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Basic Virus Machines
Luis Valencia-Cabrera1 Mario J. Pérez-Jiménez1
Xu Chen2 Beizhan Wang2 Xiangxiang Zeng3
1Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla.
Avda. Reina Mercedes s/n, 41012 Sevilla, Spain.
[email protected],[email protected]
2School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China.
[email protected],[email protected]
3Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China.
August 18, 2015. Valencia
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Contents
1 Introduction
2 Virology
3 Virus Machines
4 Examples
5 Computational power
6 Conclusion
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Contents
1 Introduction
2 Virology
3 Virus Machines
4 Examples
5 Computational power
6 Conclusion
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Motivation
Starting point
Several bio-inspired paradigms: cellular automata, genetic algorithms, DNA
computing, membrane computing...
There is no perfect model of computation for every purpose and problem.
New inspiration, new ideas
Animals or plants have survived the evolution process for many years...
Why animals or plants, or even bacteria? Why living cells? Are they more
successful than other processing units present in Nature?
Let us observe Nature again with a microscope...
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Motivation
Starting point
Several bio-inspired paradigms: cellular automata, genetic algorithms, DNA
computing, membrane computing...
There is no perfect model of computation for every purpose and problem.
New inspiration, new ideas
Animals or plants have survived the evolution process for many years...
Why animals or plants, or even bacteria? Why living cells? Are they more
successful than other processing units present in Nature?
Let us observe Nature again with a microscope...
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Motivation
Starting point
Several bio-inspired paradigms: cellular automata, genetic algorithms, DNA
computing, membrane computing...
There is no perfect model of computation for every purpose and problem.
New inspiration, new ideas
Animals or plants have survived the evolution process for many years...
Why animals or plants, or even bacteria? Why living cells? Are they more
successful than other processing units present in Nature?
Let us observe Nature again with a microscope...
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
A progressive view on us...
A human being...
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
A progressive view on us...
Ok, it is a Matryoshka, a russian nesting doll, but let us imagine...
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
A progressive view on us...
Some systems, organs, tissues, many cells...
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
A progressive view on us...
Oh, wait... bacteria? Yes, 10 times more bacteria than human cells
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
A progressive view on us...
What was that? Is it a virus? Yes, 10 times more viruses than bacteria!
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
A progressive view on us...
But... That implies 100 times more viruses than human cells!
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
A progressive view on us...
Many people would consider it a success, but even if you disagree...
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
A progressive view on us...
Should not we at least observe viruses and their behaviour?
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Contents
1 Introduction
2 Virology
A brief introduction to virology
Inspiration and informal description
3 Virus Machines
4 Examples
5 Computational power
6 Conclusion
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Contents
1 Introduction
2 Virology
A brief introduction to virology
Inspiration and informal description
3 Virus Machines
4 Examples
5 Computational power
6 Conclusion
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
A bit about viruses...
Definition
Parasitic biological agent
No reproduction, but replication using host machinery
Every known species is infected by viruses
Easy transmission in various ways
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Parasitic activity
Pacific coexistence =⇒ Lysogenic cycle
Use and abuse =⇒ Lytic cycle
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Contents
1 Introduction
2 Virology
A brief introduction to virology
Inspiration and informal description
3 Virus Machines
4 Examples
5 Computational power
6 Conclusion
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Inspiration for a computational model
Some ideas
A new computational device, the Virus Machine
Processing units, hosts, connected by transmission channels
Interpretation of hosts
Host: a group of cells (part of a colony, organism, system, organ or tissue).
Each cell in the group will contain at most one virus
Only interested in the group, not individual cells
Only one type of virus
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Transmission channels
Interpretation of channels
Arranged in a virus transmission network
Channels transmit viruses from one host to another
Weights associated with channels (number of viruses to transmit)
v......
3
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Transmission channels
Interpretation of channels
Arranged in a virus transmission network
Channels transmit viruses from one host to another
Weights associated with channels (number of viruses to transmit)
......
v v
v
3
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Transmission channels
Control
Channels are initially closed
They can be opened by control instruction units
An instruction-channel control network enables the opening of the channels to
replicate one virus and transmit its copies
Only one instruction active per step, in the sequence given by an instruction
transfer network
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Contents
1 Introduction
2 Virology
3 Virus Machines
4 Examples
5 Computational power
6 Conclusion
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Formal definition
A Virus Machine Π of degree (p,q), with p ≥ 1,q ≥ 1, is a tuple (Γ,H, I,DH ,DI ,GC ,n1, . . . ,np, i1,hout ):
Γ = {v}H = {h1, . . . ,hp}, I = {i1, . . . , iq} ordered sets, v /∈ H∪ I, H∩ I = /0
DH = (H∪{hout},EH ,wH) weighted directed graph, EH ⊆ H× (H∪{hout}),
(h,h) /∈ EH ,out-degree(hout ) = 0, wH mapping from EH onto IN\{0}DI = (I,EI ,wI) weighted directed graph, EI ⊆ I× I, wI mapping from EI onto IN\{0}for each vertex ij ∈ I, out-degree(ij )≤ 2
GC = (VC ,EC) undirected bipartite graph, VC = I∪EH ,
{I,EH} the partition, all edges going between the sets I and EH
for each vertex ij ∈ I, out-degree(ij )≤ 1
nj ∈ IN (1≤ j ≤ p), i1 ∈ I
hout /∈ I∪{v}, hout denoted by h0 if hout /∈ H.
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Summary
A heterogeneous network
DH , virus transmission network:
directed graph
weights associated with edges
hosts and channels
DI , instruction transfer network:
weighted directed graph
sequence of instructions
GC , instruction-channel control network:
undirected bipartite graph
linking instructions in DI with channels in DH
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Interpretation
p hosts h1, · · · ,hp ,
⇒ with nj viruses, 1≤ j ≤ p
q control instruction units i1, · · · , iq
hout output region
⇒ a host or the environment
Arcs from DH : transmission channels
Channels initially closed
⇒ opened by edges {ij ,(hs,hs′)} in GC
2
2
HD
ID
CG
2
33
4
*
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Semantics (I) - Configurations
Instantaneous description (configuration):
Ct = (a1,t , . . . ,ap,t ,ut ,et)
with a1,t , . . . ,ap,t ,et natural numbers, ut ∈ I∪{#}, # /∈ H∪{h0}∪ I.
