P Sys ems-Based Compu ing Polynomials
Wi h In ege Coe icien s: Design and
Fo mal Ve i ica ion
Ming Zhu, Gexiang Zhang ,
Membe ,
IEEE
, Qiang Yang, Haina Rong, Wei ao Yuan,
and Ma io J. Pé ez-Jiménez
Abs ac
—Au oma ic design o mechanical p ocedu es
sol ing abs ac p oblems is a ele an scien i ic challenge.
In pa icula , au oma ic design o memb anes sys ems pe -
o ming some p e ixed asks is an impo an and use ul
esea ch opic in he a ea o Na u al Compu ing. In his
con ex , de e minis ic memb ane sys ems we e designed in
o de o cap u e he alues o polynomials wi h na u al num-
be s coe icien s. Following ha wo k, his pape ex ends
he p e ious esul o polynomials wi h in ege numbe s
coe icien s.Speci ically,a de e minis ic ansi ionP sys em
using p io i ies in he weak in e p e a ion, associa ed wi h
an a bi a y such kind polynomial, is p esen ed. The con-
igu a ion o he unique compu a ion o he sys em will be
encoded by means o wo dis inguished objec s, he alues
o he polynomial o na u al numbe s. The desc ip i e com-
pu a ional esou ces equi ed by he designed memb ane
sys em a e also analyzed.
Index
Te ms
—Memb ane compu ing, P sys ems, au o-
ma ic design o memb ane sys ems, polynomials wi h
in ege coe icien s.
I. INTRODUCTION
MEMBRANE compu ing is a apidly g owingb anch
o na u al compu ing ini ia ed in [1], which abs ac s
compu ing models om he a chi ec u e and he unc ioning
o li ing cells, as well as om he o ganiza ion o cells in
issues, o gans ( he b ain included), o o he highe -deg ee
s uc u es. In he pas wen y yea s, se e al classes o com-
pu ing models (called P sys ems) we e in oduced, inspi ed
This wo k was suppo ed in pa by he Scien i ic Resea ch Fund o
Sichuan P o incial Science and Technology Depa men unde G an s
2015JY0257 and 2017FZ0010, in pa by he Na ional Na -u al Science
Founda ion o China unde G an s 61672437, 61702428, and
61373047, in pa by he Sichuan Science and Technology P o-g am
unde G an s 2018GZ0185, 2018GZ0085, and 2017GZ0159, and in
pa by he Fundamen al Resea ch Funds o he Cen al Uni e si ies
unde G an A0920502051619-36.
(Co esponding
au ho :
Gexiang
Zhang.)
M. Zhu and Q. Yang a e wi h he College o Con ol Enginee ing,
Chengdu Uni e si y o In o ma ion Technology, Chengdu 610225, China
(e-mail: [email p o ec ed]; [email p o ec ed]).
G. Zhang, H. Rong, and W. Yuan a e wi h he School o Elec ical Engi-
nee ing, Sou hwes Jiao ong Uni e si y, Chengdu 610031, China (e-mail:
[email p o ec ed]; [email p o ec ed]; [email p o ec ed]).
M. J. Pé ez-Jiménez is wi h he Depa men o Compu e Science and
A i icial In elligence, Uni e si y o Se illa, 41012 Se illa, Spain (e-mail:
[email p o ec ed]).
Fig. 1. Schema ic g aph showing he oolbox o a i icial neu al ne wo ks.
Fig. 2. Schema ic g aph showing he aim o au oma ic design o
P sys ems.
om biological ac s o mo i a ed om ma hema ical o com-
pu e science poin s o iew [2], [3]. Many P sys em classes
a eable osimula e egis e machines and he e o e hey
a e compu a ionally comple e, ha is, hey a e equi alen in
powe o Tu ing machines [4]–[9]. I is well known ha
some P sys ems a e e icien , in he sense ha hey ha e
he abili y o sol e compu a ionally ha d p oblems by making
use o an exponen ial wo kspace c ea ed in a na u al way,
in polynomial ime [10]–[12]. Memb ane compu ing models
ha e been used in a ious applica ions like in he a eas o
app oxima e op imiza ions, sys ems and syn he ic biology and
eal-li e complex p oblems [13]–[19].
Like he oolbox o a i icial neu al ne wo ks (ANN) o
p oducing success ul ANNs sa is ying use s’ equi emen s,
which is shown in Fig. 1, he au oma ic design o P sys ems is
o de elop a me hodology o gene a ing success ul P sys ems
mee ing designe s’ equi emen s, as shown in Fig. 2.
This is a e y complica ed and challenging ask. So a ,
he me hods epo ed in he li e a u e can be classi ied in o wo
g oups: heu is ic and easoning echniques [20]. The i s ype
o me hods ocused on he use o heu is ic algo i hms, such
as gene ic algo hms (GAs) and quan um-inspi ed e olu iona y
algo i hm (QIEAs), o make a popula ion o P sys ems e ol e
owa d a success ul one [15]. This kind o me hods began
om he selec ion o an app op ia e subse om a edundan
se o e olu ion ules o design a cell-like P sys em, whe e a
memb ane s uc u e and ini ial objec s we e p e-de ined and
ixed in he p ocess o design [15], [21]–[24]. In [21], a gene ic
algo i hm was employed o design a P sys em o calcula e 42.
In [22], a bina y encoding echnique was p esen ed o deno e
an e olu ion ule se o a P sys em and a QIEA was used
o make a popula ion o P sys ems e ol e owa d success ul
ones. This me hod success ully sol ed he design o P sys-
ems o compu e 42and n2( o na u al numbe s n≥2).
In [23], an e alua ion app oach conside ing non-de e minism
and hal ing penal y ac o s and a gene ic algo i hm wi h
he bina y encoding echnique in [22] we e in oduced o
design P sys ems o 42,n2and he gene a ion o he lan-
guage {a2nb3n|n>1}. In hese s udies men ioned abo e,
a speci ic edundan e olu ion ule se was designed o a
