Super Virus Machines: Faster Virus Transmission, More Efficiency Using Superchannels

Ramírez-de-Arellano-Marrero et al. 2025 | https://doi.org/10.34133/icomputing.0103 1 RESEARCH ARTICLE Super Virus Machines: Faster Virus Transmission, More Efficiency Using Superchannels Antonio Ramírez-de-Arellano-Marrero1,2*, Luis Valencia-Cabrera1,2, David Orellana-Martín1,2, and Mario J. Pérez-Jiménez1,2 1Computer Science and Artificial Intelligence, University of Seville, Seville, Spain. 2SCORE Laboratory, I3US, Universidad de Sevilla, Seville, Spain. *Address correspondence to: [email protected] Surpassing the classical computing architecture is one of the great challenges of computer science today. The branch that approaches it from a theoretical point of view, inspired by nature, is called natural computation. Within this field, a paradigm arises, called virus machines (VMs), inspired by the propagation and replication of the biological structure of viruses. This work introduces a novel extension to the young computing paradigm of VMs, the super VMs. This extension can develop models with a new kind of channel called superchannel. In addition, several VMs are constructed to generate natural number sets and compute basic arithmetic functions, improving the basic VMs in both time and memory cost (such as hosts and instructions). Introduction The branch of bioinspired computing [ 1 ] belongs to the field of unconventional computing [ 2 ], which is dedicated to exploring computing models inspired by natural phenomena and simulat - ing various natural structures. It has been used not only for theoretical aspects (computational complexity theory through these devices [ 3 ]) but also for practical scopes (for instance, population dynamics modeling [ 4 ]) with biologic implementations (such as cellular computing [ 5 ]). Awareness of the spreading mechanisms and range of effects of viral infections dramatically increased during the COVID19 pandemic. This biological structure has shown interesting properties, prompting exploration from a computational perspective. Intriguing properties, including the necessity of a host for replication, the replication process itself, and the high mutation coefficient, can be studied in the context of computing. For a more in-depth exploration of biological aspects, refer to Dimmock et al. [ 6 ]. A new computational model, drawing inspiration from this biological architecture, has been developed: virus machines (VMs) [ 7 ], where Turing universality was first proved by simulating register machines; other techniques to prove this can be found in the works of Romero-Jiménez et al. [ 8 , 9 ]. Within this computing framework, viruses propagate and replicate among hosts via channels governed by an instruction graph. This approach offers a compelling perspective for tackling mathematical challenges. As demonstrated in their work [ 10 ], the authors devised devices based on this model for computing pairing functions, with their accuracy established through rigorous mathematical proof. In addition, VMs have resulted in an interesting scope for attacking cryptosystems, as the authors presented in a paper [ 11 ]. Moreover, in another work [ 12 ], the authors presented multiple instances of VMs that are designed to generate, compute, and recognize functions and sets of natural numbers; the latter also paves the way for developing a new theoretical complexity theory using VMs. Recently, the authors compared VMs with other neural-like models in another work [ 13 ], more precisely with spiking neural P systems [ 14 ]. Nevertheless, all the references mentioned have the same conclusion: the time efficiency of VMs has to be enhanced. For instance, the authors [ 15 ] proposed parallel VMs, where a set of