research article generation of a reconfigurable logical...

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2013 Article ID 250593 4 pageshttpdxdoiorg1011552013250593

Research ArticleGeneration of a Reconfigurable Logical Cell UsingEvolutionary Computation

I Campos-Cantoacuten1 L M Torres-Trevintildeo2 E Campos-Cantoacuten3 and R Femat3

1 Facultad de Ciencias Universidad Autonoma de San Luis Potosı Alvaro Obregon 64 78000 San Luis Potosı SLP Mexico2 Centro de Innovacion Investigacion y Desarrollo en Ingenierıa Electronica Universidad Autonoma de Nuevo LeonKm 10 de la Nueva Carretera al Aeropuerto Internacional de Monterrey PIIT Monterrey 66600 Apodaca NL Mexico

3 Division de Matematicas Aplicadas Instituto Potosino de Investigacion Cientıfica y Tecnologica Camino a la Presa San Jose 2055Colonia Lomas 4 Seccion 78216 San Luis Potosı SLP Mexico

Correspondence should be addressed to I Campos-Canton icamposfcienciasuaslpmx

Received 26 October 2012 Revised 12 February 2013 Accepted 13 February 2013

Academic Editor Gualberto Solıs-Perales

Copyright copy 2013 I Campos-Canton et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In nature an interesting topic is about how a cell can be reconfigured in order to achieve a different task Another interesting topicis about the learning process that seems to be a trial and error process In this work we present mechanisms about how to producea reconfigurable logical cell based on the tent map The reconfiguration is realized by modifying its internal parameters generatingseveral logical functions in the same structure The logical cell is built with three blocks the initial condition generating functionthe tent map and the output function Furthermore we propose a reconfigurable structure based on a chaotic system and anevolutionary algorithm is used in order to tune the parameters of the cell via trial and error process

1 Introduction

Adaptation in nature is a relevant research area that hasmany applications in artificial systems which can be used forthe benefit of society Particularly in biology a neuron canreconfigure itself to develop different tasks using the samestructure however this procedure is a mysteryThe processesof adaptation learning and coupling between them havebeen research pursuits therein and the understanding of thisprocesses can help to build artificial devicesThe act of joiningliving tissue with electronics has long been imagined in theworld of science fiction but cybernetic organisms are nowone step closer to reality thanks to work emerging fromresearchers that have built tiny electronic meshes out of sili-con nanowires and have used them as scaffolds to grow nerveheart and muscle tissue

Chaotic dynamical systems are commonly used in com-munication systems like the generation of pseudorandomsignals and encryption among others Chaotic systems arerecognized by the richness of their dynamics however these

systems could be very sensible to perturbations or a combi-nation of an infinite number of instabilities Chaotic systemsof one dimension may generate a remarkable variety ofbehaviors if they are observed as a function of time due todifferent initial conditions or their parameters [1] The use ofone-dimensional mapping is feasible as the logistic map forinstance or the tent map to generate logical functions [2ndash4]

Nowadays the searching for alternative solutions in thehardware used in computational systems is a very importantaspect in the area [5] one of them is the development ofreconfigurable hardware where the chaotic circuits are candi-dates to perform these types of tasks [6ndash8] As expected thechaotic elements might generate different logical functionswith reconfiguration capabilities [9 10] Other alternativesconsist of architectures that use programmable gates circuits(FPGArsquos) whose configuration is made by rewiring multiplestatic gates of single purpose that is a single logic gate has afixed configuration In this work a tent map 1 is used to builda reconfigurable cell that generates different logical basic

2 Discrete Dynamics in Nature and Society

functions where an evolutionary algorithm is used to adjustits parameters [11ndash13]

119891 (119909) = 120583119909 if 119909 lt 05120583 (1 minus 119909) if 119909 ge 05

(1)