speci ic compu a ional ask. This was de eloped in [15], [24]
by applying one p e-de ined edundan e olu ion ule se o
design mul iple di e en P sys ems, each o which execu es
a compu a ion ask. In [24], an au oma ic design me hod o a
cell-like P sys em amewo k o pe o ming i e basic a i h-
me ic ope a ions (addi ion, sub ac ion, mul iplica ion, di ision
and powe ) was p esen ed. In [15], a common edundan se
o e olu ion ules was applied o design success ul P sys ems
o ul illing eigh compu a ional asks, i.e., eigh compu ing
se s o na u al numbe s: 2(n−1),2n−1, n2,1
2[n(n−1)],
n(n−1),(n−1)2+2n+2, a2nb3nand 1
2(3n−1),(n>1o 2).
A signi ican de elopmen in his opic is he wo k in [25]
in which a cell-like hal ing P sys em o 42was designed
by uning memb ane s uc u es, ini ial objec s and e olu ion
ules. In ha wo k, a gene ic algo i hm wi h a bina y encod-
ing echnique was discussed o codi y he h ee ing edien s
o a P sys em, he memb ane s uc u e, ini ial objec s and
e olu ion ules. Following his wo k, an au oma ic design
me hod, Pe mu a ion Penal y Gene ic Algo i hm (PPGA), o
a de e minis ic and non-hal ing memb ane sys em by uning
memb ane s uc u es, ini ial objec s and e olu ion ules was
p oposed in [26]. The main ideas o PPGA a e he in oduc ion
o he pe mu a ion encoding echnique o a memb ane sys em,
a penal y unc ion e alua ion app oach o a candida e mem-
b ane sys em and a gene ic algo i hm o making a popula ion
o P sys ems e ol e owa d a success ul one ul illing a
gi en compu a ional ask. A cell-like memb ane sys em o
compu ing he squa e o n2( o na u al numbe s n≥1) was
success ully designed. In addi ion, he au oma ic design o he
minimal memb ane sys ems wi h espec o hei memb ane
s uc u es, alphabe , ini ial objec s and e olu ion ules o ul ill
he gi en ask we e also discussed in [26]. The second ype
o me hods use easoning echniques o ul ill he design
o a P sys em. In [27], a easoning me hod o design a
k-deg ee (k ≥ 2) polynomial P sys em was epo ed by ana-
lyzing he syn ax and seman ics o cell-like P sys ems.
In he s udy o [27], de e minis ic ansi ion P sys ems o
compu ing polynomials wi h na u al numbe coe icien s we e
designed. The nume ical alues o such polynomials, p(n),
o n ∈ N, a e always posi i e and he P sys ems compu -
ing p(n) handle only posi i e numbe s h ough he mul iplic-
i y o objec s in an usual manne . In his pape , he wo k
in [27] is ex ended o conside he design o de e minis ic
ansi ion P sys ems o compu ing polynomials wi h in ege
coe icien s, whe e he nume ical alues o p(n), o n ∈ N,
may be posi i e o nega i e and, consequen ly, he P sys ems
compu ing p(n) mus p ocess in ege numbe s by using na u al
numbe s in he mul iplici y o objec s. This ask is much mo e
challenging.
The aim o his pape is o ind a “minimal” such P sys em
compu ing an a bi a y polynomial wi h in ege coe icien s.
He e he concep “minimal” e e s o some syn ac ical ing e-
dien s associa ed wi h P sys ems: he memb ane s uc u e has
only one memb ane and he numbe o objec s used is e y
es ic i e.
The es pa s o his pape a e o ganized as ollows.
Sec ion II ecalls some p elimina ies needed in he ollowing
sec ions, including he speci ic a ian o memb ane sys ems
conside ed in his wo k. The main concep o polynomial
wi h in ege coe icien s compu ed by a de e minis ic an-
si ion P sys em is de ined in Sec ion III. The design and
o mal e i ica ion o a de e minis ic P sys em associa ed wi h
an a bi a y polynomial whose coe icien s a e in ege num-
be s, is p esen ed in Sec ion III-A. The desc ip i e compu a-
ional esou ces equi ed by he designed k-deg ee polynomial
P sys em is analyzed in Sec ion IV. The compa ison wi h
me aheu is ic app oaches is discussed in Sec ion V. Finally,
conclusions and u u e wo k a e gi en in Sec ion VI.
II. PRELIMINARIES
In his sec ion, some gene al concep s a e b ie ly desc ibed
in o de o make he wo k sel -con ained.
A.
Alphabe
and
Mul ise s
An alphabe is a non-emp y se and hei elemen s a e
called symbols.As ing u o e is an o de ed ini e sequence
o symbols, ha is, a mapping om a na u al numbe n∈N
on o . The numbe nis called he leng h o he s ing u
and i is deno ed by |u|. The emp y s ing (wi h leng h 0) is
deno ed by λ.Amul ise o e an alphabe is a mapping
om on o he se o na u al numbe s N. Fo each symbol
a∈, he na u al numbe (a)is called he mul iplici y o
symbol ain mul ise . We deno e by M() he se o all
mul ise s o e .
B.
Roo ed
T ee
An undi ec ed g aph G is an o de ed pai (V,E),whe eV
is a se whose elemen s a e called nodes and E={{x,y}|
x,y∈V,x= y}whose elemen s a e called edges.Apa h o
leng h k≥1 om x∈V o y∈Vis a sequence (x0,...,xk)
such ha x0=xand xk=y.I x0=xk hen we say ha he
pa h is a cycle. An undi ec ed g aph is connec ed i e e y pai
o nodes is connec ed by a pa h. An undi ec ed g aph wi h
no cycle is said o be acyclic.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.
C.
T ansi ion
P
Sys ems
The basic model o memb ane sys ems was in oduced
by Gh. P˘aun in i s seminal pape [1]. A ansi ion P sys em
o deg ee q≥1 is a uple
=(, μ, M1,...,Mq,(R1,ρ
