Compu a ional Comple eness o P Sys ems
Using Maximal Va ian s o he
Se De i a ion Mode
A iom Alhazo 1, Rudol F eund2, and Se gey Ve lan3
1Ins i u e o Ma hema ics and Compu e Science
Academy o Sciences o Moldo a
Academiei 5, Chi¸sin˘au, MD-2028, Moldo a
E-mail: [email p o ec ed]
2Facul y o In o ma ics, TU Wien
Fa o i ens aße 9-11, 1040 Wien, Aus ia
E-mail: [email p o ec ed]
3LACL, Uni e si ´e Pa is Es – C ´e eil Val de Ma ne
61, a . G´en´e al de Gaulle, 94010, C ´e eil, F ance
Email: [email p o ec ed]
Summa y. We conside P sys ems only allowing ules o be used in a mos one copy
in each de i a ion s ep, especially he a ian o he maximally pa allel de i a ion mode
whe e each ule may only be used a mos once. Mo eo e , we also conside he de i a ion
mode whe e om hose se s o ules only hose a e aken which ha e he maximal numbe
o ules. We check he compu a ional comple eness p oo s o se e al a ian s o P sys ems
and show ha some o hem e en li e ally s ill hold ue o he o hese wo new se
de i a ion modes. Mo eo e , we es ablish wo new esul s o P sys ems using a ge
selec ion o he ules o be chosen oge he wi h hese wo new se de i a ion modes.
1 In oduc ion
Memb ane sys ems wi h symbol objec s a e a heo e ical amewo k o pa allel
dis ibu ed mul ise p ocessing. Usually, mul ise s o ules a e applied in pa allel
o he objec s in he unde lying con igu a ion; o example, in he maximally
pa allel de i a ion mode (abb e ia ed max), a non-ex endable mul ise o ules is
applied o he cu en con igu a ion. In his pape we now conside a ian s o hese
de i a ion modes, whe e each ule is only used in a mos one copy, i.e., we conside
se s o ules o be applied in pa allel, o example, in he se -maximally pa allel
de i a ion mode (abb e ia ed smax) we apply non-ex endable se s o ules, and in
ano he de i a ion mode we apply se s o ules which con ain a maximal numbe
o applicable ules (abb e ia ed max ule).
60 A. Alhazo , R. F eund, and S. Ve lan
Taking se s o ules ins ead o mul ise s is a qui e na u al es ic ion and i
a ises om di e en mo i a ions, e.g., i ing a maximal se o ansi ions in Pe i
Ne s [5, 8] o op imizing an implemen a ion o FPGA simula o s [13]. A na u al
ques ion a ises conce ning he powe o se -based modes in con as o mul ise -
based ones. The i s a emp o go in o his di ec ion was done in [10] whe e i
was shown ha in some cases he compu a ional comple eness esul s es ablished
o he max-mode also hold o he smax-mode.
In his pape we con inue his line o esea ch and we show ha o se e al
a ian s o P sys ems he p oo s o compu a ional comple eness o max can be
aken o e e en li e ally o smax and e en ually e en o max ule, bu on he
o he hand he e a e also a ian s o P sys ems whe e he de i a ion modes smax
and max ule yield e en s onge esul s han he max-mode.
2 Va ian s o P Sys ems
In his sec ion we ecall he well-known de ini ions o se e al a ian s o P sys ems
as well as some a ian s o de i a ion modes and also in oduce he a ian s o se
de i a ion modes conside ed in he ollowing.
A (cell-like) P sys em is a cons uc
Π= (O, C, µ, w1, . . . , wm, R1, . . . , Rm, O, I) whe e
•Ois he alphabe o objec s,
•C⊂Ois he se o ca alys s,
•µis he memb ane s uc u e (wi h mmemb anes),
•w1, . . . , wma e mul ise s o objec s p esen in he m egions o µa he begin-
ning o a compu a ion,
•R1, . . . , Rma e ini e se s o ules, associa ed wi h he egions o µ,
• Ois he label o he memb ane egion om which he ou pu s a e aken (in
he gene a i e case)
• Iis he label o he memb ane egion whe e he inpu s a e pu a he beginning
o a compu a ion (in he accep ing case).
O= 0/ I= 0 indica es ha he ou pu /inpu is aken om he en i onmen .
I a ule u→ has a leas wo objec s in u, hen i is called coope a i e,
o he wise i is called non-coope a i e.Ca aly ic ules a e o he o m ca →c ,
whe e c∈Cis a special objec which ne e e ol es and ne e passes h ough a
memb ane, i jus assis s objec a o e ol e o he mul ise .
In ca aly ic P sys ems we use non-coope a i e as well as ca aly ic ules. In a
pu ely ca aly ic P sys em we only allow ca aly ic ules.
In he maximally pa allel de i a ion mode (abb e ia ed by max), in any com-
pu a ion s ep o Πwe choose a mul ise o ules om R, de ined as he union
o he se s R1, . . . , Rm, in such a way ha no u he ule can be added o i so
ha he ob ained mul ise would s ill be applicable o he exis ing objec s in he
egions 1, . . . , m.
Maximal Va ian s o he Se De i a ion Modes 61
2.1 Se De i a ion Modes
The basic se de i a ion mode is de ined as he de i a ion mode whe e in each
de i a ion s ep a mus one copy o each ule may be applied in pa allel wi h he
o he ules; his a ian o a basic de i a ion mode co esponds o he asynch onous
mode wi h he es ic ion ha only hose mul ise s o ules a e applicable which
con ain a mos one copy o each ule, i.e., we conside se s o ules:
Appl(Π, C, se ) ={R∈Appl(Π, C, asyn)| |R| ≤1 o each ∈ R}
In he se -maximally pa allel de i a ion mode ( his de i a ion mode is abb e i-
a ed by smax o sho ), in any compu a ion s ep o Πwe choose a non-ex endable
mul ise Ro ules om Appl(Π, C, se ); ollowing he no a ions elabo a ed in [7],
we de ine he mode smax as ollows:
Appl(Π, C, smax) ={R∈Appl(Π, C, se )| he e is no R0∈Appl(Π, C, se )
such ha R0⊃R}
The smax-de i a ion mode co esponds o he min1-mode wi h he disc e e pa -
i ioning o ules (each ule o ms i s own pa i ion), see [7].
