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1 On the Number of Deendent Variables for Incomletely Secied Multile-Valued Functions Tsutomu Sasao Deartment of Comuter Science and Electronics Kyushu Institute of Technology Iizua 8-85, Jaan Abstract This aer considers the minimization of deendent variables in functions with many don't cares. It also derives the conditions for almost all randomly generated function to be redundant in at least one variable. Exerimental results suort the validity of the aroach. I Introduction Logic minimization often means to reduce the number of roducts to reresent the given function []. However, in the case of incomletely secied functions (i.e., functions with don't cares), at least two roblems exist [7] The rst one is to reduce the number of the roducts to reresent the function, and the second one is to reduce the number of deendent variables. The rst roblem is imortant when the given function is realized as a sum-of-roducts exression (SOP) [], an EXOR sum-of-roducts exressions (ESOP) [], etc. The second roblem is imortant when the given function is realized as a ROM or a RAM, since the number of the inut variables only determines the cost of the realization. Examle. Consider the four-variable logic function shown in Fig.., where the blan cells denote don't cares. The SOP with the minimum number of roducts is F = x x 4 _ x x 3 ; while the SOP with the minimum number of deendent variables is F = x x _ x x 4 _ x x 4 Note that F has two roducts and deends on four variables, while F has three roducts and deends on only three variables. As shown in this examle, the minimization of the number of roducts is dierent from the minimization of the number of deendent variables. In this aer, we will consider the minimization of the number of deendent variables. The rest of the aer is organized as follows In Section II, we will give denitions and basic roerties. In Section III, we will formulate the minimization roblem of the deendent variables in the function f x x x 3 x 4 Figure. Four-variable function with don't cares. P n! Q,whereP = f; ;;g, Q = f; ;;q g. And we will show the minimization algorithm. In Section IV, we will consider random functions f P n! Q, and derive formulas to redict the number of redundant variables when the ercentage of secied mintermsisvery small. For examle, when =, q =,n = 7, and 5 minterms are maed to zeros, 5 minterms are maed to ones, and other minterms are maed to don't cares, the formula redicts that at least one variable is redundant in 9.6% of the functions. In the area of nowledge engineering, more than 99% of the entries are don't cares [6, 8, 9], and this formula is useful to redict the number of deendent variables. In Section V, we will comare the analysis with exerimental results. II Denitions and Basic Proerties Denition. An incomletely secied multile-valued function (function for short) f is a maing D! Q, where D P n, P = f; ;;g, and Q = f; ;;q g. Denition. An incomletely secied multilevalued function is reresented byasetofcharacteristic functions F i, where F i (a) = i f(a) = i (i = ; ;;q ). Note that F i F j = (i 6= j). If a P n D, then the value of f(a) is unsecied, and is denoted by d (don't care). Denition.3 [] Two-valued variables are often reresented by x i (i = ; ;;n). Multile-valued variables are reresented byx i (i =; ;;n). X i

2 Table. x x x 3 x 4 f taes one of values in P. A literal is dened asxi S = if X i S, andxi S =otherwise, where S P. For two-valued case, X =x, X = x, andx f;g =. A roduct of literals is a roduct term, andasumof roducts is a sum-of-roducts exression (SOP). If all the roducts are rime and no roduct can be removed from the SOP without changing the function, then it is an irredundant sum-of-roducts exression (ISOP). Examle. Consider the function in Examle.. In this case, = q =and n =4.Table. also shows this function. The characteristic functions are F =x x x 3 x 4 _ x x x 3 x 4 _ x x x 3 x 4 and F =x x x 3x 4 _ x x x 3x 4 _ x x x 3x 4 Examle. Consider the function in Table.. In this case, =3, q =4and n =4. The characteristic functions are F =X X X 3 X 4 _ X X X 3 X 4 ; F =X X X 3 X 4 _ X X X 3 X 4 ; F =X X X 3 X 4 _ X X X 3 X 4 ; and F 3 =X X X 3 X 4 _ X X X 3 X 4 Lemma. Let f be a function P n! Q. f is exanded withresect to X as follows f = X f _ X f X f ; where f i = f(jx = i). Denition.4 f deends on X i if there exists a air of vectors a=(a ;a ;;a i ;;a n ) and b=(a ;a ;;b i ;;a n ); such that both f(a) and f(b) are secied, and f(a) 6= f(b). Table. X X X 3 X 4 f 3 3 x x x 3 x 4 Figure. Four-variable function with don't cares. If f deends on X i, then X i is essential in f, andx i must aear in any exression for f. Denition.5 Two functions f and g P n! Q are comatible when the following condition holds For any a P n,ifboth f(a) and g(a) are secied, then f(a) =g(a). Denition.6 X is non-essential in f i f, f,,andf are comatible each other. Theorem. If X is non-essential in f, thenf can be reresented byanexression without X. Denition.7 A set of variables fx i g is redundant if f can be reresented without using fx i g. Examle.3 Consider the function f in Fig... It is easy to verify that all the variables are non-essential. Note that f can be reresented asf = x _ x 3 or F = x 8 x 4. The redundant sets of variables are fx ;x 4 g in F and fx ;x 3 g in F. Essential variables must aear in every exression for f, while non-essential variables, may aear in some exressions and not in others. Examle.4 Consider the function f in Table.. Since f(; ; ; )=and f(; ; ; )=, f deends on X,andX is essential. Since f(; ; ; ) = 3 and

3 f(; ; ; ) =, f deends on X,andX is essential. However, X 3 and X 4 are non-essential. To reresent f, both X and X are necessary. In addition, either X 3 or X 4 is necessary. This fact will be shown in Examle 3.. III Minimization of Deendent Variables In this section, we will consider the roblem to reresent a given function by exressions with the minimum number of variables. For two-valued logic functions, this roblem has been considered by Halatsis-Gaitanis [5], Brown [], and others [7, 3, 4]. Here, we will consider the cases of multile-valued functions P n! Q, where P = f; ;; g and Q = f; ;;q g. The following is an extension of Halatsis-Gaitanis's method. Algorithm 3. (Minimization of Deendent Variables) ) Exress the characteristic functions F i (i = ; ;;q ) by SOPs F i = t(f _ i) = r(i; ); where t(f i ) denotes the number of roducts in the SOP for F i,andr(i; ) denotes the -th roduct in F i. ) For each air of the characteristic functions, F i and F j (i 6= j), do the following For each air of roducts [r(i; ); r(j; `)], associate a sum-of-literals s(i; j; ; `) dened by n_ s(i; j; ; `) = y m ; m= where y m = if S m \ T m 6=, y m = x m if S m \ T m =, r(i; ) =X S XS XSn n, r(j; `) = X T XT XTn n,andm =; ;;n. 3) Dene a Boolean function R = q ^ q ^ t(f ^ i) t(f j ) ^ i= j=i+ = `= s(i; j; ; `) 4) Reresent R as an ISOP. The roduct with the minimal number of literals corresonds to the set of minimal deendent variables. (Correctness of the Algorithm) Consider two characteristic functions F i and F j,where i 6= j. Let F i have a roduct c i = X S XS XS n n, and F j have aroductc j = X T XT XT n n. Let I = f; ;; ng. ) By denition of the characteristic functions, there exists at least one variable x m such that S m \T m =, andm I. If there is no such m, then S m \ T m 6= for all m I. So, there exists a vector a such that F i (a) =F j (a) =. This contradicts the denition of the characteristic functions. ) Let L I and, for all m L, S m \ T m =, where c i = X S XS XS n n and c j = X T XT XT n n. Without loss of generality, assume that L = f; ; 3;;g ( n). In this case, we claim that at least one variable in fx ;x ;;x g is necessary to distinguish c i from c j. This corresonds to s(i; j; ; `) in the algorithm. 3) This condition must hold for all the air of the roducts (c i ;c j ) for dierent characteristic functions F i and F j.thus, the Boolean function R in the algorithm reresents the condition. 4) Therefore, each roduct of the ISOP for R reresents the condition necessary to distinguish F i from F j. 5) Since s(i; j; ; `) shows the minimum condition to distinguish F i from F j,each roduct in the ISOP corresonds to the set of minimal deendent variables. (End of the Correctness) Examle 3. Consider the function in Examle.. ) SOPs for the characteristic functions are given. Each roduct is labeled as follows r(; )=x x x 3 x 4 ; r(; )=x x x 3 x 4 ; r(; 3)=x x x 3 x 4 ; r(; )=x x x 3x 4; r(; )=x x x 3x 4; r(; 3)=x x x 3 x 4 ) For each air of roducts, nd the variables whose roducts are null. Note that in r(; ) and r(; ), only the roduct is null in x. So, we have Similarly, we have s(; ; ; ) = x s(; ; ; )=x ; s(; ; ; 3)=x _ x _ x 4 ; s(; ; ; )=x 3 _ x 4 For the air of r(; ) and r(; ), the roducts are null in all variables. So, we have s(; ; ; ) = x _ x _ x 3 _ x 4 3