1),...,(Rq,ρ
q), iou ),
whe e:
–is a ini e alphabe .
–μis a oo ed ee.
–M1,...,Mqa e mul ise s o e .
–Ri,1≤i≤q, is a ini e se o e olu ion ules o
he ollowing o ms: (a) [u]i→ 1[ 2[ 3]j]i;and
(b) [u]i→ 1[ 2[ 3]j]iδ,whe ei,j∈{1,...,q},
i= j,u, 1, 2, 3∈M() and δis a dis inguished
symbol such ha δ/∈.
–ρi,1≤i≤q, is an s ic pa ial o de o e Ri.
–iou ∈{0,1,...,q}.
A ansi ion P sys em =(, μ, M1,...,Mq,
(R1,ρ
1),...,(Rq,ρ
q), iou ),o deg eeq≥1 can be iewed
as a se o qmemb anes injec i ely labeled by 1,...,q,
a anged in a hie a chical s uc u e μgi en by a oo ed ee
whose oo is called he skin memb ane o he sys em, and
wi h an en i onmen labeled by 0 such ha : (a) M1,...,Mq
a e mul ise s o e he wo king alphabe  ep esen ing he
objec s ini ially placed in he qmemb anes o he sys em;
(b) Ri,1≤i≤n, is he se o ules associa ed wi h
memb ane i,andρip o ides p io i ies be ween ules in Ri,
in such a manne ha i ( 1, 2)∈ρiwe say ha ule 1has
a highe p io i y han 2and we deno e i by 1> 2;and
(c) iou ∈{1,...,q} ep esen s a dis inguished memb ane
( he ou pu memb ane).
Acon igu a ion a an ins an o a ansi ion P sys em
is desc ibed by he memb ane s uc u e a ins an and all
mul ise s o objec s o e associa ed wi h all he memb anes
p esen in he sys em. The ini ial con igu a ion o he sys em
is (μ, M1,··· ,Mq). Gi en a ansi ion P sys em ,wesay
ha con igu a ion C yields con igu a ion C +1in one ansi ion
s ep, i we can pass om C o C +1by applying he ules om
R1,...,Rqsynch onously, in a non-de e minis ic maximally
pa allel manne . This means he ollowing: he objec s o
e ol e in a ansi ion s ep and he ules by which hey e ol e
a e chosen in a non-de e minis ic manne , bu in such a
way ha in each memb ane we ha e a maximally pa allel
applica ion o ules (a each ansi ion s ep a mul ise o ules
which is maximal is applied, no u he applicable ule can be
added). A compu a ion o is a ( ini e o in ini e) sequence
o con igu a ions such ha : (a) he i s e m o he sequence is
he ini ial con igu a ion o he sys em; (b) each non- i s e m
o he sequence is ob ained om he p e ious con igu a ion by
applying ules o he sys em in a non-de e minis ic maximally
pa allel manne ; and (c) i he sequence is ini e hen he las
e m o he sequence is a con igu a ion, whe e no ule o he
sys em is applicable o i .
I is wo h poin ing ou ha in his pape he p io i y
be ween ules is used in he weak in e p e a ion, ha is,ina
ansi ion s ep a ule is used always when objec s exis , which
we e no used by a ule o a highe p io i y. In his pape
we deal wi h de e minis ic ansi ion P sys ems, whe e he e
is only one compu a ion s a ing om an ini ial con igu a ion.
Besides, only ules o he ype [u]i→[ ]iwill be used and
hey a e b ie ly deno ed by u→ when he memb ane being
wo ked wi h is unde s ood.
Le us conside wo auxilia y unc ions +and − om he
se o in ege numbe s Zin o he se o na u al numbe s N,
de ined as ollows:
+(x)=xi x≥0
0i x<0 −(x)=0i x≥0
−xi x<0
I is wo h poin ing ou ha o each in ege numbe x ∈ Z
we ha e +(x) ≥ 0, −(x) ≥ 0, +(x) + −(x) =|x| and
+(x) − −(x) = x.
III. DETERMINISTIC TRANSITION PSYSTEMS
COMPUTING POLYNOMIALS WITH
INTEGER COEFFICIENTS
In his sec ion we de ine he meaning o compu ing a
polynomial p(n) whose coe icien s a e in ege numbe s, by a
de e minis ic ansi ion P sys em p(n) associa ed wi h i . The
idea is he ollowing: o each na u al numbe ∈ N he
alue p( ) will be compu ed/encoded by he con igu a ion
C +1 o he unique compu a ion o p(n). Fo ha , ou
dis inguished objec s (o1, o2, p1, p2) will be conside ed in he
wo king alphabe o p(n), in such a manne ha o1, o2 will
be used o encode/ ep esen in ege numbe s by means o hei
mul iplici ies, and p1, p2 will be used as hei co esponding
ansi ion compu ing objec s.
De ini ion 1: Le p(n) be a polynomial wi h in ege numbe s
coe icien s. We say ha p(n) is compu ed by a de e minis ic
ansi ion P sys em
p(n)=(, μ, M1,...,Mq,(R1,ρ
1),...,(Rq,ρ
q), iou )
i he ollowing holds:
•The wo king alphabe has ou dis inguished objec s:
o1,o2( he ou pu objec s) and p1,p2( ansi ion compu -
ing objec s).
•Fo each ∈N, a con igu a ion C +1 he con en o he
ou pu memb ane labeled by iou encodes he alue p( )
h ough he mul iplici y o objec s o1and o2as ollows:
(a) I p( )≥0 hen he mul iplici y o o1is p( )and
he mul iplici y o o2is 0; and (b) i p( )<0 hen he
mul iplici y o o2is −p( )and he mul iplici y o o1is 0.
A.
Design
In his sec ion, a de e minis ic ansi ion P sys em p(n)
o deg ee 1 ha compu es, in he sense o De ini ion 1,
he polynomial p(n)=a0+a1·n···+ak·nko deg ee k≥1,
wi h in ege coe icien s ai∈Z,0≤i≤k, is designed.
I is easy o check ha o each na u al numbe ∈N he
ollowing holds:
p( +1)−p( )
=[a11
0+a22
0+···+ak−1k−1
0+akk
0]· 0
+[a22
1+···+ak−1k−1
1+akk
1]· 1
.............................................
+[ak−1k−1
k−2+akk
k−2]· k−2
+[akk
k−1]· k−1
Le us deno e:
a0
k=a11
0+a22
0+···+ak−1k−1
0+akk
0
a1
k=a22
1+···+ak−1k−1
1+akk
1
.......................................
ak−2
k=ak−1k−1
k−2+akk
k−2
ak−1
k=akk
k−1
Then, p( +1)−p( )=a0
k+a1
k· +a2
k· 2+···+ak−2
k· k−2+
ak−1
k· k−1=
k−1