The de i a ion mode max ulesmax is a special a ian whe e only a maximal
se o ules is allowed o be applied. Bu i can also seen as he a ian o he basic
se mode whe e we jus ake a se o applicable ules wi h he maximal numbe o
ules in in i , hence, we will also call i he max ule de i a ion mode. Fo mally we
ha e:
Appl(Π, C, max ule) ={R∈Appl(Π, C, se )| he e is no R0∈Appl(Π, C, se )
such ha |R0|>|R|}
As usual, wi h all hese a ian s o de i a ion modes as de ined abo e, we
conside hal ing compu a ions. We may gene a e o accep o e en compu ing
unc ions o ela ions. The inpu s/ou pu s may be mul ise s o s ings, de ined in
he well-known way.
2.2 The His o y o he smax-De i a ion Mode
In [13], a pape on as P sys ems simula o s using FPGA, he p oblem o he
unbounded max-mode was conside ed as oo di icul o be pa allelized on his
ha dwa e. In he ques o an e icien solu ion, he au ho s p oposed o es ic o
he case o he maximal pa allelism whe e each ule can be applied a mos once.
The mos impo an ad an age o his a ian was ha he mul ise o applicable
ules could be ep esen ed as a bina y s ing, i.e., an encoding as a numbe . Mo e-
o e , he pape showed ha in many in e es ing cases i is possible o ep esen
he language o co esponding bina y s ings a each s ep by an au oma on. Then
he p oblem o he simula ion o a P sys em could be sol ed as ollows:
62 A. Alhazo , R. F eund, and S. Ve lan
•Find he size So he se o mul ise s o applicable ules ( he size o he language
o bina y s ings).
•Take a andom numbe k∈ {1..S}and chose he s ing ep esen ing k.
This algo i hm allowed o ob ain a speed-up o magni ude 105.
The ad an ages o he se -maximally pa allel de i a ion mode o e he un-
bounded maximally pa allel de i a ion mode a e:
•A compac ep esen a ion o he applicable mul ise s o ules as bina y
s ings/numbe s is ob ained.
•Mos o he compu a ional comple eness esul s s ill hold.
•Simple analysis o he beha io is possible.
•Only a bounded numbe o (mul i)se s o ules has o be compu ed o each
de i a ion s ep.
In [10], he se -maximally pa allel de i a ion mode was called la maximal
pa allel de i a ion mode, and, o example, P sys ems wi h p omo e s a e shown
o be compu a ionally comple e using his la maximal pa allel de i a ion mode
wi h non-coope a i e ules.
2.3 Examples
In he maximally pa allel mode, we in addi ion need a ge o ule o label ag ee-
men o ob ain a2n|n≥0, o he wise only {an|n≥1}can be ob ained.
1
en i onmen (0)
2
Ini ial mul ise : a
1 : a→a(he e)a(he e)
2 : a→a(in)
a ge / ule/ label ag eemen :
he same ule is used o all symbols a
Fig. 1. Example o a P sys em.
Maximal Va ian s o he Se De i a ion Modes 63
In he se -maximally pa allel mode smax, we in addi ion need a ge o ule
o label ag eemen o ob ain {an|n≥1}, o he wise only {a}can be ob ained,
because:
•I 2 : a→a(in) is used in he i s s ep, hen ais ob ained.
•I 1 : a→a(he e)a(he e) is applied a leas once, hen om he second s ep
on i has o be applied in ini ely o en, as only one copy o acan be sen in o
memb ane 2 by he second ule 2 : a→a(in).
The same a gumen s hold o he de i a ion mode max ule.
3 Sympo /An ipo P Sys ems
Asympo /an ipo P sys em is a cons uc
Π= (O, E, µ, w0, w1, . . . , wm, R1, . . . , Rm, O, I) whe e
•Ois he alphabe o objec s,
•E⊆Ois he se o objec s being a ailable in he en i onmen in an unbounded
numbe ,
•µis he memb ane s uc u e (wi h mmemb anes),
•w0is he ini e mul ise o objec s o e O Ep esen in he en i onmen a
he beginning o a compu a ion,
•w1, . . . , wma e he mul ise s o objec s p esen in he m egions o µa he
beginning o a compu a ion,
•R1, . . . , Rma e ini e se s o sympo and/o an ipo ules, associa ed wi h
he memb anes o µ,
• O, Iis he label o he memb ane egion om which he ou pu s a e aken/ he
inpu s a e pu in.
E e y ule is o he o m (u, ou ; , in) wi h u, ∈O∗and u 6=λ; i u=λ
o =λ hen his ule is called a sympo ule, o he wise i is called an an ipo
ule. The applica ion o a ule (u, ou ; , in)∈Rimeans sending ou u om egion
iand aking in o i om he su ounding egion.
Fo (u, ou ; , in), max {|u|,| |} is called i s weigh and |u |is called i s size;
ob iously, o sympo ules weigh and size a e he same.
The amilies o se s Yγ,δ (Π), Y∈ {N, Ps},δ∈ {gen, acc}, and γ∈
{sequ, asyn, max, smax, max ule, . . . }, compu ed by sympo /an ipo P sys ems
wi h a mos mmemb anes, sympo ules wi h maximal weigh as well
as an ipo ules wi h maximal weigh wand maximal size sa e deno ed by
Yγ,δOPm(sym , an iw,s).
3.1 Accep ing An ipo P Sys ems
Theo em 1. Fo Y∈ {N, Ps},β∈ {max, smax, max ule},
Yβ,accDOPm(an i2,3) = Y RE.

64 A. Alhazo , R. F eund, and S. Ve lan
P oo . Le M= (m, B, l0, lh, P) be an a bi a y de e minis ic egis e machine.
We now cons uc an an ipo P sys em simula ing M. The numbe in egis e
is ep esen ed by he co esponding numbe o symbol objec s o .
•An ADD-ins uc ion p: (ADD( ), q) is simula ed by he ule (p, ou ;o q, in).
•A SUB-ins uc ion p: (SUB( ), q, s) is simula ed by he ollowing ules
1. (p, ou ;p0p00, in);
2. (p0, ou ; ˜p, in) as well as (p00o , ou ; ¯p, in) which is execu ed in pa allel i and
only i he egis e is no emp y;
3. (˜pp00, ou ;s, in) (i egis e was emp y),
(˜p¯p, ou ;q, in) (i egis e was no emp y).
