Heuristic Set-Covering-Based Postprocessing for Improving the Quine-McCluskey Method

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1 Heuristic Set-Coverig-Based Postprocessig for Improvig the Quie-McCluskey Method Miloš Šeda Abstract Fidig the miimal logical fuctios has importat applicatios i the desig of logical circuits. This task is solved by may differet methods but, frequetly, they are ot suitable for a computer implemetatio. We briefly summarise the well-kow Quie-McCluskey method, which gives a uique procedure of computig ad thus ca be simply implemeted, but, eve for simple examples, does ot guaratee a optimal solutio. Sice the Petrick extesio of the Quie-McCluskey method does ot give a geerally usable method for fidig a optimum for logical fuctios with a high umber of values, we focus o iterpretatio of the result of the Quie-McCluskey method ad show that it represets a set coverig problem that, ufortuately, is a NP-hard combiatorial problem. Therefore it must be solved by heuristic or approximatio methods. We propose a approach based o geetic algorithms ad show suitable parameter settigs. Keywords Boolea algebra, Karaugh map, Quie-McCluskey method, set coverig problem, geetic algorithm. I. INTRODUCTION OR miimisatio of logical fuctios, laws of the Boolea F algebra ad the Karaugh maps are mostly used. The Karaugh maps represet a very efficiet graphical tool for miimisig logical fuctios with o more tha 6 variables. However, their use is based o visual recogitio of adjacet cells ad, therefore, the method is ot suitable for automated processig o computers. A direct applicatio of the Boolea algebra laws is ot restricted i this way, but there is o geeral algorithm defiig the sequece of their applicatio ad thus this approach is ot suitable for computer implemetatio either. With the growig stregth of computatioal techiques, further ivestigatios were focused o a algorithm-based techique for simplifyig Boolea logic fuctios that could be used to hadle a large umber of variables. The well-kow method usable o computers is the algorithm proposed by Edward J. McCluskey, professor of electrical egieerig at Uiversity of Stadford, ad philosopher Willard va Orma Quie from Harvard Uiversity [3], [8]. We will assume that the umber of variables may be high but restricted i the sese that all 2 rows of the correspodig Mauscript received Jue 5, This work was supported i part by the Miistry of Educatio, Youth ad Sports of the Czech Republic uder research pla MSM "Simulatio Modellig of Mechatroic Systems". Miloš Šeda works i the Istitute of Automatio ad Computer Sciece, Faculty of Mechaical Egieerig, Bro Uiversity of Techology, Techická 2896/2, CZ Bro, Czech Republic (phoe: ; fax: ; seda@fme.vutbr.cz). truth table may be saved i a memory ad thus may be processed. If this coditio is ot satisfied, the we could oly work with a selected umber of truth table rows. This approach is possible, e.g., i image filters [9]. Of course, this is iacceptable i real logic cotrol applicatios where all iput combiatios must be tested. I the ext cosideratios we will assume the full truth table. II. QUINE-MCCLUSKEY METHOD The Quie-McCluskey method [3], [7], [8] is a exact algorithm based o systematic applicatio of the distributive law, complemet law (2) ad idempotece law (3), i.e. the laws as follows: x.( y + z) = xy + xz x+ x = (2) x + x = x (3) From ad (2), we ca easily derive the uitig theorem (4) xy+ xy = x.( y+ y) = x. = x (4) I geeral, this meas that two expressios with the same set of variables differig oly i oe variable with complemetary occurrece may be reduced to oe expressio give by their commo part, e.g. f = + = xzw( x + x) = xzw.= xzw For ext cosideratios we will eed the followig otios: Literal is a variable or its egatio. Term is a cojuctio (= product) of literals. Miterm is a cojuctio of literals icludig every variable of the fuctio exactly oce i true or complemeted form. For example, if f ( xyz,, ) = xyz+ xy, the x yz ad xy are terms ad miterms are represeted by cojuctios x yz, xyz ad xyz because xy = xy( z+ z) = xyz+ xyz. Disjuctive ormal form (DNF) of a formula f is a formula f such that f is equivalet to f ad f has the form of a disjuctio of cojuctios of literals. Caoical (or complete) disjuctive ormal form (CDNF) is a DNF where all cojuctios are represeted by miterms. Now we ca describe a skeleto of the Quie-McCluskey method:. Miterms of a give Boolea fuctio i CDNF are divided ito groups such that miterms with the same umber of egatios are i the same group. 2. All pairs of terms from adjacet groups (i.e. groups whose umber of egatios differ by oe) are compared. 39