i=0
ai
k· i, ha is,p( +1)=p( )+
k−1

i=0
ai
k· i.
De ini ion 2: Le p(n)=a0+a1·n+ ··· + ak·nk
be a polynomial o deg ee k≥1, wi h in ege coe icien s
ai∈Z,0≤i≤k. We associa e p(n)wi h he de e minis ic
ansi ion P sys em p(n)=(, μ, M1,(R1,ρ
1), iou )o
deg ee 1, de ined as ollows:
•={o1,o2,p1,p2,b1,b2,b3,···bk}
•μ=[]
1
•M1=o +(a0)
1o −(a0)
2b1
•R1is he se o he ollowing e olu ion ules:
1≡b1→p +(a0
k)
1p −(a0
k)
2b(0
0)
1b(1
0)
2b(2
0)
3···b(k−2
0)
k−1b(k−1
0)
k
2≡b2→p +(a1
k)
1p −(a1
k)
2b(1
1)
2b(2
1)
3···b(k−2
1)
k−1b(k−1
1)
k
3≡b3→p +(a2
k)
1p −(a2
k)
2b(2
2)
3···b(k−2
2)
k−1b(k−1
2)
k
.
.
.
− k−1≡bk−1→p +(ak−2
k)
1p −(ak−2
k)
2b(k−2
k−2)
k−1b(k−1
k−2)
k
k≡bk→p +(ak−1
k)
1p −(ak−1
k)
2b(k−1
k−1)
k
k+1≡p1p2→λ
k+2≡p1o2→λ
k+3≡p2o1→λ
k+4≡p1→o1
k+5≡p2→o2
•ρ1is he se o p io i ies ela ion among ules in R1:
{( k+1, k+2), ( k+1, k+3), ( k+2, k+4), ( k+2, k+5),
( k+3, k+4), ( k+3, k+5)}which can be in o mally
desc ibed as: k+1>{ k+2, k+3}>{ k+4, k+5}.
•iou =1.
B.
Fo mal
Ve i ica ion
We show in his subsec ion ha he memb ane sys em p(n)
associa ed wi h he polynomial p(n), designed in he p e ious
sec ion, compu es he alues p( )acco ding o De ini ion 1,
o each ∈N.
Theo em 1: Le p(n)=a0+a1·n+···+ak·nkbe a
polynomial o deg ee k≥1 such ha ai∈Z,0≤i≤k.
Le p(n)be he de e minis ic ansi ion P sys em conside ed
in De ini ion 1. Fo each ≥0, a con igu a ion C +1 he
con en o memb ane labeled by 1 is he ollowing mul ise :
{o +(p( ))
1o −(p( ))
2pk−1
i=0 +(ai
k)· i
1pk−1
i=0 −(ai
k)· i
2
b1b( +1)
2b( +1)2
3··· b( +1)k−1
k}
P oo : Le us p o e he esul by induc ion on .
Le us s a wi h he base case =0. A he ini ial
con igu a ion C0, he con en o memb ane labeled by 1 is
he mul ise o +(a0)
1o −(a0)
2b1. Then, con igu a ion C0yields
con igu a ion C1by applying ule 1once. Thus, a con igu-
a ion C1 he con en o memb ane labeled by 1 is he mul-
ise o +(a0)
1o −(a0)
2p +a0
k
1p −a0
k
2b1b2b3··· bk. Because
o p(0)=a0= +(a0)− −(a0), he esul holds o =0.
By induc ion hypo hesis, le us assume he esul holds
o ≥0, ha is, a con igu a ion C +1 he con en o
memb ane labeled by 1 is he mul ise
{o +(p( ))
1o −(p( ))
2pk−1
i=0 +(ai
k)· i
1pk−1
i=0 −(ai
k)· i
2
b1b( +1)
2b( +1)2
3··· b( +1)k−1
k}.
In o de o ob ain he con en o memb ane labeled by 1 a
con igu a ion C +2, le us analyze all he possible cases ha
may happen:
Case 1:
k−1