instructions could be activated simultaneously; on the other hand, a matrix representation was defined by Pérez-Segura et al. [ 16 ] for future graphics processing unit implementations. That is why there is an interesting scope in the study of possible extensions in these devices. From a biological perspective [ 17 ], the life cycle of a virus involves a specific series of complex processes tailored to each virus–host interaction, but it typically consists of 5 stages. If we focus on the second stage, that is, the replication of the genome, these replications do not occur one by one; they occur in parallel. It may be interesting to be inspired by this behavior, by setting that all viruses from one host can be replicated and transmitted simultaneously to another host. In this work, an extension of these computing models taking this inspiration, called super virus machines (SVMs), is proposed to enhance time efficiency. The novelty of this extension is a new kind of channels called superchannels, which allows more transmission and replication of viruses simultaneously. This simple but powerful extension shows better results in time efficiency for computing functions and for generating natural number sets. Citation: Ramírez-de-ArellanoMarreroA, Valencia-CabreraL, Orellana-MartínD, Pérez-JiménezMJ. Super Virus Machines: Faster Virus Transmission, More Efficiency Using Superchannels. Intell. Comput. 2025;4:Article 0103. https://doi. org/10.34133/icomputing.0103 Submitted 15 December 2023 Revised 8 July 2024 Accepted 22 August 2024 Published 21 March 2025 Copyright © 2025 Antonio Ramírezde-Arellano-Marrero etal. Exclusive licensee Zhejiang Lab. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution License (CC BY 4.0). Downloaded from https://spj.science.org at Universidad de Sevilla on May 25, 2025 Ramírez-de-Arellano-Marrero et al. 2025 | https://doi.org/10.34133/icomputing.0103 2 This paper is organized in the following manner: to begin, we explore the idea of a fundamental VM, and then we proceed to explore the expansion of SVMs. Then, a visual illustration is provided that displays the new associated semantics. Subsequently, a few SVMs are created to produce natural number sets and compute basic arithmetic operations, improving the basic VMs in both speed and fundamental components (such as hosts, instructions, and number of viruses). Finally, the paper concludes with some observations on the potential opportunities presented by these devices. Materials and Methods Basic VMs This computational model, drawing inspiration from the propagation and replication of viruses among hosts, was initially introduced by Valencia-Cabrera et al. [ 7 ]. Subsequently, the framework is formally outlined. Definition 1. A virus machine (VM for short) of degree (a,b) , a,b≥1 , is defined as the following tuple Π= (O,H,I,D H ,D I, GC,m 1 , … ,mp,i 1 ,hout) , where • O={o} is the singleton alphabet • H = { h 1 ,…,ha } and I = { i 1 ,…,ib } are ordered sets of hosts and instructions such that H∩I=� and hout ∉H , which represents the environment and is usually denoted by h0 • D H= ( H∪ { hout } ,EH,wH ) is a weighted directed graph (WDG for short), where EH ⊆H× ( H∪ { h out}) with no self-arcs; in addition, the out -degree ( h out) = 0 , and wH is a mapping from EH onto ℕ�{0} • D I= ( I,EI,wI ) is a WDG, where EI⊆I×I , wI is a mapping from EI onto ℕ�{0} , and the out-degree of each instruction is less than or equal to 2 • G C= ( VC⊆ { I∪EH } ,EC ) is an undirected bipartite graph between I and EH , such that for all i∈I , there is a unique ( h,h� ) ∈EH and ( i, ( h,h′ )) is in EC • i1∈I is the initial instruction and mk∈ℕ are the initial number of viruses, for each k , 1≤k≤a , where ℕ is the set of natural numbers Formally, a VM Π of degree (a,b) is a tuple that can be viewed as an ordered set of b hosts labeled as h1,…,ha , where each host hk initially contains mk viruses, and an ordered set of b instructions labeled with i1,…,ib∈I . In this work, the symbol hout ∉H is used to denote the output region, also written as h0 , which signifies the environment. The arcs ( hk,hk′ ) in the WDG DH represent the channels that enable the movement of viruses from one host hk∈H to another host or area hk �∈H∪ { h0 } . The computation of a VM begins with the activation of instruction i1 . At any given time, only one instruction il can be activated. If il is connected to the channel ( hk,hk′ ) with a weight of wk,k′ and is activated at time 𝜏≥0 , then that channel is opened, allowing a virus to be transmitted and multiplied wk,k′ times from hk to hk′ . As a result, one virus is depleted at hk , and wk,k′ viruses arrive at hk′ . Unless otherwise stated, all channels are closed by default. In the directed graph DI , the arcs represent the paths of instructions and each is associated with a distinct weight. In contrast, the undirected bipartite graph GC represents the relationship between instructions and channels, where an edge { il, ( hk,hk′ )} indicates a control linkage between instruction il and channel ( hk,hk′ ) . Let us move to the semantics of VMs. We say a configuration t at an instant 𝜏≥0 of a VM is defined as  𝜏= ( p1,𝜏,…,p a ,𝜏,l𝜏,p0,𝜏 ) , where p0, 𝜏 ,p1, 𝜏 , … ,pa, 𝜏∈ ℕ ; these are the numbers of viruses contained in environment h0 and in hosts h1,h2,…,ha , and l𝜏 ∈ I ∪{# } , with # ∉H∪ { h 0} ∪I . If l𝜏≠# , then it will be activated in the following step 𝜏+1 ; otherwise, no instruction will be activated and a halting configuration is reached. The initial configuration of the VM is 0 = ( m 1 ,…,ma,i 1 ,0 ) . It is stated that a nonhalting configuration  𝜏= ( a1,𝜏,…,ap,𝜏,l𝜏,a0,𝜏 ) leads to the configuration  𝜏+1= ( a1,𝜏+1,…,ap,𝜏+1,l𝜏+1,a0,𝜏+1 ) through a single transition step if there exists a transformation from 𝜏 to 𝜏+1 as described below. 1. Assume that the control instruction unit l𝜏 is attached to a channel ( hs,hs′ ) . If as,𝜏≥1 , then as+1,𝜏=as,𝜏−1 and as � +1,𝜏=as � ,𝜏+ws,s� ; if as,𝜏 =0 , then as+1,𝜏=as,𝜏 and as � +1,𝜏=as � ,𝜏 . If l𝜏 is not attached to any channel, then there is no transmission. 2. Object l𝜏+1∈I∪{#} is obtained depending on the out-degree of l𝜏 as follows: • If out -degree ( l 𝜏) = 2 and ( l𝜏,l𝜏� ) ∈E I and ( l𝜏,l𝜏�� ) ∈E I , where 𝜏≠𝜏′ , then the following holds: (a) If l𝜏 is not attached to any channel, then l𝜏+1 will be either l𝜏′ or l𝜏′′ , selected in a nondeterministic way. (b) If l𝜏 is attached to a channel ( hs,hs′ ) and as,𝜏≥1 , then l𝜏+1 is max{ w𝜏 , 𝜏′,w𝜏 , 𝜏′′ } ; otherwise, l𝜏+1 is min{ w𝜏 , 𝜏′,w𝜏 , 𝜏′′ } . In both cases, when w𝜏,𝜏 � =w𝜏,𝜏�� , either l𝜏+1=l𝜏� or l𝜏+1=l𝜏�� are selected nondeterministically. • If out -degree ( l𝜏 ) = 1 , then, l𝜏+1 = l𝜏� , being ( l𝜏,l𝜏� ) ∈EI. • If out -degree ( l 𝜏) = 0 , then l𝜏+1=# , and 𝜏+1 is a halting configuration. A computation of a VM is a sequence of configurations such that (a) the first term is the initial configuration 0 of the system; (b) for each 𝜏≥1 , the tth term of the sequence is obtained from the previous term in one transition step; and (c) if it is a halting computation, then the last term is a halting configuration. Super VMs This section defines the syntax and discusses the semantics associated with the extension, highlighting the main differences from the basic VMs. Definition 2. An SVM of degree (a,b) , a,b≥1 , is a tuple where now D H= ( H,EH=ENH ∪ESH,wH ) , where ENH is the set of normal channels and ESH is a new set of superchannels. The semantics associated are similar to basic VMs, but if a superchannel is open, not only a virus is transmitted