Linea Time Solu ion o P ime Fac o iza ion by
Tissue P Sys ems wi h Cell Di ision
Xingyi Zhang1, Yunyun Niu2, Linqiang Pan2, Ma io J. P´e ez-Jim´enez3
1School o Compu e Science and Technology
Anhui Uni e si y, 230039 He ei, China
[email p o ec ed]
2Key Labo a o y o Image P ocessing and In elligen Con ol
Depa men o Con ol Science and Enginee ing
Huazhong Uni e si y o Science and Technology, 430074 Wuhan, China
[email p o ec ed], [email p o ec ed]
3Depa 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
[email p o ec ed]
Summa y. P ime ac o iza ion is use ul and c ucial o public-key c yp og aphy, and i s
applica ion in public-key c yp og aphy is possible only because p ime ac o iza ion has
been p esumed o be di icul . A polynomial- ime algo i hm o p ime ac o iza ion on a
quan um compu e is gi en by P. W. Sho in 1997. In his wo k, a linea - ime solu ion
o p ime ac o iza ion is gi en on a kind o biochemical compu a ional de ices – issue
P sys ems wi h cell di ision, ins ead o physical compu a ional de ices.
1 In oduc ion
In ma h, p ime ac o iza ion is he b eaking down o a composi e numbe in o
smalle p imes, which when mul iplied oge he equal he o iginal in ege . Cu -
en ly, hough he p ime ac o iza ion p oblem is no known o be NP-hand, no
e icien algo i hm is publicly known. I is gene ally conside ed in ac able. The
p esumed compu a ional ha dness o his p oblem is a he hea o se e al algo-
i hms in c yp og aphy such as RSA [15].
Many a eas o ma hema ics and compu e science ha e been b ough o bea on
he p ime ac o iza ion p oblem, including ellip ic cu es, algeb aic numbe heo y,
and quan um compu ing. A polynomial- ime algo i hm o p ime ac o iza ion on
a quan um compu e is gi en by P. W. Sho in 1997 [16]. This will ha e signi ican
implica ions o c yp og aphy i a la ge quan um compu e is e e buil . Howe e ,
be o e a p ac ical quan um compu e appea s, i is s ill o in e es o ind any
easonable compu a ional de ices o sol ing p ime ac o iza ion p oblem. In his
wo k, we shall gi e a linea - ime solu ion o p ime ac o iza ion on a class o
356 X. Zhang e al.
biochemical compu a ional de ices – issue P sys ems wi h cell di ision, ins ead o
physical compu a ional de ices.
Tissue P sys ems wi h cell di ision is a class o compu a ional de ices in mem-
b ane compu ing. Memb ane compu ing is an eme gen b anch o na u al com-
pu ing, which is inspi ed by he s uc u e and he unc ioning o li ing cells, as
well as he o ganiza ion o cells in issues, o gans, and o he highe o de s uc-
u es. The de ices in memb ane compu ing, called P sys ems, p o ide dis ibu ed
pa allel and non-de e minis ic compu ing models. Since Gh. P˘aun in oduced he
i s P sys em in [12], his a ea is hea ily in es iga ed. Please e e o [13] o an
in oduc ion o memb ane compu ing, and e e o [17] o u he bibliog aphy.
In o mally, a P sys em consis s o a memb ane s uc u e, in he compa men s
o which one places mul ise s o objec s which e ol e acco ding o gi en ules in
a synch onous, non-de e minis ic, maximally pa allel manne . Tissue P sys ems
a e a class o P sys ems, whe e memb anes a e placed in he nodes o a g aph.
I is a ne o p ocesso s dealing wi h symbols and communica ing hese symbols
along channels speci ied in ad ance. The communica ion among cells is based on
sympo /an ipo ules, which was in oduced o P sys ems in [11]. Sympo ules
mo e objec s ac oss a memb ane oge he in one di ec ion, whe eas an ipo ules
mo e objec s ac oss a memb ane in opposi e di ec ions. This model has wo bio-
logical inspi a ions (see [9]): in e cellula communica ion and coope a ion be ween
neu ons. In [14], issue P sys ems a e endowed wi h he abili y o ge ing new cells
based on he mi osis o cellula di ision, hus ob aining he abili y o gene a ing
an exponen ial amoun o wo kspace in polynomial ime. Such a ian o issue P
sys ems is called issue P sys ems wi h cell di ision.
Tissue P sys ems wi h cell di ision we e widely in es iga ed o sol ing NP-
comple e p oblems. Some o hem deal wi h non-nume ical NP-comple e decision
p oblems, such as SAT p oblem [14], 3-colo ing p oblem [2], e ex co e [4]. O he s
deal wi h nume ical NP-comple e decision p oblems, ha is, decision p oblems
whose ins ances consis o se s o sequences o in ege numbe s, such as subse
sum [3], pa i ion p oblem [5]. Al hough p ime ac o iza ion we shall conside is a
nume ical p oblem, i is nei he a decision p oblem no an op imiza ion p oblem.
In his wo k, we shall cons uc a amily o issue P sys ems wi h cell di ision,
which can decompose in ege numbe s in a linea ime wi h espec o he leng h
o bina y ep esen a ion o he in ege o be ac o ed. As a esul o compu a ion,
a p ime numbe is sen o a p e ixed ou pu memb ane, ins ead o yes o no.
