MASSIVE PARALLEL NETWORKS OF EVOLUTIONARY

PROCESSORS AS NP-PROBLEM SOLVERS

Nuria G´omez Blas, Luis F. de Mingo and Eugenio Santos

Dept. Organizaci´on y Estructura de la Informaci´on, Escuela Universitaria de Inform´atica

Universidad Polit´ecnica de Madrid, Crta. de Valencia km. 7, 28031 Madrid, Spain

Keywords:

Natural Computing, Networks of Evolutionary Processors, Symbolic Systems.

Abstract:

This paper presents a new connectionist model that might be used to solve NP-problems. Most well known

numeric models are Neural Networks that are able to approximate any function or classify any pattern set

provided numeric information is injected into the net. Concerning symbolic information new research area

has been developed, inspired by George Paun, called Membrane Systems. A step forward, in a similar Neural

Network architecture, was done to obtain Networks of Evolutionary Processors (NEP). A NEP is a set of

processors connected by a graph, each processor only deals with symbolic information using rules. In short,

objects in processors can evolve and pass through processors until a stable conﬁguration is reach. Despite their

simplicity, we show how the latter networks might be used for solving an NP-complete problem, namely the

3-colorability problem, in linear time and linear resources (nodes, symbols, rules).

1 INTRODUCTION

Natural sciences, and especially biology, represents a

rich source of modeling paradigms. Well-deﬁned ar-

eas of artiﬁcial intelligence (genetic algorithms, neu-

ral networks), mathematics, and theoretical computer

science (L systems, DNA computing) are massively

inﬂuenced by the behavior of various biological en-

tities and phenomena. In the last decades or so,

new emerging ﬁelds of so-called ”natural comput-

ing” (Zandron, 2002; Paun, 2002; Ciobanu et al.,

2005) identify new (unconventional) computational

paradigms in different forms. There are attempts

to deﬁne and investigate new mathematical or theo-

retical models inspired by nature (Robinson, 1992),

as well as investigations into deﬁning programming

paradigms that implement computational approaches

suggested by biochemical phenomena. Especially

since Adleman’s experiment (Adleman, 1994), these

investigations received a new perspective. One hopes

that global system-level behavior may be translated

into interactions of a myriad of components with sim-

ple behavior and limited computing and communica-

tion capabilities that are able to express and solve, via

various optimizations, complex problems otherwise

hard to approach.

The origin of networks of evolutionary processors

is a basic architecture for parallel and distributed sym-

bolic processing, related to the Connection Machine

(Hillis, 1985) as well as the Logic Flow paradigm

(Errico and Jesshope, 1994), which consists of sev-

eral processors, each of them being placed in a node

of a virtual complete graph, which are able to han-

dle data associated with the respective node. All the

nodes send simultaneously their data and the receiv-

ing nodes handle also simultaneously all the arriv-

ing messages, according to some strategies, see, e.g.,

(Fahlman et al., 1983; Hillis, 1985).

2 MASSIVE PARALLEL NEPS

Let V an alphabet over a set of symbols. A string x of

length m belonging to an alphabet V is the sequence

of symbols a

··· a

where the symbol a

∈ V for all

1 ≤ i ≤ m. The set of all strings that belong to V is

denoted by V

∗

and the empty string is denoted by ε.

A network of evolutionary processors (Castel-

lanos et al., 2001; Castellanos et al., 2003) of size n

(NEP for short) is a construct Σ = {V, N

, N

, ··· , N

where V is an alphabet and processors N

are con-

nected within a graph.

A processor is deﬁned by N

= {M

, A

, PI

, PO

}

is the i-th evolutionary processor of the network. The

parameters of every processor are:

588

Gómez Blas N., F. de Mingo L. and Santos E. (2008).

MASSIVE PARALLEL NETWORKS OF EVOLUTIONARY PROCESSORS AS NP-PROBLEM SOLVERS.

