Sol ing P oblems Th ough a Single Memb ane
Sys em
Da id O ellana-Ma ´ın, Luis Valencia-Cab e a,
Agus ´ın Riscos-N´u˜nez, Ma io J. P´e ez-Jim´enez
Resea ch G oup on Na u al Compu ing
Depa men o Compu e Science and A i icial In elligence
Uni e sidad de Se illa
A da. Reina Me cedes s/n, 41012 Se illa, Spain
{do ellana,l alencia,a iscosn,ma pe }@us.es
Summa y. The ape o a de e minis ic Tu ing machine con ains an unbounded numbe
o cells. Thanks o ha , a single machine can sol e decision p oblems wi h an in ini e
numbe o ins ances. Ne e heless, in he amewo k o memb ane compu ing, adi-
ionally a “solu ion” o an abs ac decision p oblem consis s o a amily o memb ane
sys ems (whe e each sys em o he amily is associa ed wi h a ini e se o ins ances o
he p oblem o be sol ed). An in e es ing ques ion is o analyze he possibili y o ind
a single memb ane sys em able o deal wi h he in ini ely many ins ances o a decision
p oblem.
In his con ex , i is undamen al o de ine p ecisely how he ins ances o he p oblem
a e in oduced in o he sys em. In his pape , wo di e en me hods a e conside ed.
The i s one elies on a p e-compu ing p ocess, whe e a polynomial- ime compu able
unc ion will be in cha ge o p oducing a mul ise o objec s associa ed wi h he ins ance
o be sol ed. On he o he hand, he second one assumes ha he inpu alphabe o he
sys em is equal o he alphabe o ins ances, and he e o e ins ances a e di ec ly in o-
duced in he ini ial con igu a ion o he sys em. Polynomial complexi y classes associa ed
wi h hese wo app oaches a e in oduced and some complexi y aspec s a e s udied.
1 In oduc ion
In he 17 h B ains o ming Week on Memb ane Compu ing, an appa en ly innocen
p oblem was p esen ed by he au ho s: he ONLY-ONE-OBJECT p oblem. The goal
is o build a sys em able o dis inguish whe he in a gi en egion, a a gi en
momen , he e is only one copy o an objec , o i he mul iplici y o he objec
is s ic ly g ea e han one. Besides, he no ion o e icien sol abili y by means
o a single ecognize pola iza ionless P sys em wi h ac i e memb anes, wi hou
dissolu ion ules and using di ision o elemen a y and non-elemen a y memb anes,
was p oposed. Following a easoning based on he dependency g aph echnique, a
440 D. O ellana-Ma ´ın e al.
nega i e answe o he p e ious ques ion was concluded (i.e. he p oblem is no
sol able in he p oposed amewo k).
In some sense, he p e ious ques ion links up wi h o he s ha we e p oposed by
P. Sos´ık [17], which aise he possibili y o being able o sol e P-comple e p oblems
o NP-comple e p oblems by means o a single memb ane sys em. Speci ically,
wo “open p oblems” we e “ o mula ed” in [17], exp essed in an in o mal way as
ollows:
•Open P oblem 1. Is he e any known s anda d model o P sys em capable o
sol ing a P-comple e p oblem in polynomial ime wi hou he use o amilies,
i.e., all ins ances a e sol ed by he same P sys ems?
•Open P oblem 2. How o design a na u al (no much “ex ao dina y”) model
o P sys em capable o sol ing an NP-comple e p oblem in polynomial ime
wi hou he use o amilies?
O cou se, hese ques ions should be exp essed in a o mal way and hei answe s
will depend on he de ini ions gi en abou wha sol ing a decision p oblem h ough
a single memb ane sys em means.
Fo ins ance, wo possible de ini ions could be conside ed acco ding o he
way o en e ing he inpu inside he memb ane sys em: (a) by using p ecompu ed
esou ces ( ha is, wai ing o a polynomial ime p io o he ini ial s ep o he
compu a ion, o calcula e which is he inpu mul ise ha has o be p o ided
o he sys em); o (b) by di ec ly in oducing he inpu mul ise wi hou any
p ep ocessing, ha is, ee o ex e nal esou ces.
Fo a comp ehensi e in oduc ion o memb ane sys ems, we e e he eade
o [12, 15].