i=0
ai
k· i≥ −(p( ))
In his case, p( +1)−p( )≥ −(p( )) and he ollowing
holds:
(a)
k−1

i=0
+(ai
k)· i−
k−1

i=0
−(ai
k)· i≥ −(p( )). Indeed, i is
su icien o no ice ha ai
k= +(ai
k)− −(ai
k), o 0≤
i≤k−1.
(b)
k−1

i=0
+(ai
k)· i≥
k−1

i=0
−(ai
k)· i. Indeed, (b) ollows
om (a) ecalling ha −(p( )) ≥0.
(c) p( +1)≥0. Indeed, i p( )≥0 henp( +1)≥p( )+
−(p( )) ≥0, and i p( )<0 hen −(p( )) =−p( ),
so p( +1)≥p( )+ −(p( )) =0.
The e o e, in his case con igu a ion C +1yields con igu a-
ion C +2as ollows:
(1) F om (b) we deduce ha a con igu a ion C +1in mem-
b ane labeled by 1 he e a e mo e copies o objec p1
han copies o objec p2,so ule k+1≡p1p2→λwill
be applied
k−1

i=0
−(ai
k)· i imes, consuming all copies
o p2and “ emaining”
k−1

i=0
+(ai
k)· i−
k−1

i=0
−(ai
k)· i
copies o p1wi hou e ol ing.
(2) F om (a) we deduce ha a con igu a ion C +1in
memb ane labeled by 1 he e a e mo e copies o he
“ emaining” objec p1 han copies o objec o2,so ule
k+2≡p1o2→λwill be applied −(p( )) imes,
consuming all copies o o2and “ emaining”
k−1

i=0
+(ai
k)· i−
k−1

i=0
−(ai
k)· i− −(p( ))
copies o p1wi hou e ol ing.
(3) Rule k+4≡p1→o1will be applied
α=
k−1

i=0
+(ai
k)· i−
k−1

i=0
−(ai
k)· i− −(p( ))
imes, consuming all copies o p1and p oducing α
copies o o1.
(4) Fo each j,1≤j≤k, ule j≡bj→p +(aj−1
k)
1
p −(aj−1
k)
2b(j−1
j−1)
j...b(k−1
j−1)
kwill be applied ( +1)j−1
imes.
All he p e ious ules a e applied in pa allel in one ansi ion
s ep. Thus, in his case, a con igu a ion C +2 he con en
o memb ane labeled by 1 is he mul ise which con ains
objec s o1,o2,p1,p2and bj,(1≤j≤k)wi h he ollowing
mul iplici ies:
–Mul iplici y o o1: +(p( +1)).
Indeed, a e execu ion o ules om (2), αnew copies
o o1a e p oduced. Thus, he o al numbe o copies o o1
will be:
+(p( )) +
k−1

i=0
+(ai
k)· i−
k−1

i=0
−(ai
k)· i− −(p( ))
=p( )+
k−1

i=0
ai
k· i=p( +1)(c)
= +(p( +1)).
–Mul iplici y o o2:0.
Indeed, a e execu ion o ules om (2), all copies o
objec o2a e consumed and any new copies o objec o2
a e p oduced o any ule applied in his ansi ion s ep.
Thus, a con igu a ion C +2in memb ane labeled by 1 he
mul iplici y o o2is 0.
–Mul iplici y o p1:
k−1

i=0
+(ai
k)·( +1)i.
Indeed, a e execu ion o ules om (1), (2) and (3), all
copies o objec p1a e consumed bu by applying ules
om (4), he o al numbe o copies o p1p oduced is
k

j=1
+(aj−1
k)·( +1)j−1=
k−1

i=0
+(ai
k)·( +1)i.
–Mul iplici y o p2:
k−1

i=0
−(ai
k)·( +1)i.
Indeed, a e execu ion o ules om (1), (2) and (3), all
copies o objec p2a e consumed bu by applying ules
om (4), he o al numbe o copies o p2p oduced is
k

j=1
−(aj−1
k)·( +1)j−1=
k−1

i=0
−(ai
k)·( +1)i.
–Mul iplici y o objec bj, o each j,1≤j≤k:
( +2)j−1.
Indeed, om (3) he o al numbe o copies o objec bj
p oduced is
j−1

s=0j−1
s( +1)s=[( +1)+1]j−1=( +2)j−1
Hence, in his case he esul holds o +1.
Case 2: 0≤
k−1

i=0
ai
k· i< −(p( ))
In his case, he ollowing holds:
(a) p( )<0andp( +1)<0. Indeed, on he one hand, as
−(p( )) > 0weha e −(p( )) =−p( ). On he o he
hand,
p( +1)=p( )+
k−1

i=0
ai
k· i=− −(p( )) +
k−1

i=0
ai
k<0.
(b) 0≤
k−1

i=0
+(ai
k)· i−
k−1

i=0
−(ai
k)· i< −(p( )). Indeed,
i su ices o no ice ha ai
k= +(ai
k)− −(ai
k), o 0≤
i≤k−1.
(c) p( +1)=k−1

i=0
+(ai
k)· i−
k−1

i=0
−(ai
k)· i− −(p( )).
Indeed, as −(p( )) > 0weha ep( )<0and
−(p( )) =−p( ).So,
p( +1)=p( )+
k−1

i=0
ai
k· i
=− −(p( )) +
k−1

i=0
+(ai
k)· i−
k−1

i=0
−(ai
k)· i.
The e o e, in his case con igu a ion C +1yields con igu a ion
C +2as ollows:
(1) F om (b) we deduce ha a con igu a ion C +1in mem-
b ane labeled by 1 he e a e mo e copies o objec p1
han copies o objec p2,so ule k+1≡p1p2→λwill
be applied
k−1

