Ramí ez-de-A ellano-Ma e o e al. 2025 | h ps://doi.o g/10.34133/icompu ing.0103 1
RESEARCH ARTICLE
Supe Vi us Machines: Fas e Vi us
T ansmission, Mo e E iciency
Using Supe channels
An onio Ramí ez-de-A ellano-Ma e o1,2*, Luis Valencia-Cab e a1,2,
Da id O ellana-Ma ín1,2, and Ma io J. Pé ez-Jiménez1,2
1Compu e Science and A i icial In elligence, Uni e si y o Se ille, Se ille, Spain. 2SCORE Labo a o y,
I3US, Uni e sidad de Se illa, Se ille, Spain.
*Add ess co espondence o: a ami ezdea [email protected]
Su passing he classical compu ing a chi ec u e is one o he g ea challenges o compu e science
oday. The b anch ha app oaches i om a heo e ical poin o iew, inspi ed by na u e, is called na u al
compu a ion. Wi hin his ield, a pa adigm a ises, called i us machines (VMs), inspi ed by he p opaga ion
and eplica ion o he biological s uc u e o i uses. This wo k in oduces a no el ex ension o he young
compu ing pa adigm o VMs, he supe VMs. This ex ension can de elop models wi h a new kind o
channel called supe channel. In addi ion, se e al VMs a e cons uc ed o gene a e na u al numbe se s
and compu e basic a i hme ic unc ions, imp o ing he basic VMs in bo h ime and memo y cos (such
as hos s and ins uc ions).
In oduc ion
The b anch o bioinspi ed compu ing [ 1 ] belongs o he ield o
uncon en ional compu ing [ 2 ], which is dedica ed o explo ing
compu ing models inspi ed by na u al phenomena and simula
-
ing a ious na u al s uc u es. I has been used no only o
heo e ical aspec s (compu a ional complexi y heo y h ough
hese de ices [ 3 ]) bu also o p ac ical scopes ( o ins ance,
popula ion dynamics modeling [ 4 ]) wi h biologic implemen a-
ions (such as cellula compu ing [ 5 ]).
Awa eness o he sp eading mechanisms and ange o e ec s
o i al in ec ions d ama ically inc eased du ing he COVID-
19 pandemic. This biological s uc u e has shown in e es ing
p ope ies, p omp ing explo a ion om a compu a ional pe -
spec i e. In iguing p ope ies, including he necessi y o a hos
o eplica ion, he eplica ion p ocess i sel , and he high mu a-
ion coe icien , can be s udied in he con ex o compu ing.
Fo a mo e in-dep h explo a ion o biological aspec s, e e o
Dimmock e al. [ 6 ].
A new compu a ional model, d awing inspi a ion om his
biological a chi ec u e, has been de eloped: i us machines
(VMs) [ 7 ], whe e Tu ing uni e sali y was i s p o ed by simu-
la ing egis e machines; o he echniques o p o e his can be
ound in he wo ks o Rome o-Jiménez e al. [ 8 , 9 ]. Wi hin his
compu ing amewo k, i uses p opaga e and eplica e among
hos s ia channels go e ned by an ins uc ion g aph. This
app oach o e s a compelling pe spec i e o ackling ma he-
ma ical challenges. As demons a ed in hei wo k [ 10 ], he
au ho s de ised de ices based on his model o compu ing
pai ing unc ions, wi h hei accu acy es ablished h ough ig-
o ous ma hema ical p oo . In addi ion, VMs ha e esul ed in
an in e es ing scope o a acking c yp osys ems, as he au ho s
p esen ed in a pape [ 11 ]. Mo eo e , in ano he wo k [ 12 ], he
au ho s p esen ed mul iple ins ances o VMs ha a e designed
o gene a e, compu e, and ecognize unc ions and se s o na u-
al numbe s; he la e also pa es he way o de eloping a new
heo e ical complexi y heo y using VMs. Recen ly, he au ho s
compa ed VMs wi h o he neu al-like models in ano he wo k
[ 13 ], mo e p ecisely wi h spiking neu al P sys ems [ 14 ].
Ne e heless, all he e e ences men ioned ha e he same
conclusion: he ime e iciency o VMs has o be enhanced. Fo
ins ance, he au ho s [ 15 ] p oposed pa allel VMs, whe e a se
o ins uc ions could be ac i a ed simul aneously; on he o he
hand, a ma ix ep esen a ion was de ined by Pé ez-Segu a e al.
[ 16 ] o u u e g aphics p ocessing uni implemen a ions. Tha
is why he e is an in e es ing scope in he s udy o possible
ex ensions in hese de ices.
F om a biological pe spec i e [ 17 ], he li e cycle o a i us
in ol es a speci ic se ies o complex p ocesses ailo ed o each
i us–hos in e ac ion, bu i ypically consis s o 5 s ages. I we
ocus on he second s age, ha is, he eplica ion o he genome,
hese eplica ions do no occu one by one; hey occu in pa al-
lel. I may be in e es ing o be inspi ed by his beha io , by
se ing ha all i uses om one hos can be eplica ed and
ansmi ed simul aneously o ano he hos .
