Logical manipulations and design of tributary networks in the arithmetic spectral domain C.-H.Chang B.J. Falkowski Indexing terms: Arithmetic transform, Composite arithmetic spectra, NPN classification, Boolean functions Abstract: Formulae to represent composite arithmetic spectra of switching functions for basic logic connectives of such functions are shown. In contrast to the Walsh spectral domain, no complex dyadic convolution is involved in the calculation of composite arithmetic spectra, and the reintroduction of the transformation matrix has been avoided in the final formulae. Other important operations used in classification and optimisation of standard and tributary logical network have also been analysed in the arithmetic spectral domain. These operations include spectral decomposition, input and output negations, permutations of input variables, substitution of an input variable by a logical operation with some input variables or by the output of the function and the variable itself. Based on the introduced formulae, a new method to design tributary networks through operations on arithmetic spectra is shown. 1 Introduction In many applications of computer engineering and sci- ence, where logic functions need to be analysed or syn- thesised, it is useful to transform such functions to the corresponding spectral domain that provides various new insights into solving some important problems. The most popular transforms used in the design of logic networks are Walsh, Haar, Reed-Muller and arithmetic transforms [l-161. The renewed interest in applications of spectral methods in logic synthesis is caused by their excellent testability design and the development of efficient methods of calculating differ- ent spectra from reduced representations of both spec- tra and original functions in the form of arrays of cubes or decision diagrams [9-121. Most research work and applications of spectral techniques in logic design were done for either Walsh or Reed-Muller transforms. In this paper we will concentrate on the arithmetic 0 IEE, 1998 IEE Proceedings online no. 19982202 Paper received 7th November 1997 C.-H. Chang is with the School of Engineering, French-Singapore Institute, Nanyang Polytechnic, 12 Science Centre Road, Singapore 609098 B.J. Falkowski is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Block S1, Nanyang Avenue, Singapore 639798 transform. The latter transform is important as it allows the conversion of the analysis of Boolean and other discrete functions into the standard arithmetic operations of addition, subtraction and multiplication. It has been shown that in many cases, the resulting arithmetic expansion improves greatly the analysis of the original Boolean function [5, 14, 161. For example, important problems of satisfiability, tautology and equivalence, which are encountered in computer-aided design (CAD) processes such as synthesis, analysis and testing of digital circuits, can be easily performed in the arithmetic rather than the Boolean domain. The arith- metic transform has also been successfully employed in the testing [5, 131 and derivation of probabilistic expressions for Boolean networks [14, 171. In this paper, the methods of finding the arithmetic spectrum of AND, OR and XOR of an arbitrary number of Boolean functions directly from the spectra of composite functions have been developed. Up to now this problem was solved in the arithmetic spectral domain only for pairs of functions [14]. In contrast to the Walsh spectral domain [8, 151, we show that no complex dyadic convolution is involved in the calcula- tion of the composite arithmetic spectra, and the reintroduction of the transformation matrix can be avoided in the final formulae. Additionally, essential operations used in the classification and optimisation of the logical network are rigorously analysed in the arithmetic spectral domain. Some of these operations involve input and output negations, permutations of input variables and substitution of an input variable by a logical operation with some input variables or by the output of the function and the variable itself. The resulting algebraic expressions provide an insight into a framework for developing logic synthesis and analysis methods. As an example of such a methodology, the design of tributary networks in the arithmetic spectral domain is presented. Many of the equivalent opera- tions in the arithmetic domain appear to be simpler and more convenient for computer implementation than their spectral Walsh or Boolean counterparts, which should inspire more usage of the arithmetic transform for design-automation tools. 2 Generalised arithmetic spectrum An n- variables Boolean function F(X) F(x,, x2, ..., x?) can be expressed by a canonical generalised arithmetic expansion RW(x,, x2, ..., xJ of 2n terms [4, 5, 9-1 1, 13, 14, 16-17]: 2"-1 R~(x) = rYaZ,'aT ... 2: (1) 2=o 341 IEE Pvoc.-Comput. Digit. Tech., Vol. 145, No. 5, September 1998
Transcript
Logical manipulations and design of tributary networks in the arithmetic spectral domain
Abstract: Formulae to represent composite arithmetic spectra of switching functions for basic logic connectives of such functions are shown. In contrast to the Walsh spectral domain, no complex dyadic convolution is involved in the calculation of composite arithmetic spectra, and the reintroduction of the transformation matrix has been avoided in the final formulae. Other important operations used in classification and optimisation of standard and tributary logical network have also been analysed in the arithmetic spectral domain. These operations include spectral decomposition, input and output negations, permutations of input variables, substitution of an input variable by a logical operation with some input variables or by the output of the function and the variable itself. Based on the introduced formulae, a new method to design tributary networks through operations on arithmetic spectra is shown.