i=0
−(ai
k)· i imes, consuming all copies
o p2and “ emaining”
k−1

i=0
+(ai
k)· i−
k−1

i=0
−(ai
k)· i
copies o p1wi hou e ol ing.
(2) Con igu a ion C +1in memb ane labeled by 1 he e
a e mo e copies o objec o2 han copies o
objec p1, hen ule k+2≡p1o2→λ
will be applied
k−1

i=0
+(ai
k)· i−
k−1

i=0
−(ai
k)· i imes,

consuming all copies o p1and “ emaining” −(p( ))−
k−1

i=0
+(ai
k)· i−
k−1

i=0
−(ai
k)· icopies o o2wi hou
e ol ing.
(3) Fo each j,1≤j≤k, ule j≡bj→p +(aj−1
k)
1
p −(aj−1
k)
2b(j−1
j−1)
j...b(k−1
j−1)
kwill be applied ( +1)j−1
imes.
All he p e ious ules a e applied in pa allel in one ansi ion
s ep. Thus, in his case, a con igu a ion C +2 he con en
o memb ane labeled by 1 is he mul ise which con ains
objec s o1,o2,p1,p2and bj,(1≤j≤k)wi h he ollowing
mul iplici ies:
–Mul iplici y o objec o1: +(p( +1)).
Indeed, he applied ules do no a ec o objec o1and
om (a) we deduce ha +(p( +1)) =0= +(p( )).
–Mul iplici y o objec o2: −(p( +1)).
Indeed, a e execu ion o he ci ed ules, he mul iplici y
o o2is
−(p( )) −k−1

i=0
+(ai
k)· i−
k−1

i=0
−(ai
k)· i
(b)
=−p( +1)(c)
= −(p( +1)).
–Mul iplici y o objec p1:
k−1

i=0
+(ai
k)·( +1)i.
Indeed, a e execu ion o he ules om (1) and (2),
he mul iplici y o p1is 0, bu a e he applica ion o
ules om (3) i s mul iplici y becomes
k

i=1
+(ai−1
k)·( +1)i−1=
k−1

i=0
+(ai
k)·( +1)i
–Mul iplici y o objec p2:
k−1

i=0
−(ai
k)·( +1)i.
Indeed, because a e execu ion o he ules
om (1) and (2), he mul iplici y o p2is 0, bu a e
he applica ion o ules om (3) i s mul iplici y becomes
k

i=1
−(ai−1
k)·( +1)i−1=
k−1

i=0
−(ai
k)·( +1)i
–Mul iplici y o objec bj, o each j,1≤j≤k:
( +2)j−1.
Indeed, om (3) he o al numbe o copies o objec bj
p oduced is
j−1

s=0j−1
s( +1)s=[( +1)+1]j−1=( +2)j−1
Hence, in his case he esul holds o +1.
Case 3: k−1

i=0
ai
k· i<0∧ +(p( )) +
k−1

i=0
ai
k· i≤0
In his case, he ollowing holds:
(a)
k−1

i=0
+(ai
k)−
k−1

i=0
−(ai
k)· i<0.
Indeed, i su ices o bea in mind ha
k−1

i=0
ai
k· i<0, and
ai
k= +(ai
k)− −(ai
k), o 0≤i≤k−1.
(b) +(p( )) ≤−k−1

i=0
+(ai
k)· i−
k−1

i=0
−(ai
k)· i
Indeed, i is enough o no ice ha
+(p( )) ≤−
k−1

i=0
ai
k· i
=−
k−1

i=0
+(ai
k)· i−
k−1

i=0
−(ai
k)· i
(c) p( +1)≤0.
Indeed, i p( )≥0 henp( )= +(p( )) ≤−
k−1

i=0
ai
k· i
and p( +1)=p( )+
k−1

i=0
ai
k· i;i p( )<0 henp( +1)=
p( )+
k−1

i=0
ai
k· i<0.
The e o e, in his case con igu a ion C +1yields
con igu a ion C +2as ollows:
(1) F om (a) we deduce ha a con igu a ion C +1in mem-
b ane labeled by 1 he e a e mo e copies o objec p2
han copies o objec p1,so ule k+1≡p1p2→λwill
be applied
k−1

i=0
+(ai
k)· i imes, consuming all copies
o p1and “ emaining”
k−1

i=0
−(ai
k)· i−
k−1

i=0
+(ai
k)· i
copies o p2wi hou e ol ing.
(2) F om (b) we deduce ha a con igu a ion C +1in mem-
b ane labeled by 1 he e a e mo e copies o objec p2
han copies o objec o1,so ule k+3≡p2o1→λ
will be applied +(p( )) imes, consuming all copies
o o1and “ emaining” − +(p( )) +
k−1

i=0
ai
k· icopies
o p2wi hou e ol e bu hese copies mus e ol e by
means o ule k+5.
(3) Rule k+5≡p2→o2will be applied
− +(p( )) +
k−1

i=0
ai
k· i imes, consuming all copies
o p2and p oducing − +(p( )) +
k−1

i=0
ai
k· inew
copies o objec o2.
(4) Fo each j,1 ≤j≤k, ule j≡
bj→p +(aj−1
k)
1p −(aj−1
k)
2b(j−1
j−1)
j...b(k−1
j−1)
kwill be applied
( +1)j−1 imes.
All he p e ious ules a e applied in pa allel in one ansi ion
s ep. Thus, in his case, a con igu a ion C +2 he con en
o memb ane labeled by 1 is he mul ise which con ains
objec s o1,o2,p1,p2and bj,(1≤j≤k)wi h he ollowing
mul iplici ies:
–Mul iplici y o objec o1: +(p( +1)).
Indeed, om (2) all copies o objec o1a e consumed,
bu +(p( +1)) (c)
=0.
–Mul iplici y o objec o2: −(p( +1)).
Indeed, a e execu ion o he ules, i s mul iplici y will
be
−(p( )) − +(p( )) +
k−1