In his wo k, an ex ension o hese compu ing models aking
his inspi a ion, called supe i us machines (SVMs), is p oposed
o enhance ime e iciency. The no el y o his ex ension is a new
kind o channels called supe channels, which allows mo e ans-
mission and eplica ion o i uses simul aneously. This simple bu
powe ul ex ension shows be e esul s in ime e iciency o
compu ing unc ions and o gene a ing na u al numbe se s.
Ci a ion: Ramí ez-de-A ellano-
Ma e oA, Valencia-Cab e aL,
O ellana-Ma ínD, Pé ez-JiménezMJ.
Supe Vi us Machines: Fas e Vi us
T ansmission, Mo e E iciency Using
Supe channels. In ell. Compu .
2025;4:A icle 0103. h ps://doi.
o g/10.34133/icompu ing.0103
Submi ed 15 Decembe 2023
Re ised 8 July 2024
Accep ed 22 Augus 2024
Published 21 Ma ch 2025
Copy igh © 2025 An onio Ramí ez-
de-A ellano-Ma e o e al. Exclusi e
licensee Zhejiang Lab. No claim o
o iginal U.S. Go e nmen Wo ks.
Dis ibu ed unde a C ea i e
Commons A ibu ion License
(CC BY 4.0).
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Ramí ez-de-A ellano-Ma e o e al. 2025 | h ps://doi.o g/10.34133/icompu ing.0103 2
This pape is o ganized in he ollowing manne : o begin, we
explo e he idea o a undamen al VM, and hen we p oceed o
explo e he expansion o SVMs. Then, a isual illus a ion is p o-
ided ha displays he new associa ed seman ics. Subsequen ly,
a ew SVMs a e c ea ed o p oduce na u al numbe se s and com-
pu e basic a i hme ic ope a ions, imp o ing he basic VMs in
bo h speed and undamen al componen s (such as hos s, ins uc-
ions, and numbe o i uses). Finally, he pape concludes wi h
some obse a ions on he po en ial oppo uni ies p esen ed by
hese de ices.
Ma e ials and Me hods
Basic VMs
This compu a ional model, d awing inspi a ion om he p op-
aga ion and eplica ion o i uses among hos s, was ini ially
in oduced by Valencia-Cab e a e al. [ 7 ]. Subsequen ly, he
amewo k is o mally ou lined.
De ini ion 1. A i us machine (VM o sho ) o deg ee
(a,b)
,
a,b≥1
, is de ined as he ollowing uple
Π=
(O,H,I,D
H
,D
I,
GC,m
1
,
…
,mp,i
1
,hou )
, whe e
•
O={o}
is he single on alphabe
•
H
=
{
h
1
,…,ha
}
and
I
=
{
i
1
,…,ib
}
a e o de ed se s o
hos s and ins uc ions such ha
H∩I=�
and
hou ∉H
,
which ep esen s he en i onmen and is usually deno ed
by
h0
•
D
H=
(
H∪
{
hou
}
,EH,wH
)
is a weigh ed di ec ed g aph
(WDG o sho ), whe e
EH
⊆H×
(
H∪
{
h
ou })
wi h no
sel -a cs; in addi ion, he
ou
-deg ee
(
h
ou )
=
0
, and
wH
is a mapping om
EH
on o
ℕ�{0}
•
D
I=
(
I,EI,wI
)
is a WDG, whe e
EI⊆I×I
,
wI
is a map-
ping om
EI
on o
ℕ�{0}
, and he
ou -deg ee
o each
ins uc ion is less han o equal o 2
•
G
C=
(
VC⊆
{
I∪EH
}
,EC
)
is an undi ec ed bipa i e
g aph be ween
I
and
EH
, such ha o all
i∈I
, he e is a
unique
(
h,h�
)
∈EH and
(
i,
(
h,h′
))
is in
EC
•
i1∈I
is he ini ial ins uc ion and
mk∈ℕ
a e he ini ial
numbe o i uses, o each
k
,
1≤k≤a
, whe e
ℕ
is he
se o na u al numbe s
Fo mally, a VM
Π
o deg ee
(a,b)
is a uple ha can be iewed
as an o de ed se o
b
hos s labeled as
h1,…,ha
, whe e each hos
hk
ini ially con ains
mk
i uses, and an o de ed se o
b
ins uc-
ions labeled wi h
i1,…,ib∈I
. In his wo k, he symbol
hou ∉H
is used o deno e he ou pu egion, also w i en as
h0
,
which signi ies he en i onmen . The a cs
(
hk,hk′
)
in he WDG
DH
ep esen he channels ha enable he mo emen o i uses
om one hos
hk∈H
o ano he hos o a ea
hk
�∈H∪
{
h0
}
.
The compu a ion o a VM begins wi h he ac i a ion o ins uc-
ion
i1
. A any gi en ime, only one ins uc ion
il
can be ac i a ed.
I
il
is connec ed o he channel
(
hk,hk′
)
wi h a weigh o
wk,k′
and is ac i a ed a ime
𝜏≥0
, hen ha channel is opened, allow-
ing a i us o be ansmi ed and mul iplied
wk,k′
imes om
hk
o
hk′
. As a esul , one i us is deple ed a
hk
, and
wk,k′
i uses
a i e a
hk′
. Unless o he wise s a ed, all channels a e closed by
de aul .