In Proceedings of the Tenth International Conference on Enterprise Information Systems - AIDSS, pages 588-592

DOI: 10.5220/0001721405880592

 SciTePress

• M

is a ﬁnite set of evolution rules of one of the

following forms only (basic rules)

– a → b, a, b ∈ V (substitution rules),

– a → ε, a ∈ V (deletion rules),

– ε → a, a ∈ V (insertion rules),

More clearly, the set of evolution rules of any pro-

cessor contains either substitution or deletion or

insertion rules. Context information can be added

to M

in the following way (directional context

rules):

– ac → bc, a, b ∈ V, c ∈ V

∗

(left substitution

rules),

– ca → cb, a, b ∈ V, c ∈ V

∗

(right substitution

rules),

– ab → a, b ∈ V, a ∈ V

∗

(right deletion rules),

– ba → a, b ∈ V, a ∈ V

∗

(left deletion rules),

– a → ab, b ∈ V, a ∈ V

∗

(right insertion rules),

– a → ba, b ∈ V, a ∈ V

∗

(left insertion rules),

Or even a more general and non directional con-

text can be expressed in M

(context rules):

– dac → dbc, a, b ∈ V, c, d ∈ V

∗

(substitution

rules),

– abc → ac, b ∈ V, a, c ∈ V

∗

(deletion rules),

– ac → abc, b ∈ V, a, c ∈ V

∗

(insertion rules),

• A

is a ﬁnite set of strings over V. The set A

the set of initial strings in the i-th node. Actu-

ally, in what follows, we consider that each string

appearing in a node of the net at any step has an

arbitrarily large number of copies in that node, so

that we shall identify multisets by their supports.

• PI

and PO

are subsets of V representing the in-

put and output ﬁlter respectively. These ﬁlters

are deﬁned by membership condition, namely a

string w ∈ V

∗

can pass the input ﬁlter (the out-

put ﬁlter) if ∀x ∈ PI

, w = axb where a, b ∈ V

∗

(∀x ∈ PO

, w = axb where a, b ∈ V

∗

We write ρ

(w) = true, if w can pass the input ﬁl-

ter of the node processor i and ρ

(w) = false, oth-

erwise. We write τ

(w) = true, if w can pass the

output ﬁlter of the node processor i and τ

(w) =

false, otherwise.

A NEP with some of these rules is denoted by

NEP

(basic rules), NEP

(directional rules) and

NEP

(contex rules).

By a conﬁguration of an NEP as above we mean

an n-tuple C = (L

, L

, ··· , L

), with L

⊆ V

∗

for all

0 ≤ i ≤ 6. A conﬁguration represents the sets of

strings (remember that each string appears in an ar-

bitrarily large number of copies) which are present in

any node at a given moment; clearly the initial con-

ﬁguration of the network is C

= (A

, A

, ··· , A

2.1 Dynamics of MPNEP

A conﬁguration can change either by an evolutionary

step or by a communicating step. Computation steps

can be deﬁned in a controlled way, that is, ﬁrst an evo-

lutionary step and then a communicating step; or in

a parallel way, that is, evolution and communication

take place at the same time.

When changing by an evolutionary step, each

component L

of the conﬁguration is changed in ac-

cordance with the evolutionary rules associated with

the node i. Formally, we say that the conﬁguration

= (L

, L

, ··· ., L

) directly changes for the conﬁg-

uration C

= (L

′

, L

′

, ··· , L

′

) by an evolutionary step,

written as

⇒ C

if L

′

is the set of strings obtained by applying the

rules of R

to the strings in L

as follows:

• If the same substitiution rule may replace different

occurrences of the same symbol within a string,

all these occurrences must be replaced within dif-

ferent copies of that string. The result is the mul-

tiset in which every string that can be obtained ap-

pears in an arbitrarily large number of copies.

• Unlike their common use, deletion and insertion

rules are applied only to the end of the string.