2 3. If compared terms differ oly i oe literal (ad the uitig theorem may be applied) the these terms are removed ad a term reduced to a commo part is carried to the ext iteratio. 4. All terms are used oly oce, i.e. if ecessary the idempotece law is applied. 5. If the umber of terms i the ew iteratio is ozero, the steps 2-5 are repeated, otherwise the algorithm fiishes. The result of the algorithm is represeted by the terms that were ot removed, i.e. that could ot be simplified. Now, we will apply the Quie-McCluskey method to a simple logical fuctio from [5] ad, by meas of the Karaugh map, we will show that it will ot fid its miimal form. Example Miimise the followig logical fuctio i CDNF. Solutio: If we divide the miterms ito groups by the umber of their egatios, the we get this iitial iteratio: (0) If we compare all pairs of terms from adjacet groups i the iitial iteratio (0) ad place i frames terms that ca be simplified ad divide the resultig terms i the ext iteratio ito groups agai, the we get: (0) Applyig the previous steps to iteratio, we get: (2) f = x yzw+ xy zw+ + + x yzw+ xy zw+ xyz w ; x yzw ; ; ; ; ; ; ; ; x yzw ; ; ; ; ; ; ; x yz ; xzw ; xy z ; y zw ; xy w ; xyz; xzw; xzw ; y zw ; x y z ; xyz ; xzw ; xyz ; yzw ; xyw ; xyz; xzw; xzw ; y zw ; x y z ; xz ; yz I iteratio (2) we have oly oe group of terms ad thus o simplified terms ca be geerated ad the algorithm fiishes. Its result f m is give by a disjuctio of those terms that caot be simplified. fm = xz + y z + xyw + xyz + xzw Let us cosider the same example ad solve it usig the Karaugh map. For the give Boolea fuctio f, we get the followig map x w z Fig. Karaugh map From this map, we get two possible solutios of a miimal logical fuctio depedig o the way of coverig the first two cells with logical i the last colums. a) f mi = xz + yz + xyw + xyz b) f2mi = xz + yz + xzw + xyz We ca see that the result gaied usig Karaugh map icludes a lower umber of terms tha the result of computatio by meas of Quie-McCluskey method. S. R. Petrick proposed a modificatio of Quie-McCluskey method that tries to remove the resultig redudacy [6]. The Quie-McCluskey simplificatio techique with special modificatios is preseted i [2]. Before describig the secod phase of computatio, we will defie several otios. (i) A Boolea term g is a implicat of a Boolea term f if: each literal i g is also obtaied i f (i.e. if g has the form g(x,..., x ) the f has the form f(x,..., x, y,, y m )), ad for all combiatios of literals the implicatio g f has a value of True. We kow that the implicatio is defied by Table. From this defiitio, we get that g is a implicat of f if ad oly if g f. I our case, that meas that the secod property is satisfied if g(x,..., x ) f(x,..., x, y,, y m ) for each selectio of x,..., x ad each selectio of y,, y m. TABLE I IMPLICATION g f g f Therefore, we ca cosider oly those values of variables at which g is true ad, for all these cases, f must also be true. (ii) The terms resultig from the Quie-McCluskey method are called prime implicats. y 40