In he di ec ed g aph
DI
, he a cs ep esen he pa hs o
ins uc ions and each is associa ed wi h a dis inc weigh . In
con as , he undi ec ed bipa i e g aph
GC
ep esen s he ela-
ionship be ween ins uc ions and channels, whe e an edge
{
il,
(
hk,hk′
)}
indica es a con ol linkage be ween ins uc ion
il
and channel
(
hk,hk′
)
.
Le us mo e o he seman ics o VMs. We say a con igu a ion
a an ins an
𝜏≥0
o a VM is de ined as
𝜏=
(
p1,𝜏,…,p
a
,𝜏,l𝜏,p0,𝜏
)
,
whe e
p0,
𝜏
,p1,
𝜏
,
…
,pa,
𝜏∈
ℕ
; hese a e he numbe s o i uses
con ained in en i onmen
h0
and in hos s
h1,h2,…,ha
, and
l𝜏
∈
I
∪{#
}
, wi h
#
∉H∪
{
h
0}
∪I . I
l𝜏≠#
, hen i will be
ac i a ed in he ollowing s ep
𝜏+1
; o he wise, no ins uc ion
will be ac i a ed and a hal ing con igu a ion is eached. The ini ial
con igu a ion o he VM is
0
=
(
m
1
,…,ma,i
1
,0
)
. I is s a ed
ha a nonhal ing con igu a ion
𝜏=
(
a1,𝜏,…,ap,𝜏,l𝜏,a0,𝜏
)
leads o he con igu a ion
𝜏+1=
(
a1,𝜏+1,…,ap,𝜏+1,l𝜏+1,a0,𝜏+1
)
h ough a single ansi ion s ep i he e exis s a ans o ma ion
om
𝜏
o
𝜏+1
as desc ibed below.
1. Assume ha he con ol ins uc ion uni
l𝜏
is a ached
o a channel
(
hs,hs′
)
. I
as,𝜏≥1
, hen
as+1,𝜏=as,𝜏−1
and
as
�
+1,𝜏=as
�
,𝜏+ws,s�
; i
as,𝜏
=0
, hen
as+1,𝜏=as,𝜏
and
as
�
+1,𝜏=as
�
,𝜏
. I
l𝜏
is no a ached o any channel, hen
he e is no ansmission.
2. Objec
l𝜏+1∈I∪{#}
is ob ained depending on he
ou -deg ee
o
l𝜏
as ollows:
• I
ou
-deg ee
(
l
𝜏)
=
2
and
(
l𝜏,l𝜏�
)
∈E
I
and
(
l𝜏,l𝜏��
)
∈E
I
,
whe e
𝜏≠𝜏′
, hen he ollowing holds: (a) I
l𝜏
is no
a ached o any channel, hen
l𝜏+1
will be ei he
l𝜏′
o
l𝜏′′
, selec ed in a nonde e minis ic way. (b) I
l𝜏
is
a ached o a channel
(
hs,hs′
)
and
as,𝜏≥1
, hen
l𝜏+1
is
max{
w𝜏
,
𝜏′,w𝜏
,
𝜏′′
}
; o he wise,
l𝜏+1
is
min{
w𝜏
,
𝜏′,w𝜏
,
𝜏′′
}
.
In bo h cases, when
w𝜏,𝜏
�
=w𝜏,𝜏��
, ei he
l𝜏+1=l𝜏�
o
l𝜏+1=l𝜏��
a e selec ed nonde e minis ically.
• I
ou
-deg ee
(
l𝜏
)
=
1
, hen,
l𝜏+1
=
l𝜏�
, being
(
l𝜏,l𝜏�
)
∈EI.
• I
ou
-deg ee
(
l
𝜏)
=
0
, hen
l𝜏+1=#
, and
𝜏+1
is a
hal ing con igu a ion.
A compu a ion o a VM is a sequence o con igu a ions
such ha (a) he i s e m is he ini ial con igu a ion
0
o
he sys em; (b) o each
𝜏≥1
, he
h
e m o he sequence is
ob ained om he p e ious e m in one ansi ion s ep; and
(c) i i is a hal ing compu a ion, hen he las e m is a hal ing
con igu a ion.
Supe VMs
This sec ion de ines he syn ax and discusses he seman ics
associa ed wi h he ex ension, highligh ing he main di e ences
om he basic VMs.
De ini ion 2. An SVM o deg ee
(a,b)
,
a,b≥1
, is a uple
whe e now
D
H=
(
H,EH=ENH ∪ESH,wH
)
, whe e
ENH
is he
se o no mal channels and
ESH
is a new se o supe channels.
The seman ics associa ed a e simila o basic VMs, bu i a
supe channel is open, no only a i us is ansmi ed bu also
all i uses om ha hos , eplica ing all o hem by he weigh
o he supe channel.
Analogously as in o he wo ks wi h VMs, SVMs in compu -
ing a e p ope ly de ined.