2 The Complexi y Class PMC1p
R
Fi s , le us de ine a solu ion o a decision p oblem h ough a single memb ane
sys em allowing he possibili y o use (ex e nal) p ecompu ed esou ces o p o id-
ing he inpu mul ise o he sys em. In o he wo ds, we assume ha he e is an
a ailable de ice able o execu e he unc ion ha compu es he inpu mul ise ,
and his p ocess should be pe o med be o e he compu a ion o he memb ane
sys em s a s.
De ini ion 1. Le Rbe a class o ecognize memb ane sys em. Le X= (IX, θX)
be a decision p oblem. We say ha p oblem Xis sol able in polynomial ime by a
single memb ane sys em Π om Rwi h p ecompu ed esou ces, deno ed by X∈
PMC1p
R, i he ollowing hold:
•The e exis s a polynomial encoding cod om X o Πp o iding a “ easonable
encoding scheme” which maps p oblem ins ances in o he mul ise s desc ibing
hem [3]; ha is, he e exis s a polynomial ime compu able unc ion, cod, whose
domain is IXsuch ha o e e y ins ance u∈IX,cod(u)is a mul ise o e
he inpu alphabe o Π.
Sol ing P oblems Th ough a Single Memb ane Sys em 441
•The sys em Πis polynomially bounded wi h ega d o (X, cod); ha is, he e
exis s a polynomial p( )such ha o each ins ance u∈IX, e e y compu a ion
o he sys em Πwi h inpu mul ise cod(u)pe o ms a mos p(|u|)s eps.
•The sys em Πis sound wi h ega d o (X, cod); ha is, o each ins ance u∈
IX, i he e exis s an accep ing compu a ion o he sys em Πwi h inpu mul ise
cod(u) hen θX(u)=1.
•The sys em Πis comple e wi h ega d o (X, cod); ha is, o each ins ance
u∈IXsuch ha θX(u) = 1, e e y compu a ion o he sys em Πwi h inpu
mul ise cod(u)is an accep ing compu a ion.
In his de ini ion, he inpu mul ise ha is alloca ed in o he ini ial con igu a ion
o he sys em is p ecompu ed by means o a polynomial- ime compu able unc ion.
P oposi ion 1. I Ris a class o ecognize memb ane sys ems, hen
P⊆PMC1p
R⊆PMCR
P oo . In o de o show ha P⊆PMC1p
R, le X= (IX, θX) be a decision p oblem
in class P. Le us conside he de e minis ic ecognize (cell-like) memb ane sys em
Π={Γ, Σ, µ, M1,R, iin}o deg ee 1 de ined as ollows:
•Γ=Σ={yes,no}.
•µ= [ ]1.
• M1=∅
• R ={[yes ]1→yes [ ]1; [ no ]1→no [ ]1}
•iin = 1.
Le us conside cod as he map whose domain is IXde ined as ollows: o e e y
u∈IX,cod(u) = {yes}i θX(u) = 1, and cod(u) = {no}, o he wise. Since X∈P,
cod is a polynomial- ime unc ion. Then, we ha e:
•The sys em Πis polynomially bounded wi h ega d o (X, cod): o e e y in-
s ance u∈IX, he compu a ion o Πwi h inpu mul ise cod(u) pe o ms 1
ansi ion s ep.
•Fo e e y ins ance u∈IX, he compu a ion o he sys em Πwi h inpu mul ise
cod(u) is an accep ing compu a ion i and only i θX(u) = 1.
This de ini ion can be easily adjus ed o any class o ecognize memb ane
sys ems R, in such a way ha we ha e X∈PMC1p
R. Then, we conclude ha
P⊆PMC1p
R.
In o de o show ha PMC1p
R⊆PMCR, le X= (IX, θX) be a decision
p oblem such ha X∈PMC1p. Le Π0a memb ane sys em om Rsol ing X
acco ding o De ini ion 1, being cod0apolynomial encoding om X o Πassocia ed
wi h ha solu ion. Le us conside he amily Π={Π( )| ∈N}de ined as ollows
Π( ) = Π0, o each ∈N. Le us conside he polynomial encoding (cod, s) om
he p oblem X o he amily Πde ined as ollows: cod =cod0and s(u) = 0, o
each u∈IX. Then i is easy o check ha he amily Πis polynomially uni o m
442 D. O ellana-Ma ´ın e al.
by Tu ing machines, polynomially bounded wi h ega d o (X, cod, s), and sound
and comple e wi h ega d o (X, cod, s). Thus, X∈PMCR.