1 Introduction
In many applications of computer engineering and sci- ence, where logic functions need to be analysed or syn- thesised, it is useful to transform such functions to the corresponding spectral domain that provides various new insights into solving some important problems. The most popular transforms used in the design of logic networks are Walsh, Haar, Reed-Muller and arithmetic transforms [l-161. The renewed interest in applications of spectral methods in logic synthesis is caused by their excellent testability design and the development of efficient methods of calculating differ- ent spectra from reduced representations of both spec- tra and original functions in the form of arrays of cubes or decision diagrams [9-121. Most research work and applications of spectral techniques in logic design were done for either Walsh or Reed-Muller transforms. In this paper we will concentrate on the arithmetic
0 IEE, 1998 IEE Proceedings online no. 19982202 Paper received 7th November 1997 C.-H. Chang is with the School of Engineering, French-Singapore Institute, Nanyang Polytechnic, 12 Science Centre Road, Singapore 609098 B.J. Falkowski is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Block S1, Nanyang Avenue, Singapore 639798
transform. The latter transform is important as it allows the conversion of the analysis of Boolean and other discrete functions into the standard arithmetic operations of addition, subtraction and multiplication. It has been shown that in many cases, the resulting arithmetic expansion improves greatly the analysis of the original Boolean function [5, 14, 161. For example, important problems of satisfiability, tautology and equivalence, which are encountered in computer-aided design (CAD) processes such as synthesis, analysis and testing of digital circuits, can be easily performed in the arithmetic rather than the Boolean domain. The arith- metic transform has also been successfully employed in the testing [5, 131 and derivation of probabilistic expressions for Boolean networks [14, 171.
In this paper, the methods of finding the arithmetic spectrum of AND, OR and XOR of an arbitrary number of Boolean functions directly from the spectra of composite functions have been developed. Up to now this problem was solved in the arithmetic spectral domain only for pairs of functions [14]. In contrast to the Walsh spectral domain [8, 151, we show that no complex dyadic convolution is involved in the calcula- tion of the composite arithmetic spectra, and the reintroduction of the transformation matrix can be avoided in the final formulae. Additionally, essential operations used in the classification and optimisation of the logical network are rigorously analysed in the arithmetic spectral domain. Some of these operations involve input and output negations, permutations of input variables and substitution of an input variable by a logical operation with some input variables or by the output of the function and the variable itself. The resulting algebraic expressions provide an insight into a framework for developing logic synthesis and analysis methods. As an example of such a methodology, the design of tributary networks in the arithmetic spectral domain is presented. Many of the equivalent opera- tions in the arithmetic domain appear to be simpler and more convenient for computer implementation than their spectral Walsh or Boolean counterparts, which should inspire more usage of the arithmetic transform for design-automation tools.
2 Generalised arithmetic spectrum
An n- variables Boolean function F(X) F(x,, x2, ..., x?) can be expressed by a canonical generalised arithmetic expansion RW(x,, x2, ..., xJ of 2n terms [4, 5, 9-1 1, 13, 14, 16-17]:
where iJ = 0 or 1 is the jth bit (j = I , 2, ..., n) of the binary representation of i, 2; = I and xjl = kj. Each literal xj appears in either positive (xj) or negative (zj) form, but not in both forms simultaneously throughout the expansion. The index I of the ith coefficient rIw is formed by the concatenation of those bit numbers j that satisfy $ = 1. The polarity number U, is a binary n- tuple formed by writing a 0 for a positive literal and a 1 for a negative literal. The order of the spectral coeffi- cient rIw is the number of literals present in its associ- ated product term. A generalised arithmetic spectrum RW in polarity cu is a vector composed of all 2" coeffi- cients rr"' in Hadamard ordering of the index I , i.e. R"
The generalised arithmetic transformation matrix T = [rf, r?, e , rd 3 r y , .'., r g ..." IT. of order N = 2" in polarity U is given by [l I]
n
T; = @t"2 = &d 7L @ 6 ' h - I @ * . . t u 2 63 L J I (2) 3=1
where
coj = 0 or 1 (j= 1, 2, ..., n) and conconu;l-l .. u2col = U.
The generalised arithmetic spectrum R" = [T#] Y, where Y is the truth vector in some coding. In R-cod- ing the truth vector is represented by its original values: 0 for false minterins, 1 for true minterms and the DC minterms are represented by 0.5. In S-coding, the false, true and don't care minterms are represented by 1, -1 and 0, respectively. All derivatioiis and equations valid for R-coding can be rewritten to S-coding using the fol- lowing relationship: R = &(J - 9, where J is a N x 1 column vector whose elements are defined by j , = 1, j , = 0 (1 5 i 5 N - I).
3 operations in the arithmetic spectral domain
In the following, the mathematical formulation of the composite spectra of switching functions pertinent to some logical connectives are derived from spectral data of their individual constituent functions. Definition I : The spectrum confined by the set c9 of m bound variables (m s n), denoted by RI, is a bound spectrum generated from R in such a way that all the coefficients whose subindices do not involve all the bound variables from the set @ are equal to 0, i.e.
( 3 ) where p . . denotes all the possible subindices involving either none or some but not all the bound variables, 4.. denotes all the possible subindices not involving any bound variable. When p . . = 0 or 4.. = 0, rp, = ro or, r4,, - ro, riq,, = ri, rjq,, = rj etc. The i, j , ..., 1 are subscripts corresponding to any bound variable from the set c9. Hence there are at most 2"" non zero coefficients in the bound spectrum. Tlworem 1: Let Rf and Rg be the spectra of the n-varia- ble Boolean functions F(x ) and G(X), respectively. The composite spectrum of F(x> A G(X), denoted by Rfg, is
(4) where ' A ' is a logical AND operator, N = 2" and rI;=l x,+ is the set of bound variables defined in definition 1, where x,O = 1 and xrl = xr and i, = 0 or 1 is the rth bit (Y = 1, 2, ..., n) of the binary n-tuple of i. Proof: Let f and g be the R-coded truth vector of Boolean functions F(x> and G(X), respectively. The R- coded truth vector of the composite function F((x) A
G(X) is f. g, where the symbol '.' denotes the dot prod- uct. For simplicity, f . g will be written as fg. Then, fg = DB where Df is a diagonal matrix with the elements along the leading diagonal being the minterms off. Rfg = T N D ~ = TNDf TN-'Rg where N = 2" and TU and TN-I are forward and inverse arithmetic trasformation matrices of order N.