(1)
Π=(
Γ,H,I,D
H
,D
I
,G
C
,m
1
,…,m
b
,I
0
,h
ou ),
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Ramí ez-de-A ellano-Ma e o e al. 2025 | h ps://doi.o g/10.34133/icompu ing.0103 3
De ini ion 3. An SVM wi h inpu o deg ee
(a,b,c)
,
a≥1
,
b≥1
,
c≥1
, is he uple
Π=
(Γ,H,H
c
,I,D
H
,D
I
,G
C
,m
1
,…
,
mb,i1,hou )
, whe e
•
(
Γ,H,I,DH,DI,GC,m
1
,…,mb,i
1
,hou
)
is an SVM o
deg ee
(a,b)
•
H
c=
{
hp1,…,hpc
}
⊆
H
is he o de ed se o
c
inpu
hos s and
hou ∉Hc
Le
Π
be an SVM wi h inpu o deg ee
(a,b,c)
; hen, he
ini ial con igu a ion o
Π
wi h inpu
(
𝛼
1
,…,𝛼
c)
∈ℕ
c
is
(
n1,…,n
p1
+𝛼1,…,n
pc
+𝛼
c
,…,n
a
,i1
, 0)
, deno ed by
Π+
(
𝛼
1,
…
,
𝛼
c)
p oceeds as s a ed in he p e ious sec ion. The ou -
come o a compu a ion ha e mina es is he coun o i uses
dispa ched o he ou pu a ea ( he en i onmen ) h oughou he
compu a ion.
De ini ion 4. I is said ha a pa ial unc ion
g:ℕ
−→
ℕ
,
≥1
, is compu ed by an SVM
Π
wi h
inpu hos s, i o each
(
n
1
,…,n
)
∈ℕ
, is e i ied ha , i
g(
n
1
,…,n
)
is well de ined
and is equal o
y∈ℕ
, hen any compu a ion o
Π+(
n
1
,…,n
)
hal s and he esul is
y
; o he wise, e e y compu a ion does
no hal .
To cla i y he no el y o he ex ension, an example o an
SVM is explici ly p esen ed. Mo eo e , he compu a ions o he
de ice a e explained s ep by s ep.
Le SVM
Πex
o deg ee
(3, 5)
be de ined as
whe e
•
DH
=
(
H,E
H
=E
NH
∪E
SH
,w
H)
, whe e
E
NH =
{(
h
1
,3
),
(
h
3
,h
0)}
,
E
SH =
{(
h
1
,h
2)
,
(
h
2
,h
3)}
, and
wH(
h
1
,h
2)
=
2
,
wH(
h
2
,h
3)
=
3
, and
wH(
h
1
,h
3)
=w
H(
h
3
,h
0)
=
1
•
D
I=
(
I,EI,wI
)
, whe e
E
I=
{(
i1,i2
)
,
(
i1,i3
)
,
(
i2,i3
)
,
(
i3,i4
),
(
i
3
,i
5)(
i
5
,i
4)(
i
5
,i
5)}
and
wI(
i
1
,i
2)
=w
I(
i
5
,i
5)
=
2
and
wI(
i
�
,i
��)
=
1
o he wise
•
GC
=
(
I∪E
H
,E
C)
, whe e
E
C=
{{
i1,
(
h1,h3
)}
,
{
i2,
(
h1,h2
)} ,
{
i
3
,
(
h
2
,h
3)}
,
{
i
4
,
(
h
3
,h
0)}
,
{
i
5
,
(
h
3
,h
0)}}
I is isually p esen ed in Fig. 1 ; he squa es ep esen he
hos s, he numbe inside hem is he ini ial amoun o i uses,
he channels be ween hem a e double lines, and he supe chan-
nels a e iple lines. The ins uc ions a e d awn as blue do s ha
a e connec ed by black single a ows, ep esen ing he ins uc-
ion g aph. The ins uc ion–channel g aph is ep esen ed by
ed dashed lines. Fo be e unde s anding, he weigh ed a cs
wi h weigh 1 a e no d awn.
The ini ial con igu a ion o SVM
Πex
is
C
0=
(
3, 1, 0, i1,0
)
,
om which ins uc ion
i1
is ac i a ed, opening he channel
(
h
1
,h
3)
; hence, one i us is ansmi ed o hos
h3
and he sub-
sequen ins uc ion p oceeds along he pa h wi h he g ea es
weigh in he ins uc ion g aph, ha is, ins uc ion
i2
. The ol-
lowing con igu a ion is
C
1=
(
2, 1, 1, i2,0
)
. Now, ins uc ion
i2
opens supe channel
(
h
1
,h
2)
; hus, all i uses om
h1
pass
h ough i and a e eplica ed by he weigh o he a c, ha
is, 2, and he ollowing ins uc ion is
i3
since no o he pa h is
possible. These s eps can also be ollowed in Table 1 . The
nex con igu a ion is
C
2=
(
0, 5, 1, i3,0
)
; ins uc ion
i3
opens
he supe channel
(
h
2
,h
3)
and he nex ins uc ion has o be
nonde e minis ically chosen, as he e a e se e al pa hs o maxi-
mum weigh :
• I ins uc ion
i4
is ac i a ed, hen he con igu a ion is
C
3=
(
0, 0, 16, i4,0
)
, so only a i us eaches he en i on-
men om
h3
and a hal ing con igu a ion is eached, as
he e is no o he possible pa h om
i4
; he hal ing con-
igu a ion is
C4=(0, 0, 15, #, 1)
(Table 2).