Thus, a deletion rule a → ε can be applied only

to a string which ends by a, say wa, leading to the

string w, and an insertion rule ε → a applied to a

string x consists of adding the symbol a to the end

of x, obtaining xa. If context rules are used, that

is ab → a or abc → ac then the deletion point is

deﬁned by context information b or ac.

• If more than one rule, no matter its type, applies

to a string, all of them must be used for different

copies of that string.

More precisely, since an arbitrarily large number

of copies of each string is available in every node, af-

ter a evolutionary step in each node one gets an arbi-

trarily large number of copies of any string which can

be obtained by using any rule in the set of evolution

rules associated with that node. By deﬁnition,if L

empty for some 0 ≤ i ≤ 6, then L

′

is empty as well.

When changing by a communication step, each

node processor sends all copies of the strings it has

which are able to pass its output ﬁlter to all the other

node processors and receives all copies of the strings

sent by any node processor providing that they can

pass its input ﬁlter.

Formally, we say that the conﬁguration C

, L

, ··· , L

) directly changes for the conﬁguration

MASSIVE PARALLEL NETWORKS OF EVOLUTIONARY PROCESSORS AS NP-PROBLEM SOLVERS

589

= (L

′

, L

′

, ··· , L

′

) by a communication step, writ-

ten as

⊢ C

if for every 0 ≤ i ≤ n,

′

= L

\ {w ∈ L

|τ

(w) = true} ∪

[

j=0, j6=i

{x ∈ L

|τ

(x) = true∧ ρ

(x) = true}

A parallel computation step among two conﬁgura-

tions C

and C

, represented by C

|= C

, if the appli-

cation of the evolutionary and communication steps

in parallel, that is,

|= C

= (C

⊢ C

)||(C

⇒ C

)

Let Γ = (V, N

, N

, ··· , N

) be an NEP. By a par-

allel computation in Γ we mean a sequence of conﬁg-

urations C

, C

, ··· , where C

is the initial conﬁg-

uration andC

|= C

i+1

for all i ≥ 0. MPNEP is a NEP

with parallel computation – MPNEP is a NEP

–.

If the sequence is ﬁnite, we have a ﬁnite computa-

tion. The result of any ﬁnite computation is collected

in a designated node called the output node. If one

considers the output node of the network as being the

node N

, and if C

, C

, ··· , C

is a computation, then

the set of strings existing in the node N

at the last

step – the 0-th component of C

– is the result of this

computation. The time complexity of the above com-

putation is the number of steps, that is t.

2.2 MPNEP vs. NEP

Let Γ = (V, N

, N

, ··· , N

) be an NEP

. By a con-

trolled computation in Γ we mean a sequence of con-

ﬁgurations C

, C

, ··· , where C

is the initial con-

ﬁguration and C

⇒ C

2i+1

and C

2i+1

⊢ C

2i+2

for all

i ≥ 0.

Theorem 2.1. Problems solved using a NEP

can

be solved using a MPNEP = NEP

, where X =

{b, d, c}.

Proof. Given a processor N

= {A

, R

, PI

, PO

} be-

longing to a NEP

it is possible to transform it on

a processor N

′

= {A

, R

′

, PI

, PO

′

} that behaves in a

same way within a MPNEP = NEP

in the follow-

ing way:

• Given a rule r

∈ R

with the notation A → B, with

1≤ k ≤ p, each rule r

′

∈ R

′

has the form A → BX

• Given the ouput ﬁlter PO

, PO

′

= PO

k=1

With these new sets R

′

and PO

′

the parallel com-

putation of a NEP behavesin the same way that a con-

trolled computation since τ(w) = false until all rules

are applied.

Theorem 2.2. Problems solved using a NEP

can

be solved using a NEP

and problems solved using

a NEP

can be solved using a NEP

, where X =

{p, c}.