3 The Complexi y Class PMC1
R
The second de ini ion e e s o he case in which he inpu mul ise is di ec ly
in oduced inside he sys em as i is (“ ee” o ex e nal dependencies o esou ces),
and hus he inpu alphabe should be chosen so ha he sys em is able o “ ead”
he ins ances o he p oblem o be sol ed.
De ini ion 2. Le Rbe a class o ecognize memb ane sys ems. Le X= (IX, θX)
be a decision p oblem such ha IXis a language o e a ini e alphabe ΣX. We
say ha p oblem Xis sol able in polynomial ime by a single memb ane sys em Π
om R ee o ex e nal esou ces, deno ed by X∈PMC1
R, i he ollowing hold:
•The inpu alphabe o Πis ΣX.
•The sys em Πis polynomially bounded wi h ega d o X; ha is, he e exis s
a polynomial p( )such ha o each ins ance u∈IX, e e y compu a ion o he
sys em Πwi h inpu mul ise upe o ms a mos p(|u|)s eps.
•The sys em Πis sound wi h ega d o X; ha is, o each ins ance u∈IX,
i he e exis s an accep ing compu a ion o he sys em Πwi h inpu mul ise u
hen θX(u)=1.
•The sys em Πis comple e wi h ega d o X; ha is, o each ins ance u∈IX
such ha θX(u)=1, e e y compu a ion o he sys em Πwi h inpu mul ise u
is an accep ing compu a ion.
P oposi ion 2. Le Rbe a class o ecognize memb ane sys ems. Then we ha e
PMC1
R⊆PMC1p
R.
P oo . Le us assume ha X∈PMC1
R. Le Π0a memb ane sys em om R
whose inpu alphabe is ΣX( he wo king alphabe o he p oblem X) such ha i
is polynomially bounded, sound and comple e wi h ega d o X. Le us conside
he polynomial encoding cod om X o Π0de ined as ollows: cod(u) = u, o
e e y ins ance u∈IX. Then, Π0is polynomially bounded, sound and comple e
wi h ega d o (X, cod). Thus, X∈PMC1p
R.
4 Decision P oblems wi h a Fini e Numbe o Ins ances
In his sec ion, we wo k wi h decision p oblems whose se o ins ances is a ini e
se .
P oposi ion 3. Le T(so) he class o all ecognize ansi ion P sys ems which
make use o send-ou communica ion ules only. Then, i X= (IX, θX)is a deci-
sion p oblem whose se o ins ances is a ini e se , hen X∈PMC1
T(so).
Sol ing P oblems Th ough a Single Memb ane Sys em 443
P oo . Le X= (IX, θX) be a decision p oblem whose se o ins ances IXis a
ini e language o e he alphabe ΣX. Le us conside he ecognize ansi ion P
sys em Π= (Γ, Σ, µ, M1,R1, iin), de ined as ollows:
•The wo king alphabe is Γ=ΣX∪ {yes,no}and he inpu alphabe Σis ΣX.
•The memb ane s uc u e is µ= [ ]1and he ini ial mul ise is M1=∅.
•The se R1o ules is
{[u]1→yes [ ]1|θX(u)=1}∪{[u]1→no [ ]1|θX(u)=0}
•The inpu memb ane is labelled by 1.
Ob iously, memb ane sys em Πbelongs o he class T(so) and i sol es p oblem
X, acco ding o De ini ion 2.
4.1 The logic ga e p oblems
De ini ion 3. A Boolean unc ion o a i y n≥1is a o al unc ion om {0,1}n
o {0,1}.
Usually, in his con ex , Boolean alues 0, 1 can be associa ed wi h alse and
ue. Speci ically, alue 0 is associa ed wi h he logical alue alse (deno ed by
0∗) and alue 1 is associa ed wi h he logical alue ue (deno ed by 1∗). The
logical connec i e ¬can be conside ed as a una y Boolean unc ion and he logical
connec i es ∧,∨can be conside ed as bina y Boolean unc ions.
Any Boolean exp ession ϕwhose se o a iables is V a (ϕ) = {x1, . . . , xn}, can
be iewed as he n-a y Boolean unc ion e i ying he ollowing: o any uple
( 1, . . . , n)∈ {0,1}nwe ha e ( 1, . . . , n) = 1 i and only i σ(ϕ) = ue, whe e
σis he u h assignmen ( ∗
1, . . . , ∗
n).