As can be seen immedia ely, in each s ep only di e en ules can be applied,
each o hem only once. Hence, he p oo elabo a ed o he max-mode li e ally
also wo ks o he de i a ion modes smax and max ule wi hou any es ic ions
as well. u
4 P Sys ems wi h An i-Ma e
Fo any objec a(ma e ), we conside i s an i-objec (an i-ma e ) a−and he
co esponding (coope a i e) annihila ion ule aa−→λ. This ule is assumed o
exis in all memb anes.
In he ollowing, we assume hese annihila ion ules o ha e (weak) p io i y o e
all o he ules, i.e., o he ules may only be applied i objec s canno be bound by
an annihila ion ule any mo e.
This ype o ules is abb e ia ed by an im/p i, indica ing ma e /an i-ma e
annihila ion ules ha ing weak p io i y. Fo u he esul s we e e o [1].
4.1 Ma e /An i-Ma e Annihila ion Rules Ha ing P io i y
The ma e /an i-ma e annihila ion ules a e so powe ul ha we only need he
minimum numbe o ca alys s, i.e., ze o (ca (0) = ncoo).
Theo em 2. [1] Fo any n≥1,Y∈ {N, Ps},δ∈ {gen, acc, au },α∈ {acc, au },
Z∈ {Fun, Rel}, and β∈ {max, smax, max ule},
Yβ,δOPn(ncoo, an im/p i) = Y RE and
ZYβ,αOPn(ncoo, an im/p i) = ZY RE.
4.2 De e minis ic Ma e /An i-Ma e Accep ing P Sys ems
In he accep ing case, we can e en simula e he ac ions o a de e minis ic egis e
machine in a de e minis ic way, i.e., o each con igu a ion o he sys em, he e
can be a mos one mul ise o ules applicable o i . Ye he p oo exhibi ed in
[1], e en ul ills he condi ion ha e e y ule is only applied a mos once.
Maximal Va ian s o he Se De i a ion Modes 65
Theo em 3. Fo any n≥1,Y∈ {N, Ps}, and β∈ {max, smax, max ule},
Yβ,de accOPn(ncoo, an im/p i) = Y RE and
FunYβ,de accOPn(ncoo, an im/p i) = F unY RE.
P oo . We only show how he SUB-ins uc ions o a egis e machine M=
(m, B0, l0, lh, P) can be simula ed in a de e minis ic way wi hou in oducing a
ap symbol and he e o e causing in ini e loops by hem:
Le B={l|l: (SUB ( ), l0, l00)∈P}and, o e e y egis e ,
˜
M =n˜
l|l: (SUB ( ), l0, l00)∈Po,
˜
M −=n˜
l−|l: (SUB ( ), l0, l00)∈Po,
ˆ
M =nˆ
l|l: (SUB ( ), l0, l00)∈Po,
ˆ
M −=nˆ
l−|l: (SUB ( ), l0, l00)∈Po.
We now ake he ules a −→˜
M −ˆ
M and he annihila ion ules a a −→λ
o e e y egis e as well as ˆ
lˆ
l−→λand ˜
l˜
l−→λ o all l∈B. Then a SUB-
ins uc ion l1: (SUB ( ), l2, l3), wi h l1∈B,l2, l3∈B0, 1 ≤ ≤m, is simula ed
by
l1→¯
l1a
−,
¯
l1→ˆ
l1−(˜
M {˜
l1}),
ˆ
l1−→l2(˜
M − {˜
l1−}), and
˜
l1−→l3(ˆ
M − {ˆ
l1−}).
The symbol ˆ
l1−gene a ed by he second ule is elimina ed again and eplaced
by ˜
l1−i a −is no annihila ed.
Again, he p oo elabo a ed o he max-mode li e ally also wo ks o he
de i a ion modes smax and max ule wi hou any es ic ions as well. u
5 Ca aly ic and Pu ely Ca aly ic P Sys ems
We now in es iga e p oo s elabo a ed o ca aly ic and pu ely ca aly ic P sys ems
wo king in he max-mode o he smax-mode.
5.1 Compu a ional Comple eness o Ca aly ic P Sys ems
We i s check he cons uc ion o simula ing a egis e machine M=
(d, B, l0, lh, R) by a ca aly ic P sys em Π, wi h m≤dbeing he numbe o dec e-
men able egis e s, elabo a ed in [3] o he max-mode, and a gue why i wo ks
o he smax-mode, oo.
Fo all d egis e s, nicopies o he symbol oia e used o ep esen he alue
niin egis e i, 1 ≤i≤d. Fo each o he mdec emen able egis e s, we ake
66 A. Alhazo , R. F eund, and S. Ve lan
a ca alys ciand wo speci ic symbols di, ei, 1 ≤i≤m, o simula ing SUB-
ins uc ions on hese egis e s. Fo e e y l∈B, we use pl, and also i s a ian s
¯pl,ˆpl,˜pl o l∈BSUB, whe e BSUB deno es he se o labels o SUB-ins uc ions.
Π= (O, C, µ = [ ]1, w1=c1. . . cmd1. . . dmp1w0, R1, = 1),
O=C∪D∪E∪Σ
∪ {#}∪{pl|l∈B}∪{¯pl,ˆpl,˜pl|l∈BSUB},
C={ci|1≤i≤m},
D={di|1≤i≤m},
E={ei|1≤i≤m},
Σ={oi|1≤i≤d},
R1={pj→o pkDm, pj→o plDm|j: (ADD( ), k, l)∈R}
∪ {pj→ˆpje Dm, , pj→¯pjDm, ,ˆpj→˜pjD0
m, ,
¯pj→pkDm,˜pj→pkDm|j: (SUB( ), k, l)∈R}
∪ {c o →c d , c d →c , c ⊕m1e →c ⊕m1|1≤ ≤m},
∪ {d →#, c e →c #|1≤ ≤m}
∪ {#→#}.
He e ⊕m1 o < m simply is + 1, whe eas o =mwe de ine m⊕m1 = 1;
w0s ands o addi ional inpu p esen a he beginning.
Usually, e e y ca alys ci,i∈ {1, . . . , m}, is kep busy wi h he symbol di
using he ule cidi→ci, as o he wise he symbols diwould ha e o be apped by
he ule di→#, and he ap ule # →# hen en o ces an in ini e non-hal ing
compu a ion.
In he smax-de i a ion mode only one ap ule #→#will be ca ied
ou , bu his is he only di e ence!