Let PN = TNDfTN-I = TNQM For n = 1, the R-coded truth vector can be partitioned into two halves such that f = r0f1lT, then
Let Rf = [R$ R f T . Since Rfo = R$ and Rf = Rd + R{,
where Pi = Rlf. For n = 2, f = cfo f1 f2 f3 lTand
Q4 = DfT;1= [ Qa O ] Q'Z Q'Z where
IEE Proc -Comput Digit Tech, Val 145, No 5, September 1.998
of generation of the matrix PN from the vector Rf for the case N = 16 is illustrated in Fig. 1, the matrix P,, is then used to calculate R f g . Hence, RLT = [l -1 -1 1 -1 1.5 1 -1.5 -1 1 2 -2 1 -1.5 -1 1.5IT.
Theorem 2: The composite spectra of OR of m func- tions, F1 v F2 v ... v F, is given by
m RflVf2Vf3V Vf,n = Rfz - Rf*.fj
t=1 2#3
+ R f z f J f L + . . . + ( - - l )m- lRf l fZ f m
Z#J#k,Z#k
(5) where i, j , k E (1, 2, ..., m ] .
Prooj The composite spectrum of OR of two func- tions, R A v f 2 = Rfl + R f 2 - R f f 2 is given in [14]. The log-
For higher n, it follows that the order N matrix PN can be generated recursively from the column vector Rf by partitioning the resulting matrix after every iteration into 2”-* submatrices, each of order 2i, where i is the number of iterations. During each iteration, a new sub- matrix of equal dimension is generated towards the right of each original submatrix. The newly generated odd submatrices are zero matrices of order 2l-l and the newly generated even submatrices are equal to the sum- mation of their respective left and top left submatrices. Since PN is a triangular matrix and the ith column of PN is equal to Rq zxkil R f g = P N R g = Z: g;’ (zth column of PN x ith element of R g ) = R.H.S. of eqn. 4. Example 1: Let the R-coded minterm vectors of the four variable Boolean functions F(‘(X> and G(X) be: f = [ l o 0 0 1 1 0 0 0 0 1 I O 1 1 0IT, g = [ l o 1 0 0 0.5 0 1 1 0 1 0 1 0 1 0.5IT where the superscript ‘T’ denotes the matrix transpose. The zero polarity arithmetic spectra o f F a n d Gare Rf= [l -1 -1 1 0 1 0 - 1 -1 1 2 - 1 0 0 0 -1IT and RS = [l -1 0 0 -1 1.5 0 0.5 0 0 0 0 1 -1.5 0 OIT, respectively.
The spectrum of F bound by variables xIx3 can be calculated from eqn. 3 as follows:
N submatrices of order 1 N I 2 submatrices of order 2
....
Fig. 1
IEE Psoc -Camput Digit Tech, Vol 145 N o 5, September 1998
Genesation o f the matrix PNfrom the vector Rf
349
ical OR of the R-coded truth vectors f i and f2 can be expressed by arithmetic addition and dot product between vectors as follows: f i v f2 = f i + f i -j&
Assuming the expression
+ (- 1 ) P - l f l .f2 . . . f p
is true for any p 2 2, then for p + 1,
i=l \1=1 / \z=1 /
+ . * . + (-l)"-'fifi . . ' f p
- . f ~ + l cfi -c.fifj + f i f J f k ( p i=l i # j + J # k i # k
+ . . ' + (-1)"-'f1f2.. ' f P
P+ 1
= Cfi - CfZfJ + c f z f j f k i=l i#j i f j # k , i # k
+ . . ' + (-1)"f1f:! * . . fP+l
Hence, by mathematical induction, the above expres- sion is true for all p t 2. Multiplying both sides of the above expression by the arithmetic transformation matrix, eqn. 5 is obtained. Theorem 3: The composite spectrum Rfl@hO-@fm of XOR of y?.z functions, F, 0 F2 0 ... 0 Fm is given by:
R f I @ f Z @ f 3 @ " ' @ . f m
m Rfz - 2 Rf.-fj + 4 R f i f i f k
i= 1 i#j i f j f k ,a# k
+ . . . + ( -2)m-1Rfl fZ- . . f~n
(6) where i, j , k E { 1, 2, ..., m}. Prooj! Consider the XOR of two functions, f i 0 f2-= Multiplying both sides of the equation by the arithme- tic _transformation matrix TN, Rf@f2 = RhA + Rhfi. Rf1f2 = PN x Rf2 where P, is the matrix described in the proof of theorem 1. The spectrum of the comple- ment function Rf2 is equal to J - Rf2, where J is an N x 1 column vector whose elements are defined by
= P,(J - Rh). .Since PNJ = first column of P, = Rfl,
hf2 v h5 = hf2 + h.5 - 6mf-25) = fl3 + Lfi .