3 I the Petrick modificatio of the Quie-McCluskey method, a table of prime implicats give by the results of the first phase is built. It has prime implicats i the heads of colums, miterms from a give CNDF are i the heads of rows ad, i cells represetig itersectios of rows ad colums, a selected symbol (e.g. ) is placed if the prime implicat i questio is a subset of a correspodig miterm. We say that prime implicats cover miterms. I our example the we get Table II as follows. TABLE II COVERING OF MINTERMS BY PRIME IMPLICANTS xz yz x yw xyz xzw x yzw Although the prime implicats caot be simplified, some of them ca be redudat ad thus may be omitted. However, symbols i colums of cosidered prime implicats must be spread ito all rows, otherwise the disjuctio of prime implicats would ot express the iitial caoical (complete) disjuctive ormal form, i.e., there would be a o-covered miterm. If there is a row i which occurs oly oce, the the prime implicat i the correspodig colum caot be omitted. Such implicats are called essetial prime implicats. I our example there are three essetial prime implicats: xz, yz ad xyz. Now we will simplify Table II by deletig colums of essetial prime implicats ad rows that cotai their symbols. We get Table 3. TABLE III x yw xzw It ca be easily see that, i simplified Table III, the miterm x yzw may be covered by the prime implicat x yw or by xzw. Hece we get two miimal disjuctive forms: fmi = xz + yz + xyw + xyz ad f = xz + yz + xzw + xyz, 2mi which agrees with the previous solutio by the Karaugh map. Although computatios by the Quie-McCluskey method with the Petrick modificatio have a wider use tha the approach based o the Karaugh map, however, this improved method has also its restrictios. The mai problem is i the procedure of coverig miterms by prime implicats. It ca be easily foud if the umber of the essetial prime implicats is high ad thus the table of prime implicats will be very reduced. I geeral, we caot expect this, ad it is eve possible that oe of the prime implicats is essetial. From the combiatorial optimisatio theory, it is kow that the set coverig problem is NP-hard ad its time complexity grows expoetially with the growig iput size. It ca be show that, for a CNDF with miterms, the upper boud of the umber of prime implicats is 3 /. If =32, we ca get more tha 6.5 * 0 5 prime implicats. To solve such a large istace of the set coverig problem is oly possible by heuristics (stochastic or determiistic) [], [2], [4], [6], [0], [], [4], [5], [20], [22] or by approximatio [7], [8] methods. However, they do ot guaratee a optimal solutio. A formal defiitio of the problem ad possible solutio by a geetic algorithm is proposed i the ext sectio. III. SET COVERING PROBLEM (SCP) The set coverig problem (SCP) is the problem of coverig the rows of a m-row, -colum, zero-oe matrix (a ij ) by a subset of colums at a miimal cost. Defiig x j = if colum j (with cost c j =>0) is i the solutio ad x j =0 otherwise, the SCP is: Miimise cx j j (5) subject to aij xj, i =, 2,..., m, (6) x j {0,}, j =, 2,..., (7) Costraits (6) esure that each row is covered by at least oe colum ad (7) is a costrait guarateeig itegers. I geeral, the cost coefficiets c j are positive itegers. Here, i the applicatio of the SCP for the secod phase of the Quie-McCluskey method, we assume all c j equal to because we try to miimise the umber of the coverig colums. This special case of SCP is called a uicost SCP. SCP has, besides Quie-McCluskey method, a wide rage of applicatios, for example vehicle ad crew schedulig [2], facilities locatio [], assembly lie balacig ad Boolea expressio simplificatio. There are umber of other combiatorial problems that ca be formulated as, or trasformed to, SCP such as the graph colourig problem [] ad the idepedet vertex set problem [7]. A fuzzy versio of SCP is studied i [3] ad [9]. Sice we use, for solvig the SCP, a modified versio of the geetic algorithm (GA) proposed by Beasley ad Chu [2], [4] for the o-uicost SCP, we summarise the basic 4

4 properties of GAs IV. GENETIC ALGORITHM The skeleto for GA ca be described as follows: geerate a iitial populatio ; evaluate fitess of idividuals i the populatio ; repeat select parets from the populatio; recombie (mate) parets to produce childre ; evaluate fitess of the childre ; replace some or all of the populatio by the childre util a satisfactory solutio has bee foud ; Sice the priciples of GAs are well-kow, we will oly deal with GA parameter settigs for the problems to be studied. Now we describe the geeral settigs. Idividuals i the populatio (chromosomes) are represeted as biary strigs of legth, where a value of 0 or at the i-th bit (gee) implies that x i = 0 or i the solutio respectively. The populatio size N is usually