Any Boolean unc ion o a i y n≥1 has associa ed a decision p oblem X =
(IX , θX ), in a na u al way, as ollows: IX ={0,1}nand θX (x1, . . . , xn) =
(x1, . . . , xn), o each (x1, . . . , xn)∈ {0,1}n.
As an in e es ing case o Boolean unc ions, we conside he ollowing decision
p oblems associa ed wi h logic ga es.
•NOT-GATE= ({0,1}, θ), whe e θ(0) = 1 and θ(1) = 0.
•OR-GATE= ({0,1} × {0,1}, θ), whe e θ(u) = 0, i u= (0,0), and θ(u) = 1,
o he wise.
•AND-GATE= ({0,1} × {0,1}, θ), whe e θ(u) = 1, i u= (1,1), and θ(u) = 0,
o he wise.
P oposi ion 4. Le T(nc, so) he class o all ecognize ansi ion P sys ems which
makes only use o non-coope a i e send-ou communica ion ules. Then, p oblems
NOT-GATE,OR-GATE,AND-GATE belong o PMC1
T(nc,so).
P oo . Le us conside he P sys em Π= (Γ, Σ, µ, M1,R1, iin) de ined as ollows:

444 D. O ellana-Ma ´ın e al.
•The wo king alphabe is Γ=Σ∪ {yes,no}and he inpu alphabe Σ={0,1}
.
•The memb ane s uc u e is µ= [ ]1and he ini ial mul ise is M1=∅.
•The se R1o ules is {[ 0 ]1→yes ; [ 1 ]1→no [ ]1}.
•The inpu memb ane is labelled by 1.
Ob iously, sys em Π om class T(nc, so) sol es he NOT-GATE p oblem, acco ding
o De ini ion 2.
Wi h espec o he OR-GATE p oblem, le us conside he P sys em Π=
(Γ, Σ, µ, M1,R1, iin) de ined as ollows:
•The wo king alphabe Γ=Σ∪ {yes,no}and he inpu alphabe Σ={0,1} ×
{0,1}.
•The memb ane s uc u e is µ= [ ]1and he ini ial mul ise is M1=∅.
•The se R1o ules is {[ (u, ) ]1→yes [ ]1|(u, )∈ {0,1}×{0,1}, u + ≥1}
∪ {[ (0,0) ]1→no [ ]1}.
•The inpu memb ane is labelled by 1.
Ob iously, sys em Π om class T(nc, so) sol es he OR-GATE p oblem, acco d-
ing De ini ion 2. In a simila way, i can be shown ha he AND-GATE p oblem
belongs o PMC1
T(nc,so).
I is wo h poin ing ou ha in his kind o solu ion by means o a single
memb ane sys em using non-coope a i e ules, he ep esen a ion o he ins ances
is specially ele an . Fo ins ance, in he case o he OR-GATE p oblem and he
AND-GATE p oblem, he coope a ion in he ules o he sys em can be a oided
because he se o ins ances is desc ibed by symbols o he language {0,1}×{0,1}.
5 The NONE-OBJECT P oblem
In his sec ion, we conside he NONE-OBJECT p oblem which in o mally co e-
sponds o he ask o de e mining whe he he e is any inpu objec o no in he
sys em. Fo mally, le X= (IX, θX) be he decision p oblem de ined as ollows:
IX={∅} ∪ {an|n∈N, n ≥1}, θX(∅)=1,and θX(an) = 0 o each n≥1
Tha is, he p oblem Xdis inguishes wo ypes o si ua ions: absence o objec s
on one hand, and a leas one copy o objec a, on he o he hand.
Theo em 1. Le T(nc, e , so, dis, p ) he class o all non-coope a i e ecognize P
sys ems which makes use o minimal p oduc ion in objec e olu ion ules ( ha is,
only one objec in he igh -hand side o he ule), send-ou communica ion ules,
dissolu ion ules and p io i ies. Then, NONE-OBJECT∈PMC1
T(nc,e ,so,dis,p ).
P oo . Le us conside he sys em Π om T(nc, e , so, dis, p ) de ined as ollows:
•The wo king alphabe is Γ={a, b, c}and he inpu alphabe is Σ={a}.