Evolutionary computation is a paradigm inspired bythe natural selection theory proposed by Charles Darwinwhere several evolutionary processes can be accounted for incomputing including three of them in the developing of anevolutionary algorithm the fitness assignation the inclusionof diversity and the selection of more fitted individuals thatcan reproduce and have offspring In evolutionary computa-tion the same process that occurs in nature is emulated butin a simple way Genetic algorithms genetic programmingand estimation of distribution algorithms are examples ofthis kind of algorithms [11 12] In this work an easy imple-mentation of an estimation of distribution algorithm calledEvonorm [13] is used for tuning the parameters of the recon-figurable logic cell This counts as the actual way for tuningthe parameters usually made by trial and error process Usingthis algorithm generated all the sixteen logic functions con-sidering two inputs

Although previous paragraphs seem to be uncorrelatedthe possibility of designing reconfigurable logic cells by elec-tronic circuits allows one (i) to explain how a same structureof artificial systems (circuits) can be reconfigured to developdifferent tasks (ii) to incorporate chaotic dynamics as thebasis of reconfiguration at the artificial systems (circuits) (iii)to define alternative architectures distinct from FPGArsquos suchthat the reconfiguration goes beyond rewiring and (iv) toexploit evolutionary computation towards the optimizationof parameters in the reconfiguration process The maincontribution of this paper is on a reconfigurable logic cellThis reconfigurable logic cell can represent several logicalfunctions all in the same structure changing specific param-eters The setting of these parameters is made by an evolu-tionary algorithm In Section 2 a reconfigurable logic cell ispresented In Section 3 the Evonorm evolutionary algorithmis described and used to set the parameters of reconfigurablelogic cell proposed In Section 4 the proposal architecture isused to generate different logical functions Conclusion andfuture work are given in the last section

2 Reconfigurable Dynamical Logic Cell

In our proposal the reconfigurable logical cell is constitutedby an architecture consisting of three blocks (i) a generatorof initial conditions (ii) a tent map and (iii) an output blockEach block obeys the following

(i) the initial condition of the reconfigurable logic cell isdefined by the following equation

1199090= (119909119904+ 119861119880) mod 1 (2)

where119909119904isin (0 1) is an ignition seed119861 = [119887

11198872] isin 1198772

and 119880 = [11990611199062]119879 with 119906

1 1199062isin 0 1

(ii) the tent map yields the first iteration using (1) that is1199091= 119891 (119909

119904+ 11988711199061+ 11988721199062) (3)

1199061

1199062

119909119904

119910

Logic cell

1199091199090Eq (2) Eq (1) Eq (4)

Figure 1 Block diagram of the logical cell

(iii) the output block is a function119910(0 1) rarr 0 11199091997891rarr

0 1 generating a bistable response as follows

119910 (119909) = 1 if 119909

1gt 120573

0 otherwise(4)

where 120573 is a threshold reference signal to determinethe output of the system a logical zero or a logicalone

Figure 1 illustrates a block diagram of a reconfigurablelogic cell defined by (1) (2) and (4) Considering the inputs1199061 1199062 the reconfigurable logic cell generates 16 logic func-

tions Without a loss of generality we explain two functions11989114and 1198918(OR ampAND logic gates) of these 16 whose outputs

satisfy the true Table 1

21 Function 11989114

(OR) For the function 11989114 the recon-

figurable logical cell is tuned by considering the followingparameters 120583 = 2 119887

1= 1198872= 02 120573 = 05 and 119909

119904= 01 These

parameters satisfy the response shown in Table 2 column119884OR

The behavior of the reconfigurable logic cell is describedin what follows If the initial condition is 119909

0= 01 + 025119906

1+

0251199062and inputs 119906

1= 0 119906

2= 0 then 119909

0= 01 and the

first iteration of the tent map 1 is 119891(119909) = 02 Thus the outputdefined by 4 is119910 = 0 Repeating the exercise with 119906

1= 0 119906

2=

1 we have 1199090= 035 119891(119909) = 07 and 119910 = 1 Following this

way all the possible combinations are considered with twologic inputs to get the results illustrated in column 119884OR(119909) ofTable 2 so the logical cell under the parameters defined abovecan generate the behavior of a logical function OR