Role of the control instruction unit ut :
If ut ∈ I, ut will be activated at step t + 1
Otherwise (i.e. ut = #) no instr. will be enabled (halting configuration)
Initial configuration: C0 = (n1, . . . ,np, i1,0) (no virus in the output region hout )
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Semantics (II) - Virus replication and transmission
Transition step from Ct = (a1,t , . . . ,ap,t ,ut ,et) to
Ct+1 = (a1,t+1, . . . ,ap,t+1,ut+1,et+1):
(a) Ct a non halting config., so ut ∈ I is activated.
(b) Let us assume ut attached to channel (hs,hs′). Then it will be opened and:
If as,t ≥ 1, 1 virus consumed from hs, ws,s′ copies of v produced in hs′
If as,t = 0, no transmission
(c) If ut not attached to any channel (hs,hs′), then there is no transmission of virus.
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Semantics (III) - Next instruction
(d) Object ut+1 ∈ I∪{#} is obtained as follows:
If out− degree(ut ) = 2 , (ut ′ and ut ′′ such that (ut ,ut ′) ∈ EI and (ut ,ut ′′) ∈ EI ).
If ut attached to (hs,hs′), as,t ≥ 1 :
⇒ ut+1 the instruction with weight: max{wt,t ′ ,wt,t ′′}⇒ if wt,t ′ = wt,t ′′ , non-deterministic selection
If ut attached to (hs,hs′), as,t = 0 :
⇒ ut+1 the instruction with weight: min{wt,t ′ ,wt,t ′′}⇒ if wt,t ′ = wt,t ′′ , non-deterministic selection, then is the instruction
If ut not attached to a channel ⇒ non-deterministic selection.
If out− degree(ut ) = 1 , deterministic, ut+1 the instruction verifying
(ut ,ut+1) ∈ EI .
If out− degree(ut ) = 0 , ut+1 is #, Ct+1 halting configuration.
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Virus Machine with input
A Virus Machine with input of degree (p,q, r),p ≥ 1,q ≥ 1, r ≥ 1, is a tuple
(Γ,H,Hr , I,DH ,DI ,GC ,n1, . . . ,np, i1,hout )
where:
(Γ,H, I,DH ,DI ,GC ,n1, . . . ,np, i1,hout ) a Virus Machine of degree (p,q).
Hr = {hi1 , . . . ,hir } ⊆ H an ordered set of r input hosts, hout /∈ Hr .
Initial configuration of Π with input (α1, . . . ,αr ):
(n1, . . . ,ni1 + α1, . . . ,nir + αr , . . . ,np, it ,0).
A computation of Π with input (α1, . . . ,αr ), denoted by Π + (α1, . . . ,αr ), starts with
the initial config. (n1, . . . ,ni1 + α1, . . . ,nir + αr , . . . ,np, it ,0) and proceeds as stated.
.
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Contents
1 Introduction
2 Virology
3 Virus Machines
4 Examples
A Virus Machine working in the computing mode
A Virus Machine working in the accepting mode
5 Computational power
6 Conclusion
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Contents
1 Introduction
2 Virology
3 Virus Machines
4 Examples
A Virus Machine working in the computing mode
A Virus Machine working in the accepting mode
5 Computational power
6 Conclusion
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
An example of Virus Machine with input:
A (deterministic) virus machine with input working in the computing mode
Specifically, a machine ΠSub simulating the subtraction of two natural numbers
Input: a pair of natural numbers (n1,n2) with n1 ≥ n2 supplied to the system
Output: the natural number n1− n2
Computation of virus machine Π + (α1, . . . ,αr ) in computing mode
Total number n of viruses sent to the environment during the computation.
We say that number n is computed by the virus machine Π + (α1, . . . ,αr ).
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Let us consider the virus machine with input of degree (3,4,2)
ΠSub = (Γ,H,H2, I,DH ,DI,GC,0,0,0, i1,hout)
working in the computing mode, defined as follows:
Γ = {v}, H = {h1,h2,h3}, H2 = {h1,h2}, I = {i1, i2, i3, i4}, hout = h0.
DH = ({h0,h1,h2,h3},EH ,wH), with EH = {(h1,h0),(h1,h3),(h2,h3)},w1,0 = w1,3 = w2,3 = 1.
DI = ({i1, i2, i3, i4},EI ,wI), with EI = {(i1, i2),(i1, i3),(i2, i1),(i3, i3),(i3, i4)},w1,2 = w3,3 = 2, w1,3 = w2,1 = w3,4 = 1.
GC = (I ∪ EH ,EC), with EC = {{i1,(h2,h3)},{i2,(h1,h3)},{i3,(h1,h0)}}.
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
H = {h1,h2,h3}H2 = {h1,h2}I = {i1, i2, i3, i4}hout = h0
DH = ({h0,h1,h2,h3},EH ,wH ), with
EH = {(h1,h0),(h1,h3),(h2,h3)},w1,0 = w1,3 = w2,3 = 1.
DI = ({i1, i2, i3, i4},EI ,wI), with
EI = {(i1, i2),(i1, i3),(i2, i1),(i3, i3),(i3, i4)},w1,2 = w3,3 = 2, w1,3 = w2,1 = w3,4 = 1.
GC = (I ∪ EH ,EC), with
EC = {{i1,(h2,h3)},{i2,(h1,h3)},{i3,(h1,h0)}}.
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 1: n1 = n2 = 0
C0 = (0,0,0, i1,0)
C1 = (0,0,0, i3,0)
C2 = (0,0,0, i4,0)
C3 = (0,0,0,#,0)
Output: the natural number 0
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 1: n1 = n2 = 0
C0 = (0,0,0, i1,0)
C1 = (0,0,0, i3,0)
C2 = (0,0,0, i4,0)
C3 = (0,0,0,#,0)
Output: the natural number 0
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 1: n1 = n2 = 0
C0 = (0,0,0, i1,0)
C1 = (0,0,0, i3,0)
C2 = (0,0,0, i4,0)
C3 = (0,0,0,#,0)
Output: the natural number 0
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 1: n1 = n2 = 0
C0 = (0,0,0, i1,0)
C1 = (0,0,0, i3,0)
C2 = (0,0,0, i4,0)
C3 = (0,0,0,#,0)
Output: the natural number 0
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 1: n1 = n2 = 0
C0 = (0,0,0, i1,0)
C1 = (0,0,0, i3,0)
C2 = (0,0,0, i4,0)
C3 = (0,0,0,#,0)
Output: the natural number 0
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 2: n1 > n2 = 0
C0 = (n1,0,0, i1,0)
C1 = (n1,0,0, i3,0)
C2 = (n1− 1,0,0, i3,1)
. . . . . . . . . . . . . . . . . .