but also all viruses from that host, replicating all of them by the weight of the superchannel. Analogously as in other works with VMs, SVMs in computing are properly defined. (1) Π=( Γ,H,I,D H ,D I ,G C ,m 1 ,…,m b ,I 0 ,h out), Downloaded from https://spj.science.org at Universidad de Sevilla on May 25, 2025 Ramírez-de-Arellano-Marrero et al. 2025 | https://doi.org/10.34133/icomputing.0103 3 Definition 3. An SVM with input of degree (a,b,c) , a≥1 , b≥1 , c≥1 , is the tuple Π= (Γ,H,H c ,I,D H ,D I ,G C ,m 1 ,… , mb,i1,hout) , where • ( Γ,H,I,DH,DI,GC,m 1 ,…,mb,i 1 ,hout ) is an SVM of degree (a,b) • H c= { hp1,…,hpc } ⊆ H is the ordered set of c input hosts and hout ∉Hc Let Π be an SVM with input of degree (a,b,c) ; then, the initial configuration of Π with input ( 𝛼 1 ,…,𝛼 c) ∈ℕ c is ( n1,…,n p1 +𝛼1,…,n pc +𝛼 c ,…,n a ,i1 , 0) , denoted by Π+ ( 𝛼 1, … , 𝛼 c) proceeds as stated in the previous section. The outcome of a computation that terminates is the count of viruses dispatched to the output area (the environment) throughout the computation. Definition 4. It is said that a partial function g:ℕr −→ ℕ , r≥1 , is computed by an SVM Π with r input hosts, if for each ( n 1 ,…,n r) ∈ℕ r , is verified that, if g( n 1 ,…,n r) is well defined and is equal to y∈ℕ , then any computation of Π+( n 1 ,…,n r) halts and the result is y ; otherwise, every computation does not halt. To clarify the novelty of the extension, an example of an SVM is explicitly presented. Moreover, the computations of the device are explained step by step. Let SVM Πex of degree (3, 5) be defined as where • DH = ( H,E H =E NH ∪E SH ,w H) , where 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 ) , where 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 otherwise • GC = ( I∪E H ,E C) , where 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)}} It is visually presented in Fig. 1 ; the squares represent the hosts, the number inside them is the initial amount of viruses, the channels between them are double lines, and the superchannels are triple lines. The instructions are drawn as blue dots that are connected by black single arrows, representing the instruction graph. The instruction–channel graph is represented by red dashed lines. For better understanding, the weighted arcs with weight 1 are not drawn. The initial configuration of SVM Πex is C 0= ( 3, 1, 0, i1,0 ) , from which instruction i1 is activated, opening the channel ( h 1 ,h 3) ; hence, one virus is transmitted to host h3 and the subsequent instruction proceeds along the path with the greatest weight in the instruction graph, that is, instruction i2 . The following configuration is C 1= ( 2, 1, 1, i2,0 ) . Now, instruction i2 opens superchannel ( h 1 ,h 2) ; thus, all viruses from h1 pass through it and are replicated by the weight of the arc, that is, 2, and the following instruction is i3 since no other path is possible. These steps can also be followed in Table 1 . The next configuration is C 2= ( 0, 5, 1, i3,0 ) ; instruction i3 opens the superchannel ( h 2 ,h 3) and the next instruction has to be nondeterministically chosen, as there are several paths of maximum weight: • If instruction i4 is activated, then the configuration is C 3= ( 0, 0, 16, i4,0 ) , so only a virus reaches the environment from h3 and a halting configuration is reached, as there is no other possible path from i4 ; the halting configuration is C4=(0, 0, 15, #, 1) (Table 2). • In case the instruction is i5 , the configuration is C3= ( 0, 0, 16, i5,0 ) ; from this point, a single virus is dispatched to the environment, and the subsequent instruction remains i5 , adhering to the path of greatest weight. This (2) Π ex = ( O={o},H= { h1,h2,h3 } , I= { i1,…,i5 } ,DH,DI,GC, 3, 1, 0, i1,h0 ), Fig. 1. Example