jo 1, jj = 0 (1 5 i 5 N - 1) [9, 111. Thus, Rflf2
Rfd1 = Rf1 - Rjlf2. Similarly, Rf2fi = Rf2 - Rfh. Rfl@h = Rfl - Rfh + Rf2 - @if2 = Rfl + Rf2 - 2Rhf2, By repeating the above process for each additional function until m functions are XORed together, eqn. 6 is obtained Theorem 4: The composite spectra Rf@f2, RflVf2 and Rff2 are related by
(7 ) Rfl@fZ = RflvfZ - RfIfZ
Proof: Trivial
350
Example 2: Consider the four variable Boolean func- tions F(x> and G(X) from example 1. By theorem 2, Rfvg= Rf + Rg- Rfg= [I -1 0 0 0 1-1 I O 0 0 I O 0 1 -2.5IT. By theorem 4, Rf@g = Rfvg - Rfg = [0 0 I -1 1 -0.5 -2 2.5 1 -1 -2 3 -1 1.5 2 -4IT. Since don't care minterms are coded as an arithmetic average between the logical true and false values [3, 9-11], for incom- pletely specified functions, the resulting composite spectra are the spectra of an arithmetic average of com- posite functions generated from all possible assign- ments of don't cares in each individual function. From the result of Rfg obtained in example 1, it can be veri- fied that Rfvg - Rfg = R@g.
In what follows, the essential operations used in prin- cipal algebraic classification methods such as negation of input variables, permutation of input variables and substitution of an input variable by logical operation of this variable with an other input variable are described in the arithmetic spectral domain.
Theorem 5: Negating m (m 5 n) input variables xi, xj, .,. of an n-variable Boolean function F(X) is equivalent to changing the original spectrum in polarity zero to the spectrum in polarity U*, where each bit q* in the binary representation U* for r = 1, 2, ..., n is equal to 1 if the corresponding input variable x, is negated and 0 otherwise. The spectrum of the function Rf<w*> is related to the spectrum Rf as follows:
where g.. denotes all possible subindices having none of the negated variables, ig.. and jq.. denote all possible subindices having some but not all the negated varia- bles, ijq.. denotes all possible subindices having all the negated variables. When 4.. = 0, rq,, = ro, rig,, = ri, rj+ = rj, rvq,, = rv, Qv denotes the set of all the negated varia- bles and Qj denotes the set of all the negated variables without those that appear in the subindices of the coef- ficient being calculated. rIIQ, denotes the spectral coeffi- cient rI bound by the set @.
When only one input variable xi is negated, the cal- culation of the spectrum R@*> is reduced to
Alternatively, the spectrum Rf<,*' can also be gener- ated by multiplying the dyadic convolution of the min- term vector Lo, by the transpose of the zero-polarity arithmetic transformation matrix TNT, where Lon is obtained from the minterm vector f by swapping mint- erms mi and miodw*. The symbol 'Od) refers to dyadic addition [5, 111. Since TNT is an upper triangular matrix, the resulting spectral coefficients can be expressed as
1.
mjedw* x aij (10) $4 = I
j=0
where I is the index of the coefficient being calculated and av is the entry in row i, column j of TN. The row number i is also the decimal equivalent of the posi- tional notation of the index I.
However, eqn. 10 still contains some unnecesary operations owing to imbedded zero entries in the upper triangular section of TN A more compact expression is possible by the following corollary:
IEE Puoc.-Comput. Digit. Tech., Vol. 145, No 5, September 1998
3 i O I
- - c (-1)C:=,Jr@I, m3 (11) 3 @ d m * c O I
where mJ is the jth R-coded minterm and j CO Z denotes that j is the zero subnumber of the index I. For integers i , j E (0, 1, ..., 2“ - l} CO i if and only if j, = 0 or ir = 1 for each r = 1, 2, ..., n [12]. Example 3: From example 1, Rf = [I -1 -1 1 0 1 0 -1 -1 1 2 -1 0 0 0 -1IT. The spectrum Rf<’> resulting from negating input variables x1 and x3 can be calculated by either theorem 5 or corollary 1. A sample calculation of some selected coefficients by both methods is shown below.
The coefficients whose subindices do not have the negated variables are ro, r2, r4, and ~24. The coefficients whose subindices have some but not all the negated variables are rl, r3, r12, ~14, ~23, Y ~ ~ , r124 and r234. The coefficients whose subindices have all the negated vari- ables are r I3 , ~ 1 2 3 , ~ 1 3 4 and r1234. Using eqn. 8, the coef- ficients ~24, ~ 1 2 4 and ~ 1 2 3 from each of the above categories are calculated as follows: r;j’ = r1234/x x3 = r24 + Ti24 + r234 + r1234 = 2 - 1 + o - I = 0; r&z = -r12341x1 = -(r,24 + Y1234) = -(-1-1) = 2; ~72; = ~ 1 2 3 -1.