4 Similarly, we have s(; ; ; 3)=x _ x 3 ; s(; ; 3; )=x _ x _ x 4 ; s(; ; 3; )=x 4 ; s(; ; 3; 3)=x 3) R is obtained asthelogical roduct of s(i; j; ; `), and we have R=x x (x _ x _ x 4 )(x 3 _ x 4 )(x _ x _ x 3 _ x 4 ) (x _ x 3 )(x _ x _ x 4 )x 4 x =x x x 4 This means that the function can be reresented by the SOP with variable set fx ;x ;x 4 g F = x x x 4 _ x x x 4 _ x x x 4 Examle 3. Consider the function in Examle.. ) SOPs for characteristic functions are given in Examle.. Each roduct is labeled as follows r(; )=X X X 3 X 4 ; r(; )=X X X 3 X 4 ; r(; )=X X X 3 X 4 ; r(; )=X X X 3 X 4 ; r(; )=X X X 3 X 4 ; r(; )=X X X 3 X 4 ; r(3; )=X X X 3 X 4 ; r(3; )=X X X 3 X 4 ) For each air of roducts, nd the variables whose roducts are null s(; ; ; )=x s(; ; ; )=x _ x _ x 3 _ x 4 s(; ; ; )=x _ x 4 s(; ; ; )=x 3 _ x 4 s(; ; ; )=x _ x 3 _ x 4 s(; ; ; )=x _ x 3 s(; ; ; )=x _ x 3 _ x 4 s(; ; ; )=x _ x 3 _ x 4 s(; 3; ; )=x s(; 3; ; )=x _ x s(; 3; ; )=x _ x _ x 4 s(; 3; ; )=x _ x 4 s(; ; ; )=x 3 _ x 4 s(; ; ; )=x _ x _ x 3 s(; ; ; )=x s(; ; ; )=x _ x 4 s(; 3; ; )=x s(; 3; ; )=x s(; 3; ; )=x _ x _ x 3 _ x 4 s(; 3; ; )=x _ x 3 _ x 4 s(; 3; ; )=x _ x 3 _ x 4 s(; 3; ; )=x _ x 3 _ x 4 s(; 3; ; )=x _ x _ x 3 s(; 3; ; )=x _ x _ x 3 3) R is obtained as the logical roduct of s(i; j; ; `), and we have R = x x (x 3 _x 4 ). Thus, the function can be reresented by the SOPs with variable set fx ;X ;X 3 g IV F =X X X 3 _ X X X 3 ; F =X X X 3 _ X X X 3 ; and F 3 =X X X 3 _ X X X 3 Or, the SOPs with the variable set fx ;X ;X 4 g F =X X X 4 _ X X X 4 ; F =X X X 4 _ X X X 4 ; and F 3 =X X X 4 _ X X X 4 Analysis of Random Multile-Valued Functions In this Section, we will estimate the number of redundant variables in random incomletely secied multile-valued functions. To mae the roblem easy to analyze, we use the following Assumtion 4. In a random multile-valued function f, if the robability of aearing i in f is, then the robability of aearing i in the sub-function f(jx j = ) is also, where i f; ;;q g and f; ;; g. This assumtion imlies that n, the number of the inut variables, is suciently large. Furthermore, for simlicity, we assume that <. Thus, in the function table for f, most entries are don't cares, and only a fraction of the entries are secied. Theorem 4. Let f P n! Q, where P = f; ;; g and Q = f; ;;q g be arandom function such that the robability taing the value i (i =; ;;q ) is. Then, the robability that the variable X i is redundant in f is given by (n; ; q; ) = M ; where = q( + ) (q ), =q, and M = n. 4

5 X= X= X=- f f f - Figure 4. A ma of -valued function. (Proof) Consider the ma shown in Fig. 4., where f is exanded with resect to X f = X f _ X f X f ; where f i = f(jx = i). To bex non-essential, sub-functions f i (i = ; ;; ) must be comatible with each other. To be these sub-functions comatible, cells that corresond to f(j; a) (j = ; ;; ) must be comatible with each other, where a is a vector in P n. In Fig. 4., these cells are hatched. These cells are comatible when the following conditions are satised All the cells are don't cares. The robability of this case is given by, where =q. Only one cell taes value i (i =; ; ;;q ) and others are don't cares (dc's). The robability of this case is given by. Two cells tae value i and others are dc's. The robability of this case is given by. All the cells tae value i. The robability of this case is given by. The robability that these cells are comatible is derived as follows a) All the cells are dc's. b) At least one cell is and others dc's. c) At least one cell is and others dc's. d) At least one cell is q and others dc's. Since