Up o now, besides he e a e wo polynomial- ime solu ions o p ime ac o -
iza ion by P sys ems wi h ac i e memb anes [6, 10], one well known polynomial
algo i hm ha sol es ac o iza ion p oblem is based on quan um compu e [16]. As
he case o quan um compu e , he solu ion gi en in his wo k indica es how pow-
e ul issue P sys ems wi h cell di ision can be, al hough a his momen nobody
knows how o build a biochemical compu e .
The pape is o ganized as ollows. In Sec ion 2, some p elimina ies a e ecalled.
The o mal de ini ion o issue P sys ems wi h cell di ision is gi en in Sec ion
3. A amily o issue P sys ems ha uni o mly sol e he ac o iza ion p oblem is
Fac o iza ion by Tissue P Sys ems wi h Cell Di ision 357
p esen ed in Sec ion 4, wi h a sho o e iew o he compu a ion and he necessa y
esou ces. Conclusions and commen s a e p esen ed in Sec ion 5.
2 P elimina ies
An alphabe Σis a non-emp y se , whose elemen s a e called symbols. An o de ed
sequence o symbols is a s ing. The numbe o symbols in a s ing uis he leng h
o he s ing, and i is deno ed by |u|. As usual, he emp y s ing (wi h leng h 0)
will be deno ed by λ. The se o s ings o leng h nbuil wi h symbols om he
alphabe Σis deno ed by Σnand Σ∗=∪n≥0Σn. A language o e Σis a subse
om Σ∗.
Amul ise mo e a se Ais a pai (A, ), whe e :A→Nis a mapping. I
m= (A, ) is a mul ise , hen i s suppo is de ined as supp(m) = {x∈A| (x)>
0}and i s size is de ined as Px∈A (x). A mul ise is emp y ( esp. ini e) i i s
suppo is he emp y se ( esp. ini e).
I m= (A, ) is a ini e mul ise o e A, and supp(m) = {a1, . . . , ak}, hen
i will be deno ed as m={{a (a1)
1, . . . , a (ak)
k}}. Tha is, supe sc ip s indica e
he mul iplici y o each elemen . I (x) = 0 o any x∈A, hen his elemen is
omi ed.
3 Tissue P Sys ems wi h Cell Di ision
In [8, 9], he i s de ini ion o he model o issue P sys ems was p oposed, whe e
he memb ane s uc u e did no change along he compu a ion. We now shall
in oduce a model o issue P sys ems wi h cell di ision based on he cell-like
model o P sys ems wi h memb anes di ision [14]. The biological inspi a ion o
his model is clea : ali e issues a e no s a ic ne wo k o cells, since new cells a e
gene a ed by memb ane ission in a na u al way.
The main ea u es o his model, om he compu a ional poin o iew, a e
ha cells a e no pola ized ( he con a y holds in he cell-like model o P sys ems
wi h ac i e memb anes, see [13]); he cells ob ained by di ision ha e he same
labels as he o iginal cell and i a cell is di ided, i s in e ac ion wi h o he cells o
wi h he en i onmen is blocked du ing he di ision p ocess. In some sense, his
means ha while a cell is di iding i closes i s communica ion channels wi h o he
cells and wi h he en i onmen .
Fo mally, a ( unc ion) compu ing issue P sys em wi h cell di ision o deg ee
q≥1 and o de (m, n), m≥1, n ≥1, is a uple o he o m
Π= (Γ, Σ, Λ, w1, . . . , wq,E,R, iin, iou ),
whe e:
1. Γis he alphabe o objec s;
358 X. Zhang e al.
2. Σ={a1, . . . , am}is an o de ed inpu alphabe s ic ly con ained in Γ;
3. Λ={b1, . . . , bn}is an o de ed ou pu alphabe con ained in Γ;
4. w1, . . . , wqa e s ings o e Γ, desc ibing he ini ial mul ise s o objec s placed
in he cells o he sys em a he beginning o he compu a ion;
5. E ⊆ Γis he se o objec s in he en i onmen in a bi a ily copies each;
6. Ris a ini e se o ules o he ollowing o ms:
(a) (i, u/ , j), o i, j ∈ {0,1,2, . . . , q}, i 6=j,u, ∈Γ∗;
Communica ion ules; 1,2,· · · , q iden i y he cells o he sys em, 0 is he
en i onmen ; when applying a ule (i, u/ , j), he objec s o he mul ise
ep esen ed by ua e sen om egion i o egion jand simul aneously he
objec s o he mul ise a e sen om egion j o egion i(|u|+| |is
called he leng h o he communica ion ule (i, u/ , j));
(b) [a]i→[b]i[c]i, whe e i∈ {1,2, . . . , q},a, b, c ∈Γ, and i6=iou ;
Di ision ules; in eac ion wi h an objec a, he cell is di ided in o wo
cells wi h he same label; all he objec s in he o iginal cells a e eplica ed
and copies o hem a e placed in each o he new cells, wi h he excep ion
o he objec a, which is eplaced by he objec bin he i s new cell and
by cin he second one; he ou pu cell canno be di ided;
7. iin ∈ {1,2, . . . , q}is he inpu cell;
8. iou ∈ {0,1,2, . . . , q}is he ou pu cell.
The ules o a sys em as abo e a e used in he non-de e minis ic maximally
pa allel manne . In each s ep, all cells which can e ol e mus e ol e in a maximally
pa allel way (in each s ep we apply a mul ise o ules which is maximal, no u he
ule can be added). This way o applying ules has only one es ic ion when a cell
is di ided, he di ision ule is he only one which is applied o ha cell in ha
s ep; he objec s inside ha cell do no e ol e by means o communica ion ules.
