠTM is D–consis en ; ha is, o each k∈N+,ΠTM(k)is a
de e minis ic alid decision P sys em.
2. The amily ΠTM is TM–uni o m; ha is, he e exis s a de e minis ic Tu ing
machine, TM, which cons uc s ΠTM(k)in polynomial ime s a ing om
k≥1( he e exis s a polynomial p(k)depending on TM such ha o each
k,TM(k)hal s in less han p(k)s eps and i s ou pu is ΠTM(k)).
3. The amily ΠTM is polynomially bounded; ha is, he e exis s a polynomial
p(k), depending on TM, such ha e e y compu a ion o ΠTM(k)always
hal s in less han p(k)s eps.
4. The amily ΠTM is TM–sound; ha is, he Tu ing machine TM accep s
( esp. ejec s) he inpu s ing ai1...a
iki and only i ΠTM(k)accep s ( esp.
ejec s) g(ai1...a
ik)(gis a sui able polynomial encoding o s ings by mul-
ise s).
No e 1. The ac ha he amily ΠTM is D–consis en has he consequence
ha o each k≥1, he P sys em ΠTM(k) has a polynomial size in he ollowing
sense: he size o he wo king alphabe , he numbe o memb anes, he size o
he ini ial mul ise s, and he sum o he leng hs o all he ules, is bounded by
k , o some cons an depending on TM.
No e 2. A sui able polynomial encoding, g, o s ings by inpu mul ise s o
ΠTM(k) means he ollowing: he e exis s a Tu ing machine, TM, and a poly-
nomial q(k) depending on TM such ha o each inpu da a wo TM we ha e
ha TM(w) hal s in less han q(|w|) s eps and i s ou pu is g(w) (an inpu
mul ise o he P sys em ΠTM(|w|)).
Theo em 1. Each de e minis ic Tu ing machine can be simula ed in polynomial
ime by a amily o de e minis ic alid decision P sys ems.

P oo . We conside de e minis ic Tu ing machines ollowing [6].
Suppose we ha e QTM ={qN,q
Y,q
0,...,q
n},ΓTM ={B, -, a1,...,a
m},
ΣTM ={a1,...,a
p}, wi h p≤m, and δTM(qi,a
j)=(qQ(i,j),a
A(i,j),D(i, j)) as
se o s a es, wo king alphabe , inpu alphabe and ansi ion unc ion o TM,
espec i ely. We deno e aB=Band a0=-.
We cons uc a amily o de e minis ic decision P sys ems ΠTM =(ΠTM(1),
ΠTM(2),...,Π
TM(k),...) which simula es TM as ollows: o each k∈N, he
decision P sys em ΠTM(k) is:
•Inpu alphabe : Σk={a, i:a∈ΣTM,1≤i≤k}
•Wo king alphabe : Γk={a, i:a∈ΣTM,0≤i≤k}∪{ i:1≤i≤k}∪
{s−
i,s
+
i,s
i:i∈{T1,T
2,F,S,1,...,9}} ∪
{qN,q
Y,h,h
,YES,NO}∪
{qi:0≤i≤n}∪{bi,b

i,b

i,c
i:0≤i≤m}
•Memb anes s uc u e: µΠ=[
1]1.
•Inpu memb ane: iΠ=1.
•Ini ial mul ise s: M1={{q0,b
0,s
−
T1,s
−
T2,s
−
F,s
−
S,s
−
1,...,s
−
9,s
T1}}.
•E olu ion ules: R=R0∪R1∪R2∪R3∪R4, whe e:
•R0=R0,1∪R0,2∪R0,3∪R0,4,wi h
R0,1≡sT1s−
T1→s+
T1>s
−
T1→s−
T1>ai,j→ai,j j(1≤i≤p,1≤j≤k)
s+
T1→s−
T1sT2
R0,2≡sT2s−
T2→s+
T2>s
−
T2→s−
T2>
2
1s+
T2→s−
T2sF>···>
>
2
ks+
T2→s−
T2sF>
1...
ks+
T2→s−
T2sS>s
+
T2→s−
T2sF
R0,3≡sFs−
F→s+
F>s
−
F→s−
F>




s−
T1s−
T2s−
Ss−
1...s
−
9s+
Fq0b0→(NO, ou )
ai,j→λ(1≤i≤p,1≤j≤k)
j→λ(1≤j≤k)
R0,4≡