Sol ing P oblems Th ough a Single Memb ane Sys em 445
•The memb ane s uc u e µis µ= [ [ ]2]1and he ini ial mul ise s a e M1=∅
and M2={c}.
•The se Ro ules o Πis he ollowing:
{[a→b]2; [ b]2→no; [ c]2→yes; [ yes]1→yes [ ]1; [ no]1→no [ ]1}
•The se o p io i ies Pamong ules o Πis he ollowing:
([ a→b]2,[c]2→yes); ([ b]2→no,[c]2→yes)
•The inpu memb ane is labelled by 2.
Then, he ollowing hold:
•Fo each na u al numbe n≥1, he sys em Πwi h inpu mul ise {an}is
de e minis ic, he compu a ion o Π+{an}pe o ms h ee ansi ion s eps
and i is a ejec ing compu a ion.
•The sys em Πwi h inpu mul ise ∅is de e minis ic, he compu a ion o Π+∅
pe o ms wo ansi ion s eps and i is an accep ing compu a ion.
Thus, NONE-OBJECT∈PMC1
T(nc,e ,so,dis,p ).
6 The ONLY-ONE-OBJECT P oblem
In his sec ion, he p oblem o elling apa “one” om “mo e- han-one” objec is
conside ed. Fo mally, le X= (IX, θX) be he decision p oblem de ined as ollows:
IX={an|n∈N, n ≥1}and θX(an) = 1 i and only i n= 1
Tha is, he p oblem Xdis inguishes he case when he e is only one copy o objec
a om he es o possible cases wi h se e al copies o ha objec . We deno e ha
p oblem as he ONLY-ONE-OBJECT p oblem. Ob iously, he ONLY-ONE-OBJECT p ob-
lem belongs o class Psince i is easy o design a de e minis ic Tu ing machine sol -
ing ha p oblem which akes wo compu a ion s eps. Thus, ONLY-ONE-OBJECT∈P.
Bea ing in mind ha o e e y class Ro ecognize memb ane sys ems, we ha e
we P⊆PMC1p
R, we deduce ha ONLY-ONE-OBJECT∈PMC1p
R.
I is easy o p o e ha ONLY-ONE-OBJECT∈PMC1
T(nc,e ,so,dis,p ), bu he ol-
lowing esul shows ha his p oblem canno be sol ed by a memb ane sys em
om AM0(−d, +ne) wi hou using p ecompu ed esou ces, being AM0(−d, +ne)
he class o pola iza ionless P sys ems wi hou dissolu ion ules and wi h di ision
ules o elemen a y and non-elemen a y memb anes.
Theo em 2. The e does no exis a ecognize memb ane sys em Π0∈
AM0(−d, +ne)sol ing he ONLY-ONE-OBJECT p oblem in a polynomial ime by
a single memb ane sys em and ee o esou ces. Tha is, ONLY-ONE-OBJECT/∈
PMC1
AM0(−d,+ne).
446 D. O ellana-Ma ´ın e al.
P oo . (Reasoning by educ io ad absu dum) Le us assume ha he e exis s a
ecognize memb ane sys em Π0 om AM0(−d, +ne) e i ying he ollowing:
(a) The inpu alphabe o Π0is he single on {a}.
(b) E e y compu a ion o Π0wi h inpu mul ise {a}is an accep ing compu a ion.
(c) E e y compu a ion o Π0wi h inpu mul ise {an}, o each n > 1, is a ejec ing
compu a ion.
Le us deno e by GΠ0+{a}( espec i ely, GΠ0+{an}, o each n > 1) he dependency
g aph1associa ed wi h he sys em Π0+{a}( esp. Π0+{an}). Then, we ha e:
•Fo each n > 1, GΠ0+{a}=GΠ0+{an}. Indeed, in bo h g aphs he e is only one
edge s a ing om s, speci ically, he edge {s, (a, iin)}, and he es o edges
a e gi en by he ules o Π0, due o Π0∈ AM0(−d, +ne).
•A compu a ion o Π0+{a}is an accep ing compu a ion i and only i he e
exis s a pa h in GΠ0+{a} om s o (yes, en ).
•Fo each n > 1, a compu a ion o Π0+{an}is an accep ing compu a ion i
and only i he e exis s a pa h in GΠ0+{an} om s o (yes, en ).