set betwee ad 2. May empirical results have show that populatio sizes i the rage [50, 200] work quite well for most problems. Iitial populatio is obtaied by geeratig radom strigs of 0s ad s i the followig way: First, all bits i all strigs are set to 0, ad the, for each of the strigs, radomly selected bits are set to util the solutios (represeted by strigs) are feasible. The fitess fuctio correspods to the objective fuctio to be maximised or miimised. There are three most commoly used methods of selectio of two paret solutio for reproductio: proportioate selectio, rakig selectio, ad touramet selectio. The touramet selectio is perhaps the simplest ad most efficiet amog these three methods. We use the biary touramet selectio method where two idividuals are chose radomly from the populatio. The more fit idividual is the allocated a reproductive trial. I order to produce a child, two biary touramets are held, each of which produces oe paret. The recombiatio is provided by the uiform crossover operator, which has a better recombiatio potetial tha do other crossover operators as the classical oe-poit ad twopoit crossover operators. The uiform crossover operator works by geeratig a radom crossover mask B (usig Beroulli distributio) which ca be represeted as a biary strig B = b b 2 b 3 b - b where is the legth of the chromosome. Let P ad P 2 be the paret strigs P [],...,P [] ad P 2 [],...,P 2 [] respectively. The the child solutio is created by lettig: C[i] = P [i] if b i = 0 ad C[i] = P 2 [i] if b i =. Mutatio is applied to each child after crossover. It works by ivertig M radomly chose bits i a strig where M is experimetally determied. We use a mutatio rate of 5/ as a lower boud o the optimal mutatio rate. It is equivalet to mutatig five radomly chose bits per strig. Whe v child solutios have bee geerated, the childre will replace v members of the existig populatio to keep the populatio size costat, ad the reproductive cycle will restart. As the replacemet of the whole paret populatio does ot guaratee that the best member of a populatio will survive ito the ext geeratio, it is better to use steady-state or icremetal replacemet which geerates ad replaces oly a few members (typically or 2) of the populatio durig each geeratio. The least-fit member, or a radomly selected member with below-average fitess, are usually chose for replacemet. Termiatio of a GA is usually cotrolled by specifyig a maximum umber of geeratios t max or relative improvemet of the best objective fuctio value over geeratios. Sice the optimal solutio values for most problems are ot kow, we choose t max V. GENETIC ALGORITHM FOR SCP The chromosome is represeted by a -bit biary strig S where is the umber of colums i the SCP. A value of for the j-th bit implies that colum j is i the solutio ad 0 otherwise. Sice the SCP is a miimisatio problem, the lower the fitess value, the more fit the solutio is. The fitess of a chromosome for the uicost SCP is calculated by (8). f ( S) = S[ j] (8) The biary represetatio causes problems with geeratig ifeasible chromosomes, e.g. i iitial populatio, i crossover ad/or mutatio operatios. To avoid ifeasible solutios a repair operator is applied. Let I = {,, m} = the set of all rows; J = {,, } = the set of all colums; α i = {j J a ij =} = the set of colums that cover row i, i I; β j = {i I a ij =} = the set of rows covered by colum j, j J; S = the set of colums i a solutio; U = the set of ucovered rows; w i = the umber of colums that cover row i, i I i S. The repair operator for the uicost SCP has the followig form: iitialise w i : = S α i, i I ; iitialise U : = { i w i = 0, i I } ; for each row i i U (i icreasig order of i) do begi fid the first colum j (i icreasig order of j) i α i that miimises / U β j ; S : = S + j ; w i : = w i +, i β j ; U : = U β j ed ; for each colum j i S (i decreasig order of j) do if w i 2, i β j the begi S : = S j ; w i : = w i, i β j ed ; { S is ow a feasible solutio to the SCP ad cotais o redudat colums } Iitialisig steps idetify the ucovered rows. For statemets are greedy heuristics i the sese that i the st 42