Only du ing he simula ion o SUB-ins uc ions on egis e he co esponding
ca alys c is le ee o dec emen ing o o ze o-checking in he second s ep o
he simula ion, and in he dec emen case bo h c and i s “coupled” ca alys c ⊕m1
a e needed o be ee o speci ic ac ions in he hi d s ep o he simula ion.
Fo he simula ion o ins uc ions, we use:
Dm=Qi∈[1..m]di,
Dm, =Qi∈[1..m] { }di,
D0
m, =Qi∈[1..m] { , ⊕m1}di.
The HALT-ins uc ion labeled lhis simply simula ed by no in oducing he
co esponding s a e symbol plh, i.e., eplacing i by λ, in all ules de ined in R1.
Each ADD-ins uc ion j: (ADD( ), k, l), o ∈ {1, . . . , d}, can easily be
simula ed by he ules pj→o pkDmand pj→o plDm; in pa allel, he ules
cidi→ci, 1 ≤i≤m, ha e o be ca ied ou , as o he wise he symbols diwould
ha e o be apped by he ules di→#.
Each SUB-ins uc ion j: (SUB( ), k, l), is simula ed as shown in he able
lis ed below ( he ules in b acke s [ and ] a e hose o be ca ied ou in case o a
w ong choice):
Maximal Va ian s o he Se De i a ion Modes 67
Simula ion o he SUB-ins uc ion j: (SUB( ), k, l) i
egis e is no emp y egis e is emp y
pj→ˆpje Dm, pj→¯pjDm,
c o →c d [c e →c #] c should s ay idle
ˆpj→˜pjD0
m, ¯pj→pkDm
c d →c [d →#] [d →#]
˜pj→pkDm
c ⊕m1e →c ⊕m1
In he i s s ep o he simula ion o each ins uc ion (ADD-ins uc ion, SUB-
ins uc ion, and e en HALT-ins uc ion) due o he in oduc ion o Dmin he
p e ious s ep (we also s a wi h ha in he ini ial con igu a ion) e e y ca alys
c is kep busy by he co esponding symbol d , 1 ≤ ≤m.
Based on he cons uc ion elabo a ed in [3] and ecalled abo e in sum we ha e
ob ained he ollowing esul :
Theo em 4. Fo any egis e machine M= (d, B, l0, lh, R), wi h m≤dbeing he
numbe o dec emen able egis e s, we can cons uc a ca aly ic P sys em
Π= (O, C, µ = [ ]1, w1, R1, = 1)
wo king in he max- o he smax-de i a ion mode and simula ing he compu a ions
o Msuch ha
|R1| ≤ ADD1(R)+2×ADD2(R)+5×SUB(R)+5×m+ 1,
whe e ADD1(R)deno es he numbe o de e minis ic ADD-ins uc ions in R,
ADD2(R)deno es he numbe o non-de e minis ic ADD-ins uc ions in R, and
SUB(R)deno es he numbe o SUB-ins uc ions in R.
5.2 Compu a ional Comple eness o Pu ely Ca aly ic P Sys ems
Fo he pu ely ca aly ic case, one addi ional ca alys cm+1 is needed o be used
wi h all he non-coope a i e ules. Un o una ely, in his case a sligh ly mo e
complica ed simula ion o SUB-ins uc ions is needed, a esul es ablished in [12],
whe e o ca aly ic P sys ems
|R1| ≤ 2×ADD1(R)+3×ADD2(R)+6×SUB(R)+5×m+ 1,
and o pu ely o ca aly ic P sys ems
|R1| ≤ 2×ADD1(R)+3×ADD2(R)+6×SUB(R)+6×m+ 1,
is shown. Ye also his p oo li e ally wo ks o he smax-de i a ion mode as well,
wi h he only excep ion ha he ap ule # →# is ca ied ou a mos once.
74 A. Alhazo , R. F eund, and S. Ve lan
In he ollowing sec ions, we now u n ou a en ion o models o P sys ems
whe e he de i a ion mode smax yields di e en , in ac , s onge esul s han he
de i a ion mode max.
8 A omic P omo e s and Inhib o s
As shown in [11], P sys ems wi h non-coope a i e ules and a omic inhibi o s
a e no compu a ionally comple e when he maximally pa allel de i a ion mode
is used. P sys ems wi h non-coope a i e ules and a omic p omo e s can a leas
gene a e PsET0L. On he o he hand, al eady in [10], he compu a ional com-
ple eness o P sys ems wi h non-coope a i e ules and a omic p omo e s has been
shown. In he ollowing we will es ablish a new p oo o he simula ion o a egis-
e machine whe e he o e all numbe o p omo e s only depends on he numbe
o dec emen able egis e s o he egis e machine. Mo eo e , we also show a new
p e y su p ising esul , es ablishing compu a ional comple eness o P sys ems
wi h non-coope a i e ules and a omic inhibi o s, and he numbe o inhibi o s
again only depends on he numbe o dec emen able egis e s o he simula ed
egis e machine. Finally, in bo h cases, i he egis e machine is de e minis ic,
hen he P sys em is de e minis ic, oo.
8.1 A omic P omo e s
We now es ablish ou new p oo o he compu a ional comple eness o P sys ems
wi h non-coope a i e ules and a omic p omo e s when using he de i a ion mode
smax; he o e all numbe o p omo e s only is 5mwhe e mis he numbe o
dec emen able egis e s o he simula ed egis e machine.
Theo em 8. Fo any egis e machine M= (d, B, l0, lh, R), wi h m≤dbeing
he numbe o dec emen able egis e s, we can cons uc a P sys em wi h a omic
inhibi o s
Π= (O, µ = [ ]1, w1=l0, R1, = 1)
wo king in he smax- o max ule-de i a ion mode and simula ing he compu a ions
o Msuch ha
|R1| ≤ ADD1(R)+2×ADD2(R)+5×SUB(R)+7×m,
whe e ADD1(R)deno es he numbe o de e minis ic ADD-ins uc ions in R,
ADD2(R)deno es he numbe o non-de e minis ic ADD-ins uc ions in R, and
SUB(R)deno es he numbe o SUB-ins uc ions in R; mo eo e , he numbe o
a omic inhibi o s is 5m. Finally, i he egis e machine is de e minis ic, hen he
P sys em is de e minis ic, oo.

Maximal Va ian s o he Se De i a ion Modes 75
P oo . The numbe s o objec s o ep esen he con en s o he egis e s , 1 ≤
≤d; mo eo e , we deno e BSUB ={p|p: (SUB( ), q, s)∈R}.