Cn1+1 = (0,0,0, i3,n1)
Cn1+2 = (0,0,0, i4,n1)
Cn1+1 = (0,0,0,#,n1)
Output: the natural number n1
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 2: n1 > n2 = 0
C0 = (n1,0,0, i1,0)
C1 = (n1,0,0, i3,0)
C2 = (n1− 1,0,0, i3,1)
. . . . . . . . . . . . . . . . . .
Cn1+1 = (0,0,0, i3,n1)
Cn1+2 = (0,0,0, i4,n1)
Cn1+1 = (0,0,0,#,n1)
Output: the natural number n1
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 2: n1 > n2 = 0
C0 = (n1,0,0, i1,0)
C1 = (n1,0,0, i3,0)
C2 = (n1− 1,0,0, i3,1)
. . . . . . . . . . . . . . . . . .
Cn1+1 = (0,0,0, i3,n1)
Cn1+2 = (0,0,0, i4,n1)
Cn1+1 = (0,0,0,#,n1)
Output: the natural number n1
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 2: n1 > n2 = 0
C0 = (n1,0,0, i1,0)
C1 = (n1,0,0, i3,0)
C2 = (n1− 1,0,0, i3,1)
. . . . . . . . . . . . . . . . . .
Cn1+1 = (0,0,0, i3,n1)
Cn1+2 = (0,0,0, i4,n1)
Cn1+1 = (0,0,0,#,n1)
Output: the natural number n1
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 2: n1 > n2 = 0
C0 = (n1,0,0, i1,0)
C1 = (n1,0,0, i3,0)
C2 = (n1− 1,0,0, i3,1)
. . . . . . . . . . . . . . . . . .
Cn1+1 = (0,0,0, i3,n1)
Cn1+2 = (0,0,0, i4,n1)
Cn1+1 = (0,0,0,#,n1)
Output: the natural number n1
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 2: n1 > n2 = 0
C0 = (n1,0,0, i1,0)
C1 = (n1,0,0, i3,0)
C2 = (n1− 1,0,0, i3,1)
. . . . . . . . . . . . . . . . . .
Cn1+1 = (0,0,0, i3,n1)
Cn1+2 = (0,0,0, i4,n1)
Cn1+1 = (0,0,0,#,n1)
Output: the natural number n1
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 2: n1 > n2 = 0
C0 = (n1,0,0, i1,0)
C1 = (n1,0,0, i3,0)
C2 = (n1− 1,0,0, i3,1)
. . . . . . . . . . . . . . . . . .
Cn1+1 = (0,0,0, i3,n1)
Cn1+2 = (0,0,0, i4,n1)
Cn1+1 = (0,0,0,#,n1)
Output: the natural number n1
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 2: n1 > n2 = 0
C0 = (n1,0,0, i1,0)
C1 = (n1,0,0, i3,0)
C2 = (n1− 1,0,0, i3,1)
. . . . . . . . . . . . . . . . . .
Cn1+1 = (0,0,0, i3,n1)
Cn1+2 = (0,0,0, i4,n1)
Cn1+1 = (0,0,0,#,n1)
Output: the natural number n1
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 3: n1 > n2 > 0
C0 = (n1,n2,0, i1,0)
C1 = (n1,n2− 1,1, i2,0)C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (n1− n2 + 1,0,2n2− 1, i2,0)C2n2 = (n1− n2,0,2n2, i1,0)C2n2+1 = (n1− n2,0,2n2, i3,0)C2n2+2 = (n1− n2− 1,0,2n2, i3,1). . . . . . . . . . . . . . . . . .Cn2+n1+1 = (0,0,2n2, i3,n1− n2)Cn2+n1+2 = (0,0,2n2, i4,n1− n2)Cn2+n1+3 = (0,0,2n2,#,n1− n2)
Output: the natural number n1− n2
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 3: n1 > n2 > 0
C0 = (n1,n2,0, i1,0)C1 = (n1,n2− 1,1, i2,0)
C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (n1− n2 + 1,0,2n2− 1, i2,0)C2n2 = (n1− n2,0,2n2, i1,0)C2n2+1 = (n1− n2,0,2n2, i3,0)C2n2+2 = (n1− n2− 1,0,2n2, i3,1). . . . . . . . . . . . . . . . . .Cn2+n1+1 = (0,0,2n2, i3,n1− n2)Cn2+n1+2 = (0,0,2n2, i4,n1− n2)Cn2+n1+3 = (0,0,2n2,#,n1− n2)
Output: the natural number n1− n2
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 3: n1 > n2 > 0
C0 = (n1,n2,0, i1,0)C1 = (n1,n2− 1,1, i2,0)C2 = (n1− 1,n2− 1,2, i1,0)
. . . . . . . . . . . . . . . . . .C2n2−1 = (n1− n2 + 1,0,2n2− 1, i2,0)C2n2 = (n1− n2,0,2n2, i1,0)C2n2+1 = (n1− n2,0,2n2, i3,0)C2n2+2 = (n1− n2− 1,0,2n2, i3,1). . . . . . . . . . . . . . . . . .Cn2+n1+1 = (0,0,2n2, i3,n1− n2)Cn2+n1+2 = (0,0,2n2, i4,n1− n2)Cn2+n1+3 = (0,0,2n2,#,n1− n2)
Output: the natural number n1− n2
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 3: n1 > n2 > 0
C0 = (n1,n2,0, i1,0)C1 = (n1,n2− 1,1, i2,0)C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .
C2n2−1 = (n1− n2 + 1,0,2n2− 1, i2,0)C2n2 = (n1− n2,0,2n2, i1,0)C2n2+1 = (n1− n2,0,2n2, i3,0)C2n2+2 = (n1− n2− 1,0,2n2, i3,1). . . . . . . . . . . . . . . . . .Cn2+n1+1 = (0,0,2n2, i3,n1− n2)Cn2+n1+2 = (0,0,2n2, i4,n1− n2)Cn2+n1+3 = (0,0,2n2,#,n1− n2)
Output: the natural number n1− n2
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 3: n1 > n2 > 0
C0 = (n1,n2,0, i1,0)C1 = (n1,n2− 1,1, i2,0)C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (n1− n2 + 1,0,2n2− 1, i2,0)
C2n2 = (n1− n2,0,2n2, i1,0)C2n2+1 = (n1− n2,0,2n2, i3,0)C2n2+2 = (n1− n2− 1,0,2n2, i3,1). . . . . . . . . . . . . . . . . .Cn2+n1+1 = (0,0,2n2, i3,n1− n2)Cn2+n1+2 = (0,0,2n2, i4,n1− n2)Cn2+n1+3 = (0,0,2n2,#,n1− n2)
Output: the natural number n1− n2
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Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 3: n1 > n2 > 0
C0 = (n1,n2,0, i1,0)C1 = (n1,n2− 1,1, i2,0)C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (n1− n2 + 1,0,2n2− 1, i2,0)C2n2 = (n1− n2,0,2n2, i1,0)
C2n2+1 = (n1− n2,0,2n2, i3,0)C2n2+2 = (n1− n2− 1,0,2n2, i3,1). . . . . . . . . . . . . . . . . .Cn2+n1+1 = (0,0,2n2, i3,n1− n2)Cn2+n1+2 = (0,0,2n2, i4,n1− n2)Cn2+n1+3 = (0,0,2n2,#,n1− n2)
Output: the natural number n1− n2
1h
2h
1i
2i
3i
2
3h
4i
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 3: n1 > n2 > 0
C0 = (n1,n2,0, i1,0)C1 = (n1,n2− 1,1, i2,0)C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (n1− n2 + 1,0,2n2− 1, i2,0)C2n2 = (n1− n2,0,2n2, i1,0)C2n2+1 = (n1− n2,0,2n2, i3,0)
C2n2+2 = (n1− n2− 1,0,2n2, i3,1). . . . . . . . . . . . . . . . . .Cn2+n1+1 = (0,0,2n2, i3,n1− n2)Cn2+n1+2 = (0,0,2n2, i4,n1− n2)Cn2+n1+3 = (0,0,2n2,#,n1− n2)
Output: the natural number n1− n2
1h
2h
1i
2i
3i
2
3h
4i
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 3: n1 > n2 > 0
C0 = (n1,n2,0, i1,0)C1 = (n1,n2− 1,1, i2,0)C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (n1− n2 + 1,0,2n2− 1, i2,0)C2n2 = (n1− n2,0,2n2, i1,0)C2n2+1 = (n1− n2,0,2n2, i3,0)C2n2+2 = (n1− n2− 1,0,2n2, i3,1)
. . . . . . . . . . . . . . . . . .Cn2+n1+1 = (0,0,2n2, i3,n1− n2)Cn2+n1+2 = (0,0,2n2, i4,n1− n2)Cn2+n1+3 = (0,0,2n2,#,n1− n2)
Output: the natural number n1− n2
1h
2h
1i
2i
3i
2
3h
4i
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 3: n1 > n2 > 0
C0 = (n1,n2,0, i1,0)C1 = (n1,n2− 1,1, i2,0)C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (n1− n2 + 1,0,2n2− 1, i2,0)C2n2 = (n1− n2,0,2n2, i1,0)C2n2+1 = (n1− n2,0,2n2, i3,0)C2n2+2 = (n1− n2− 1,0,2n2, i3,1). . . . . . . . . . . . . . . . . .
Cn2+n1+1 = (0,0,2n2, i3,n1− n2)Cn2+n1+2 = (0,0,2n2, i4,n1− n2)Cn2+n1+3 = (0,0,2n2,#,n1− n2)
Output: the natural number n1− n2
1h
2h
1i
2i
3i
2
3h
4i
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 3: n1 > n2 > 0
C0 = (n1,n2,0, i1,0)C1 = (n1,n2− 1,1, i2,0)C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (n1− n2 + 1,0,2n2− 1, i2,0)C2n2 = (n1− n2,0,2n2, i1,0)C2n2+1 = (n1− n2,0,2n2, i3,0)C2n2+2 = (n1− n2− 1,0,2n2, i3,1). . . . . . . . . . . . . . . . . .Cn2+n1+1 = (0,0,2n2, i3,n1− n2)
Cn2+n1+2 = (0,0,2n2, i4,n1− n2)Cn2+n1+3 = (0,0,2n2,#,n1− n2)
Output: the natural number n1− n2
1h
2h
1i
2i
3i
2
3h
4i
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 3: n1 > n2 > 0
C0 = (n1,n2,0, i1,0)C1 = (n1,n2− 1,1, i2,0)C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (n1− n2 + 1,0,2n2− 1, i2,0)C2n2 = (n1− n2,0,2n2, i1,0)C2n2+1 = (n1− n2,0,2n2, i3,0)C2n2+2 = (n1− n2− 1,0,2n2, i3,1). . . . . . . . . . . . . . . . . .Cn2+n1+1 = (0,0,2n2, i3,n1− n2)Cn2+n1+2 = (0,0,2n2, i4,n1− n2)
Cn2+n1+3 = (0,0,2n2,#,n1− n2)
Output: the natural number n1− n2
1h
2h
1i
2i
3i
2
3h
4i
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 3: n1 > n2 > 0
C0 = (n1,n2,0, i1,0)C1 = (n1,n2− 1,1, i2,0)C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (n1− n2 + 1,0,2n2− 1, i2,0)C2n2 = (n1− n2,0,2n2, i1,0)C2n2+1 = (n1− n2,0,2n2, i3,0)C2n2+2 = (n1− n2− 1,0,2n2, i3,1). . . . . . . . . . . . . . . . . .Cn2+n1+1 = (0,0,2n2, i3,n1− n2)Cn2+n1+2 = (0,0,2n2, i4,n1− n2)Cn2+n1+3 = (0,0,2n2,#,n1− n2)
Output: the natural number n1− n2
1h
2h
1i
2i
3i
2
3h
4i
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 3: n1 > n2 > 0
C0 = (n1,n2,0, i1,0)C1 = (n1,n2− 1,1, i2,0)C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (n1− n2 + 1,0,2n2− 1, i2,0)C2n2 = (n1− n2,0,2n2, i1,0)C2n2+1 = (n1− n2,0,2n2, i3,0)C2n2+2 = (n1− n2− 1,0,2n2, i3,1). . . . . . . . . . . . . . . . . .Cn2+n1+1 = (0,0,2n2, i3,n1− n2)Cn2+n1+2 = (0,0,2n2, i4,n1− n2)Cn2+n1+3 = (0,0,2n2,#,n1− n2)
Output: the natural number n1− n2
1h
2h
1i
2i
3i
2
3h
4i
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 4: n1 = n2 > 0
C0 = (n1,n2,0, i1,0)
C1 = (n1,n2− 1,1, i2,0)
C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (1,0,2n2− 1, i2,0)
C2n2 = (0,0,2n2, i1,0)
C2n2+1 = (0,0,2n2, i3,0)
C2n2+2 = (0,0,2n2, i4,0)
C2n2+3 = (0,0,2n2,#,0)
Output: the natural number 0
1h
2h
1i
2i
3i
2
3h
4i
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 4: n1 = n2 > 0
C0 = (n1,n2,0, i1,0)
C1 = (n1,n2− 1,1, i2,0)
C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (1,0,2n2− 1, i2,0)
C2n2 = (0,0,2n2, i1,0)
C2n2+1 = (0,0,2n2, i3,0)
C2n2+2 = (0,0,2n2, i4,0)
C2n2+3 = (0,0,2n2,#,0)
Output: the natural number 0
1h
2h
1i
2i
3i
2
3h
4i
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 4: n1 = n2 > 0
C0 = (n1,n2,0, i1,0)
C1 = (n1,n2− 1,1, i2,0)
C2 = (n1− 1,n2− 1,2, i1,0)
. . . . . . . . . . . . . . . . . .C2n2−1 = (1,0,2n2− 1, i2,0)
C2n2 = (0,0,2n2, i1,0)
C2n2+1 = (0,0,2n2, i3,0)
C2n2+2 = (0,0,2n2, i4,0)
C2n2+3 = (0,0,2n2,#,0)
Output: the natural number 0
1h
2h
1i
2i
3i
2
3h
4i
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 4: n1 = n2 > 0
C0 = (n1,n2,0, i1,0)
C1 = (n1,n2− 1,1, i2,0)
C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .
C2n2−1 = (1,0,2n2− 1, i2,0)
C2n2 = (0,0,2n2, i1,0)
C2n2+1 = (0,0,2n2, i3,0)
C2n2+2 = (0,0,2n2, i4,0)
C2n2+3 = (0,0,2n2,#,0)
Output: the natural number 0
1h
2h
1i
2i
3i
2
3h
4i
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 4: n1 = n2 > 0
C0 = (n1,n2,0, i1,0)
C1 = (n1,n2− 1,1, i2,0)
C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (1,0,2n2− 1, i2,0)
C2n2 = (0,0,2n2, i1,0)
C2n2+1 = (0,0,2n2, i3,0)
C2n2+2 = (0,0,2n2, i4,0)
C2n2+3 = (0,0,2n2,#,0)
Output: the natural number 0
1h
2h
1i
2i
3i
2
3h
4i
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 4: n1 = n2 > 0
C0 = (n1,n2,0, i1,0)
C1 = (n1,n2− 1,1, i2,0)
C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (1,0,2n2− 1, i2,0)
C2n2 = (0,0,2n2, i1,0)
C2n2+1 = (0,0,2n2, i3,0)
C2n2+2 = (0,0,2n2, i4,0)
C2n2+3 = (0,0,2n2,#,0)
Output: the natural number 0
1h
2h
1i
2i
3i
2
3h
4i
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 4: n1 = n2 > 0
C0 = (n1,n2,0, i1,0)
C1 = (n1,n2− 1,1, i2,0)
C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (1,0,2n2− 1, i2,0)
C2n2 = (0,0,2n2, i1,0)
C2n2+1 = (0,0,2n2, i3,0)
C2n2+2 = (0,0,2n2, i4,0)
C2n2+3 = (0,0,2n2,#,0)
Output: the natural number 0
1h
2h
1i
2i
3i
2
3h
4i
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 4: n1 = n2 > 0
C0 = (n1,n2,0, i1,0)
C1 = (n1,n2− 1,1, i2,0)
C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (1,0,2n2− 1, i2,0)
C2n2 = (0,0,2n2, i1,0)
C2n2+1 = (0,0,2n2, i3,0)
C2n2+2 = (0,0,2n2, i4,0)
C2n2+3 = (0,0,2n2,#,0)
Output: the natural number 0
1h
2h
1i
2i
3i
2
3h
4i
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 4: n1 = n2 > 0
C0 = (n1,n2,0, i1,0)
C1 = (n1,n2− 1,1, i2,0)
C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (1,0,2n2− 1, i2,0)
C2n2 = (0,0,2n2, i1,0)
C2n2+1 = (0,0,2n2, i3,0)
C2n2+2 = (0,0,2n2, i4,0)
C2n2+3 = (0,0,2n2,#,0)
Output: the natural number 0
1h
2h
1i
2i
3i
2
3h
4i
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 1: subtraction
Computation of ΠSub with input (n1,n2)
Case 4: n1 = n2 > 0
C0 = (n1,n2,0, i1,0)
C1 = (n1,n2− 1,1, i2,0)
C2 = (n1− 1,n2− 1,2, i1,0). . . . . . . . . . . . . . . . . .C2n2−1 = (1,0,2n2− 1, i2,0)
C2n2 = (0,0,2n2, i1,0)
C2n2+1 = (0,0,2n2, i3,0)
C2n2+2 = (0,0,2n2, i4,0)
C2n2+3 = (0,0,2n2,#,0)
Output: the natural number 0
1h
2h
1i
2i
3i
2
3h
4i
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Contents
1 Introduction
2 Virology
3 Virus Machines
4 Examples
A Virus Machine working in the computing mode
A Virus Machine working in the accepting mode
5 Computational power
6 Conclusion
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
An example of Virus Machine with input:
A (deterministic) virus machine with input working in the accepting mode
Specifically, a machine ΠEqual evaluating the equality of two natural numbers
Input: a pair of natural numbers (n1,n2) supplied to the system
Output: yes, if and only if n1 = n2 (no otherwise)
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation of virus machine Π + (α1, . . . ,αr ) in accepting mode
All computations halt and either:
for each computation some viruses are sent to the output region,
or
for each computation no virus is sent to the output region.
Result:
yes, if viruses sent to output in any computation (last component of halting config. 6= 0)
no otherwise
If output is yes (resp. no), input (α1, . . . ,αr ) is accepted (resp. rejected).
Each halting computation is either an accepting or a rejecting computation.