of a super virus machine (SVM) Πex . Table 1. First steps in the computation of SVM Πex Configuration h1 h2 h3 Instruction h0 C0 3 1 0 i1 0 C1 2 1 1 i2 0 C2 0 5 1 i3 0 Table 2. First branch in the computation of SVM Πex Configuration h1 h2 h3 Instruction h0 C3 0 0 16 i4 0 C4 0 0 15 # 1 Downloaded from https://spj.science.org at Universidad de Sevilla on May 25, 2025 Ramírez-de-Arellano-Marrero et al. 2025 | https://doi.org/10.34133/icomputing.0103 4 cycle continues until host h3 is depleted, resulting in the configuration C19 = ( 0, 0, 0, i 5 , 16 ) . After this, the next instruction is i4 as no virus transmission has occurred; there is no virus transmitted again, and the halting configuration is C21 =(0, 0, 0, #, 16) (Table 3). Results and Discussion Function computing mode For this subsection, SVMs are studied in function computing mode and are studied showing several examples that improve the efficiency of the basic computing paradigm examples presented by Ramírez-de-Arellano et al. [ 10 ]. Computing the addition function The SVM with input of degree (2, 3, 2) , that is, 2 hosts, 3 instructions, and 2 input hosts, Πsum , computing the addition function is illustrated in Fig. 2 . For any input (a , b)∈ ℕ 2 , the initial configuration is C = ( a,b,i 1 ,0 ) . From here, instruction i1 opens the superchannel ( h 1 ,h 0) ; hence, the viruses from h1 are sent to the environment, leading to the configuration C1 = ( 0, b,i 2 ,a ) . Analogously, instruction i2 opens the superchannel ( h 2 ,h 0) and a halting configuration is reached C2=(0, 0, #, a+b) . Thus, after 2 transition steps, the machine halts and returns a+b . Computing the multiplication function The SVM with input of degree (2, 5, 2) , Πmult , computing the multiplication function is depicted in Fig. 3 . For each input (a,b)∈ℕ , the machine verification will focus on the following invariant of Πmult +(a,b) : This invariant formula shows that for each virus in host h1 , b viruses are sent to the environment. The proof will be carried out by induction; for k=0 , the verification is trivial since the initial configuration is C0 = ( a,b, 0, 0, i 1 ,0 ) , which is exactly 𝜑(0) . For the inductive step, suppose that for each k , with 0≤k<a , 𝜑(k) is true; by the induction hypothesis, the follow - ing configuration is reached: C k (2 ⋅b +3) = ( a−k,b,k, 0, i 1 ,b⋅k ) . From this, we have That is, exactly 𝜑(k+1) ; thus, the formula is verified for each 0≤k≤a . In particular, it is true for k=a ; then 𝜑 (a)≡Ca (2 b +3) = ( a−a,b,a, 0, i 1 ,a⋅b ) . From here, as host h1 is empty, the following instruction is i5 , leading to the halting configuration C2+ a (2 b +3) =( 0, b,a, 0, #, a ⋅ b) . Therefore, the machine halts and returns a⋅b . Computing the power of 2 function The SVM with input of degree (4, 4, 2) , Πpow2 , computing the function f(n)=2n is depicted in Fig. 4 . As in the previous machine, the verification will focus on the invariant that holds the machine Πpow2 ; for each ( a,b )∈ ℕ 2 , the following invariant holds: This invariant shows that for each virus in host h1 , one power of 2 is increased in host h2 . Focusing now in the proof of the formula, for the case k=0 , it is direct, as the initial configuration is C0 = ( a, 1, 0, 0, i 1 ,0 ) , which is 𝜑(0) . For the inductive step, suppose that 𝜑 is true for each 0≤k<a , then the following configuration is reached: (3) 𝜑 (k)≡Ck⋅ (2 b +3) = ( a−k,b,k, 0, i 1 ,b⋅k ) , for 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 ) for 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 branch in the computation of SVM Πex Configuration h1 h2 h3 Instruction 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 computing the addition function. Downloaded from https://spj.science.org at Universidad de Sevilla