O={o |1≤ ≤d}∪{o0
, c , c0
, c00
, c000
|1≤ ≤m}
∪(B {lh})∪ {p0, p00, p000 |p∈BSUB}
The symbols om {o0
, c , c0
, c00
, c000
|1≤ ≤m}a e used as p omo e s.
An ADD-ins uc ion p: (ADD( ), q, s) is simula ed by he wo ules p→qo
and p→so .
A SUB-ins uc ion p: (SUB( ), q, s) is simula ed in ou s eps as ollows:
1. p→p0c ;
2. p0→p00c0
;o →o0
|c ,c →λ;
3. p00 →p000c000
,c0
→c00
|o0
,o0
→λ;
4. p000 →q|c00
,p000 →s|c0
,c0
→λ|c000
,c00
→λ,c000
→λ.
As inal ule we could use lh→λ, ye we can omi his ule and eplace e e y
appea ance o lhin all ules as desc ibed abo e by λ.u
8.2 A omic Inhib o s
We now show ha e en P sys ems wi h non-coope a i e ules and a omic p omo -
e s using he de i a ion mode smax can simula e any egis e machine needing
only 2m+ 1 inhibi o s whe e mis he numbe o dec emen able egis e s o he
simula ed egis e machine.
Theo em 9. Fo any egis e machine M= (d, B, l0, lh, R), wi h m≤dbeing
he numbe o dec emen able egis e s, we can cons uc a P sys em wi h a omic
inhibi o s
Π= (O, µ = [ ]1, w1=l0, R1, = 1)
a P sys em wi h a omic inhibi o s Π= (O, µ = [ ]1, w1=l0, R1, = 1) wo king
in he smax- o max ule-de i a ion mode and simula ing he compu a ions o M
such ha
|R1| ≤ ADD1(R)+2×ADD2(R)+5×SUB(R)+3×m+ 1,
whe e ADD1(R)deno es he numbe o de e minis ic ADD-ins uc ions in R,
ADD2(R)deno es he numbe o non-de e minis ic ADD-ins uc ions in R, and
SUB(R)deno es he numbe o SUB-ins uc ions in R; mo eo e , he numbe o
a omic inhibi o s is 2m+ 1. Finally, i he egis e machine is de e minis ic, hen
he P sys em is de e minis ic, oo.
P oo . The numbe s o objec s o ep esen he con en s o he egis e s , 1 ≤
≤d. The symbols d p e en he egis e symbols o , 1 ≤ ≤m, om e ol ing.
76 A. Alhazo , R. F eund, and S. Ve lan
O={o |1≤ ≤d}∪{o0
|1≤ ≤m}∪{d |0≤ ≤m}
∪(B {lh})∪ {p0, p00,˜p|p∈BSUB}
We deno e D=Qm
i=1 diand D =Qm
i=1,i6= di.
An ADD-ins uc ion p: (ADD( ), q, s) is simula ed by he wo ules p→qo D
and p→so D.
A SUB-ins uc ion p: (SUB( ), q, s) is simula ed in ou s eps as ollows:
1. p→p0D ;
2. p0→p00Dd0; in pa allel, he ollowing ules a e used:
o →o0
|¬d ,dk→λ, 1 ≤k≤m;
3. p00 →˜pD |¬o0
;o0
→λ,d0→λ;
again, in pa allel he ules dk→λ, 1 ≤k≤m, a e used;
4. p00 →qD |¬d0, ˜p→sD.
As inal ule we could use lh→λ, ye we can omi his ule and eplace e e y
appea ance o lhin all ules as desc ibed abo e by λ.u
9 P Sys ems wi h Ta ge Selec ion
In P sys ems wi h a ge selec ion, all objec s on he igh -hand side o a ule mus
ha e he same a ge , and in each de i a ion s ep, o each egion a (mul i)se o
ules – non-emp y i possible – ha ing he same a ge is chosen. We show ha
o P sys ems wi h a ge selec ion in he de i a ion mode smax no ca alys is
needed any mo e, and wi h max ule, we e en ob ain a de e minis ic simula ion o
de e minis ic egis e machines.
Theo em 10. Fo any egis e machine M= (d, B, l0, lh, R), wi h m≤dbeing
he numbe o dec emen able egis e s, we can cons uc a P sys em wi h non-
coope a i e ules wo king in he smax-de i a ion mode and simula ing he compu-
a ions o M.
P oo . As usual, we ake an a bi a y egis e machine Mwi h d egis e s sa -
is ying he ollowing condi ions: he ou pu egis e s a e m+ 1,· · · , d, and hey
a e ne e dec emen ed; mo eo e , egis e s 1,· · · , m a e emp y in any eachable
hal ing con igu a ion. Clea ly, hese condi ions do no es ic he gene ali y. We
cons uc he ollowing P sys em Πsimula ing M.
The co ec beha io o he objec associa ed o he simula ed ins uc ion o
Mis he ollowing. In he dec emen case, we ha e in + 2, ou ,in2, idle, ou ,in2,
he e,ou ,he e (9 s eps in o al), whe eas in he ze o- es case, we ha e he same
as be o e, excep ha he ou h and he i h s eps a e ou and he e ins ead o idle
and ou , espec i ely. In case o an inc emen ins uc ion, we ge he e,he e,he e,
he e,in2,he e,ou ,he e (8 s eps in o al). We ema k ha he i s ou s eps
a e ca ied ou in he skin, while he las ou s eps epea he cases o ze o- es
and dec emen .
Maximal Va ian s o he Se De i a ion Modes 77
The alue o each egis e is ep esen ed by he mul iplici y o objec s o in he
skin. Fo e e y dec emen able egis e , he e is a ule sending o in o egion +2.
Howe e , his ule may only be applied sa ely in he i s s ep o he simula ion o
he SUB ins uc ion, as o he wise some o he objec will also en e he same egion
as # (ei he one o e,e0,e00, ˆe, ˆe0, which we will in he ollowing e e o as he
gua ds, o an objec associa ed o he label o he simula ed ins uc ion, which we
will in he ollowing call a p og am symbol) o cing an unp oduc i e compu a ion,
see he ules in b acke s in he ables below.