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Let us consider the virus machine with input of degree (3,5,2) in accepting mode
ΠEqual = (Γ,H,H2, I,DH ,DI,GC,0,0,0, i1,hout)
working in the computing mode, defined as follows:
Γ = {v}, H = {hE1 ,hE
2 ,hE3 }, H2 = {hE
1 ,hE2 }, I = {iE1 , iE2 , iE3 , iE4 , iE5 }, hout = h0.
DH = ({h0,hE1 ,hE
2 ,hE3 },EH ,wH), where EH = {(hE
1 ,hE3 ),(hE
3 ,h0),(hE2 ,hE
3 )} and
w1,3 = w2,3 = w3,0 = 1.
DI = ({iE1 , iE2 , iE3 , iE4 , iE5 },EI ,wI), where EI = {(iE1 , iE2 ),(iE1 , iE3 ),(iE2 , iE1 ),(iE2 , iE4 ),
(iE3 , iE4 ), (iE3 , iE5 )} and w1,3 = w3,5 = w2,4 = 1, w1,2 = w2,1 = w3,4 = 2.
GC = (VC ,EC), where VC = I ∪ EH and the set of edges is:
EC = {{iE1 ,(hE1 ,hE
3 )},{iE2 ,(hE2 ,hE
3 )},{iE3 ,(hE2 ,hE
3 )},{iE5 ,(hE3 ,h0)}}.
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
H = {hE1 ,hE
2 ,hE3 }
H2 = {hE1 ,hE
2 }I = {iE1 , iE2 , iE3 , iE4 , iE5 }hout = h0
DH = ({h0,hE1 ,hE
2 ,hE3 },EH ,wH ), with
EH = {(hE1 ,hE
3 ),(hE3 ,h0),(hE
2 ,hE3 )},
w1,3 = w2,3 = w3,0 = 1.
DI = ({iE1 , iE2 , iE3 , iE4 , iE5 },EI ,wI), with
EI = {(iE1 , iE2 ),(iE1 , iE3 ),(iE2 , iE1 ),(iE2 , iE4 ),
(iE3 , iE4 ), (iE3 , iE5 )}, w1,3 = w3,5 = w2,4 = 1,
w1,2 = w2,1 = w3,4 = 2.
GC = (VC ,EC), with VC = I ∪ EH ,
EC = {{iE1 ,(hE1 ,hE
3 )},{iE2 ,(hE2 ,hE
3 )},{iE3 ,(hE2 ,hE
3 )},{iE5 ,(hE3 ,h0)}} 2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation. Initial configuration
Given two positive integer numbers n1,n2, the virus
machine ΠEqual with input (n1,n2) determines
whether or not n1 = n2.
Starting point: numbers n1 and n2 are encoded in
the number of viruses in input hosts h(E)1 and h
(E)2
Initial configuration: C0 = (n1,n2,0, iE1 ,0).
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 > n2
C0 = (n1,n2,0, iE1 ,0)
C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n2−1 = (n1− n2,1,2n2− 1, iE2 ,0)C2n2 = (n1− n2,0,2n2, iE1 ,0)C2n2+1 = (n1− n2− 1,0,2n2 + 1, iE2 ,0)C2n2+2 = (n1− n2− 1,0,2n2 + 1, iE4 ,0)C2n2+3 = (n1− n2− 1,0,2n2 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 > n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)
C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n2−1 = (n1− n2,1,2n2− 1, iE2 ,0)C2n2 = (n1− n2,0,2n2, iE1 ,0)C2n2+1 = (n1− n2− 1,0,2n2 + 1, iE2 ,0)C2n2+2 = (n1− n2− 1,0,2n2 + 1, iE4 ,0)C2n2+3 = (n1− n2− 1,0,2n2 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 > n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0)
. . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n2−1 = (n1− n2,1,2n2− 1, iE2 ,0)C2n2 = (n1− n2,0,2n2, iE1 ,0)C2n2+1 = (n1− n2− 1,0,2n2 + 1, iE2 ,0)C2n2+2 = (n1− n2− 1,0,2n2 + 1, iE4 ,0)C2n2+3 = (n1− n2− 1,0,2n2 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 > n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .
in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n2−1 = (n1− n2,1,2n2− 1, iE2 ,0)C2n2 = (n1− n2,0,2n2, iE1 ,0)C2n2+1 = (n1− n2− 1,0,2n2 + 1, iE2 ,0)C2n2+2 = (n1− n2− 1,0,2n2 + 1, iE4 ,0)C2n2+3 = (n1− n2− 1,0,2n2 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 > n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n2−1 = (n1− n2,1,2n2− 1, iE2 ,0)C2n2 = (n1− n2,0,2n2, iE1 ,0)C2n2+1 = (n1− n2− 1,0,2n2 + 1, iE2 ,0)C2n2+2 = (n1− n2− 1,0,2n2 + 1, iE4 ,0)C2n2+3 = (n1− n2− 1,0,2n2 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 > n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n2−1 = (n1− n2,1,2n2− 1, iE2 ,0)
C2n2 = (n1− n2,0,2n2, iE1 ,0)C2n2+1 = (n1− n2− 1,0,2n2 + 1, iE2 ,0)C2n2+2 = (n1− n2− 1,0,2n2 + 1, iE4 ,0)C2n2+3 = (n1− n2− 1,0,2n2 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 > n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n2−1 = (n1− n2,1,2n2− 1, iE2 ,0)C2n2 = (n1− n2,0,2n2, iE1 ,0)
C2n2+1 = (n1− n2− 1,0,2n2 + 1, iE2 ,0)C2n2+2 = (n1− n2− 1,0,2n2 + 1, iE4 ,0)C2n2+3 = (n1− n2− 1,0,2n2 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 > n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n2−1 = (n1− n2,1,2n2− 1, iE2 ,0)C2n2 = (n1− n2,0,2n2, iE1 ,0)C2n2+1 = (n1− n2− 1,0,2n2 + 1, iE2 ,0)
C2n2+2 = (n1− n2− 1,0,2n2 + 1, iE4 ,0)C2n2+3 = (n1− n2− 1,0,2n2 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 > n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n2−1 = (n1− n2,1,2n2− 1, iE2 ,0)C2n2 = (n1− n2,0,2n2, iE1 ,0)C2n2+1 = (n1− n2− 1,0,2n2 + 1, iE2 ,0)C2n2+2 = (n1− n2− 1,0,2n2 + 1, iE4 ,0)