on May 25, 2025 Ramírez-de-Arellano-Marrero et al. 2025 | https://doi.org/10.34133/icomputing.0103 5 That is, exactly 𝜑(k+1) ; thus, the formula is verified for each 0≤k≤a . In particular, it is true for k=a . From the configuration 𝜑 (a)=C 3 a= ( a−a,2 a ,a, 0, i 1 ) , as host h1 is empty, there is no virus transmission, and the next activated instruction will be i4 , and the configuration is C3 a+ 1 = ( 0, 2 a ,a, 0, i 4 ,0 ) . Instruction i4 opens the superchannel ( h 2 ,h 0 ) , and a halting configuration is reached, C3a+2= ( 0, 0, a,0,#,2 a) . Hence, after 3a+2 steps, the machine halts and returns 2a . Generating natural number sets In this section, the generating mode of SVMs is presented in addition to several VMs that generate a set of classic numbers, using less resources than the basic ones presented by Ramírez-de-Arellano et al. [ 12 ]. As in basic VMs, a generating mode can be defined as the number of viruses that reach the output host during the entire computation. We say that a natural number set A is generated by SVM ΠA if and only if • for each natural number a∈A , there exists a computation of ΠA that generates A , and • for each computation of ΠA , the number generated x is in A . Generating even numbers The SVM of degree (2, 3) , Πeven , generating even natural numbers is depicted in Fig. 5 . Fig. 3. SVM Πmult computing the multiplication function. Fig. 4. SVM Πpow2 computing the function f(x)=2x . Downloaded from https://spj.science.org at Universidad de Sevilla on May 25, 2025 Ramírez-de-Arellano-Marrero et al. 2025 | https://doi.org/10.34133/icomputing.0103 6 First, let us see that the only channel attached to the environment has weight 2, so any computation of the machine will generate an even number. Let us now focus on the other inclusion; that is, for any even number n=2m , there exists a halting computation of Πeven that generates n=2m . If m=0 , from instruction i1 , the nondeterministic decision is going to i3 ; as h1 is empty in the next step, the following halting configuration is reached: C2(0, 1, #, 0) . For m>0 , we have That is, the machine generates n=2m for any natural number even m∈ℕ . Generating square numbers The SVM of degree (5, 9) , Πsquare , generating natural square numbers is depicted in Fig. 6 . For generating the number 0, the computation, which chooses instruction i9 in the first step, reaches the halting configuration C2 = ( 0, 0, 0, 2, 0, i 9 ,0 ) . Let us now focus on generating the number n2 for each n>0 . For this purpose, the verification will be focused on proving the following invariant: where 𝛽k = 𝛼0 + 𝛼1 +…+ 𝛼k and 𝛼j=2j+7 , for each j>0 . The idea behind this formula is to increase the output host by the amount of viruses necessary to reach the next square number. Moving on to the proof, for k=0 , it is direct if i2 is chosen nondeterministically, 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) , for each 0 ≤k<n 2, Fig. 5. SVM Πeven generating the set { 2m | m∈ℕ } . Fig. 6. SVM Πsquare generating the set { n 2| n∈ℕ } . Downloaded from https://spj.science.org at Universidad de Sevilla on May 25, 2025 Ramírez-de-Arellano-Marrero et al. 2025 | https://doi.org/10.34133/icomputing.0103 7 Let us focus on the inductive step; that is, let us suppose that the formula is verified for any 0≤k<n2 −1 . Then, the formula is true also for k+1 , and by the induction hypothesis, we have Hence, 𝜑(k+1) is verified. As 𝜑(k) is true for each 0≤k<n2 , in particular, it is true for n−1 , leading to the following configuration, 𝜑(n−1)≡ C 1+𝛽 n−1 = ( n−1, 0, 0, 2, 0, i2,(n−1) 2) , from which a halting configuration can be reached similarly to before: C 2+𝛽 n= ( n+1, 0, 0, 2, 0, #, n 2) ; that is, after 2+𝛽n steps, the machine halts and generates n2 , which is the