If eqn. 11 is used to calculate these three coefficients, we have r$> = m5 - m7 - m13 + mI5 = 1 - 0 - 1 + 0 = 0; r;24> = -m5 + m4 + ml - mg + mI3 - m12 - mi5 + mi4
The remaining coefficients are calculated in a similar way using either eqn. 8 or eqn. 11 and the complete spectrum, R<5> = [l, 0, -1, 0, -1, 1, I , -1, 0, -1, 0, 2, 0,
Theorem 6: Permutation of any pair of input variables x, - xJ, i r j , is equivalent to the interchange of NI4 pairs of spectral coefficients as follows:
where 4.. denotes subindices of coefficients not having the variables being interchanged, i.e. xi and xj. When 4. . = 0, riq,, = Ti and rjq., = rj. All coefficients that do not have the variables being interchanged and all the coeffi- cients rgq,, having both such variables remain the same.
Table 1 summarises the changes in the arithmetic spectrum for the substitution of an input variable xi with either a variable xj (j # i) or with some logical operations between xi and xj. In Table 1, the spectral coefficients without and with the superscript ‘*’ repre- sent the spectral coefficients of the function before and after the substitution, respectively, and 4.. denotes all possible subindices that do not have both variables xi and xi. From the Table, it is noticed that the function obtained by substituting xi with xi A xi is the same as
that obtained by substituting xi with xi v xj. In addi- tion, the spectra obtained by substituting xi with x; 0 xi and by substituting xi with xi v xj differ only in those coefficients whose indices have both xi and x. In each of these substitutions, if the negated variable gj instead of xi is used, theorem 5 is applied to obtain the spec- trum of the function F(xl, x2, ..., xi, xj, ...) prior to the application of the corresponding formula provided in Table 1. In general, each substitution introduces several don’t cares in the truth vector of the function. NI2 don’t care minterms are introduced for the cases of replacement of xi with xj,or with x i 0 xj, while NI4 don’t care minterms are introduced for the cases of replacement of xi with xi v xi or with xi A xj.
4 Spectral decomposition and olperations between input variables and the output
In disjunctive decomposition and synthesis of multiple level logical networks [2, 3, 161, post processors are usually extracted by combining an output with one or more input variables, as shown in Fig. 2. Therefore we shall investigate the arithmetic spectrum of the function obtained under such operations. We consider not only the positive but also the negated variables.
=;d function core P l *n
function U Fig.2 variables
Combining the output of u core functionf with one or more input
Theorem 7: The relationships expressing the 2*-’ gener- alised arithmetic coefficients of the cofactor FZi or Fx, and the 2” generalised arithmetic coefficients of the original n-variable Boolean function F(x) can be derived as follows:
r;., = rj.,, ri.. = rj.. + rij.. , ry,. = rj.. if wi = 1 (13)
where mi is the ith bit of the polarity number o, r) , , and r‘),, are the corresponding coefficients of the cofactors Fxi, and Fxi, respectively and j . . denotes all possible subindices not involving i, including j.. = 0. When j . . = 0 r . . = r .
ry,, = rj.. + rij. if wi = O
> 8.. 1
Table 1: Arithmetic spectra for the substitution xi with xi and with xi op xi where op E {A, v, @I and j # i
Replacing x i by ro* rq,,* r: riq..* ri” rjq..* riy rijq..
Proof: Consider the generalised arithmetic transforma- tion matrix TGW> in eqn. 2. If the most significant bit of the polarity number 0, = 0, the generalised arithme- tic spectrum R is given by:
where M' and M" are the minterm vectors of the cofac- tors Fin and F,, respectively, x, is the most significant variable, R' and R" are arithmetic spectra of the cofac- tors FZn and FXn, respectively, wt = <0,_10n02 ... col>. It is obvious that the subindices of the coefficients in R' and R" do not involve a.
Since the generalised arithmetic coefficients are arranged in natural ordering, the subindices of the upper 2n-1 elements of R do not involve n. Thus, rl.. = rti,,. The lower 2+' elements of the R are coefficients whose subindices involve n, i.e. rnj.. = rt; - r ) , , . Adding PI;., and rnj,,, we obtain Y " ~ , , = rj.. + rnj,. .
Similarly, when mn = 1, we have
Hence, Y],, = rnj,, and rnj,. = r;,, - r'),,. Adding rj., and rnj,,, we obtain a),, = ai.. + a, ,,,.
Owing to the nature of natural ordering, interchang- ing any input variables .xi and xj, i z j requires the per- mutation of 2n-2 pairs of coefficients, such that ri - vi, rili - rll,, r;k[ - rjk[ and so on. All coefficients whose subindices involve both i and j or do not involve both i and j are not affected. By performing all the necessary permutations caused by interchanging variables x, and xi, such that the minterm vector M' represents the cofactor FZL and M the cofactor FsL and replacing the spectrum R with the one that has the coefficient values permuted according to the rule described above, eqn. 13 is derived with subscripts n of the above equa- tions replaced by i. Theorem 8: The arithmetic spectrum of the function R f A X i A X j A X / c A ... generated from the Boolean product between the output of the boolean function F(x) and m (m s n) of its input variables xi, xi, xk, ... etc., is given by
All coefficients whose subindices do not have any of the selected variables xl, xi, xk, ... etc., are equal to 0. Proof: Consider an n-variable function G(X) = xi A xj A
xk A .,. formed as the conjunction of m (m 5 n) input variables xi, xi, xi<, ... etc. Only one coefficient of Rg whose index is formed by the concatenation of the sub- scripts of all the dependent variables is equal to 1 and all the remaing coefficients are equal to 0. By theorem 1,
(14) R f A X , A X , A X k A . . . -
- Rflz,e,zk...