C2n2+3 = (n1− n2− 1,0,2n2 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 > n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n2−1 = (n1− n2,1,2n2− 1, iE2 ,0)C2n2 = (n1− n2,0,2n2, iE1 ,0)C2n2+1 = (n1− n2− 1,0,2n2 + 1, iE2 ,0)C2n2+2 = (n1− n2− 1,0,2n2 + 1, iE4 ,0)C2n2+3 = (n1− n2− 1,0,2n2 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 > n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n2−1 = (n1− n2,1,2n2− 1, iE2 ,0)C2n2 = (n1− n2,0,2n2, iE1 ,0)C2n2+1 = (n1− n2− 1,0,2n2 + 1, iE2 ,0)C2n2+2 = (n1− n2− 1,0,2n2 + 1, iE4 ,0)C2n2+3 = (n1− n2− 1,0,2n2 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 < n2
C0 = (n1,n2,0, iE1 ,0)
C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,n2− n1 + 1,2n1− 1, iE2 ,0)C2n1 = (0,n2− n1,2n1, iE1 ,0)C2n1+1 = (0,n2− n1,2n1, iE3 ,0)C2n1+2 = (0,n2− n1− 1,2n1 + 1, iE4 ,0)C2n1+3 = (0,n2− n1− 1,2n1 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 < n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)
C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,n2− n1 + 1,2n1− 1, iE2 ,0)C2n1 = (0,n2− n1,2n1, iE1 ,0)C2n1+1 = (0,n2− n1,2n1, iE3 ,0)C2n1+2 = (0,n2− n1− 1,2n1 + 1, iE4 ,0)C2n1+3 = (0,n2− n1− 1,2n1 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 < n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0)
. . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,n2− n1 + 1,2n1− 1, iE2 ,0)C2n1 = (0,n2− n1,2n1, iE1 ,0)C2n1+1 = (0,n2− n1,2n1, iE3 ,0)C2n1+2 = (0,n2− n1− 1,2n1 + 1, iE4 ,0)C2n1+3 = (0,n2− n1− 1,2n1 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 < n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,n2− n1 + 1,2n1− 1, iE2 ,0)C2n1 = (0,n2− n1,2n1, iE1 ,0)C2n1+1 = (0,n2− n1,2n1, iE3 ,0)C2n1+2 = (0,n2− n1− 1,2n1 + 1, iE4 ,0)C2n1+3 = (0,n2− n1− 1,2n1 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 < n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,n2− n1 + 1,2n1− 1, iE2 ,0)
C2n1 = (0,n2− n1,2n1, iE1 ,0)C2n1+1 = (0,n2− n1,2n1, iE3 ,0)C2n1+2 = (0,n2− n1− 1,2n1 + 1, iE4 ,0)C2n1+3 = (0,n2− n1− 1,2n1 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 < n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,n2− n1 + 1,2n1− 1, iE2 ,0)C2n1 = (0,n2− n1,2n1, iE1 ,0)
C2n1+1 = (0,n2− n1,2n1, iE3 ,0)C2n1+2 = (0,n2− n1− 1,2n1 + 1, iE4 ,0)C2n1+3 = (0,n2− n1− 1,2n1 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 < n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,n2− n1 + 1,2n1− 1, iE2 ,0)C2n1 = (0,n2− n1,2n1, iE1 ,0)C2n1+1 = (0,n2− n1,2n1, iE3 ,0)
C2n1+2 = (0,n2− n1− 1,2n1 + 1, iE4 ,0)C2n1+3 = (0,n2− n1− 1,2n1 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 < n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,n2− n1 + 1,2n1− 1, iE2 ,0)C2n1 = (0,n2− n1,2n1, iE1 ,0)C2n1+1 = (0,n2− n1,2n1, iE3 ,0)C2n1+2 = (0,n2− n1− 1,2n1 + 1, iE4 ,0)
C2n1+3 = (0,n2− n1− 1,2n1 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 < n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,n2− n1 + 1,2n1− 1, iE2 ,0)C2n1 = (0,n2− n1,2n1, iE1 ,0)C2n1+1 = (0,n2− n1,2n1, iE3 ,0)C2n1+2 = (0,n2− n1− 1,2n1 + 1, iE4 ,0)C2n1+3 = (0,n2− n1− 1,2n1 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 < n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,n2− n1 + 1,2n1− 1, iE2 ,0)C2n1 = (0,n2− n1,2n1, iE1 ,0)C2n1+1 = (0,n2− n1,2n1, iE3 ,0)C2n1+2 = (0,n2− n1− 1,2n1 + 1, iE4 ,0)C2n1+3 = (0,n2− n1− 1,2n1 + 1,#,0)
Output: no virus sent, so input (n1,n2) is rejected.
Thus, the virus machine answers that n1 = n2 is no.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 = n2
C0 = (n1,n2,0, iE1 ,0)
C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,1,2n1− 1, iE2 ,0)C2n1 = (0,0,2n1, iE1 ,0)C2n1+1 = (0,0,2n1, iE3 ,0)C2n1+2 = (0,0,2n1, iE5 ,0)C2n1+3 = (0,0,2n1,#,1)
Output: one virus sent, so input (n1,n2) is accepted.
Thus, the virus machine answers that n1 = n2 is yes
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 = n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)
C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,1,2n1− 1, iE2 ,0)C2n1 = (0,0,2n1, iE1 ,0)C2n1+1 = (0,0,2n1, iE3 ,0)C2n1+2 = (0,0,2n1, iE5 ,0)C2n1+3 = (0,0,2n1,#,1)
Output: one virus sent, so input (n1,n2) is accepted.
Thus, the virus machine answers that n1 = n2 is yes
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 = n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0)
. . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,1,2n1− 1, iE2 ,0)C2n1 = (0,0,2n1, iE1 ,0)C2n1+1 = (0,0,2n1, iE3 ,0)C2n1+2 = (0,0,2n1, iE5 ,0)C2n1+3 = (0,0,2n1,#,1)
Output: one virus sent, so input (n1,n2) is accepted.
Thus, the virus machine answers that n1 = n2 is yes
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 = n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .
in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,1,2n1− 1, iE2 ,0)C2n1 = (0,0,2n1, iE1 ,0)C2n1+1 = (0,0,2n1, iE3 ,0)C2n1+2 = (0,0,2n1, iE5 ,0)C2n1+3 = (0,0,2n1,#,1)
Output: one virus sent, so input (n1,n2) is accepted.