Thei labels p ecisely iden i y he ules which can be applied o hem.
A con igu a ion o issue P sys em wi h cell di ision is desc ibed by all mul i-
se s o objec s o e Γassocia ed wi h all he cells p esen in he sys em and he
mul ise o objec s o e Γ− E associa ed wi h en i onmen . The ini ial con igu a-
ion o he sys em Πwi h inpu w∈Σ∗is he uple (w1, w2, . . . , wiin w, . . . , wq;
∅); ha is, he co esponding con igu a ion a e adding he mul ise w o he
con en o he inpu cell iin. The compu a ion s a s om he ini ial con igu a-
ion and p oceeds as de ined abo e. When he e is no ule can be applied, he
compu a ion s ops. Only hal ing compu a ions gi e a esul . I C={Ci}i< is
a hal ing compu a ion, whe e Cia e con igu a ions, hen he esul o compu-
a ion Ou pu (C)=(C −1
b1(iou ), C −1
b2(iou ), . . . , C −1
bn(iou )), whe e C −1
bj(iou ),
1≤j≤n, is he mul iplici y o objec bjin he egion iou in he hal ing con igu-
a ion C −1.
Fo a unc ion , we deno e he domain o by D( ) and he ange o by
R( ). Fo a issue P sys em wi h cell di ision Πha ing o de ed inpu alphabe
Σ={a1, a2, . . . , am}and o de ed ou pu alphabe Λ={b1, b2, . . . , bn}, and pa ial
unc ion :Nm→Nn, unc ion is encoded in a una y no a ion in he ollowing
Fac o iza ion by Tissue P Sys ems wi h Cell Di ision 359
way: (α1, . . . , αm)∈D( ) is exp essed by aα1
1aα2
2. . . aαm
m; (β1, . . . , βn)∈R( ) is
exp essed by bβ1
1bβ2
2. . . bβn
n.
De ini ion 1. We say ha a pa ial unc ion :Nm→Nnis compu ed in poly-
nomial ime by a amily Π={Π( )| ∈N}o issue P sys ems wi h cell di ision
in una y encoding i he ollowing holds:
•The 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( ) om ∈N.
•The e exis a polynomial- ime compu able unc ion so e he domain D( )o
unc ion such ha :
− o each u= (α1, . . . , αm)∈D( ),s(u)is a na u al numbe and aα1
1. . . aαm
m
is an inpu mul ise o he sys em Π(s(u));
− he amily Πis polynomially bounded wi h ega d o ( , s), ha is, he e
exis s a polynomial unc ion p, such ha o each u= (α1, . . . , αm)∈D( )
e e y compu a ion o Π(s(u)) wi h inpu aα1
1. . . aαm
mis hal ing and, mo e-
o e , i pe o ms a mos p(|u|)s eps;
− he amily Πis sound wi h ega d o ( , s), ha is, o each u= (α1, . . . ,
αm)∈D( ), i he e exis s a compu a ion Co Π(s(u)) wi h inpu
aα1
1. . . aαm
msuch ha Ou pu (C) = (β1, . . . , βn), hen (u) = (β1,···, βn);
− he amily Πis comple e wi h ega d o ( , s), ha is, o each u= (α1, . . . ,
αm)∈D( ), i (u)=(β1,· · · , βn), hen e e y compu a ion Co Π(s(u))
wi h inpu aα1
1. . . aαm
mhas Ou pu (C) = {β1, . . . , βn}.
In he De ini ion 1, he inpu and ou pu a e encoded in una y no a ion. How-
e e , in classical complexi y heo y, based upon Tu ing machine, swi ching om
bina y o una y encoding gene ally co esponds o simpli y he p oblem. In his
wo k, bina y encoding is used o in ege ac o iza ion p oblem. In wha ollows,
we will gi e he de ini ion ha a unc ion is compu ed by a amily o P sys ems
wi h cell di ision in bina y encoding. In he case o bina y encoding, he inpu
alphabe is no asked o be o de ed, and no ou pu alphabe is ixed.
A( unc ion) compu ing issue P sys em wi h cell di ision wi h inpu o deg ee
q≥1 is a uple o he o m
Π= (Γ, Σ, w1, . . . , wq,E,R, iin, iou ),
whe e:
1. Γis he alphabe o objec s;
2. Σis an (un-o de ed) inpu alphabe s ic ly con ained in Γ;
3. w1, . . . , wqa e s ings o e Γ, desc ibing he ini ial mul ise s o objec s placed
in he cells o he sys em a he beginning o he compu a ion;
4. E ⊆ Γis he se o objec s in he en i onmen in a bi a ily copies each;
5. Ris a ini e se o ules o he ollowing o ms:
(a) (i, u/ , j), o i, j ∈ {0,1,2, . . . , q}, i 6=j,u, ∈Γ∗;

360 X. Zhang e al.
(b) [a]i→[b]i[c]i, whe e i∈ {1,2, . . . , q},a, b, c ∈Γ, and i6=iou ;
6. iin ∈ {1,2, . . . , q}is he inpu cell;
7. iou ∈ {0,1,2, . . . , q}is he ou pu cell.