RfAg = R ~ T ~ + Rfl rf + Rf Is2r$ 0 x1
352
+ RJ /,,,,rp2 + . . . + ~f I 3: 1 x2.. .z ,L e a . . .n - - RS l x . z , z l c . . .
Corollary 2: The arithmetic spectrum R f A X i A z P z k " - of the function generated from the Boolean product between the output of the Boolean function F(x> and the negation of m (m 5 n) of its input variables xi, xj, xi<, ... etc., is given by
f A Z A Z , A Z k A . . . - - (--l)Qr:..
(15)
rfAZ,AZ,AZkA.-. - 4.. - T i . . ; rp. .q.:
where p.. denotes all possible subindices having only a (a .s nz) selected variables, 4.. denotes all possible subindices that do not have any of the selected varia- bles. When 4.. = 0, rq, = ro, r,, = rp,.. Theorem 9: The arithmetic spectrum of the function RfVXiVX,VXkV ... generated from the Boolean sum between the output of the Boolean function F(X) and m (wz 5 n) of its input variables xi, xi, xk, ... etc., is given by
,fVx,Vx,vxkV... -
fVX,VXjVXkV... - f
Tp..,.. fVXiVX, VXkV... = (-I)",{,.
4 . . - T i . .
rp.. - ( - l ) " ( T o - 1)
for vq.. f o (16)
where p . . denotes all possible subindices having only a (a 5 m) selected variables, 4.. denotes all possible subindices that do not have any of the selected varia- bles. When 4.. = 0, r+ = yo, rp,,4 = r *,,. Proof: The spectrum of the complement function R I is equal to J - Rf. Applying eqn. 15 to the complement function, we have
where ijk.. denotes the subindex of exactly m selected variables, 4.. denotes all possible subindices that do not have any of the selected variables. Theorem 10: The arithmetic spectrum of the function Rf@x@x~ox~o- generated from the Boolean antivalence (XOR) between the output of the Boolean function F(X) and m (m I n) of its input variables x i , x,, xk, ... etc., is given by
(18) where p . . denotes all possible subindices having only a (a 5 m) selected variables, 4 . . denotes all possible subindices that do not have any of the selected varia- bles, and @e denotes the set of subindices having exactly e selected variables from those of p.. . When 4 . . - 0, r4,, = ro, rp,.9 = rp,,, rsq,, = Y,. The value of the con- stant C depends on whether the subindices of the coef- ficients have the non selected variables. C = (-2)a-1 if q.. = 0 and C = 0 if q.. z 0. When s E Qo, r, = ro and rsq.. = rq.:
Proof: Consider an n-variable function G(X) = xi 0 xi 0 xk 0 ... formed as the conjunction of m (m s n) input variables xi, xj, Xk, ... etc. The coefficients o f Rg whose subindices have only a (a 5 m) selected variables are equal to (-2)"' and the remaining coefficients are all equal to 0.
-
Using theorem 1, we have a
e=O
where s E Be. Since r,g = (-2)e-' and
The term that has all the selected variables $ for e = a cancels out in the summation.
By theorem 3, Rf@y@x~@xk@ = Rf + Rg - 2Rh. Since all coefficients r4 and rp of Rg, whose subindices have any o f the non selected variables are equal to 0, ~ q f o x , ~ ~ ~ ~ ~ k ~ = rf Consider the case where 4. . = 0, Rg = (-2)a-l = C . Hq ence, .
s @Xx ex, @Xk CE TP
@-I r 1
Consider the case where 4 . . * 0, C = Rg = 0. Hence, f @Xt @XI @Xk@
r p . . 4..
IEE Proc.-Comput. Digit Tech., Vol. 145, No. 5, September 1998
r 1
Since 9! :9 . . = %Da rsq.. f
(-1)a 2 2 0 - e r!q.. r f c B X ; c B x , @ X 1 ; @ - =
S E 9 e 1 p..q.. e=O
Corollary 4: The arithmetic spectrum of the function R f @ x @ z ~ @ x k O - generated from the Boolean antivalence (XOR) between the output of the Boolean function F(x) and the negation of m (m s n) of its input varia- bles xi, xj, xk, ... etc., is given by:
R.f@zt@z,@zk@. . . - - J - R ~ @ X ~ @ X J @ Z ~ @ . . - if is odd
R f @ % c @ % ~ @ E k @ ' . ' - - R f @ X z @ X , @ X k @ " - if m is even
where J is an N x 1 column vector whose elements are defined byjo = 1, ji = 0 (1 I i I N - 1).
In general, if the input variables that are logically connected with the output of the function consist of both non-negated and negated variables, the spectrum of the function obtained under such operations is gen- erated in two phases: first calculate the spectrum of the function due to the set of non negated variables fol- lowed by the spectrum of the new function with logical operations applied to it and the set of negated varia- bles. Additionally, the DeMorgan theorem can be used in conjunction with the above theorems and corollaries to calculate the spectrum of the function if NAND or NOR gates are used as the logical connections. Example 4: The same Boolean function F(X, from example 1 is used again here. We will calculate the arithmetic spectrum of the function obtained under the following operations: the Boolean product, sum and antivalence among the input variables xI, x3 and the output of F. For simplicity, let us i.dentify the coeffi- cients of the new function by the superscript * and the coefficients of the original function I: are denoted by rI without any superscript.