• In case he ins uc ion is
i5
, he con igu a ion is
C3=
(
0, 0, 16, i5,0
)
; om his poin , a single i us is dispa ched
o he en i onmen , and he subsequen ins uc ion
emains
i5
, adhe ing o he pa h o g ea es weigh . This
(2)
Π
ex =
(
O={o},H=
{
h1,h2,h3
}
,
I=
{
i1,…,i5
}
,DH,DI,GC, 3, 1, 0, i1,h0
),
Fig. 1. Example o a supe i us machine (SVM)
Πex
.
Table 1. Fi s s eps in he compu a ion o SVM
Πex
Con igu a ion
h1
h2
h3
Ins uc ion
h0
C0
3 1 0
i1
0
C1
2 1 1
i2
0
C2
0 5 1
i3
0
Table 2. Fi s b anch in he compu a ion o SVM
Πex
Con igu a ion
h1
h2
h3
Ins uc ion
h0
C3
0 0 16
i4
0
C4
0 0 15
#
1
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Ramí ez-de-A ellano-Ma e o e al. 2025 | h ps://doi.o g/10.34133/icompu ing.0103 4
cycle con inues un il hos
h3
is deple ed, esul ing in he
con igu a ion
C19
=
(
0, 0, 0, i
5
, 16
)
. A e his, he nex
ins uc ion is
i4
as no i us ansmission has occu ed;
he e is no i us ansmi ed again, and he hal ing con-
igu a ion is
C21 =(0, 0, 0, #, 16)
(Table 3).
Resul s and Discussion
Func ion compu ing mode
Fo his subsec ion, SVMs a e s udied in unc ion compu ing
mode and a e s udied showing se e al examples ha imp o e
he e iciency o he basic compu ing pa adigm examples p e-
sen ed by Ramí ez-de-A ellano e al. [ 10 ].
Compu ing he addi ion unc ion
The SVM wi h inpu o deg ee
(2, 3, 2)
, ha is, 2 hos s, 3 ins uc-
ions, and 2 inpu hos s,
Πsum
, compu ing he addi ion unc ion
is illus a ed in Fig. 2 .
Fo any inpu
(a
,
b)∈
ℕ
2
, he ini ial con igu a ion is
C
=
(
a,b,i
1
,0
)
. F om he e, ins uc ion
i1
opens he supe chan-
nel
(
h
1
,h
0)
; hence, he i uses om
h1
a e sen o he en i on-
men , leading o he con igu a ion
C1
=
(
0, b,i
2
,a
)
. Analogously,
ins uc ion
i2
opens he supe channel
(
h
2
,h
0)
and a hal ing
con igu a ion is eached
C2=(0, 0, #, a+b)
. Thus, a e 2 ansi-
ion s eps, he machine hal s and e u ns
a+b
.
Compu ing he mul iplica ion unc ion
The SVM wi h inpu o deg ee
(2, 5, 2)
,
Πmul
, compu ing he
mul iplica ion unc ion is depic ed in Fig. 3 .
Fo each inpu
(a,b)∈ℕ
, he machine e i ica ion will ocus
on he ollowing in a ian o
Πmul +(a,b)
:
This in a ian o mula shows ha o each i us in hos
h1
,
b
i uses a e sen o he en i onmen . The p oo will be ca ied
ou by induc ion; o
k=0
, he e i ica ion is i ial since
he ini ial con igu a ion is
C0
=
(
a,b, 0, 0, i
1
,0
)
, which is exac ly
𝜑(0)
. Fo he induc i e s ep, suppose ha o each
k
, wi h
0≤k<a
,
𝜑(k)
is ue; by he induc ion hypo hesis, he ollow
-
ing con igu a ion is eached:
C
k
(2
⋅b
+3)
=
(
a−k,b,k, 0, i
1
,b⋅k
)
.
F om his, we ha e
Tha is, exac ly
𝜑(k+1)
; hus, he o mula is e i ied o
each
0≤k≤a
. In pa icula , i is ue o
k=a
; hen
𝜑
(a)≡Ca
(2
b
+3)
=
(
a−a,b,a, 0, i
1
,a⋅b
)
. F om he e, as hos
h1
is
emp y, he ollowing ins uc ion is
i5
, leading o he hal ing
con igu a ion
C2+
a
(2
b
+3)
=(
0, b,a, 0, #, a
⋅
b)
. The e o e, he
machine hal s and e u ns
a⋅b
.
Compu ing he powe o 2 unc ion
The SVM wi h inpu o deg ee
(4, 4, 2)
,
Πpow2
, compu ing he
unc ion
(n)=2n
is depic ed in Fig. 4 .
As in he p e ious machine, he e i ica ion will ocus on
he in a ian ha holds he machine
Πpow2
; o each
(
a,b
)∈
ℕ
2
,
he ollowing in a ian holds:
This in a ian shows ha o each i us in hos
h1
, one powe
o 2 is inc eased in hos
h2
. Focusing now in he p oo o he
o mula, o he case
k=0
, i is di ec , as he ini ial con igu a-
ion is
C0
=
(
a, 1, 0, 0, i
1
,0
)
, which is
𝜑(0)
. Fo he induc i e
s ep, suppose ha
𝜑
is ue o each
0≤k<a
, hen he ollow-
ing con igu a ion is eached:
(3)
𝜑
(k)≡Ck⋅
(2
b
+3)
=
(
a−k,b,k, 0, i
1
,b⋅k
)
, o each 0 ≤k≤a
.