In seman ics, P sys ems ha ing un-o de ed alphabe s is he same wi h P sys-
ems wi h o de ed inpu and ou pu alphabe s excep o he way o encoding
inpu and ou pu . In he una y encoding, he sizes o o de ed inpu and ou pu
alphabe s a e ela ed wi h he dimensions o domain and ange o unc ion ha
is compu ed. Speci ically, an o de ed inpu alphabe {a1, . . . , am}and an o de ed
ou pu alphabe {b1, . . . , bn}can encode each unc ion whose domain ( esp. ange)
is a subse o Nm0(m0≤m) ( esp. Nn0(n0≤n)). In he bina y encoding, he size
o alphabe is ela ed wi h bo h inpu alue and ou pu alue. Fo example, o
he unc ion (x)=22x(x∈N) and an inpu n, he leng h o inpu nin bina y
exp ession is blg nc+ 1, and he leng h o ou pu (n) in bina y exp essions is
2n+ 1, which is an exponen ial unc ion wi h espec o blg nc+ 1. Fo unc ions
such as (x) = 22x, maybe, we need exponen ial (wi h espec o he inpu size)
la ge alphabe o encode he unc ion in P sys ems, hence we canno cons uc
a amily o P sys ems wi h cell di ision in polynomial ime by Tu ing machine
o compu e unc ions such as (x)=22x. I depends on he p ope y o unc ion
whe he a unc ion can be compu ed by issue P sys ems wi h cell di ision in
bina y encoding.
Fo p ime ac o iza ion p oblem, he ac o s a e less han he in ege o be
ac o ed. In ac enables us o ind a easonable bina y encoding o p ime ac o -
iza ion p oblem. Speci ically, we shall use he me hod om [7] o encode bina y
numbe s by mul ise s o objec s. Le xk−1,· · · , x1, x0(wi h k≥1) be he bina y
ep esen a ion o in ege x≥0, ha is, x=Pk−1
i=0 xi2i. We use he objec s om
he ollowing alphabe Ak, o k≥1:
Ak={hb, ji | b∈ {0,1}, j ∈ {1,2,· · · , k}}.
Objec s hb, jiis used o ep esen bi bin posi ion jin he bina y encoding o an
in ege numbe . Hence, o ep esen he abo e numbe xwe will use he ollowing
mul ise (ac ually, a se ) o objec s:
hxk−1, k −1i,· · · ,hx1,1i,hx0,0i.
Le us ema k ha he alphabe Akdepends on he leng h o he bina y ep e-
sen a ion o he numbe x. Mo eo e , i is clea ha wi h Akwe can ep esen all
in ege numbe s in he ange 0,1,· · · ,2k−1. In o de o dis inguish be ween he
objec s ha ep esen he bi s o di e en in ege s Aand B, a leading label A, B
a e used o ma k each elemen in he mul ise . To his aim, he alphabe Akis
modi ied as ollows:
A0
k={hl, b, ji | l∈ {A, B}, b ∈ {0,1}, j ∈ {1,2,· · · , k}}.
In his way, he i- h bi o A( ha is, ai) and he j- h bi o B( ha is, bj) a e
ep esen ed by he objec s hA, ai, iiand hB, bj, ji, espec i ely.
Fac o iza ion by Tissue P Sys ems wi h Cell Di ision 361
In gene al, we gi e he ollowing de ini ion ha a unc ion is compu ed by P
sys ems wi h cell di ision in bina y encoding.
De ini ion 2. We say ha a pa ial unc ion :N→Nis compu ed in polynomial
ime by a amily Π={Π( )| ∈N}o issue P sys ems wi h cell di ision in bina y
encoding i he ollowing holds:
•The 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( ) om ∈N.
•The e exis s a pai (cod, s)o polynomial- ime compu able unc ions o e he
domain D( )o unc ion such ha :
− o each u∈D( ),s(u)is a na u al numbe and cod(u)is an inpu mul ise
o he sys em Π(s(u));
− he amily Πis polynomially bounded wi h ega d o ( , cod, s), ha is,
he e exis s a polynomial unc ion p, such ha o each u∈D( )e e y
compu a ion o Π(s(u)) wi h inpu cod(u)is hal ing and, mo eo e , i pe -
o ms a mos p(|u|)s eps;
− he amily Πis sound wi h ega d o ( , cod, s), ha is, o each u∈D( ),
i he e exis s a compu a ion Co Π(s(u)) wi h inpu cod(u)and he objec s
in egion iou in he las con igu a ion o Cencode (β1,· · · , βq)∈Nq, hen
(u) = (β1,· · · , βq);
− he amily Πis comple e wi h ega d o ( , cod, s), ha is, o each u∈
D( ), i (u)=(β1,· · · , βq)∈Nq, hen in e e y compu a ion o Π(s(u))
wi h inpu cod(u), he objec s in egion iou in he las con igu a ion encode
(β1,· · · , βq).
4 A Linea Time Solu ion o he Fac o iza ion P oblem
When we discuss he p ime ac o iza ion p oblem, i is necessa y o dis inguish wo
di e en e sions o he p oblem: decision p oblem e sion and unc ion p oblem
e sion.
The decision p oblem e sion o p ime ac o iza ion can be o mula ed as “is
na composi e numbe ?” (o equi alen ly: “is na p ime numbe ?”). This e sion
is na u al and use ul because mos well-s udied complexi y classes a e de ined
as classes o decision p oblems, no unc ion p oblems. Bu he decision p oblem
e sion o p ime ac o iza ion is much easie han he p oblem o inding he ac o s
o n. Speci ically, i can be sol ed in polynomial ime (wi h espec o he numbe
o digi s o n) wi h he AKS p imali y es [1].