The R-coded minterm vector o f f / \ x1 A x3 is [0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 OlT. Pre-multiplying this vector by T16 yields the spectrum [0 0 0 0 0 1 0 -1 0 0 0 0 0 0 0 OlT which is the same as that calculated using theo- rem 8.
r4 = I x -1 = -I, r1234 = (-112 ~ 2 4 = 1 x 2 = 2. The R-coded minterm vector off v x1 v x3 is [I 1 0 1
1 1 1 1 0 1 1 1 1 1 1 1IT Pre-multiplying this vector by TI6 yields the spectrum [1 0 -1 1 0 0 1 -1 -1 1 2 -2 1 -1 -2 2IT which is the same as that calculated by theo- rem 9.
For q.. 0, C = (-2)"'. rl* = (-1)'(2r0 + r l ) + 1 = -1 x (2 x 1 -1) + 1 = 0, r3* = (-1)'(2ro + r3) + 1 = -1 x (2 x 1 + 0) + 1 = -1 r13* = (-1)~[2~ ro + 2(rl + r3) + r13] + (-2) = 4 x 1 + 2 x (-1 + 0) + 1-2 = 1. For q,, z 0, C = 0, rI2* = (-1)l(2r2 + rlz) = -1 x (2 x -1 + 1) = 1, r14* = (- 1)'(2r4 + rI4) = -1 x (2 x -I + I> = I, ~23* = (-11'(2rz + r23) = -1 x (2 x -1 + 0) = 2, r34* = (-1)l(2r4 + r34) = -1 x (2 x -1 + 0) =,2, ~ 1 2 4 * = (-1)l(2~24 + ~ 1 2 4 ) = -1 x (2 x 2 - 1) = -3, r234 = (-1)1(2r24 + r234) = -1 x (2 x 2 + 0 ) = -4, rlz3* = (-I)2[22r2 + ?(rl2 + rZ3) + rlZ3] = 4 x -I + 2 x (1 + 0) - 1 = -3, r134 = (-1)2[22v4 + :(~14 + r34) + r134] = 4 x -1 + 2 x (1 + 0) + 0 = -2, TI234 = (-1)2[22r24 + 2(r124 + r234) + r1234] = 4 x 2 + 2 x (-1 + 0 ) - 1 = 5.
The R-coded minterm vector off 0 x1 0 x3 is [l 1 0 1 0 1 1 0 0 1 1 0 1 1 0 0IT. Pre-multiplying this vector by TI6 yields the spectrum [l 0 -1 1 -1 1 2 -3 -1 1 2 -3 2 -2 -4 SIT which is the same as that calculated using theorem 10.
'n-1 'n
Fig.3 Tributary switching network
5 Design of tributary and cascade switching networks
In this Section, we show a typical application of employing arithmetic spectral operations to the synthe- sis of a special form of cellular arrays known as the Matra cascade [l, 2, 18, 191. A Maitra cascade is a one- dimensional cellular array of two-inputs and one out- put cells. By limiting each gate's fan-out to unity, smaller transistors can be used to maximise silicon area
utilisation. The regular structure conserves physical space further by using direct short interconnections between cells. For simplicity, we will consider only the irredundant cascade in which every input is driven by a distinct Boolean variable, as shown in Fig. 3. Such a cascade network is known in the literature as a tribu- tary switching network (TRIB) [l, 181. Later on, addi- tional conditions are included to yield an irredundant cascade with increasing dimensions for some non TRIB realisable networks.
Not all switching functions are TRIB realisable. Prior to the conduction of the TRIB-realisable test, the function under the test must be nondegenerate, i.e. it must not contain any vacuous variable. The detection and elimination of vacuous variables from the arithme- tic spectrum of a degenerate Boolean function have already been discussed in [9] and will not be repeated here. Given a nondegenerate Boolean function F and a logical connective op, testing of TRIB realisation and extraction of logic gates are carried out concurrently by examining if F = xi op G where x i E Support(F) and Support(G) = Support(q - kj, (see Fig. 4).
The conditions in Fig. 4 can be proved by Shannon and Davio decompositions [6]. By expressing the arith- metic spectrum through the edge valued binary deci- sion diagram (EVBDD) [9], the extraction conditions can be easily inspected and the residue function can be generated as the side effect of the successful test. When compared with the extraction algorithm for the Walsh spectrum given in [I], our method is much simpler and more elegant. If at any stage of the synthesis, the func- tion or its residue function fails to meet all the above three requirements, the given function is not TRIB realisable. This situation can be resolved by the intro- duction of an additional dimension into the cellular array. Although the resulting network is no longer a tributary, it remains cellular and well structured. Fig. 5 shows the extraction of F1 = (xi v G) A E and F2 = (ki A G) v E. The variable extracted xi is unate and the conditions for which the extraction is possible are given as follows:
If F,, A Fx, = Fxi, the extraction is either F = (xi A G) v E or F = (xi v G) A E where Fii and Fxi are the cofactors of F around the variable Xi and xi, respec- tively, G = F,, - Fii and E = FZi. If FZi A FXi = Fxi, the extraction is either F = (Xi A (2) v E or F = (Xi v G) A
E, where G = FZi - Fxj, and E = F,,. The arithmetic spectrum of the consensus of F with respect to xi, i.e. FZi A Fxi can be obtained from the composite spectrum of the product of cofactors around xi by eqn. 5 of
TR1B_SYJWF)
if (F;l = 0) { I* test for op = A */ if (F;, = 0) G = F=, and F = x, A G is extracted; if (Fx, = 0) G = Fi, and F =FI A G is extracted;
if (Fxi = 1) G = FEi and F = x, v G is extracted; if (Fzi = 1) G = Fxl and F = ZI v G is extracted;
either G = Fz, andF =x,@ G or G = Fxl and F = f, CD G is extracted;
theorem 1. In terms of EVBDD representation, the consensus can be generated easily by an apply multipli- cation operation on the EVBDDs of the cofactors [9].