Proof. It is obvious that abc → adc where a, (c = ε) ∈

∗

is similar to ab → ad and ab → ad where (a = ε) ∈

∗

is similar to b → d.

3 COMPUTATIONAL

COMPLETENESS

Since MPNEPs with ﬁnite ﬁlters can generate regular

languages only, we shall consider in the sequel MP-

NEPs having inﬁnite regular languages as ﬁlters.

Theorem 3.1. Each recursively enumerable lan-

guage can be generated by a complete MPNEP of size

Proof. Let G = (N, T, S, P) be an arbitrary phrase-

structure grammar in the Kuroda normal form,

namely P contains only rules of the following forms:

A → a, A → BC, AB → CD, A → ε

where A, B, C, D are nonterminals and a is a terminal.

We assume that the rules A → BC and AB → CD of

P are labelled in a one-to-one manner by the labels

, r

, ··· , r

. We shall refer to the rules of the form

A → a, A → ε, A → BC, y AB → CD as rules of type

0, 1, 2, and 3, respectively. We construct the following

NEP of size 5 having a complete underlying graph:

Γ = (N ∪ T ∪V ∪ {X}, N

, N

, K

)

where V = {r

, p

, q

, s

, t

|1 ≤ i ≤ n} and

= (

0, T

∗

, (N ∪ T ∪V ∪ {X})

∗

(N ∪V ∪ {X})

(N ∪ T ∪V ∪ {X})

∗

)

= (M

, {S}, (N ∪ T)

∗

∪ (N ∪T)

∗

|1 ≤ i ≤ n}(N ∪T)

∗

(N ∪ T)

∗

({r

, s

, t

|1 ≤ i ≤ n} ∪ {X})

(N ∪ T)

∗

∪ T

∗

)

with M

= {A → X|A → ε ∈ P}

∪{A → a|A → a ∈ P} ∪

{A → r

, r

→ t

: A → BC ∈ P} ∪

{A → s

, B → p

: AB → CD ∈ P}

ICEIS 2008 - International Conference on Enterprise Information Systems

590

= ({ε → q

: A → BC},

(N ∪ T)

∗

|1 ≤ i ≤ n}(N ∪T)

∗

(N ∪ T ∪V)

∗

)

= (M

0, (N ∪ T)

∗

, t

|1 ≤ i ≤ n}

(N ∪ T)

∗

, (N ∪ T)

∗

)

with M

= {t

→ B, q

→ C|r

: A → BC ∈ P} ∪

→ C, p

→ D|r

: AB → CD ∈ P}

= ({X → ε},

0, (N ∪ T)

∗

{X}(N ∪ T)

∗

, (N ∪ T)

∗

)

Taking into accountthis NEP conﬁguration, a MP-

NEP can be deﬁned according to theorem 2.1.

• Given a rule r

∈ R

with the notation A → B, with

1≤ k ≤ p, each rule r

′

∈ R

′

has the form A → BX

• Given the ouput ﬁlter PO

, PO

′

= PO

k=1

This MPNEP behaves in the same way that a NEP

and therefore each recursively enumerable language

can be generated.

4 3-COLORABILITY PROBLEM

Theorem 4.1. The ”3-colorability problem” can be

solved in O(m+ n) time by a MPNEP of size 4m+ 1,

where n is the number of vertices and m is the number

of edges of the input graph.

Proof. Let G = ({1, 2, · ·· , n}, {e

, e

, ··· , e

}) a

graph an assume that e

= {k

, l

}, 1 ≤ k

≤ l

≤ n, 1 ≤

t ≤ m. We consider the alphabet U = V ∪V

′

∪ T ∪ A,

where V = {b, r, g}, T = {a

, a

, ··· , a

}, and A =

{

, ··· ,

We construct the following processors of a mas-

sive parallel NEP.

• A generator processor:

= {{ a

··· a

→ b

, a

→ r

, a

→ g

|1 ≤ i ≤ n},

0, {

|1 ≤ i ≤ n}}

This processor generates all possible color com-

binations, solutions or not, to the problem. And it

sends those strings to next processors.