The unc ion p oblem e sion o p ime ac o iza ion: gi en an in ege n, ind
an in ege dwi h 1 < d < n ha di ides n(o conclude ha nis p ime). I is
i ially in he class FNP, bu we do no known whe he i lies in class FP o no .
This e sion is gene ally conside ed in ac able, which means ha no polynomial-
ime (wi h espec o he ins ance size) algo i hm is known ha sol es i on e e y
362 X. Zhang e al.
ins ance; and i is he e sion sol ed by mos p ac ical implemen a ions. In his
wo k, we shall conside a es ic ed e sion o p ime ac o iza ion p oblem, based
on he ollowing wo ac s. (1) Gi en an algo i hm o in ege ac o iza ion, one
can ac o any in ege down o i s cons i uen p ime ac o s by epea ed applica-
ion o his algo i hm. (2) No all numbe s o a gi en leng h a e equally ha d o
ac o . Semip imes ( he p oduc o wo p ime numbe s) a e belie ed as he ha des
ins ances o in ege ac o iza ion o cu en ly known echniques.
P oblem 1. NAME: ac o iza ion.
– INSTANCE: a posi i e in ege numbe which is he p oduc o wo p ime num-
be s.
– OUTPUT: he p ime ac o ha is no g ea e han ano he one.
Nex , we shall cons uc a amily {Π(k)}k∈No issue P sys ems wi h cell
di ision o ac o in ege s, whe e each sys em Π(k) can decompose all numbe s o
leng h kin bina y o m, p o ided ha an app op ia e inpu mul ise is gi en. The
esolu ion is a b u e o ce algo i hm, which consis s o he ollowing s ages:
•Gene a ion S age: By di ision, all he possible pai s o in ege numbe s o
leng h kin bina y o m a e p oduced (one pai o each memb ane wi h label
2).
•P e-checking S age: In his s age, he p oduc o each pai o in ege numbe s
o leng h kis calcula ed.
•Checking S age: The sys em checks whe he o no he e exis s a pai o in ege
numbe s such ha hei p oduc equals o he numbe n o be composed.
•Ou pu S age: The sys em sends o he ou pu egion a p ime numbe .
Fo each k∈N,
Π(k) = (Γ(k), Σ(k), w1, w2,R(k),E(k), iin, iou ),
wi h he ollowing componen s:
•Γ(k) = Σ(k)∪ {ai, bi,hX, 0, ii, i, gi|0≤i≤k−1}∪
{hA, j, ii,hB, j, ii,hA0, j, ii,hB0, j, ii | 0≤i≤k−1,0≤j≤1}∪
{hAj, l, ii,hBj, l, ii | 0≤i≤k−1,0≤j≤ dlg ke+ 1,0≤l≤1}∪
{ci|0≤i≤4k+dlg 2ke+dlg ke+ 5}∪{c0
i|1≤i≤ dlg ke+ 2k+ 3}∪
{hC, 0, ii,hC, 1, ii | 0≤i≤2k−1} ∪ {hi, ji | 0≤i, j ≤k−1}∪
{di| −1≤i≤k−2}∪{ei| −1≤i≤k−1}∪{z}.
•Σ(k) = {hn, 0, ii,hn, 1, ii | 0≤i≤k−1}.
•w1={{c0}}.
•w2={{a0a1· · · ak−1b0b1···bk−1z}} ∪ {{hi, ji | 0≤i, j ≤k−1}}.
• R(k) is he se o ules:
1. Di ision ule:
1,i ≡[ai]2→[hA, 0, ii]2[hA, 1, ii]2, o 0 ≤i≤k−1;
2,i ≡[bi]2→[hB, 0, ii]2[hB, 1, ii]2, o 0 ≤i≤k−1.
Fac o iza ion by Tissue P Sys ems wi h Cell Di ision 363