sSs−
S→s+
S>s
−
S→s−
S>ai,j→ai,j−12(1≤i≤p,1≤j≤k)
ai,0→bi(1≤i≤p)
>s
+
S→s−
Ss1
•R1=R1,1∪R1,2∪R1,3,wi h
R1,1≡s1s−
1→s+
1>s
−
1→s−
1>




h→hh
bi→bib
i(0≤i≤m)
s+
1→s−
1s2
R1,2≡










hs2s−
2→s+
2>s
2→s3>s
−
2→s−
2>
>b
2
i→b
i(0≤i≤m)>




b
i→λ(0≤i≤m)
b
i→b
i(0≤i≤m)
s+
2→s−
2s2
R1,3≡s3s−
3→s+
3>s
−
3→s−
3>b2
i→λ(0≤i≤m)
s+
3→s−
3s4
•R2=R2,1∪R2,2,wi h
R2,1≡s4s−
4→s+
4>s
−
4→s−
4>h→hh
s+
4→s−
4s5
R2,2≡s5s−
5→s+
5>s
−
5→s−
5>


Rules o he ansi ion
unc ion
s+
5→s−
5s6
The ules o he ansi ion unc ion, δTM, a e he ollowing:
Case 1: s a e q , elemen as=B
Mo emen Rules
le q b
sh→qQ( ,s)b
scA( ,s),i A( , s)=B
q b
sh→qQ( ,s)b
s,i A( , s)=B
s and q b
s→qQ( ,s)b
scA( ,s),i A( , s)=B
q b
s→qQ( ,s)b
s,i A( , s)=B
igh q b
s→qQ( ,s)b
scA( ,s)h, i A( , s)=B
q b
s→qQ( ,s)b
sh, i A( , s)=B
Case 2: s a e q , no elemen
Mo emen Rules
le q h→qQ( ,s)cA( ,s),i A( , s)=B
q h→qQ( ,s),i A( , s)=B
s and q →qQ( ,s)cA( ,s),i A( , s)=B
q →qQ( ,s),i A( , s)=B
igh q →qQ( ,s)cA( ,s)h, i A( , s)=B
q →qQ( ,s)h, i A( , s)=B
To a oid conflic s, e e y ule in case 1 has highe p io i y han
any ule in case 2.
•R3=R3,1∪R3,2,wi h
R3,1≡hs6s−
6→s+
6>s
6→s7>s
−
6→s−
6>




b
i→b2
i(0≤i≤m)
ci→c2
i(0≤i≤m)
s+
6→s−
6s6
R3,2≡s7s−
7→s+
7>s
−
7→s−
7>




bib
i→λ(0≤i≤m)
ci→bi(0≤i≤m)
s+
7→s−
7s8
•R4=R4,1∪R4,2,wi h
R4,1≡s8s−
8→s+
8>s
−
8→s−
8>