Thus, the virus machine answers that n1 = n2 is yes
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 = n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,1,2n1− 1, iE2 ,0)C2n1 = (0,0,2n1, iE1 ,0)C2n1+1 = (0,0,2n1, iE3 ,0)C2n1+2 = (0,0,2n1, iE5 ,0)C2n1+3 = (0,0,2n1,#,1)
Output: one virus sent, so input (n1,n2) is accepted.
Thus, the virus machine answers that n1 = n2 is yes
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 = n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,1,2n1− 1, iE2 ,0)
C2n1 = (0,0,2n1, iE1 ,0)C2n1+1 = (0,0,2n1, iE3 ,0)C2n1+2 = (0,0,2n1, iE5 ,0)C2n1+3 = (0,0,2n1,#,1)
Output: one virus sent, so input (n1,n2) is accepted.
Thus, the virus machine answers that n1 = n2 is yes
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 = n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,1,2n1− 1, iE2 ,0)C2n1 = (0,0,2n1, iE1 ,0)
C2n1+1 = (0,0,2n1, iE3 ,0)C2n1+2 = (0,0,2n1, iE5 ,0)C2n1+3 = (0,0,2n1,#,1)
Output: one virus sent, so input (n1,n2) is accepted.
Thus, the virus machine answers that n1 = n2 is yes
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 = n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,1,2n1− 1, iE2 ,0)C2n1 = (0,0,2n1, iE1 ,0)C2n1+1 = (0,0,2n1, iE3 ,0)
C2n1+2 = (0,0,2n1, iE5 ,0)C2n1+3 = (0,0,2n1,#,1)
Output: one virus sent, so input (n1,n2) is accepted.
Thus, the virus machine answers that n1 = n2 is yes
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 = n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,1,2n1− 1, iE2 ,0)C2n1 = (0,0,2n1, iE1 ,0)C2n1+1 = (0,0,2n1, iE3 ,0)C2n1+2 = (0,0,2n1, iE5 ,0)
C2n1+3 = (0,0,2n1,#,1)
Output: one virus sent, so input (n1,n2) is accepted.
Thus, the virus machine answers that n1 = n2 is yes
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 = n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,1,2n1− 1, iE2 ,0)C2n1 = (0,0,2n1, iE1 ,0)C2n1+1 = (0,0,2n1, iE3 ,0)C2n1+2 = (0,0,2n1, iE5 ,0)C2n1+3 = (0,0,2n1,#,1)
Output: one virus sent, so input (n1,n2) is accepted.
Thus, the virus machine answers that n1 = n2 is yes
.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Example 2: equality
Computation
Case 1: n1 = n2
C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)C2 = (n1− 1,n2− 1,2, iE1 ,0). . . . . . . . . . . . . . . . . . . . . . . . . . .in turns, min{n1,n2}times,
making at least one host (h(E)1 or h
(E)2 ) become empty
C2n1−1 = (0,1,2n1− 1, iE2 ,0)C2n1 = (0,0,2n1, iE1 ,0)C2n1+1 = (0,0,2n1, iE3 ,0)C2n1+2 = (0,0,2n1, iE5 ,0)C2n1+3 = (0,0,2n1,#,1)
Output: one virus sent, so input (n1,n2) is accepted.
Thus, the virus machine answers that n1 = n2 is yes.
2
2
( )
1
Eh
( )
2
Eh
( )
3
Eh
( )
1
Ei
( )
2
Ei
( )
3
Ei
( )
4
Ei
( )
5
Ei
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Contents
1 Introduction
2 Virology
3 Virus Machines
4 Examples
5 Computational power
6 Conclusion
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Universality of Virus Machines
Notation
For each p,q,n ≥ 1, we denote by NVM(p,q,n) the family of all subsets of INcomputed by virus machines with at most p hosts, q instructions, and all hosts having at
most n viruses in any instant of each computation.
If one of the parameters p,q,n is not bounded, then it is replaced with ∗.
DefinitionsA Non-restricted Virus Machine is a virus machine where there is no restriction on
the number of hosts, instructions and viruses contained in any host along any
computation.
A Bounded Virus Machine is a virus machine in which the number of viruses
present in each host during any computation is bounded.
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Universality of Non-restricted Virus Machines
Result
Non-restricted virus machines working in the computing mode are computationally
complete (i.e., they can compute all recursively enumerable sets of natural numbers -
subsets of natural numbers which are Turing computable -)).
Theorem
NVM(∗,∗,∗) = NRE.
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Equivalence with register machines
ADD
2
rhh
2li
3li
1li1l
i1l
i
SUB
2
rh h
2li
3li
1li
OUTPUT
1h
#ihli
2
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Computational power of Bounded Virus Machines
Results
Bounded virus machines working in the computing mode:
Are not universal.
Characterize SLIN (the family of semi-linear sets of natural numbers, which is a
subset of P(N)).
Theorem
NVM(∗,∗,n) = SLIN, for all n ≥ 2.
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Contents
1 Introduction
2 Virology
3 Virus Machines
4 Examples
5 Computational power
6 Conclusion
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Conclusions
A new computability model, called Virus Machine, has been introduced.
Inspired in the replication and transmission of viruses.
Some examples have been provided for virus machines with input:
Working in the computing mode
Working in the accepting mode
The computational completeness, characterization of NRE , has been established
for non-restricted virus machines.
The characterization of SLIN for bounded virus machines has been proved.
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia
Future work
Concerning the computational power of virus machines, study how many hosts
(with unbounded contents) and instructions are required to compute a
non-semi-linear set of natural numbers.
Of definite interest is to consider parallel virus machines through different possible
mechanisms as discussed below.
Consider an instruction attached to several different channels. Two
possibilities:
Only one channel chosen, in a non-deterministic way, to be opened
All channels connected with the instruction to be opened
At each step, a non-empty set of control instructions activated
simultaneously. Then, many channels opened at that instant, allowing the
parallel transmission of viruses from different hosts.
Luis Valencia-Cabrera, Mario J. Pérez-Jiménez, Xu Chen, Beizhan Wang, Xiangxiang Zeng ( Research Group on Natural Computing, Department of Computer Science and Artificial Intelligence, University of Sevilla. Avda. Reina Mercedes s/n, 41012 Sevilla, Spain. [email protected],[email protected], School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China. [email protected],[email protected], Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China. [email protected], )Basic Virus Machines August 18, 2015. Valencia