2. Communica ion ules:
3,i ≡(1, ci/c2
i+1,0), o 0 ≤i≤2k−1;
4≡(1, c2k/z, 2);
5,i ≡(2, c2k+i/c2
2k+i+1,0), o 0 ≤i≤ dlg 2ke − 1;
6,i,j ≡(2, c2k+dlg 2kehA, j, ii/c2k+dlg 2ke+1hA0, j, ii,0),
o 0 ≤i≤k−1, 0 ≤j≤1;
7,i,j ≡(2, c2k+dlg 2kehB, j, ii/c2k+dlg 2ke+1hB0, j, ii,0),
o 0 ≤i≤k−1, 0 ≤j≤1;
8≡(2, c2k+dlg 2ke+1/c0
1c2k+dlg 2ke+2,0);
9,i ≡(2, c2k+dlg 2ke+i/c2k+dlg 2ke+i+1,0), o 2 ≤i≤ dlg ke+ 2k+ 4;
10,i ≡(2, c0
i/c0
i+1,0), o 1 ≤i≤ dlg ke+ 2k+ 2;
11,i,j,l ≡(2,hAj, l, ii/hAj+1, l, ii2,0),
o 0 ≤i≤k−1, 0 ≤j≤ dlg ke, 0 ≤l≤1;
12,i,j,l ≡(2,hBj, l, ii/hBj+1, l, ii2,0),
o 0 ≤i≤k−1, 0 ≤j≤ dlg ke, 0 ≤l≤1;
13,i,j ≡(2,hAdlg ke+1,0, iihBdlg ke+1,0, jihi, ji/hC, 0, i +ji,0),
o 0 ≤i, j ≤k−1;
14,i,j ≡(2,hAdlg ke+1,0, iihBdlg ke+1,1, jihi, ji/hC, 0, i +ji,0),
o 0 ≤i, j ≤k−1;
15,i,j ≡(2,hAdlg ke+1,1, iihBdlg ke+1,0, jihi, ji/hC, 0, i +ji,0),
o 0 ≤i, j ≤k−1;
16,i,j ≡(2,hAdlg ke+1,1, iihBdlg ke+1,1, jihi, ji/hC, 1, i +ji,0),
o 0 ≤i, j ≤k−1;
17,i ≡(2,hC, 0, iihC, 0, ii/hC, 0, ii,0), o 0 ≤i≤2k−2;
18,i ≡(2,hC, 0, iihC, 1, ii/hC, 1, ii,0), o 0 ≤i≤2k−2;
19,i ≡(2,hC, 1, iihC, 1, ii/hC, 0, iihC, 1, i + 1i,0), o 0 ≤i≤2k−2;
20,i,j ≡(2, c0
dlg ke+2k+3hC, 1, iihn, j, k −1i/λ, 0),
o k≤i≤2k−2, 0 ≤j≤1;
21,i,j ≡(2, c4k+dlg 2ke+dlg ke+5hC, j, iihn, j, ii/hX, 0, ii,0),
o 0 ≤i≤k−1, 0 ≤j≤1;
22 ≡(2,hX, 0, k −1i/dk−2,0);
23,i ≡(2, dihX, 0, ii/di−1,0), o 0 ≤i≤k−2;
24 ≡(2, d−1/ek−1,0);
25,i,j ≡(2,hAdlg ke+1, j, iihBdlg ke+1, j, iiei/hAdlg ke+1, j, ii
hBdlg ke+1, j, iiei−1,0), o 0 ≤i≤k−1, 0 ≤j≤1;
26 ≡(2, e−1/ 0,0);
27,i ≡(2,hAdlg ke+1,1, iihBdlg ke+1,0, iiei/hAdlg ke+1,1, ii
hBdlg ke+1,0, ii 0,0), o 0 ≤i≤k−1;
28,i,j ≡(2, ihBdlg ke+1, j, ii/ i+1hB0, j, ii,0), o 0 ≤i≤k−2, 0 ≤j≤1;
29,j ≡(2, k−1hBdlg ke+1, j, k −1i/hB0, j, k −1i,0), o 0 ≤j≤1;
30,i,j ≡(2,hB0, j, ii/λ, 3), o 0 ≤i≤k−1, 0 ≤j≤1;
31,i ≡(2,hAdlg ke+1,0, iihBdlg ke+1,1, iiei/hAdlg ke+1,0, ii
hBdlg ke+1,1, iig0,0), o 0 ≤i≤k−1;
32,i,j ≡(2, gihAdlg ke+1, j, ii/gi+1hA0, j, ii,0), o 0 ≤i≤k−2, 0 ≤j≤1;
370 X. Zhang e al.
1
z
16
2
c
16
3
〈
C ,
0,1
〉
〈
C ,
0,2
〉
〈
n ,
1, 1
〉
〈
X ,
0, 0
〉
〈
A
2,
0,0
〉
2
2
c
16
3
〈
C ,
0,1
〉
〈
C , 0,2
〉
〈
n ,
1, 1
〉
〈
X ,
0,0
〉
〈
A
2,
1, 0
〉
2
〈
A
2,
0,1
〉
2
〈
A
2,
0,1
〉
2
〈
B
2,
0,0
〉
2
〈
B
2,
0,1
〉
2
〈
B
2,
0,0
〉
2
〈
B
2,
0,1
〉
2
2
c
16
3
〈
C ,
0,1
〉〈
C , 0,2
〉
〈
n, 1, 1
〉
〈
X ,
0,0
〉
〈
A
2,
0,0
〉
2
2
c
16
3
〈
C ,
0,1
〉
〈
C , 0,2
〉
〈
n, 1, 1
〉
〈
X ,
0,0
〉
〈
A
2,
1,1
〉
2
〈
A
2,
1,1
〉
2
〈
A
2,
1, 0
〉
2
〈
B
2,
0,0
〉
2
〈
B
2,
0,1
〉
2
〈
B
2,
0,0
〉
2
〈
B
2,
0,1
〉
2
2
c
16
3
〈
C ,
0,1
〉
〈
C , 0,2
〉
〈
n ,
1, 1
〉
〈
X ,
0, 0
〉
〈
A
2,
0,0
〉
2
2
c
16
4
〈
C ,