• For each edge in the graph e

= {k

, l

}, we have 4

ﬁltering processors (where i = {k

, l

}):

= {

0, {g

→ g

′

, r

→ r

′

{

}, {g

′

, r

′

}}

= {

0, {g

→ g

′

, b

→ b

′

{

}, {g

′

, b

′

}}

= {

0, {b

→ b

′

, r

→ r

′

{

}, {b

′

, r

′

}}

= {

0, {r

′

→ r

, g

′

→ g

, b

′

→ b

}, {

}}

It is clear to see that we can build a MPNEP

with previous nodes in such a way that N

generates

all possible colored strings and then apply processors

, N

to ﬁlter such strings for edge e

. Re-

peating such ﬁltering process with the rest of edges

gives a valid solution to the given problem.

A MPNEP with the above architecture can solve

the 3-colorability problem of n cities with m edges.

For the ﬁrst n steps, that are evolution ones when

nothing is actually communicated, the strings will re-

main in N

until no letter in T appears in them any-

more. When this process is ﬁnished, the obtained

strings encode all possible ways of coloring the ver-

tices, satisfying or not the requirements of the prob-

lem. After this, 1 step is needed to communicate all

possible solutions to next processors. Now, for each

edge e

, MPNEP keeps only those strings which en-

codes a col- orability satisfying the condition for the

two vertices of e

. This is done by means of the nodes

, N

, ﬁnally N

in 12 steps. As one can see,

the overall time of a computation is 12m+ n+ 1. We

ﬁnish the proof by mentioning that the total number of

rules is 18m+ 3n+ 1. In conclusion, all parameters of

the network are of O(m+ n) size.

5 FINAL REMARKS

The concept of network of evolutionary processors

is based on mechanisms inspired from cell biology.

A variation of these networks called massive paral-

lel of network of evolutionary processors is also stud-

ied. These networks consists of nodes which are very

simple processors and are able to perform rules oper-

ations. These nodes are endowed with a ﬁlter which

are deﬁned by regular sets. Evolution and commu-

nication is done in a parallel way which seem to be

close to the possibilities of biological implementation

rather than NEPs since bio-operation, chemical reac-

tions, mutations etc. happen in a parallel way since

there is no inherent synchronization among them.

A new dynamic of NEP has been introduced in

this paper. It has been proof that any given NEP can

be model with an equivalent MPNEP, see theorem

2.1, and therefore the inherent computational com-

MASSIVE PARALLEL NETWORKS OF EVOLUTIONARY PROCESSORS AS NP-PROBLEM SOLVERS

591

pleteness of NEP can be incorporated in MPNEP.

This massive parallel way of operation is closer to bi-

ological operations than classical NEP.

The 3-colorability problem has been solved using

a software tool that simulates a MPNEP according to

theorem 4.1. This soft tool can be use to check other

useful theorems (Castellanos et al., 2001) in NEP re-

search area.

This paper has proposed a new dynamic of NEP

with a high parallel and non-deterministic behavior,

and therefore all research about NEP can be trans-

lated into MPNEP. For instance, MPNEP can be

modiﬁed according to (Diaz et al., 2007) in order to

move ﬁlters towards edges. Each edge is viewed as

a two-way channel such that input and output ﬁlters

coincide. Thus, the possibility of controlling the com-

putation in such networks seems to be diminished. In

spite of this these networks are still computationally

complete (Castellanos et al., 2006). Also, another

rules can be implemented in MPNEP to extend this

computing model. In (Manea et al., 2007) one re-

places the point mutations associated with each node

by the missing operation mentioned above, that of

splicing. This new processor is called splicing proces-

sor. This computing model, called accepting network

of splicing processors (shortly ANSP), is similar to

some extent to the test tube distributed systems based

on splicing.

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