Basic Virus Machines - UPVusers.dsic.upv.es/workshops/cmc16/VPCWZtalk.pdf · Many people would consider it a success, ... 41012 Sevilla, Spain.Basic Virus Machines August 18, 2015

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.

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2School of Software, Xiamen University Xiamen 361005,Fujian, People’s Republic of China.


3Department of Computer Science, Xiamen University. Xiamen 361005,Fujian, People’s Republic of China.

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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



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Contents

1 Introduction

2 Virology

3 Virus Machines

4 Examples

5 Computational power

6 Conclusion




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Contents

1 Introduction

2 Virology

3 Virus Machines

4 Examples


6 Conclusion




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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...




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Motivation

Starting point












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Motivation

Starting point












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A progressive view on us...

A human being...




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Ok, it is a Matryoshka, a russian nesting doll, but let us imagine...




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Some systems, organs, tissues, many cells...




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Oh, wait... bacteria? Yes, 10 times more bacteria than human cells




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What was that? Is it a virus? Yes, 10 times more viruses than bacteria!




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But... That implies 100 times more viruses than human cells!




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Many people would consider it a success, but even if you disagree...




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Should not we at least observe viruses and their behaviour?




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Contents

1 Introduction

2 Virology

A brief introduction to virology

Inspiration and informal description

3 Virus Machines

4 Examples


6 Conclusion




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Contents

1 Introduction

2 Virology



3 Virus Machines

4 Examples


6 Conclusion




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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




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Parasitic activity

Pacific coexistence =⇒ Lysogenic cycle

Use and abuse =⇒ Lytic cycle




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Contents

1 Introduction

2 Virology



3 Virus Machines

4 Examples


6 Conclusion




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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




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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




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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




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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




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Contents

1 Introduction

2 Virology

3 Virus Machines

4 Examples


6 Conclusion




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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.




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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




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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

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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 )




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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.




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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.




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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.

.




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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


6 Conclusion




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Contents

1 Introduction

2 Virology

3 Virus Machines

4 Examples




6 Conclusion




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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 ).




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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)}}.




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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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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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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)


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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)


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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)


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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)


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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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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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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)


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Contents

1 Introduction

2 Virology

3 Virus Machines

4 Examples




6 Conclusion




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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)




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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.




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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)}}.




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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

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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).

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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

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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,


(E)2 ) become empty




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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,


(E)2 ) become empty




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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,


(E)2 ) become empty




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[email protected]

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,


(E)2 ) become empty




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Example 2: equality

Computation

Case 1: n1 > n2



(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)



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Example 2: equality

Computation

Case 1: n1 > n2



(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)



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[email protected]

Example 2: equality

Computation

Case 1: n1 > n2



(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)



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[email protected]

Example 2: equality

Computation

Case 1: n1 > n2



(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)



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2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 > n2



(E)2 ) become empty




.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 > n2



(E)2 ) become empty



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




[email protected]

Example 2: equality

Computation

Case 1: n1 < n2

C0 = (n1,n2,0, iE1 ,0)



(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)



.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 < n2

C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)



(E)2 ) become empty




.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 < n2




(E)2 ) become empty




.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 < n2



(E)2 ) become empty




.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 < n2



(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)



.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 < n2



(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)



.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 < n2



(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)



.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 < n2



(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)



.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 < n2



(E)2 ) become empty




.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 < n2



(E)2 ) become empty



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




[email protected]

Example 2: equality

Computation

Case 1: n1 = n2

C0 = (n1,n2,0, iE1 ,0)



(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




[email protected]

Example 2: equality

Computation

Case 1: n1 = n2

C0 = (n1,n2,0, iE1 ,0)C1 = (n1− 1,n2,1, iE2 ,0)



(E)2 ) become empty




.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 = n2




(E)2 ) become empty




.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

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,


(E)2 ) become empty




.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 = n2



(E)2 ) become empty




.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 = n2



(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)



.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 = n2



(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)



.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 = n2



(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)



.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 = n2



(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)



.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 = n2



(E)2 ) become empty




.

2

2

( )

1

Eh

( )

2

Eh

( )

3

Eh

( )

1

Ei

( )

2

Ei

( )

3

Ei

( )

4

Ei

( )

5

Ei

2




[email protected]

Example 2: equality

Computation

Case 1: n1 = n2



(E)2 ) become empty



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




[email protected]

Contents

1 Introduction

2 Virology

3 Virus Machines

4 Examples


6 Conclusion




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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.




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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.




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Equivalence with register machines

ADD

2

rhh

2li

3li

1li1l

i1l

i

SUB

2

rh h

2li

3li

1li

OUTPUT

1h

#ihli

2




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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.




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Contents

1 Introduction

2 Virology

3 Virus Machines

4 Examples


6 Conclusion




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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.




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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.




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Documents

Basic Virus Machines - UPVusers.dsic.upv.es/workshops/cmc16/VPCWZtalk.pdf · Many people would consider it a success, ... 41012 Sevilla, Spain.Basic Virus Machines August 18, 2015