5 for, colums with low cost-ratios are beig cosidered first ad i the 2 d for, colums with high costs are dropped first wheever possible. VI. CONCLUSION I this paper, we studied the problem of miimisig the Boolea fuctios with a rather high umber of variables. Sice traditioal approaches based o the Boolea algebra or Karaugh maps have restrictios i the umber of variables ad the sequece of laws that could be applied is ot uique, we focus o the well-kow Quie-McCluskey method. Sice it does ot guaratee the fidig of a optimal solutio, we must apply a postprocessig phase. Ufortuately, the data resultig from the Quie-McCluskey method are i the form of a uicost set coverig problem, which is NP-hard. Therefore, for logical fuctios, the obvious Petrick s extesio of the Quie-McCluskey method caot be applied, ad heuristic or approximatio method must be used istead. We proposed a geetic algorithm-based approach ad discussed problem-orieted parameter settigs. I the future, we are goig to implemet also other stochastic heuristics, such as simulated aealig ad tabusearch, ad compare them with the geetic algorithm. [4] A. Mofroglio, Hybrid Heuristic Algorithms for Set Coverig, Computers & Operatios Research, vol. 25, pp , 998. [5] M. Ohlsso, C. Peterso ad B. Söderberg, A Efficiet Mea Field Approach to the Set Coverig Problem, Europea Joural of Operatioal Research, vol. 33, pp , 200. [6] S.K. Petrick, O the Miimizatio of Boolea Fuctios, i Proceedigs of the Iteratioal Coferece Iformatio Processig, Paris, 959, pp [7] Ch. Posthoff ad B. Steibach, Logic Fuctios ad Equatios: Biary Models for Computer Sciece. Berli: Spriger-Verlag, [8] W.V. Quie, The Problem of Simplifyig Truth Tables, America Mathematical Mothly, vol. 59, o. 8, pp , 952. [9] L. Sekaia, Evolvable Compoets. Berli: Spriger-Verlag, [20] M. Solar, V. Parada ad R. Urrutia, A Parallel Geetic Algorithm to Solve the Set-Coverig Problem, Computers & Operatios Research, vol. 29, pp , [2] S.P. Tomaszewski, I.U. Celik ad G.E. Atoiou, WWW-Based Boolea Fuctio Miimizatio, Iteratioal Joural of Applied Mathematics ad Computer Sciece, vol. 3, o. 4, pp , [22] M. Yagiura, M. Kishida ad T. Ibaraki, A 3-Flip Neighborhood Local Search for the Set Coverig Problem, Europea Joural of Operatioal Research, vol. 72, pp , REFERENCES [] L. Brotcore, G. Laporte ad F. Semet, Fast Heuristics for Large Scale Coverig-Locatio Problems, Computers & Operatios Research, vol. 29, pp , [2] J.E. Beasley P.C. Chu, A Geetic Algorithm for the Set Coverig Problem, Joural of Operatioal Research, vol. 95, o. 2, pp , 996. [3] C.I. Chiag, M.J. Hwag ad Y.H. Liu, A Alterative Formulatio for Certai Fuzzy Set-Coverig Problems, Mathematical ad Computer Modellig, vol. 42, pp , [4] P. Chu, A Geetic Algorithm Approach for Combiatorial Optimisatio Problems, PhD thesis, The Maagemet School Imperial College of Sciece, Techology ad Medicie, Lodo, 997. [5] K. Čulík, M. Skalická, I. Váňová, Logic (i Czech). Bro: VUT FE, 968. [6] P. Galiier ad A. Hertz, Solutio Techiques for the Large Set Coverig Problem, Discrete Applied Mathematics, vol. 55, pp , [7] F.C. Gomes, C.N. Meeses, P.M. Pardalos ad G.V.R. Viaa, Experimetal Aalysis of Approximatio Algorithms for the Vertex Cover ad Set Coverig Problems, Computers & Operatios Research, vol. 33, pp , [8] T. Grossma ad A. Wool, Computatioal Experiece with Approximatio Algorithms for the Set Coverig Problem, Europea Joural of Operatioal Research, vol. 0, pp. 8-92, 997. [9] M.J. Hwag, C.I. Chiag ad Y.H. Liu, Solvig a Fuzzy Set-Coverig Problem, Mathematical ad Computer Modellig, vol. 40, pp , [0] G. La ad G.W. DePuy, O the Effectiveess of Icorporatig Radomess ad Memory ito a Multi-Start Metaheuristic with Applicatio to the Set Coverig Problem, Computers & Idustrial Egieerig, vol. 5, pp , [] G. La, G.W. DePuy ad G.E. Whitehouse, A Effective ad Simple Heuristic for the Set Coverig Problem, Europea Joural of Operatioal Research, vol. 76, pp , [2] M. Mesquita ad A. Paias, Set Partitioig/Coverig-Based Approaches for the Itegrated Vehicle ad Crew Schedulig Problem, Computers & Operatios Research, i press, [3] E.J. McCluskey, Miimizatio of Boolea Fuctios, Bell System Techical Joural, vol. 35, o. 5, pp ,