1,0
〉
〈
C ,
0,1
〉
〈
C , 0,2
〉
〈
n ,
1, 1
〉
〈
n,
0,0
〉
〈
A
2,
0,1
〉
2
〈
A
2,
0,1
〉
2
〈
A
2,
1, 0
〉
2
〈
B
2,
0,1
〉
2
〈
B
2,
1, 0
〉
2
〈
B
2,
0,1
〉
2
〈
B
2,
1, 0
〉
2
2
c
16
2
〈
C , 0,2
〉
〈
X ,
0, 1
〉〈
X ,
0,0
〉
〈
A
2,
1, 1
〉
2
2
c
16
3
〈
C ,0,2
〉
〈
X ,
0, 1
〉
〈
A
2,
1,1
〉
2
〈
A
2,
0,0
〉
2
〈
A
2,
1, 0
〉
2
〈
B
2,
0,1
〉
2
〈
B
2,
1, 0
〉
2
〈
B
2,
0,1
〉
2
〈
B
2,
1, 0
〉
2
2
c
16
3
〈
C ,
0,2
〉
〈
X ,
0, 0
〉
〈
A
2,
0,1
〉
2
2
c
16
2
〈
C , 0,2
〉
〈
X ,
0, 1
〉
〈
X ,
0, 0
〉
〈
A
2,
0,1
〉
2
〈
A
2,
0,0
〉
2
〈
A
2,
1, 0
〉
2
〈
B
2,
1,1
〉
2
〈
B
2,
0,0
〉
2
〈
B
2,
1, 1
〉
2
〈
B
2,
0,0
〉
2
2
c
16
3
〈
C ,
0,1
〉
〈
X ,
0,0
〉
〈
A
2,
1, 1
〉
2
2
c
16
3
〈
C ,
1,1
〉
〈
X ,
0, 0
〉
〈
A
2,
1,1
〉
2
〈
A
2,
0,0
〉
2
〈
A
2,
1, 0
〉
2
〈
B
2,
1,1
〉
2
〈
B
2,
0,0
〉
2
〈
B
2,
1, 1
〉
2
〈
B
2,
0,0
〉
2
2
c
16
3
〈
C ,
0,1
〉
〈
C , 0,2
〉
〈
n ,
1, 1
〉
〈
X ,
0,0
〉
〈
A
2,
0,1
〉
2
2
c
16
3
〈
C ,
1,0
〉
〈
C , 0,2
〉
〈
X ,
0, 1
〉
〈
n,
0,0
〉
〈
A
2,
0,1
〉
2
〈
A
2,
0,0
〉
2
〈
A
2,
1, 0
〉
2
〈
B
2,
1, 1
〉
2
〈
B
2,
1, 0
〉
2
〈
B
2,
1, 1
〉
2
〈
B
2,
1, 0
〉
2
2
c
16
3
〈
C ,
1,1
〉
〈
X ,
0,0
〉
〈
A
2,
1, 1
〉
2
2
c
16
4
〈
C ,
1,0
〉
〈
C ,
0,1
〉
〈
C ,
0,2
〉
〈
n ,
0, 0
〉
〈
A
2,
1,1
〉
2
〈
A
2,
0,0
〉
2
〈
A
2,
1, 0
〉
2
〈
B
2,
1, 1
〉
2
〈
B
2,
1, 0
〉
2
〈
B
2,
1, 1
〉
2
〈
B
2,
1, 0
〉
2
c
'
8
c
'
8
c
'
8
c
'
8
c
'
8
c
'
8
c
'
8
c
'
8
c
'
8
c
'
8
c
'
8
〈
C ,1,0
〉
〈
n,
0,0
〉
〈
C ,
0,1
〉
〈
n ,
1, 1
〉
c
'
8
Fig. 5. The con igu a ion o sys em Π(2) o ac o ing in ege numbe 2 a s ep 19
5 Conclusions and Commen s
P ime ac o iza ion p oblem is no in i sel widely use ul p oblem. I has become
use ul only because i has been ound o be c ucial o public-key c yp og aphy,
and his applica ion is in u n possible only because hey ha e been p esumed o be
di icul . Cu en ly, no de e minis ic polynomial- ime algo i hm is known, which
can be execu ed on Tu ing machines, ha sol es he p oblem o e e y possible
ins ance. I is o in e es o explo e any possible and easonable way o sol e p ime
ac o iza ion p oblem because o i s impo ance in public-key c yp og aphy.
P ime ac o iza ion p oblem is nei he decision p oblem no op imiza ion p ob-
lem. In his wo k, i is conside ed as a unc ion p oblem, and in he amewo k
o issue P sys ems wi h cell di ision, a linea - ime solu ion o p ime ac o iza ion
p oblem is gi en. The ini ial s uc u e o he sys ems is e y simple, which consis s
o h ee cells. The sys em is ini ialized wi h inpu ing in o he ixed inpu cell he
mul ise ha exp esses he in ege numbe n o be ac o ed. A e a linea ime
wi h espec o he size o n(i. e., blg kc+ 1), we can ead ou one ac o o nin
he ou pu cell.