a) to d) are mutually disjoint, the robability is given by =q =q[( + ) ]+ Note that M = n cells exist in the sub-function f. To bex non-essential, the same thing must hold for each of these M cells. Thus, the robability that X is non-essential in f is given by (n; ; q; ) = M. Corollary 4. ) Let =and q =. We have =(+), and =. Thus, = ( ) ( ) = ; (n; ; ;)= n ) Let =3and q =3. We have =3( + ) 3 3,and = 3. Thus, =3( ) 3 ( 3) 3 = ; (n; 3; 3;)= 3n 3) Let =and q =4. We have =4( + ) 3,and = 4. Thus, = 4( 3) 3( 4) = ; (n; ; 4;)= n 4) Let =4and q =. We have =(+) 4 4, and =. Thus, =() 4 () 4 = ; (n; 4; ;)= 4n 5) Let =4and q =4. We have =4( + ) 4 3 4,and = 4. Thus, =4(3) 4 3(4) 4 = ; (n; 4; 4;)= 4n Theorem 4. Let f P n! Q, where P = f; ;; g and Q = f; ;;q g be arandom function such that the robability of taing the value i (i =; ;;q ) is. Then, the robability that f is a redundant with fx ;X ;;X g is = M, where = q( + ) (q ), = q, and M = n. (Proof) Assume that f is exanded with resect to X, X,, and X f=x X X f _ X X X f X X X f fx ;X ;;X g is a redundant setinf if subfunctions f,, f are mutually comatible. To be these sub-functions comatible, cells that corresond to f(a; ; ;;) must be comatible, where a isavector in P. Similar to the roof of Theorem 4., we have =q =q[(+) ]+ 5

6 Note that M = n cells exist in the sub-function f. When f is redundant in fx ;X ;;X g, the same thing hold for each of these M cells. Thus, the robability that fx ;X ;;X g is a redundant setis = M. Theorem 4.3 In an n variable function f, let be the robability that the set of variables is redundant. Then, we have the following ) The robability that at least one variable is redundant in f is = ( ) n ) The robability that f hasatleast one set of redundant variables with size two is = ( ) n(n)= 3) The robability that f hasatleast one set of redundant variables with size three is 3 = ( 3 ) n(n)(n)=6 (Proof) The robability thatx i is essential is. The robability that all the variables are essential is ( ) n.thus, we have = ( ) n. Similarly, we have and 3. Denition 4. A roerty A is said to hold for almost all functions if the roortion of n-variable functions that do not satisfy A tends to zero asn!. In other words, let w(n) be the number of n-variable functions that do not satisfy A. Then w(n)= n! as n!. Theorem 4.4 Let = q =. If (n)=, then at least one variable is redundant for almost all functions, where is a ositive constant. (Proof) When = (n)=,wehave = M,where M = n. Thus, = = M. Note that for suciently large n, < M << and M is very large. In this case, we have = M =( M )M ' e ' (35) By Theorem 4.3, the robability that at least one variable is redundant isgivenby = ( ) n. As n!,( ) n!. Thus, for almost all functions at least one variable is redundant. Theorem 4.5 Let = q = 3. If 3 (n+)=, then at least one variable is redundant for almost all functions, where is a ositive constant. (Proof) When = 3 (n+)=, wehave = 9M, where M =3 n. Note that = Since <, we have ' M Similar to the roof of Theorem 4.4, we can show that at least one variable is redundant for almost all functions. Theorem 4.6 Let = q =4. If n =3, thenat least one variable is redundant for almost all functions, where is a ositive constant. (Proof) When = n =3, we have = 36M, where M =4 n. Note that = Since <, we have ' M Similar to the roof of Theorem 4.4, we can show that at least one variable is redundant in f. V Exerimental Results We develoed a rogram to nd maximal redundant setsofinutvariables for incomletely secied multile-valued functions. Table 5. summarizes the results of statistical analysis and comuter simulation. For each set of arameters (; q; n; N min ), we randomly generated samle functions and found maximal numbers of redundant variables. N min denotes the number of combinations that are maed to i (i =; ;;q ). That is, for the case of q =, f has N min true minterms and N min false minterms. The number of unsecied minterms is n qn min. 5. Exeriment Suorting Theorem 4.3 In Table 5., the column headed by shows the robability that f has at least one redundant variable, and the column headed by shows the robability that f has at least one set of redundant variables with size two. The values of and were calculated by using the formulas in Theorem 4.3. The column denoted by ( =; ; ; 3) shows the number of functions that have the sets of redundant variables with maximum size. These values were obtained by comuter simulation. For examle, when =,q =,n =7andN min = 5, the statistical analysis shows that =9588 and =9779. On the other hand, the comuter simulation shows that out of functions, 86 functions have no redundantvariables (i.e., all the variables are essential); for 683 functions, the sizes of the maximal set of redundant variables are one; and for 3 6