square number. Generating powers of 2 The SVM of degree (2, 5) , Πgpow2 , generating the powers of 2 is depicted in Fig. 7 . Firstly, let us see that for each number x∈{2n|n∈ℕ} , there exists a halting computation of Πgpow2 that generates x : That is, exactly 20. Let us now see that the computation for generating the natural number of the form 22n for some n≥1 . That is, 2n , for any n∈ℕ with n≥1 . Analogously, we can verify that there exists a computation that generates 22n+1 for each natural number n∈ℕ . That is, 2n+1 , for any n∈ℕ . Lastly, it can be easily seen that the only nondeterministic decisions in the machine are in instructions i1,i2,i3 , and for all these decisions, it has been seen that only powers of 2 can be generated. Thus, for each computation of Πgpow2 , the generated number is in the set {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 generating the set { 2 n| n∈ℕ } . Table 4. Time complexity comparison between a basic VM and the proposed SVM for computing the basic functions of a given input ( a,b ) ∈ ℕ Function VM SVM Addition (sum) a + b + 3 2 Multiplication (mult) 3b(a + 1) + 2 a(2b + 3) + 2 Exponentiation (pow2) Exponential 3a + 2 Table 5. Comparison of sources needed between a basic VM and the proposed SVM for generating basic subsets of natural numbers. Note that the “-” symbol means that there is no result published. Sets Source VM SVM Even numbers (even) Hosts 2 2 Instructions 4 2 Steps Linear Linear Square numbers (square) Hosts - 5 Instructions - 9 Steps - n√n Powers of 2 (gpow2) Hosts 2 2 Instructions 7 5 Steps Exponential Linear Downloaded from https://spj.science.org at Universidad de Sevilla on May 25, 2025 Ramírez-de-Arellano-Marrero et al. 2025 | https://doi.org/10.34133/icomputing.0103 8 Discussion In this subsection, a comparison with the results presented by Ramírez-de-Arellano et al. [ 10 , 12 ] with basic VMs is discussed. For the computing mode, all of the SVMs presented have a better time efficiency (depending on the input given) compared to the basic VMs presented by Ramírez-de-Arellano et al. [ 10 ], as can be seen in Table 4 . More precisely, for a given input (a,b)∈ℕ in unary codification, the addition function decreased its time cost from a+b+3 steps in VMs to 2 steps in SVMs. Moreover, for the same input, the multiplication function presented decreased from 3b(a+1)+2 steps in VMs to a(2b+3)+2 steps. Finally, the exponentiation function was first simulated by any extension of VMs, but it can be easily assumed that the time complexity will be at least exponential in basic VMs, while in this new extension, the SVM presented has a linear time complexity. For the generating mode, all of the SVMs presented need fewer or the same hosts and instructions as the VMs presented by Ramírez-de-Arellano et al. [ 12 ], as can be seen in Table 5 . We highlight Πgpow2 , where the number of instructions was decreased from 7 to 5 instructions and the number of steps needed from exponential to linear depending on the number generated. In addition, generating the square numbers has been done only for SVMs. Conclusion In this paper, an extension of VMs’ computing paradigm is defined: SVMs; this extension can develop models with a new kind of channels called superchannels. The semantics associated are shown with a visual example. In addition, several SVMs in function computing and generating mode are designed, improving the basic VMs in both time and memory costs. However, the efficiency of this paradigm should be further improved to be able to attack NP-hard problems [ 18 ]; for them, more nature-inspired ingredients can be considered in addition to the superchannels proposed, e.g., mutation, host replication, and host death. 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