(4)
Ck(2⋅b+3)+1=
(
a−k−1, b,k+1, 0, i2,b⋅k
)
, as h1(a−k),
Ck(2⋅b+3)+2=(a−(k+1),0,k+1, 2b,i3,b⋅k),
Ck(2⋅b+3)+2+1=(a−(k+1),1,k+1, 2b−1, i4,b⋅k), as h3(2b)
,
Ck(2⋅b+3)+2+2=(a−(k+1),1,k+1, 2b−2, i3,b⋅k+1),
⋮
Ck(2⋅b+3)+2+2b=(a−(k+1),b,k+1, 2b−2b,i3,b⋅k+b),
C
k(2⋅b+3)+2b+3
⏟⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏟
(
k
+1)(2
b
+3)
=(a−(k+1),b,k+1, 0, i1,b⋅k+b
⏟⏟⏟
(k+1)⋅b
), as h3(0).
(5)
𝜑
(k)≡C
3
k=
(
a−k,2
k
,k, 0, i
1
,0
)
o each 0 ≤k≤a
.
(6)
C3k=(a−k,2
k
,k, 0, i1,0
),
C3k+1=(a−k−1, 2k,k+1, 0, i2, 00), as h1(a−k)
,
C3k+2=(a−(k+1), 0, k+1, 2⋅2k,i3,0
),
C
3k+3
⏟⏟⏟
3(
k
+1)
=(a−(k+1),2⋅2k
⏟⏟⏟
2k+1
,k+1, 0, i1, 0).
Table 3. Second b anch in he compu a ion o SVM
Πex
Con igu a ion
h1
h2
h3
Ins uc ion
h0
C3
0 0 16
i5
0
C4
0 0 15
i5
1
⋮ ⋮ ⋮ ⋮
i5
⋮
C19
0 0 0
i5
16
C20
0 0 0
i6
16
C21
0 0 0
#
16
Fig. 2. SVM
Πsum
compu ing he addi ion unc ion.
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Ramí ez-de-A ellano-Ma e o e al. 2025 | h ps://doi.o g/10.34133/icompu ing.0103 5
Tha is, exac ly
𝜑(k+1)
; hus, he o mula is e i ied o each
0≤k≤a
. In pa icula , i is ue o
k=a
. F om he con igu a-
ion
𝜑
(a)=C
3
a=
(
a−a,2
a
,a, 0, i
1
)
, as hos
h1
is emp y, he e
is no i us ansmission, and he nex ac i a ed ins uc ion
will be
i4
, and he con igu a ion is
C3
a+
1
=
(
0, 2
a
,a, 0, i
4
,0
)
.
Ins uc ion
i4
opens he supe channel
(
h
2
,h
0
)
, and a hal ing
con igu a ion is eached,
C3a+2=
(
0, 0, a,0,#,2
a)
. Hence, a e
3a+2
s eps, he machine hal s and e u ns
2a
.
Gene a ing na u al numbe se s
In his sec ion, he gene a ing mode o SVMs is p esen ed
in addi ion o se e al VMs ha gene a e a se o classic
numbe s, using less esou ces han he basic ones p esen ed by
Ramí ez-de-A ellano e al. [ 12 ].
As in basic VMs, a gene a ing mode can be de ined as he
numbe o i uses ha each he ou pu hos du ing he en i e
compu a ion. We say ha a na u al numbe se
A
is gene a ed
by SVM
ΠA
i and only i
• o each na u al numbe
a∈A
, he e exis s a compu a-
ion o
ΠA
ha gene a es
A
, and
• o each compu a ion o
ΠA
, he numbe gene a ed
x
is in
A
.
Gene a ing e en numbe s
The SVM o deg ee
(2, 3)
,
Πe en
, gene a ing e en na u al num-
be s is depic ed in Fig. 5 .
Fig. 3. SVM
Πmul
compu ing he mul iplica ion unc ion.
Fig. 4. SVM
Πpow2
compu ing he unc ion
(x)=2x
.
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Ramí ez-de-A ellano-Ma e o e al. 2025 | h ps://doi.o g/10.34133/icompu ing.0103 6
Fi s , le us see ha he only channel a ached o he en i on-
men has weigh 2, so any compu a ion o he machine will
gene a e an e en numbe . Le us now ocus on he o he inclu-
sion; ha is, o any e en numbe
n=2m
, he e exis s a hal ing
compu a ion o
Πe en
ha gene a es
n=2m
.
I
m=0
, om ins uc ion
i1
, he nonde e minis ic decision
is going o
i3
; as
h1
is emp y in he nex s ep, he ollowing hal -
ing con igu a ion is eached:
C2(0, 1, #, 0)
. Fo
m>0
, we ha e
Tha is, he machine gene a es
n=2m
o any na u al num-
be e en
m∈ℕ
.
Gene a ing squa e numbe s
The SVM o deg ee
(5, 9)
,
Πsqua e
, gene a ing na u al squa e
numbe s is depic ed in Fig. 6 .