i=0
ai
k· i
=−p( )−
k−1

i=0
ai
k· i
=−p( +1)(c)
= −(p( +1)).
–Mul iplici y o objec p1:
k−1

i=0
+(ai
k)·( +1)i.
Indeed, a e execu ion o he ules om (1) all copies
o p1a e consumed bu om (4) he p oduced copies a e
he ollowing:
k

i=1
+(ai−1
k)·( +1)i−1=
k−1

i=0
+(ai
k)·( +1)i
–Mul iplici y o objec p2:
k−1

i=0
−(ai
k)·( +1)i.
Indeed, a e execu ion o he ules om (1), (2) and (3),
all copies o objec p2a e consumed bu by applying
ules in (4) new copies o p2a e p oduced, in o al he
numbe o copies will be:
k

i=1
−(ai−1
k)·( +1)i−1=
k−1

i=0
−(ai
k)·( +1)i
–Mul iplici y o objec bj, o each j,1≤j≤k:
( +2)j−1.
Indeed, om (4) he o al numbe o copies o objec bj
p oduced is
j−1

s=0j−1
s( +1)s=[( +1)+1]j−1=( +2)j−1
Hence, in his case he esul holds o +1.
Case 4: k−1

i=0
ai
k· i<0∧ +(p( )) +
k−1

i=0
ai
k· i>0
In his case, he ollowing holds:
(a)
k−1

i=0
+(ai
k)−
k−1

i=0
−(ai
k)· i<0.
Indeed, i is enough o no ice ha
k−1

i=0
ai
k· i<0, and
ai
k= +(ai
k)− −(ai
k), o 0≤i≤k−1.
(b) +(p( )) > −k−1

i=0
+(ai
k)· i−
k−1

i=0
−(ai
k)· i.
Indeed, i su ices o bea in mind ha +(p( )) >
−
k−1

i=0
ai
k· i=−k−1

i=0
+(ai
k)· i−
k−1

i=0
−(ai
k)· i
and ai
k= +(ai
k)− −(ai
k), o 0≤i≤k−1.
(c) p( )>0andp( +1)>0.
Indeed, om he hypo hesis in his case we ha e
+(p( )) > −
k−1

i=0
ai
k· i>0. So, p( )>0and
+(p( )) =p( ). Thus,
p( +1)=p( )+
k−1

i=0
ai
k· i= +(p( )) +
k−1

i=0
ai
k· i>0
The e o e, in his case con igu a ion C +1yields con igu a ion
C +2as ollows:
(1) F om (a) we deduce ha a con igu a ion C +1in mem-
b ane labeled by 1 he e a e mo e copies o objec p2
han copies o objec p1,so ule k+1≡p1p2→λwill
be applied
k−1

i=0
+(ai
k)· i imes, consuming all copies
o p1and “ emaining”
k−1

i=0
−(ai
k)· i−
k−1

i=0
+(ai
k)· i
copies o p2wi hou e ol ing.
(2) F om (b) we deduce ha a con igu a ion C +1in mem-
b ane labeled by 1 he e a e mo e copies o objec o1
han copies o objec p2,so ule k+3≡p1o1→λ
will be applied
k−1

i=0
−(ai
k)· i−
k−1

i=0
+(ai
k)· i imes,
consuming all copies o p2and “ emaining” +(p( ))−
k−1

i=0
−(ai
k)· i−
k−1

i=0
+(ai
k)· icopies o o1wi hou
e ol ing.
(3) Fo each j,1 ≤j≤k, ule j≡bj→
p +(aj−1
k)
1p −(aj−1
k)
2b(j−1
j−1)
j...b(k−1
j−1)
kwill be applied ( +
1)j−1 imes.
All he p e ious ules a e applied in pa allel in one ansi ion
s ep. Thus, in his case, a con igu a ion C +2 he con en
o memb ane labeled by 1 is he mul ise which con ains
objec s o1,o2,p1,p2and bj,(1≤j≤k)wi h he ollowing
mul iplici ies:
–Mul iplici y o objec o1: +(p( +1)).
Indeed, om (2) we deduce ha he numbe o copies o
objec o1is:
+(p( )) −k−1

i=0
−(ai
k)· i−
k−1

i=0
+(ai
k)· i
= +(p( )) +
k−1

i=0
ai
k· i(c)
=p( )+
k−1

i=0
ai
k· i
=p( +1)(c)
= +(p( +1))
all copies o objec o1a e consumed, bu
+(p( +1)) (c)
=0.
–Mul iplici y o objec o2: −(p( +1)).
Indeed, objec o2is no in ol ed by he applica ion o
ules o each con igu a ion C +2 om con igu a ion C +1,
so he mul iplici y o o2is −(p( )) (c)
=0(c)
= −(p( +1)).
–Mul iplici y o objec p1:
k−1

i=0
+(ai
k)·( +1)i.
Indeed, om (1) all copies o objec p1a e consumed
bu om (3) he o al numbe o p oduced copies is
k

j=1
+(aj−1
k)·( +1)j−1=
k−1

i=0
+(ai
k)·( +1)i.
–Mul iplici y o objec p2:
k−1

i=0
−(ai
k)·( +1)i.
Indeed, om (1) all copies o objec p2a e consumed
bu om (3) he o al numbe o p oduced copies is
k

j=1
−(aj−1
k)·( +1)j−1=
k−1

i=0
−(ai
k)·( +1)i.
–Mul iplici y o objec bj, o each j,1≤j≤k:
( +2)j−1.
Indeed, om (3) he o al numbe o copies o objec bj
p oduced is
j−1