qY→qYs9,q
N→qNs9
qi→qis1(0≤i≤n)
s+
8→s−
8
R4,2≡s9s−
9→λ>s
−
9→s−
9>















s−
1...s
−
8→λ
qY→(Y ES, ou )
qN→(NO, ou )
h→λ
bi→λ(0≤i≤m)
Le us see ha ΠTM =(ΠTM(1),Π
TM(2),...,Π
TM(k),...) is a amily o de-
e minis ic alid decision P sys ems which simula es TM.
Ob iously i is a amily o de e minis ic alid decision P sys ems. Mo eo e ,
ΠTM is an uni o m amily. Indeed, le TM be a de e minis ic Tu ing machine
such ha he se o s a es has size n+ 2, he wo king alphabe has size m+2
and he inpu alphabe has size p(wi h p≤m). The necessa y esou ces o
cons uc ΠTM(k) a e he ollowing:
1. The size o he wo king alphabe Γkis p·k+4m+n+ 45; ha is, in he
o de θ(k·m+n).
2. The deg ee o he P sys em is 1.
3. The size o he ini ial configu a ion o each x∈Σk
TM is k+16∈θ(k).
4. The o al numbe o ules is in he o de o
O(p·k+n·m)=O(k·m+n·m)
5. The g ea es leng h o a ule is 16 ∈O(1).
Le us see now ha , o each k∈N,ΠTM(k)issound: le Lbe he language
decided by TM. In o de o decide i a s ing ai1...a
ik∈ΣTM, o leng h k,
belongs o L, we encode i by he mul ise {{ai1,1,... ,aik,k}} which is he
inpu gi en o ΠTM(k). The ules o his P sys em ha e been ca e ully chosen
in such a way ha i s wo k goes h ough he ollowing main s ages:
1. Check ha he mul ise ecei ed as inpu codes a s ing o leng h k. Fo his,
we ha e o e i y ha o each j=1,...,k he e exis s one and only one
pai whose second componen is equal o j. O he wise, he P sys em hal s
and ejec s he mul ise .
2. T ans o m he inpu mul ise in o ano he mul ise which encodes he s ing
in base 2 (in o de o speci y he symbol w i en in each cell o he ape).
3. Read he elemen in he cell scanned by he head.
4. Compu e he new elemen o be w i en in he cell, mo e he head and change
s a e (acco ding o he ansi ion unc ion).
5. E ase he old elemen and w i e he new elemen .
6. Check i a final s a e is eached: i no , epea om s age 2; i qYis he final
s a e eached, hen accep he s ing; i qNis he final s a e eached, hen
ejec he s ing.
These s ages will be ca ied ou in se e al small s eps, each o hem managed by
a g oup o ules. To a oid ules om dis inc s eps being applied oge he , we
will use s−
jas o bidding objec s and s+
jas pe mi ing objec s;i s−
jis p esen
in he P sys em, hen ules o s ep jcanno be execu ed; i , ins ead, s+
jis he
objec p esen , hen ules o s ep jmus be execu ed (i possible). O cou se, a
any ime, o each s ep only he co esponding o bidding o pe mi ing objec
will be p esen . Also, he e will always exis only one pe mi ing objec in he
sys em.
To indica e ha we wan o pe o m a s ep, we will use sjas p omo e
objec s. When sjappea s in he P sys em, i will ans o m i s co esponding
o bidding objec s−
jin o he pe mi ing one s+
j, hus allowing he ules o s ep
j o be applied. Then he pe mi ing objec will ans o m i sel again in o he
o bidding one, and in o a sui able p omo e objec .
The fi s wo s ages ake ca e o fil e ing he inpu mul ise s: hose which do
no encode a s ing o size ka e ejec ed; hose which do encode such s ings a e
ans o med in such a way ha we ha e a code in base 2 o he symbols in he
cells o he ape o he Tu ing machine. These ope a ions a e pe omed by he
ules in R0.
Fo a mul ise o co ec ly encode a s ing o size k, i has o e i y wo
condi ions: o each j=1,...,k, i has o con ain no mo e han one pai wi h
second componen equal o j; o each j=1,...,k, i has o con ain a leas
one pai wi h second componen equal o j. These wo condi ions a e checked
by ules in R0,1and R0,2, using objec s jas signals o he second componen s
o he pai s.
I he check ails, all objec s in he P sys em a e elimina ed and i sends ou
he objec NO, o ejec he mul ise . I he check passes, hen we double j imes
each pai o he o m a, j, changing hem a he end by a bj.
Fo a de ailed desc ip ion o he simula ion o he unning o he Tu ing
machine o e a co ec mul ise see [6]. We only ecall he e how we ep esen he
Tu ing machine inside he P sys em.
To ep esen he s a es we will use objec s qN,q
Y,q
0,...,q
n.
Du ing he simula ion, objec s b0,...,b
mwill ep esen , in base 2, he cells
which con ain symbols a0,...,a
m, espec i ely (no e ha , a any ime, he num-
be o non-emp y cells in he apes o he Tu ing machine is fini e); objec s
b
0,...,b

m,b

0,...,b

mwill be used as wo king copies o he p e ious objec s; he
single p ime objec s will also be use ul o indica e he symbols ead om he
cells, and objec s c0,...,c
mwill be use ul o indica e he new symbols o w i e
in o hem.
The cell scanned by he head, numbe ed om ze o, will be ep esen ed by
he objec h, in base one; objec hwill be used, when needed, as a wo king copy
o objec h; i will be used as a coun e .
Finally, le us see ha he amily ΠTM is polynomially bounded. Indeed, i
x∈Σk
TM is an inpu s ing o size ko he Tu ing machine, hen we ha e:
1. The check o a mul ise o co ec ly encode a s ing o size kneeds ou
s eps in he affi ma i e case and six s eps in he nega i e one (in his las
case, he P sys em hal s).
2. The gene a ion o he mul ise encoding, in base 2, he non-emp y cells in
he ini ial configu a ion o TM o x equi es k+ 3 s eps o he P sys em.
3. The simula ion o each ansi ion s ep o he Tu ing machine equi es a cos
in he o de o O(5j+12), whe e jis he cell being ead by he head o TM.