22 Function 1198918(AND) For this case the same parameter

values used above are considered except the 120573 parameteris now tuned to 075 Note that only one parameter isreconfiguredThe output of the logic gate is shown in Table 2column119910ANDThismeans that only one parameter is changedfor the reconfiguration of the logical cell generating from thelogical function OR to the logical function AND

Note that the selection of the parameter values can beadjusted by trial and error but this was an exhausted taskhence the evolutionary algorithms arise as a convenient alter-native We propound the Evonorm evolutionary algorithmto be used for the selection of the parameter values and thealgorithm explained in the next section

Discrete Dynamics in Nature and Society 3

Table 1 True table for OR and AND logic gates

1199061

1199062

OR AND0 0 0 00 1 1 01 0 1 01 1 1 1

Table 2 Outputs for the logical functions 11989114and 119891

8(OR and AND

logical gates)

1199061

1199062

119909119904

119891(119909) 119884OR(119909) 119884AND(119909)

0 0 01 02 0 00 1 01 07 1 01 0 01 07 1 01 1 01 08 1 1

3 Evolutionary Computation

An important question is about the existence of an efficientway to determine the values of the parameters of the reconfig-urable logic cell In this section we show how parameters canbe tuned in order that the logical functions will be generatedParticularly we illustrate the case of 2 inputs generating16 outputs An evolutionary algorithm is used to this endEssentially an evolutionary algorithm is used for searchingsolutions [11 12] Here we exploit it to tune the parametersof the reconfigurable logic cell The selected evolutionaryalgorithm is called Evonorm [13] and is based on four itemsas follows

(1) representation of individuals(2) evaluation or fitness calculation of every individual(3) selection of the best fitted individuals(4) generation of new individuals

Tracking the above four items every potential solutionis represented as an individual of a population where it isassigned an evaluation value depending on its performanceto solve the problem of tuning the parameters Specifically InEvonorm a marginal random variable is used for yielding anew population via an estimation of a normal distributionThe implementation of the algorithm requires two matricesand three vectors One of the two matrices is named 119875 isinR119879119894times119879119875 which represents the population of solutions where119879119894is the number of total individuals and 119879

119875is the number of

parametersTheothermatrix is denoted as119875119904isin R119879119904times119879119875 which

stores selected individuals where119879119904is the number of selected

individuals usually ten percent of the total population Thevector 119865119864 isin R119879119894times1 stores the evaluation value per individualThe vectors 120583 isin R119879119901times1 and 120590 isin R119879119901times1 are used for storingthe mean and standard deviation respectively of the randomvariable used per parameter The above definitions corre-spond to the first item of evolutionary algorithm EvonormThe initial population is generated randomly using a uniformdistribution function

In what follows the Evonorm evolutionary algorithmis highlighted for determining the initial conditions of the

reconfigurable logic cells Every individual 119896 of the popula-tion is evaluated as follows

Step 1 Evaluation and extraction of the logical cell parame-ters Extraction of the parameters 119909

119904= 119875(119896 1) 120583 = 119875(119896 2)

120573 = 119875(119896 3) 1198871= 119875(119896 4) and 119887

2= 119875(119896 5)

Step 2 Calculation of 119903 = sum119879119875pr=1 |119910119889pr minus 119910(119909119904) 120583 119896 1198871 1198872)|4where 119910119889pr is a vector that represents the expected outputvalues corresponding to every input combination in the cellFor example the function119891

7has the following outputs119910119889pr =

[1 1 1 0] that correspond to the inputs [00 01 10 11]respectively The maximum number of outputs is consideredto perform the normalization in this example number fouris used for the normalization of the evaluation

Step 3 Determine the evaluation as 119865119864119896= 1 minus 119903 A maximi-

zation of this function is expected

Step 4 Selection of the most fitted individuals This proce-dure sorts the individuals of matrix 119875 using the evaluationvector 119865119864 as a criterion Then a selection of 119879