s=0j−1
s( +1)s=[( +1)+1]j−1=( +2)j−1
Hence, in his case he esul holds o +1.
Co olla y 1: Le p(n)=a0+a1·n+ ··· + ak·nkbe
a polynomial o deg ee ksuch ha ai∈Z,i=0,1,...,k.
Le p(n)be he de e minis ic ansi ion P sys em conside ed
in De ini ion 2. Then, polynomial p(n)is compu ed by he
sys em p(n)acco ding wi h De ini ion 1.
P oo : F om Theo em 1 we deduce ha o each ∈N
a con igu a ion C +1 he con en o memb ane labeled by 1 is
he ollowing mul ise :
{o +(p( ))
1o −(p( ))
2pk−1
i=0 +(ai
k)· i
1pk−1
i=0 −(ai
k)· i
2
b1b( +1)
2b( +1)2
3··· b( +1)k−1
k}
In o de o know he mul iplici y o objec s o1and o2in
memb ane labeled by 1 a con igu a ion C +1, wo cases a e
dis inguished:
•I p( )≥0 hen +(p( )) =p( )and −(p( )) =0.
So, he mul iplici y o o1is +(p( )) =p( )and he
mul iplici y o o2is −(p( )) =0.
•I p( )<0 hen +(p( )) =0and −(p( )) =−p( ).
Thus, he mul iplici y o o1is +(p( )) =0and he
mul iplici y o o2is −(p( )) =−p( ).
IV. DESCRIPTIVE COMPUTATIONAL RESOURCES
In his sec ion, he desc ip i e compu a ional esou ces
equi ed by he de e minis ic ansi ion P sys em p(n) con-
side ed in De ini ion 2 which compu es polynomial p(n) wi h
in ege numbe s coe icien s, is depic ed.
•The size o he wo king alphabe : k+4.
•The ini ial numbe o objec s: 1 +|a0|.
•The numbe o ules: k+5.
•The o al numbe o objec s in ol ed in he ules is
2k+k+9+
k−1
|ai
k|.
i=0
Hence, he o al amoun o desc ip i e compu a ional esou ces
is exponen ial in he size o he polynomial.
V. DISCUSSIONS
Un il now, wo kinds o me hods ha e been epo ed in li -
e a u e o implemen au oma ic design o memb ane sys ems.
One is he easoning way p esen ed in his pape and [27],
which is called REASON. The o he is he me aheu is-
ic app oaches (META) al eady used in memb ane sys ems
design, such as gene ic algo i hms [21], [23], [25], in pa icula
Pe mu a ion Penal y Gene ic Algo i hms (PPGAs) [26], and
quan um-inspi ed e olu iona y algo i hm (QIEAs) [22], [24].
REASON and META ha e he ollowing di e ences:
•Concep : META uses a me aheu is ic app oach o
e ol e a popula ion o candida e P sys ems ( easi-
ble o in easible) owa d he success ul P sys ems, while
REASON uses induc i e me hod ( om simple o com-
plex P sys ems) o ob ain he success ul P sys ems.
A me aheu is ic app oach may be a gene ic algo i hm,
a quan um-inspi ed e olu iona y algo i hm o o he s.
•Usage: META is qui e easy o unde s and and mas e
o a beginne , while REASON sounds a qui e complex
echnique o a beginne .
•Gene a ion: META is a mo e gene al echnique han
REASON and he e o e i is possible o use META o
design di e en P sys ems. While in REASON, di e en
P sys ems a e designed by using di e en speci ic ea-
soning echniques.
•Resou ce: In REASON, i is possible o calcula e he
esou ce equi ed by a P sys em wi h espec o com-
pu ing ime and wo kspace. While in META, i is qui e
ha d o summa ize he esou ce.
•So wa e: The e alua ion o a success ul P sys em in
META is pe o med by using he well-known P sys em
simula o , P-Lingua [28]. REASON does no need any
so wa e.
•Ex endibili y: REASON can be easily ex ended om a
speci ic o a gene al P sys em, e.g., om a low-deg ee o
high-deg ee polynomial P sys em, o a kind o memb ane
sys em. This ex endibili y is no sui able o META.
VI. CONCLUSION
This pape ex ends he wo k in [27] om he au oma ic
design o de e minis ic ansi ion P sys ems o compu ing
polynomials wi h na u al numbe coe icien s o he au oma ic
design o such kind o memb ane sys ems o compu ing poly-
nomials wi h in ege coe icien s, by analyzing he syn ac ical
and seman ics ing edien s o cell-like memb ane sys ems. This
is a signi ican s ep o he p og ammabili y o memb ane
sys ems, namely how o au oma ically design a P sys em by
using p og ams so as o de elop a use ul oolbox o he
communi y o memb ane compu ing.
As u u e wo k we plan o ex end his me hod in o de o
design new a ian s o memb ane sys ems wi h he capabili y
o pe o ming mo e complex asks like inding he mini-
mal memb ane sys em, wi h espec o he numbe o used
objec s, o a gi en assignmen o like p ac ical applica ions
such as memb ane con olle s o mobile obo s. On he
o he hand, we p opose: (a) o de elop so wa e pla o ms
o simula e ansi ion P sys ems using a weak in e p e a ion
o he p io i ies as well as FPGA (Field P og ammable Ga e
A ay) based ha dwa e o implemen hem; and (b) he use
memb ane-inspi ed e olu iona y algo i hms [15], [29], [30] o
op imiza ion spiking neu al P sys ems [31] o implemen he
au oma ic design o a memb ane sys ems (including spiking
neu al P sys ems) o sol ing compu a ionally ha d p oblems.
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