Fac o iza ion by Tissue P Sys ems wi h Cell Di ision 371
1
z
16
2
c
16
3
〈
C ,
0,1
〉
〈
C ,
0,2
〉
〈
n,
1, 1
〉
〈
X ,
0, 0
〉
〈
A
2,
0,0
〉
2
2
c
16
3
〈
C ,
0,1
〉
〈
C , 0,2
〉
〈
n,
1, 1
〉
〈
X ,
0,0
〉
〈
A
2,
1, 0
〉
2
〈
A
2,
0,1
〉
2
〈
A
2,
0,1
〉
2
〈
B
2,
0,0
〉
2
〈
B
2,
0,1
〉
2
〈
B
2,
0,0
〉
2
〈
B
2,
0,1
〉
2
2
c
16
3
〈
C ,
0,1
〉
〈
C , 0,2
〉
〈
n, 1, 1
〉
〈
X ,
0,0
〉
〈
A
2,
0,0
〉
2
2
c
16
3
〈
C ,
0,1
〉
〈
C , 0,2
〉
〈
n, 1, 1
〉
〈
X ,
0,0
〉
〈
A
2,
1,1
〉
2
〈
A
2,
1,1
〉
2
〈
A
2,
1, 0
〉
2
〈
B
2,
0,0
〉
2
〈
B
2,
0,1
〉
2
〈
B
2,
0,0
〉
2
〈
B
2,
0,1
〉
2
2
c
16
3
〈
C ,
0,1
〉
〈
C , 0,2
〉
〈
n,
1, 1
〉
〈
X ,
0, 0
〉
〈
A
2,
0,0
〉
2
2
c
16
4
〈
C ,
1,0
〉
〈
C ,
0,1
〉
〈
C , 0,2
〉
〈
n,
1, 1
〉
〈
n,
0,0
〉
〈
A
2,
0,1
〉
2
〈
A
2,
0,1
〉
2
〈
A
2,
1, 0
〉
2
〈
B
2,
0,1
〉
2
〈
B
2,
1, 0
〉
2
〈
B
2,
0,1
〉
2
〈
B
2,
1, 0
〉
2
2
c
16
2
〈
C , 0,2
〉
〈
A
2,
1, 1
〉
2
2
c
16
3
〈
C ,0,2
〉
〈
A
2,
1,1
〉
2
〈
A
2,
0,0
〉
2
〈
A
2,
1, 0
〉
2
〈
B
2,
0,1
〉〈
B
2,
1, 0
〉〈
B
2,
0,1
〉
2
〈
B
2,
1, 0
〉
2
2
c
16
3
〈
C ,
0,1
〉
〈
X ,
0,0
〉
〈
A
2,
0,1
〉
2
2
c
16
2
〈
C , 0,2
〉
〈
A
2,
0,1
〉
〈
A
2,
0,0
〉
2
〈
A
2,
1, 0
〉
〈
B
2,
1,1
〉
2
〈
B
2,
0,0
〉
2
〈
B
2,
1, 1
〉
2
〈
B
2,
0,0
〉
2
2
c
16
3
〈
C ,
0,1
〉
〈
X ,
0, 0
〉
〈
A
2,
1, 1
〉
2
2
c
16
3
〈
C ,
1,1
〉
〈
X ,
0, 0
〉
〈
A
2,
1,1
〉
2
〈
A
2,
0,0
〉
2
〈
A
2,
1, 0
〉
2
〈
B
2,
1,1
〉
2
〈
B
2,
0,0
〉
2
〈
B
2,
1, 1
〉
2
〈
B
2,
0,0
〉
2
2
c
16
3
〈
C ,
0,1
〉
〈
C , 0,2
〉
〈
n,
1, 1
〉
〈
X ,
0,0
〉
〈
A
2,
0,1
〉
2
2
c
16
3
〈
C ,
1,0
〉
〈
C , 0,2
〉
〈
n,
0,0
〉
〈
A
2,
0,1
〉
2
〈
A
2,
0,0
〉
2
〈
A
2,
1, 0
〉
2
〈
B
2,
1, 1
〉
2
〈
B
2,
1, 0
〉
2
〈
B
2,
1, 1
〉
2
〈
B
2,
1, 0
〉
2
2
c
16
3
〈
C ,
1,1
〉
〈
X ,
0,0
〉
〈
A
2,
1, 1
〉
2
2
c
16
4
〈
C ,
1,0
〉
〈
C ,
0,1
〉
〈
C ,
0,2
〉
〈
n ,
0, 0
〉
〈
A
2,
1,1
〉
2
〈
A
2,
0,0
〉
2
〈
A
2,
1, 0
〉
2
〈
B
2,
1, 1
〉
2
〈
B
2,
1, 0
〉
2
〈
B
2,
1, 1
〉
2
〈
B
2,
1, 0
〉
2
c
'
8
c
'
8
c
'
8
c
'
8
c
'
8
c
'
8
c
'
8
c
'
8
c
'
8
c
'
8
c
'
8
d
0
d
0
〈
C ,1,0
〉
〈
n,
0,0
〉
〈
B' ,
0,1
〉 〈
B' ,
1, 0
〉
〈
A' ,
0,1
〉〈
A' ,
1, 0
〉
〈
C , 0,2
〉
〈
n,
1, 1
〉
c
'
8
Fig. 6. The con igu a ion o sys em Π(2) o ac o ing in ege numbe 2 a s ep 25
P sys em is a highly dis ibu ed pa allel model o compu a ion. Cu en ly,
nobody knows how o build a biochemical compu e /an a i icial issue-like com-
pu e . P sys ems may be implemen ed using molecules, cells o a la ge compu e
ne wo k such as he In e ne . Al hough i goes beyond he scope o his wo k o
discuss he implemen a ion o P sys ems, clea ly, i is o pa icula in e es and i
is a big challenging opic.
Acknowledgemen s
The wo k was suppo ed by Na ional Na u al Science Founda ion o China
(61033003, 61003038 and 30870826), Ph.D. P og ams Founda ion o Minis y o
Educa ion o China (20100142110072), Fundamen al Resea ch Funds o he Cen-
al Uni e si ies (2010ZD001), and Na u al Science Founda ion o Hubei P o ince
(2008CDB113 and 2008CDB180). Ma io J. P´e ez-Jim´enez also acknowledges he
suppo o he p ojec TIN2009-13192 o he Minis e io de Ciencia e Inno aci´on o
Spain, co inanced by FEDER unds, and he “P oyec o de Excelencia con In es i-
gado de Reconocida Val´ıa” o he Jun a de Andaluc´ıa unde g an P08-TIC04200.
372 X. Zhang e al.
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