Fo gene a ing he numbe 0, he compu a ion, which chooses
ins uc ion
i9
in he i s s ep, eaches he hal ing con igu a ion
C2
=
(
0, 0, 0, 2, 0, i
9
,0
)
. Le us now ocus on gene a ing he num-
be
n2
o each
n>0
.
Fo his pu pose, he e i ica ion will be ocused on p o ing
he ollowing in a ian :
whe e
𝛽k
=
𝛼0
+
𝛼1
+…+
𝛼k
and
𝛼j=2j+7
, o each
j>0
.
The idea behind his o mula is o inc ease he ou pu hos by
he amoun o i uses necessa y o each he nex squa e num-
be . Mo ing on o he p oo , o
k=0
, i is di ec i
i2
is chosen
nonde e minis ically,
C1
=
(
0, 0, 0, 2, 0, i
2
,0
)
, which is
𝜑(0)
.
(7)
C
0=
(
1, 0, i1,0
)
,
C
1=(0, 2, i2,0
),
C
2=(2, 0, i1,1
),
⋮
|
|
|
|
|
|
|
|
|
|
C2m=
(
m+1, 0, i1,0
),
C2m+1=(m, 2, i3,0
),
C2m+2=(0, 2, i3,2m),
(8)
𝜑
(k)≡C1+𝛽
k
=
(
k, 0, 0, 2, 0, i2,k
2)
, o each 0 ≤k<n
2,
Fig. 5. SVM
Πe en
gene a ing he se
{
2m
|
m∈ℕ
}
.
Fig. 6. SVM
Πsqua e
gene a ing he se
{
n
2|
n∈ℕ
}
.
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Ramí ez-de-A ellano-Ma e o e al. 2025 | h ps://doi.o g/10.34133/icompu ing.0103 7
Le us ocus on he induc i e s ep; ha is, le us suppose
ha he o mula is e i ied o any
0≤k<n2
−1
. Then, he
o mula is ue also o
k+1
, and by he induc ion hypo hesis,
we ha e
Hence,
𝜑(k+1)
is e i ied.
As
𝜑(k)
is ue o each
0≤k<n2
, in pa icula , i is ue
o
n−1
, leading o he ollowing con igu a ion,
𝜑(n−1)≡
C
1+𝛽
n−1
=
(
n−1, 0, 0, 2, 0, i2,(n−1)
2)
, om which a hal ing
con igu a ion can be eached simila ly o be o e:
C
2+𝛽
n=
(
n+1, 0, 0, 2, 0, #, n
2)
; ha is, a e
2+𝛽n
s eps, he machine
hal s and gene a es
n2
, which is he squa e numbe .
Gene a ing powe s o 2
The SVM o deg ee
(2, 5)
,
Πgpow2
, gene a ing he powe s o 2 is
depic ed in Fig. 7 .
Fi s ly, le us see ha o each numbe
x∈{2n|n∈ℕ}
, he e
exis s a hal ing compu a ion o
Πgpow2
ha gene a es
x
:
Tha is, exac ly 20.
Le us now see ha he compu a ion o gene a ing he na u-
al numbe o he o m
22n
o some
n≥1
.
Tha is,
2n
, o any
n∈ℕ
wi h
n≥1
.
Analogously, we can e i y ha he e exis s a compu a ion
ha gene a es
22n+1
o each na u al numbe
n∈ℕ
.
Tha is,
2n+1
, o any
n∈ℕ
.
Las ly, i can be easily seen ha he only nonde e minis ic
decisions in he machine a e in ins uc ions
i1,i2,i3
, and o all
hese decisions, i has been seen ha only powe s o 2 can be
gene a ed. Thus, o each compu a ion o
Πgpow2
, he gene a ed
numbe is in he se
{2n|n∈
ℕ
}
.
(9)
C
1+𝛽k=(k, 0, 0, 2, 0, i2,k2),
C
1+𝛽k+1=(0, 2k, 0, 2, 0, i3,k2),
C
1+𝛽k+1+1=(1, 2k−1, 0, 2, 0, i4,k2),
C
1+𝛽k+1+2=(1, 2k−2, 1, 2, 0, i3,k2),
⋮
C
1+𝛽k+1+2k=(k,2k−2k,k, 2, 0, i3,k2),
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
C1+𝛽k+2k+2=
(
k, 0, k, 2, 0, i5,k
2)
,
C1+𝛽k+2k+3=(k, 0, k, 0, 4, i6,k2),
C1+𝛽k+2k+4=(k, 0, k, 4, 0, i7,k2),
C1+𝛽k+2k+5=(k+1, 0, k, 3, 0, i8,k2),
C1+𝛽k+2k+6=(k+1, 0, 0, 2, 0, i1,k2+1),
C1+𝛽k+2k+7=(k+1, 0, 0, 2, 0, i2,k2+1+2k
⏟⏞⏞⏟⏞⏞⏟
(
k
+1)
2
).