The “co ec ” a ge selec ion o he inne egions no mally coincides wi h
ha o he p og am symbol (desc ibed abo e) and no ule is applied he e i
he p og am symbol is no he e, wi h he ollowing excep ions. In he i s s ep
o simula ing an ins uc ion, objec eexi s memb ane 2, as i is he only ule
applicable he e in his s ep. In he las s ep o simula ing an ins uc ion, objec
¯eis ew i en in o ein memb ane 2, as i is he only ule applicable he e in his
s ep. In he ou h s ep o he dec emen case, he p og am symbol is idle while
objec dis e ased. The “co ec ” a ge selec ion o he skin coincides wi h ha
o he p og am symbol, and is he e i he p og am symbol is missing in he skin.
78 A. Alhazo , R. F eund, and S. Ve lan
Π= (O, µ, w1,· · · , wm+2, R1,· · · , Rm+2) whe e
O={o |1≤ ≤d}∪{¯p, p |p∈B}∪{p0, p00 ˆp|p∈BADD}
∪ {p0, p−, p0
−, p0, p0
0, p00
0|p∈BSUB}∪{¯e, e, e0, e00,ˆe, ˆe0, d, #},
µ= [ [ ]2· · · [ ]m+2 ]1,
w1=l0, w2=e, w +2 =λ, 1≤ ≤m,
R1=
m+2
[
i=1
(R1,i,s ∪R1,i,#),
Ri=Ri,1,s ∪Ri,1,#∪Ri,i,s ∪Ri,i,#,2≤j≤m+ 2,
R1,1,s ={e→e0, e0→e00, e00 →ˆe, ˆe→ˆe0, e0→λ}
∪ {p0
0→p00
0|p∈BSUB}∪{¯p→p|p∈B}
∪ {p→˜po |p: (ADD( ), q, s)∈P}
∪ {˜p→p0, p0→p00, p00 →ˆp|p∈BADD},
R1,2,s ={p0→(p−, in2), p0→(p0, in2), p0
−→(p0
−, in2), p00
0→(p00
0, in2)
|p∈BSUB}∪{ˆp→(ˆp, in2)|p∈BADD}∪{d→(d, in2)}
R1, +2,s ={o →(o , in +2)}∪{p→(p, in +2)
|p: (SUB( ), q, s)∈P},1≤ ≤m,
R1,1,#={p0→#, p00
0→#, p0
−→#|p∈BSUB}∪{ˆp→#|p∈BADD}
∪ {#→#},
R1,2,#={p0
0→(#, in2), e00 →(#, in2)|p∈BSUB}
∪ {¯p→(#, in2)|p∈B},
R1, +2,#={x→(#, in +2} | x∈ {e, e0, e00,ˆe, ˆe0}
∪ {p0
0, p0
−|P∈BSUB}∪{¯p|P∈B}}
∪ {p→(#, in +2)|p: (SUB(i), q, s)∈P, i 6= }
∪ {p0→(#, in +2)|p∈BSUB},1≤ ≤m,
R2,1,s ={e→(e, ou )}∪{¯p→(¯p, ou )|p∈B}
∪ {p0→(p0
0, ou ), p−→(p0
−, ou )|p∈BSUB},
R2,2,s ={d→λ, ¯e→e} ∪ {| p∈B}
∪ {p00
0→¯s¯e, p0
−→¯q¯e|p: (SUB( ), q, s)∈P}
∪ {ˆp→¯q¯e, ˆp→¯s¯e|p: (ADD( ), q, s)∈P},
R2,1,#={d→(#, ou ),#→(#, ou )},
R2,2,#={p0→#|p∈BSUB}∪{¯p→#|p∈B},
R +2,1,s ={p→(p0, ou )|p∈BSUB}∪{o →(d, ou },1≤ ≤m
R +2, +2,#={#→(#, ou )}, R +1, +1,s =R +1, +1,#=∅.
Mos apping ules, gi en in b acke s in he ables below and lis ed in ule
g oups Ri,j,#abo e, a e only needed o o ce he “co ec ” a ge selec ion. The
excep ion a e some ules in s eps 4 and 5 o he simula ion o SUB ins uc ions,
Maximal Va ian s o he Se De i a ion Modes 79
needed o e i ying ha he dec emen and he ze o es ha e been pe o med
co ec ly ( he guess is made a s ep 3 by he p og am symbol, and is e lec ed
in i s subsc ip ). Indeed, i he ze o- es is chosen while dis p esen (signi ying
ha he egis e was dec emen ed), causing a a ge con lic : ei he p0o dwill
be anyway ew i en in o #. Howe e , i he dec emen is chosen while dis absen
(signi ying ha he egis e was ze o), hen p−will appea in he skin in s ep 4
ins ead o s ep 5, causing a a ge con lic : ei he p0
−o e00 will be anyway ew i en
in o #.
Below we p esen he ables desc ibing he simula ion o ins uc ions o M. An
applica ion o one o he ules gi en in b acke s leads o non-hal ing compu a ions,
no con ibu ing o he esul .
(p: (SUB( ), q, s))
+ 2 1 2
1 - o →(o , in +2)e→(e, ou )
-p→(p, in +2)
[p→(#, ini+2), i6= ]
2p→(p0, ou )e→e0-
o →(d, ou ) [e→(#, ini+2)]
3 - p0→(p−, in2) -
p0→(p0, in2)
d→(d, in2)
[p0→#]
[e0→(#, ini+2)]
1,- 1,0 2,- 2,0
4e0→e00 d→λ p0→(p0
0, ou )
[p−→(p0
−, ou )] [d→(#, ou )]
[p0→#]
5e00 →ˆe p0
0→p00
0p−→(p0
−, ou ) -
[p0
−→(p0
−, in2)] e00 →ˆe
[p0
−→#] [p0
0→(#, in )]
[e00 →(#, in )] [e00 →(#, in )]
[ o > 1] [ o > 1]
6p0
−→(p0
−, in2)p00
0→(p00
0, in2) -
[p0
−→#] [p00
0→#]
[p0
−→(#, ini+2)] [p00
0→(#, ini+2)]
7 ˆe→ˆe0p0
−→¯q¯e p00
0→¯s¯e
[ˆe→(#, ini+2)]
8 ˆe0→λ¯q→(¯q, ou ) ¯s→(¯s, ou )
[ˆe0→(#, ini+2)] [¯q→#] [¯s→#]
9 ¯q→q¯s→s¯e→e
[¯q→(#, in )] [¯s→(#, in )]

80 A. Alhazo , R. F eund, and S. Ve lan
(p: (ADD( ), q, s))
1 2
1p→˜po e→(e, ou )
2 ˜p→p0-
e→e0
3p0→p00 -
e0→e00
4p00 →ˆp-
e00 →ˆe
5 ˆp→(ˆp, in2) -
[ˆp→#]
6 ˆe→ˆe0ˆp→¯x¯e
7 ˆe0→λ¯x→(¯x, ou )
[¯x→#]
8 ¯x→x¯e→e
Auxilia y ules
+ 2 1 2
[# →(#, ou )] [# →#] [# →(#, ou )]
Nea ly hal o he s eps in he p eceding cons uc ions is needed o eleasing
he auxilia y symbol ein he i s s ep o a simula ion om memb ane 2, ye in
ou cons uc ion, eand i s de i a i es a e needed o con ol he co ec a ge
selec ion in he skin memb ane, and especially o keep he egis e objec s o om
mo ing in o memb ane + 2. u
We now show ha aking he maximal se s o ules which a e applicable, he
simula ion o SUB-ins uc ions can e en be ca ied ou in a de e minis ic way.