119894individuals is

made for generating a new population 119875119904 In this procedure is

stored the best fitted individual in vector 119868119909isin R119879119904times119879119875

Step 5 Calculate the mean and the standard deviation perparameter

120583pr =sum119879119904

119896=1(119875119904119896pr)

119879119904

120590pr =

radic(sum119879119904

119896=1(119875119904119896prminus 120583pr))

119879119904

(5)

Step 6 Generate a new population considering a marginalrandom variable with normal distribution

119875119896pr =

119873(120583pr 120590pr) 119880 ( ) gt 05

119873 (119868119909pr 120590pr) otherwise

(6)

119873(120583pr 120590pr) is a random variable that generates numberswith a normal distribution function considering a mean 120583prand a standard deviation 120590pr precalculated previously Theapproximation of a standard random variable with a normaldistribution is calculated using (7) where 119880( ) is a randomvariable with normal distribution

119873(120583 120590) = 120583 + 120590(12

sum119894=1

(119880 ( )) minus 6) (7)

Steps 1ndash6 are repeated several times in cycles called gener-ations This algorithm is capable of adjusting the parametersof the reconfigurable logical cell and generates all the logicalfunctions for two inputs 119906

1 1199062


Table 3 Logic functions

1199061

1199062

1198910

1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918

1198919

11989110

11989111

11989112

11989113

11989114

11989115

0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 10 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 11 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 11 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1

Table 4 Parameterrsquos values for generating 16 logic functions

Funcion 120583 1198871

1198872

120573 119909119904

1198910

2 025 025 085 01111198911

2 03 03 09 0451198912

2 07 015 05 01111198913

2 03 02 08 041198914

2 015 07 05 01111198915

2 02 04 05 041198916

2 045 045 025 01111198917

2 045 045 01 01111198918

2 025 025 075 01111198919

2 05 minus01 06 0311989110

2 01 025 05 011111989111

2 minus01 04 06 0311989112

2 025 01 05 011111989113

2 04 minus01 06 0311989114

2 025 025 05 011111989115

2 025 025 02 0111

4 Tuning Parameter Values by Evonorm

Evonorm algorithm is used to generate a better parametricsolution for all the logic functions considering two inputvariables as shown in Table 3

The parameter values for all the logic functions withtwo inputs were generated by an evolutionary algorithm(Table 4) For example the boolean function 119891

12is generated

by considering the parameter values 119909119904= 0111 120583 = 2

1198871= 025 119887

2= 01 and 120573 = 05 according to Table 4 The

initial condition 1199090is equal to 011 with inputs 119906

1= 0 119906

2=

0 so these value are used in (1) to get in the first iteration119891(119909) = 0222 and the output 119910 = 0 is given by (4) Repeatingthis process all the outputs are generated as illustrated inTable 4

5 Conclusion

A development of a reconfigurable logic cell based on tentmap is presented The reconfiguration is made by tuningthe parameter values 119909

119904 120573 119887

1 1198872 and 120583 The Evonorm

algorithm is a useful tool for tuning the parameter values ofthe reconfigurable logic cell and generating all the 16 booleanfunctions of two logical inputs (remaining fixed 120583 Table 4)As a future work a comparison between other evolutionaryalgorithms and the use of reconfigurable logic cells with threeor more inputs is expected

Acknowledgment

E Campos-Canton acknowledges CONACYT for the finan-cial support through Project no 181002

References

[1] E Campos-Canton R Femat and A N Pisarchik ldquoA familyof multimodal dynamic mapsrdquo Communications in NonlinearScience and Numerical Simulation vol 16 no 9 pp 3457ndash34622011

[2] S Sinha andW Ditto ldquoDynamics based computationrdquo PhysicalReview Letters vol 81 no 10 pp 2156ndash2159 1998

[3] S Sinha and W Ditto ldquoComputing with distributed chaosrdquoPhysical Review E vol 60 no 1 pp 363ndash377 1999