7 Table 5. Comarison of statistical analysis and comuter simulation. Statistical Comuter Analysis Simulation q n N min functions, the sizes of the maximal set of redundant variables are two. The formula of Theorem 4.3 shows that for 9.6% of the functions have at least one redundant variables, while comuter simulation shows that = 94 functions out of have redundant set with size one or two. In this case, the formula redicts the sizes of the redundant variable set quite well. 5. Exeriment Suorting Theorem 4.4 In Table 5., for = q =,values for N min were selected to satisfy = N min = n (n)=,where =. Thus, N min = (n+)=. Table 5. shows that as n increases, the value of increases monotonously to aroach. Similar tendency can be observed in the cases of = q = 3 and = q =4. VI Conclusion and Comments In this aer, we have ) Formulated the minimization roblem of deendent variables for incomletely secied multilevalued logic functions, and shown the algorithm. ) Derived the robability that a given set of variable is redundant in a randomly generated function. 3) Shown conditions that at least one variable is redundant in randomly generated functions. 4) Veried the usefulness of the aroach by comuter simulation. In this aer, we used statistical analysis. It is ossible to use combinatorial analysis instead []. In such a case, the estimation is accurate even when n is small. Acnowledgments This wor was suorted in art by a Grant in Aid for Scientic Research of the Ministry of Education, Science, Culture and Sorts of Jaan as well as the Taayanagi Foundation for Electronics Science and Technology. Mr. M. Matsuura edited the LaT E X le. References [] F. M. Brown, Boolean Reasoning The logic of Boolean Equations, Kluwer Academic Publishers, Boston, 99. [] R. K. Brayton, G. D. Hachtel, C. T. McMullen, and A. L. Sangiovanni-Vincentelli, Logic Minimization Algorithms for VLSI Synthesis, Boston, MA. Kluwer Academic Publishers, 984. [3] M. Fujita and Y. Matsunaga, \Multi-level logic minimization based on minimal suort and its alication to the minimization of loo-u table tye FPGAs," ICCAD-9, [4] D. L. Dietmeyer, Logic Design of Digital Systems (Second Edition), Allyn and Bacon Inc., Boston, 978. [5] C. Halatsis and N. Gaitanis, \Irredundant normal forms and minimal deendence sets of a Boolean functions," IEEE Trans. on Comuters, Vol. C-7, No., , Nov [6] Se June Hong, \R-MINI An iterative aroach for generating minimal rules from examles," IEEE Transactions on Knowledge and Data Engineering, Vol. 9, No. 5,., Set./Oct [7] Y. Kambayashi, \Logic design of rogrammable logic arrays," IEEE Transactions on Comuters, Vol. C-8, No. 9, , Set. l979. [8] M. A. Perowsi, T. Ross, D. Gadd, J. A. Goldman, and N. Song, \Alication of ESOP minimization in machine learning and nowledge discovery," Proc. IFIP WG.5 Worsho on Alications of the Reed-Muller Exansion in Circuit Design (Reed-Muller'95),. -9. [9] M. A. Perowsi, M. Mare-Sadowsa, L. Jozwia, T. Luba, G. Grygiel, M. Nowica, R. Malvi, Z. Wang, and J. S. Zhang, \Decomosition of multile-valued relations," in ISMVL-97,. 3-8, Nova Scotia, Canada, May, 997. [] T. Sasao, \On the number of incomletely secied switching functions that are redundant with one and two variables," (draft), Feb.. [] T. Sasao, \EXMIN A simlication algorithm for exclusive-or-sum-of-roducts exressions for multile-valued inut two-valued outut functions," IEEE Transactions on Comuter-Aided Design of Integrated Circuits and Systems, Vol., No. 5,. 6-63, May 993. [] T. Sasao, Switching Theory for Logic Synthesis, Kluwer Academic Publishers,