Theo em 11. Fo any egis e machine M= (d, B, l0, lh, R), wi h m≤dbeing
he numbe o dec emen able egis e s, we can cons uc a P sys em wi h non-
coope a i e ules
Π= (O, µ = [ [ ]2. . . [ ]2m+1 ]1, w1, λ, . . . , λ, R1. . . R2m+1, = 1)
wo king in he max ule-de i a ion mode and simula ing he compu a ions o M
such ha
|R1| ≤ 1×ADD1(R)+2×ADD2(R)+4×SUB(R) + 10 ×m+ 3,
whe e ADD1(R)deno es he numbe o de e minis ic ADD-ins uc ions in R,
ADD2(R)deno es he numbe o non-de e minis ic ADD-ins uc ions in R, and
SUB(R)deno es he numbe o SUB-ins uc ions in R.
Maximal Va ian s o he Se De i a ion Modes 81
P oo . The con en s o he egis e s , 1 ≤ ≤dis ep esen ed by he numbe s o
objec s o , and o he dec emen able egis e s we also use a copy o he symbol o0
o each copy o he objec o . This second copy o0
is needed du ing he simula ion
o SUB-ins uc ions o be able o dis inguish be ween he dec emen and he ze o
es case. Fo each , he wo objec s o and o0
can only be a ec ed by he ules
o →(λ, in +1) and o0
→(λ, in +1) sending hem in o he memb ane + 1
co esponding o memb ane (and a he same ime e asing hem; in ac , we
could also lea e hem in he memb ane una ec ed o e e as a ga bage). These a e
al eady wo ules, so any o he combina ion o ules wi h di e en a ge s has o
con ain a leas h ee ules.
One o he main ideas o he p oo cons uc ion is ha in he skin mem-
b ane he label po an ADD-ins uc ion is ep esen ed by he h ee objec s pand
e, e0, and he label po any SUB-ins uc ion is ep esen ed by he eigh objec s
p, e, e0, e00, d , d0
,˜
d ,˜
d
0. Hence, o each p∈(B {lh}) we de ine R(p) = pee0 o
p∈BADD and R(p) = pee0e00d d0
˜
d ˜
d
0 o p∈BSUB as well as R(lh) = λ; as
ini ial mul ise w1in he skin memb ane, we ake R(l0).
O={o |1≤ ≤d}∪{o0
|1≤ ≤m} ∪ (B {lh})
∪nd , d0
,˜
d ,˜
d
0|1≤ ≤mo∪ {e, e0, e00}
An ADD-ins uc ion p: (ADD( ), q, s) is simula ed by he ules p→R(q)o
and p→R(s)o as well as he ules e→λand e0→λ. This combina ion o h ee
ules supe cedes any combina ion o ules o →(λ, in +1) and o0
→(λ, in +1), o
some 1 ≤ ≤m.
A SUB-ins uc ion p: (SUB( ), q, s) is simula ed in wo s eps as ollows:
1. In R1, o he i s s ep we ake one o he ollowing uple o ules
p→(p, in +1), d →(λ, in +1), d0
→(λ, in +1), ˜
d →(λ, in +1),
o →(λ, in +1), o0
→(λ, in +1);
p→(p, inm+ +1), d →(λ, inm+ +1), d0
→(λ, inm+ +1),
˜
d →(λ, inm+ +1), ˜
d
0→(λ, inm+ +1);
he applica ion o he ules o →(λ, in +1), o0
→(λ, in +1) in con as o he
applica ion o he ule ˜
d
0→(λ, inm+ +1) de e mines whe he he i s o he
second uple o ules has o be chosen. He e i becomes clea why we ha e o
use he wo egis e symbols o and o0
, as we ha e o gua an ee ha he a ge
+ 1 canno be chosen i none o hese symbols is p esen , as in his case hen
only ou ules could be chosen in con as o he i e ules o he ze o es
case. On he o he hand, i some o hese symbols o and o0
a e p esen , hen
six ules a e applicable supe ceding he i e ules which could be used o he
ze o es case.
2. In he second s ep, he ollowing h ee o ou ules, again supe ceding any
combina ion o ules o →(λ, in +1) and o0
→(λ, in +1) o some 1 ≤ ≤m,
a e used in he skin memb ane:
e→λ,e0→λ,e00 →λ, and in he dec emen case also he ule ˜
d
0→λ.
82 A. Alhazo , R. F eund, and S. Ve lan
In he second s ep, we ei he ind he he symbol pin memb ane + 1, i a
symbol o oge he wi h i s copy o0
has been p esen o dec emen ing o in
memb ane m+ + 1, i no symbol o has been p esen (ze o es case).
In he dec emen case, he ollowing ule is used in R +1:p→(R(q), ou ).
In he ze o es case, he ollowing ule is used in Rm+ +1:p→(R(s), ou ).
We inally poin ou ha he simula ion o he SUB-ins uc ions wo ks de e min-
is ically, hence, al hough he P sys em i sel is no de e minis ic syn ac icly, i
wo ks in a de e minis ic way i he unde lying egis e machine is de e minis ic.
u
10 Conclusion and Fu u e Wo k
I is no e y su p ising ha he p oo s we ha e checked in he p eceding sec ions
also wo k o he de i a ion mode smax, as many cons uc ions elabo a ed o he
de i a ion mode max jus “b eak down” maximal pa allelism o nea sequen iali y
in o de o wo k o he simula ion o egis e machines. On he o he hand, we
also ha e shown ha due o his ac some a ian s o P sys ems become e en
s onge wi h he modes smax and max ule.