[4] T Munakata S Sinha and W L Ditto ldquoChaos computingimplementation of fundamental logical gates by chaotic ele-mentsrdquo IEEE Transactions on Circuits and Systems I vol 49 no11 pp 1629ndash1633 2002

[5] D Kuo ldquoChaos and its computing paradigmrdquo IEEE Potentialsvol 24 no 2 pp 13ndash15 2005

[6] KMurali S SinhaW LDitto andA R Bulsara ldquoReliable logiccircuit elements that exploit nonlinearity in the presence of anoise floorrdquo Physical Review Letters vol 102 no 10 Article ID104101 4 pages 2009

[7] W L Ditto A Miliotis K Murali S Sinha and M L SpanoldquoChaogates morphing logic gates that exploit dynamical pat-ternsrdquo Chaos vol 20 no 3 Article ID 037107 7 pages 2010

[8] K Murali A Miliotis W L Ditto and S Sinha ldquoLogic fromnonlinear dynamical evolutionrdquo Physics Letters A vol 373 no15 pp 1346ndash1351 2009

[9] I Campos-Canton E Campos-Canton J A Pecina-Sanchezand H C Rosu ldquoA simple circuit with dynamic logic architec-ture of basic logic gatesrdquo International Journal of Bifurcation andChaos vol 20 no 8 pp 2547ndash2551 2010

[10] H Peng Y Yang L Li and H Luo ldquoHarnessing piecewise-linear systems to construct dynamic logic architecturerdquo Chaosvol 18 no 3 Article ID 033101 6 pages 2008

[11] J H Holland Adaptation in Natural and Artificial Systems AnIntroductory Analysis with Applications to Biology Control andArtificial Intelligence MIT Press Ann Arbor Mich USA 1975

[12] D E GoldbergGenetic Algorithms in Search Optimization andMachine Learnin Addison-Wesley New York NY USA 1989

[13] L Torres ldquoEvonorm easy and effective implementation of esti-mation of distribution algorithmsrdquo in Research in ComputingScience A Gelbuckh and S S Guerra Eds vol 23 pp 75ndash832006

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functions where an evolutionary algorithm is used to adjustits parameters [11ndash13]

119891 (119909) = 120583119909 if 119909 lt 05120583 (1 minus 119909) if 119909 ge 05

(1)

Evolutionary computation is a paradigm inspired bythe natural selection theory proposed by Charles Darwinwhere several evolutionary processes can be accounted for incomputing including three of them in the developing of anevolutionary algorithm the fitness assignation the inclusionof diversity and the selection of more fitted individuals thatcan reproduce and have offspring In evolutionary computa-tion the same process that occurs in nature is emulated butin a simple way Genetic algorithms genetic programmingand estimation of distribution algorithms are examples ofthis kind of algorithms [11 12] In this work an easy imple-mentation of an estimation of distribution algorithm calledEvonorm [13] is used for tuning the parameters of the recon-figurable logic cell This counts as the actual way for tuningthe parameters usually made by trial and error process Usingthis algorithm generated all the sixteen logic functions con-sidering two inputs

Although previous paragraphs seem to be uncorrelatedthe possibility of designing reconfigurable logic cells by elec-tronic circuits allows one (i) to explain how a same structureof artificial systems (circuits) can be reconfigured to developdifferent tasks (ii) to incorporate chaotic dynamics as thebasis of reconfiguration at the artificial systems (circuits) (iii)to define alternative architectures distinct from FPGArsquos suchthat the reconfiguration goes beyond rewiring and (iv) toexploit evolutionary computation towards the optimizationof parameters in the reconfiguration process The maincontribution of this paper is on a reconfigurable logic cellThis reconfigurable logic cell can represent several logicalfunctions all in the same structure changing specific param-eters The setting of these parameters is made by an evolu-tionary algorithm In Section 2 a reconfigurable logic cell ispresented In Section 3 the Evonorm evolutionary algorithmis described and used to set the parameters of reconfigurablelogic cell proposed In Section 4 the proposal architecture isused to generate different logical functions Conclusion andfuture work are given in the last section