(10)
C
0=
(
1, 0, i1,0
),
C
1=(1, 0, i5,0
)
,
C2=(
0, 0, #,1
)
,
(11)
C
0=
(
1, 0, i1,0
)
,
C
1=(1, 0, i2,0
),
C
2=(0, 2, i3,0
),
C3
=
(
4, 0, i
2
,0
)
,
|
|
|
|
|
|
|
|
|
|
⋮
C2n=(0, 22n−1,i3,0
)
,
C2n+1=(22n, 0, i5,0
),
C
2
n+
2
=
(
0, 0, #, 22n
)
.
(12)
C
0=
(
1, 0, i1,0
)
,
C
1=(1, 0, i2,0
),
C
2=(0, 2, i3,0
),
C3
=
(
4, 0, i
2
,0
)
,
|
|
|
|
|
|
|
|
|
|
⋮
C2n+1=(22n, 0, i3,0
),
C2n+2=(0, 22n+1,i5,0
)
,
C
2
n+
3
=
(
0, 0, #, 22n+1
)
.
Fig. 7. SVM
Πgpow2
gene a ing he se
{
2
n|
n∈ℕ
}
.
Table 4. Time complexi y compa ison be ween a basic VM and
he p oposed SVM o compu ing he basic unc ions o a gi en
inpu
(
a,b
)
∈
ℕ
Func ion VM SVM
Addi ion (sum) a + b + 3 2
Mul iplica ion (mul ) 3b(a + 1) + 2 a(2b + 3) + 2
Exponen ia ion (pow2) Exponen ial 3a + 2
Table 5. Compa ison o sou ces needed be ween a basic VM
and he p oposed SVM o gene a ing basic subse s o na u al
numbe s. No e ha he “-” symbol means ha he e is no
esul published.
Se s Sou ce VM SVM
E en numbe s (e en) Hos s 2 2
Ins uc ions 4 2
S eps Linea Linea
Squa e numbe s
(squa e)
Hos s - 5
Ins uc ions - 9
S eps -
n√n
Powe s o 2 (gpow2) Hos s 2 2
Ins uc ions 7 5
S eps Exponen ial Linea
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Ramí ez-de-A ellano-Ma e o e al. 2025 | h ps://doi.o g/10.34133/icompu ing.0103 8
Discussion
In his subsec ion, a compa ison wi h he esul s p esen ed by
Ramí ez-de-A ellano e al. [ 10 , 12 ] wi h basic VMs is discussed.
Fo he compu ing mode, all o he SVMs p esen ed ha e a
be e ime e iciency (depending on he inpu gi en) compa ed
o he basic VMs p esen ed by Ramí ez-de-A ellano e al. [ 10 ],
as can be seen in Table 4 . Mo e p ecisely, o a gi en inpu
(a,b)∈ℕ
in una y codi ica ion, he addi ion unc ion dec eased
i s ime cos om
a+b+3
s eps in VMs o 2 s eps in SVMs.
Mo eo e , o he same inpu , he mul iplica ion unc ion p e-
sen ed dec eased om
3b(a+1)+2
s eps in VMs o
a(2b+3)+2
s eps. Finally, he exponen ia ion unc ion was i s simula ed
by any ex ension o VMs, bu i can be easily assumed ha he ime
complexi y will be a leas exponen ial in basic VMs, while in his
new ex ension, he SVM p esen ed has a linea ime complexi y.
Fo he gene a ing mode, all o he SVMs p esen ed need
ewe o he same hos s and ins uc ions as he VMs p esen ed
by Ramí ez-de-A ellano e al. [ 12 ], as can be seen in Table 5 .
We highligh
Πgpow2
, whe e he numbe o ins uc ions was
dec eased om 7 o 5 ins uc ions and he numbe o s eps
needed om exponen ial o linea depending on he numbe
gene a ed. In addi ion, gene a ing he squa e numbe s has been
done only o SVMs.
Conclusion
In his pape , an ex ension o VMs’ compu ing pa adigm is
de ined: SVMs; his ex ension can de elop models wi h a new
kind o channels called supe channels. The seman ics associ-
a ed a e shown wi h a isual example. In addi ion, se e al SVMs
in unc ion compu ing and gene a ing mode a e designed, imp o -
ing he basic VMs in bo h ime and memo y cos s.
Howe e , he e iciency o his pa adigm should be u he
imp o ed o be able o a ack NP-ha d p oblems [ 18 ]; o hem,
mo e na u e-inspi ed ing edien s can be conside ed in addi ion
o he supe channels p oposed, e.g., mu a ion, hos eplica ion,
and hos dea h. In o he pa adigms o compu a ion (e.g., in
memb ane compu ing [ 19 ]), he inclusion o hese “ing edi-
en s” has allowed memb ane sys ems o e icien ly sol e p ob-
lems e en abo e he class NP.
Acknowledgmen s
Funding: The esea ch desc ibed in his wo k was suppo ed
by he Zhejiang Lab BioBi P og am (G an No. 2022BCF05).
Au ho con ibu ions: A.R.-d.-A. and D.O.-M. concei ed he
idea and w o e he manusc ip . L.V.-C. e ised and edi ed
he manusc ip . M.J.P.-J. supe ised he esea ch and edi ed
he manusc ip .
Compe ing in e es s: The au ho s decla e ha hey ha e no
compe ing in e es s.
Da a A ailabili y
No da a we e used o his wo k.
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