Thus, bea ing in mind ha GΠ0+{a}=GΠ0+{an}we deduce ha e e y compu a-
ion o Π0+{a}is an accep ing compu a ion i and only i e e y compu a ion o
Π0+{an}, o each n > 1, is an accep ing compu a ion. Hence, condi ions (b) and
(c) a e con adic o y.

Co olla y 1. PMC1
AM0(−d,+ne)(P⊆PMC1p
AM0(−d,+ne).
7 A Ve sion o he PARITY P oblem
In his sec ion, a e sion o he PARITY p oblem is conside ed. Speci ically, le
PARITY = (IPARITY, θPARITY) be he decision p oblem de ined as ollows:
IPARITY ={an|n∈N, n ≥1}and θPARITY(an) = 1 i and only i nis e en
Tha is, he PARITY p oblem dis inguishes an e en numbe o copies o objec a
om an odd numbe o copies o ha objec . Ob iously, his e sion o he PARITY
p oblem belongs o class Psince i is easy o design a de e minis ic Tu ing machine
sol ing ha p oblem.
Theo em 3. Le T(mcmp, so, dis, p ) he class o all ecognize P sys ems which
make use o minimal coope a ion and minimal p oduc ion in objec e olu ion ules,
send-ou communica ion ules, dissolu ion ules and p io i ies. Then, PARITY∈
PMC1
T(mcmp,so,dis,p ).
1We will no ecall he o mal de ini ion he e (see [2, 18] o de ails). The dependency
g aph can be in ui i ely seen as a map o “ eac an s-p oduc ” ela ionship be ween
objec s: he nodes a e pai s (objec , egion) and o e e y ule o he sys em he e will
be an a c connec ing each objec on he le -hand-side o each objec on he igh -hand
side.
Sol ing P oblems Th ough a Single Memb ane Sys em 447
P oo . Le us conside he sys em Π om T(mcmp, so, dis, p ) de ined as ollows:
(a) The wo king alphabe is Γ={a, b}and he inpu alphabe is Σ={a}.
(b) The memb ane s uc u e is µ= [ [ ]2]1, and he ini ial mul ise s a e M1=∅
and M2=∅.
(d) The se Ro ules o Πis he ollowing:
{[a2→b]2; [ b2→b]2; [ a]2→no; [ b]2→yes}∪
{[no]1→no [ ]1; [ yes]1→yes [ ]1}
(e) The se o p io i ies Pamong ules o Πis he ollowing:
([ a2→b]2,[a]2→no); ([ b2→b]2,[a]2→no); ([ a2→b]2,[b]2→yes);
([ b2→b]2,[b]2→yes); ([ a]2→no,[b]2→yes)
( ) The inpu memb ane is labelled by 2.
Then, o each na u al numbe n≥1, he ollowing hold:
•The sys em Πwi h inpu mul ise {an}is de e minis ic.
•The compu a ion o Π+{an}pe o ms 2 + blog2(n)c ansi ion s eps.
•The na u al numbe nis odd i and only i he con igu a ion Cblog2(n)ccon ains
a copy o objec a.
•The na u al numbe nis e en i and only i he compu a ion o Π+{an}is an
accep ing compu a ion.
Thus, PARITY∈PMC1
T(mcmp,so,dis,p ).
8 Conclusions
In his wo k, he abili y o sol ing p oblems by single “s and-alone” memb ane
sys ems ins ead o amilies o memb ane sys ems is s udied. While using p e-
compu ed esou ces, i is easy o see ha p oblems om Pcan be sol ed by
a single memb ane sys em using only send-ou ules. A ques ion a ises om
he e: Wha i we canno access o a p ecompu ed encoding and we ha e he
aw ins ance as inpu ? In his pape , he powe o single memb ane sys ems ee
o p ecompu ed esou ces is also s udied, gi ing, on he one hand, solu ions o
decision p oblems by means o a single memb ane sys em sol ing hem, and on
he o he hand demons a ing he inabili y o sys ems om AM0(−d, +ne) o
sol e he ONLY-ONE-OBJECT p oblem by using he dependency g aph echnique in
a no el way.
While alking abou ecognize memb ane sys ems, we suppose ha hey can,
a leas , send an objec o he en i onmen o e u n he answe . E en wi h his
minimal de ini ion, he lowe bound o PMC1p
Rhas been demons a ed o be