2 Reconfigurable Dynamical Logic Cell

In our proposal the reconfigurable logical cell is constitutedby an architecture consisting of three blocks (i) a generatorof initial conditions (ii) a tent map and (iii) an output blockEach block obeys the following

(i) the initial condition of the reconfigurable logic cell isdefined by the following equation

1199090= (119909119904+ 119861119880) mod 1 (2)

where119909119904isin (0 1) is an ignition seed119861 = [119887

11198872] isin 1198772

and 119880 = [11990611199062]119879 with 119906

1 1199062isin 0 1

(ii) the tent map yields the first iteration using (1) that is1199091= 119891 (119909

119904+ 11988711199061+ 11988721199062) (3)

1199061

1199062

119909119904

119910

Logic cell

1199091199090Eq (2) Eq (1) Eq (4)

Figure 1 Block diagram of the logical cell

(iii) the output block is a function119910(0 1) rarr 0 11199091997891rarr

0 1 generating a bistable response as follows

119910 (119909) = 1 if 119909

1gt 120573

0 otherwise(4)

where 120573 is a threshold reference signal to determinethe output of the system a logical zero or a logicalone

Figure 1 illustrates a block diagram of a reconfigurablelogic cell defined by (1) (2) and (4) Considering the inputs1199061 1199062 the reconfigurable logic cell generates 16 logic func-

tions Without a loss of generality we explain two functions11989114and 1198918(OR ampAND logic gates) of these 16 whose outputs

satisfy the true Table 1

21 Function 11989114

(OR) For the function 11989114 the recon-

figurable logical cell is tuned by considering the followingparameters 120583 = 2 119887

1= 1198872= 02 120573 = 05 and 119909

119904= 01 These

parameters satisfy the response shown in Table 2 column119884OR

The behavior of the reconfigurable logic cell is describedin what follows If the initial condition is 119909

0= 01 + 025119906

1+

0251199062and inputs 119906

1= 0 119906

2= 0 then 119909

0= 01 and the

first iteration of the tent map 1 is 119891(119909) = 02 Thus the outputdefined by 4 is119910 = 0 Repeating the exercise with 119906

1= 0 119906

2=

1 we have 1199090= 035 119891(119909) = 07 and 119910 = 1 Following this

way all the possible combinations are considered with twologic inputs to get the results illustrated in column 119884OR(119909) ofTable 2 so the logical cell under the parameters defined abovecan generate the behavior of a logical function OR

22 Function 1198918(AND) For this case the same parameter

values used above are considered except the 120573 parameteris now tuned to 075 Note that only one parameter isreconfiguredThe output of the logic gate is shown in Table 2column119910ANDThismeans that only one parameter is changedfor the reconfiguration of the logical cell generating from thelogical function OR to the logical function AND

Note that the selection of the parameter values can beadjusted by trial and error but this was an exhausted taskhence the evolutionary algorithms arise as a convenient alter-native We propound the Evonorm evolutionary algorithmto be used for the selection of the parameter values and thealgorithm explained in the next section



1199061

1199062

OR AND0 0 0 00 1 1 01 0 1 01 1 1 1


8(OR and AND

logical gates)

1199061

1199062

119909119904

119891(119909) 119884OR(119909) 119884AND(119909)

0 0 01 02 0 00 1 01 07 1 01 0 01 07 1 01 1 01 08 1 1












119904= 119875(119896 1) 120583 = 119875(119896 2)

120573 = 119875(119896 3) 1198871= 119875(119896 4) and 119887

2= 119875(119896 5)











120583pr =sum119879119904

119896=1(119875119904119896pr)

119879119904

120590pr =

radic(sum119879119904

119896=1(119875119904119896prminus 120583pr))

119879119904

(5)