•The e a e many models o P sys ems o which he maximally pa allel de i a-
ion mode has been used, especially o showing compu a ional comple eness.
•As we ha e seen by ca e ul inspec ion o se e al p oo s o compu a ional com-
ple eness, many esul s es ablished wi h using he maximally pa allel de i a ion
mode li e ally hold ue as well o he de i a ion modes smax and max ule.
•Many o he cons uc ions wo king in he maximally pa allel de i a ion mode
ha e o be checked ca e ully i hey wo k o he de i a ion modes smax and
max ule, oo.
•Fo some p oo s ha ing been es ablished in he maximally pa allel de i a ion
mode we migh need comple ely new p oo s o p oo echniques o he se -
maximally pa allel de i a ion mode; one such example is he p oo o P sys ems
wi h a ge selec ion.
•Some a ian s o P sys ems become e en s onge wi h he mode smax; as
al eady poin ed ou by Gheo ghe P˘aun, P sys ems wi h non-coope a i e ules
and a omic p omo e s a e compu a ionally comple e wi h he smax-mode, also
see [10], and in his pape we ha e shown a new p oo o his compu a ional
comple eness esul and e en shown a simila esul o P sys ems wi h non-
coope a i e ules and a omic inhibi o s.
•On he o he hand, e en ually, some esul s s ablished in he maximally pa allel
de i a ion mode a e no alid any mo e o he se -maximally pa allel de i a ion
mode.
Maximal Va ian s o he Se De i a ion Modes 83
Re e ences
1. A. Alhazo , B. Aman, R. F eund, and Gh. P˘aun. Ma e and an i-ma e in mem-
b ane sys ems. In P oceedings o he Twel h B ains o ming Week on Memb ane
Compu ing, pages 1–26, 2014.
2. A. Alhazo and R. F eund. P sys ems wi h oxic objec s. In M. Gheo ghe, G. Rozen-
be g, A. Salomaa, P. Sos´ık, and C. Zand on, edi o s, Memb ane Compu ing - 15 h
In e na ional Con e ence, CMC 2014, P ague, Czech Republic, Augus 20–22, 2014,
Re ised Selec ed Pape s, olume 8961 o Lec u e No es in Compu e Science, pages
99–125. Sp inge , 2014.
3. A. Alhazo and R. F eund. Small ca aly ic P sys ems. In M. J. Dinneen, edi o ,
P oceedings o he Wo kshop on Memb ane Compu ing 2015 (WMC2015), (Sa el-
li e wo kshop o UCNC2015), Augus 2015, olume CDMTCS-487 o CDMTCS Re-
sea ch Repo Se ies. Cen e o Disc e e Ma hema ics and Theo e ical Compu e ,
ScienceDepa men o Compu e Science Uni e si y o Auckland, Auckland, New
Zealand, 2015.
4. A. Alhazo , R. F eund, H. Heikenw¨alde , M. Oswald, Yu. Rogozhin, and S. Ve -
lan. Sequen ial P sys ems wi h egula con ol. In E. Csuhaj-Va j´u, M. Gheo ghe,
G. Rozenbe g, A. Salomaa, and Gy. Vaszil, edi o s, Memb ane Compu ing - 13 h In-
e na ional Con e ence, CMC 2012, Budapes , Hunga y, Augus 28-31, 2012, Re ised
Selec ed Pape s, olume 7762 o Lec u e No es in Compu e Science, pages 112–127.
Sp inge , 2013.
5. H. Bu kha d. O de ed i ing in pe i ne s. Elek onische In o ma ions e a bei ung
und Kybe ne ik, 17(2/3):71–86, 1981.
6. R. F eund and Gh. P˘aun. How o ob ain compu a ional comple eness in P sys ems
wi h one ca alys . In T. Nea y and M. Cook, edi o s, P oceedings Machines, Com-
pu a ions and Uni e sali y 2013, MCU 2013, Z¨u ich, Swi ze land, Sep embe 9-11,
2013, olume 128 o EPTCS, pages 47–61, 2013.
7. R. F eund and S. Ve lan. A o mal amewo k o s a ic ( issue) P sys ems. In
G. Ele he akis, P. Ke alas, Gh. P˘aun, G. Rozenbe g, and A. Salomaa, edi o s, Mem-
b ane Compu ing. 8 h In e na ional Wo kshop, WMC 2007 Thessaloniki, G eece,
June 25-28, 2007. Re ised Selec ed and In i ed Pape s, olume 4860 o Lec u e No es
in Compu e Science, pages 271–284. Sp inge , 2007.
8. P. F isco and G. Go an. P sys ems wi h ac i e memb anes ope a ing unde minimal
pa allelism. In M. Gheo ghe, Gh. P˘aun, G. Rozenbe g, A. Salomaa, and S. Ve -
lan, edi o s, Memb ane Compu ing - 12 h In e na ional Con e ence, CMC 2011,
Fon ainebleau, F ance, Augus 23-26, 2011, Re ised Selec ed Pape s, olume 7184
o Lec u e No es in Compu e Science, pages 165–181. Sp inge , 2011.
9. K. K i hi asan, Gh. P˘aun, and A. Ramanujan. On con olled P sys ems. In
L. Valencia-Cab e a, M. Ga c´ıa-Quismondo, L. Mac´ıas-Ramos, M. Ma ´ınez-del-
Amo , Gh. P˘aun, and A. Riscos-N´u˜nez, edi o s, P oceedings 11 h B ains o ming
Week on Memb ane Compu ing, Se illa, 4–8 Feb ua y 2013, pages 137–151. F´enix
Edi o a, Se illa, 2013.
10. L. Pan, Gh. P˘aun, and B. Song. Fla maximal pa allelism in P sys ems wi h p o-
mo e s. Theo e ical Compu e Science, 2015, o appea .
11. D. Sbu lan. Fu he esul s on P sys ems wi h p omo e s/inhibi o s. In e na ional
Jou nal o Founda ions o Compu e Science, 17(1):205–221, 2006.
12. P. Sos´ık and M. Lange . Small ca aly ic P sys ems simula ing egis e machines.
Theo e ical Compu e Science, accep ed, 2015.