119875119896pr =

119873(120583pr 120590pr) 119880 ( ) gt 05

119873 (119868119909pr 120590pr) otherwise

(6)


119873(120583 120590) = 120583 + 120590(12

sum119894=1

(119880 ( )) minus 6) (7)


1 1199062



1199061

1199062

1198910

1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918

1198919

11989110

11989111

11989112

11989113

11989114

11989115

0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 10 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 11 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 11 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1


Funcion 120583 1198871

1198872

120573 119909119904

1198910

2 025 025 085 01111198911

2 03 03 09 0451198912

2 07 015 05 01111198913

2 03 02 08 041198914

2 015 07 05 01111198915

2 02 04 05 041198916

2 045 045 025 01111198917

2 045 045 01 01111198918

2 025 025 075 01111198919

2 05 minus01 06 0311989110

2 01 025 05 011111989111

2 minus01 04 06 0311989112

2 025 01 05 011111989113

2 04 minus01 06 0311989114

2 025 025 05 011111989115

2 025 025 02 0111




12is generated


1198871= 025 119887



1= 0 119906

2=


5 Conclusion


119904 120573 119887



Acknowledgment


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199061

1199062

OR AND0 0 0 00 1 1 01 0 1 01 1 1 1


8(OR and AND

logical gates)

1199061

1199062

119909119904

119891(119909) 119884OR(119909) 119884AND(119909)

0 0 01 02 0 00 1 01 07 1 01 0 01 07 1 01 1 01 08 1 1












119904= 119875(119896 1) 120583 = 119875(119896 2)

120573 = 119875(119896 3) 1198871= 119875(119896 4) and 119887

2= 119875(119896 5)











120583pr =sum119879119904

119896=1(119875119904119896pr)

119879119904

120590pr =

radic(sum119879119904

119896=1(119875119904119896prminus 120583pr))

119879119904

(5)


119875119896pr =

119873(120583pr 120590pr) 119880 ( ) gt 05

119873 (119868119909pr 120590pr) otherwise

(6)


119873(120583 120590) = 120583 + 120590(12

sum119894=1

(119880 ( )) minus 6) (7)


1 1199062



1199061

1199062

1198910

1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918

1198919

11989110

11989111

11989112

11989113

11989114

11989115

0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 10 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 11 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 11 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1


Funcion 120583 1198871

1198872

120573 119909119904

1198910

2 025 025 085 01111198911

2 03 03 09 0451198912

2 07 015 05 01111198913

2 03 02 08 041198914

2 015 07 05 01111198915

2 02 04 05 041198916

2 045 045 025 01111198917

2 045 045 01 01111198918

2 025 025 075 01111198919

2 05 minus01 06 0311989110

2 01 025 05 011111989111

2 minus01 04 06 0311989112

2 025 01 05 011111989113

2 04 minus01 06 0311989114

2 025 025 05 011111989115

2 025 025 02 0111




12is generated


1198871= 025 119887



1= 0 119906

2=


5 Conclusion


119904 120573 119887



Acknowledgment


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199061

1199062

1198910

1198911

1198912

1198913

1198914

1198915

1198916

1198917

1198918

1198919

11989110

11989111

11989112

11989113

11989114

11989115

0 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 10 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 11 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 11 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1


Funcion 120583 1198871

1198872

120573 119909119904

1198910

2 025 025 085 01111198911

2 03 03 09 0451198912

2 07 015 05 01111198913

2 03 02 08 041198914

2 015 07 05 01111198915

2 02 04 05 041198916

2 045 045 025 01111198917

2 045 045 01 01111198918

2 025 025 075 01111198919

2 05 minus01 06 0311989110

2 01 025 05 011111989111

2 minus01 04 06 0311989112

2 025 01 05 011111989113

2 04 minus01 06 0311989114

2 025 025 05 011111989115

2 025 025 02 0111




12is generated


1198871= 025 119887



1= 0 119906

2=


5 Conclusion


119904 120573 119887



Acknowledgment


References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article generation of a reconfigurable logical...

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