X i E ii E
a b Fig.5 array
Extraction of cells for a two-dimensionul irredundant cellular
If the function or cofactor is binate in all its support variables, either positive or negative Davio decomposi- tion is used as the last resort. If positive Davio decom- position is used, the function F = x, A G 0 E is extracted where G = FZI 0 FX1 and E = F,,, whereas for negative Davio decomposition, F = X i A G 0 E is extracted where G = FTl 0 FXl and E = FZI. The config- uration of this type of cascade is identical to Fig. 5b if the v operator is replaced by the 0 operator. The cas- cade that is perpendicularly disposed to the cascade of 'O' operators is called the collecting cascade in [6] and the complete structure formed this way resembles the cutpoint cellular arrays which have the advantage of being highly programmable [6]. It should be noted that the two conditions based on Davio decomposition are new and they were not examined in [l]. Intuitively, if the same operator is used in two adjacent cells in a cas- cade, they may be combined to form a single cell with an additional input. Thus if mixed cells with different fan-in are allowed, the operator of the preceding cell is given the higher priority in the TRIB realisable test of TRlB-Synth.
Table 2: Zero polarity arithmetic spectrum of the six var- iable function F
ro r1 0 0
0 0
0 0
0 0
0 0
0 0
0 0
0 0
r4 r14
r5 r15
r45 r145
r6 r16
r46 r146
%6 r156
r456 r1456
r2 0
0
0
0
r24
r2 5
r245
r2 6 0
r246 0
0
0
r256
r2456
r12 0
0 r124
r125 0
0
0
0
r1245
rl 26
r1246
r1256 0
0 r12456
r3 0
r34 1
r3 5 1
r345 -2
0 r36
r346 -1
r356 -1
r3456 2
r13 0
-1 rl 34
rl 35 0
2
0
1
r1345
rl 36
r1346
r1356 0
-2 r13456
r23 1
r234 -1
r235 -1
r2345 2
r236 -1
r2346 1
r2356 1
r23456 -2
r123 0
r1234 1
r1235 0
-2 r12345
r1236 0
r12346 -1
rl 2356 0
2 r123456
Table 3: Zero polarity arithmetic spectrum of the five variable function G,
r0 r'l r2 r12 r4 r14 0 0 1 0 1 -1
1 0 -1 0 -2 2
0 0 - 1 0 - 1 1
-1 0 1 0 2 -2
r5 r15 r25 r125 r45 r145
r6 r16 r26 r126 r46 r146
'56 r156 r256 r1256 r456 r1456
r24 r124 -1 1
2 -2
1 -1
r245 r1245
r246 r1246
r2456 r12456 -2 2
IEE Puoc.-Comput. Digit. Tech., Vo!. 145, No. 5, September 1998
Example 5 : Consider the six variable Boolean function F(xl, x2, x3, x4, .x5, xg) = Zm(l0, 12, 24, 26, 28, 30, 42, 46, 56, 58, 60, 62) from [l, 181. We will show that the same TRIB circuit can be designed in the arithmetic spectral domain. The arithmetic spectrum R f of this function is shown in Table 2. Since all coefficients that do not contain 3 as their subindex are equal to 0, theo- rem 8 suggests the extraction of the variable x3 from F by logical AND operation, Using eqn. 13, the spectral decomposition around x3 produces Rp = [0 0 ... 0IT corresponding to the case of F,, = 0. From TRIB- Synth(F), GI = FXz and the cell F = x3 A G1 is extracted. The arithmetic spectrum Rg1 of the subfunction GI is calculated using eqn. 13 and shown in Table 3 . Decom- posing the spectrum R g l around x6 using eqn. 13 pro- duces Rgl" = [0 0 ... OlT corresponding to the case of GIxh = 0. From TRIBSynth(G1), G2 =: GI,, and the cell GI = %6 A G2 is extracted. The spectrum R g 2 calculated using eqn. 13 is shown in Table 4. Comparing the spec- tra Rg1 and Rg2 , it can be verified that the relationship given by eqn. 15 is fulfilled. Decomposing the spectrum R g 2 around x2, we have R g 2 " = J8 = [l 0 0 ... OlT corresponding to the case of G2x2 =: 1. From TRIB- Synth(G2), G, = G2x2 and the cell G, = x2 v G3 is extracted. The spectrum R g 3 is calculated to be [ro rl r4 r14 r5 r15 r45 rI& = [0 0 1 -1 1 0 -2 2IT. Decomposing Rg3 around xs, we have Rg3' = J4 - 1V3" . From TRIB- Synth(G3), G4 = G3x8 and the cell G3 = x5 0 G4 is extracted. Alternatively, we can also select G4 = G3x8 and extract the cell G3 = X 5 0 G+ We choose the former case, which leads to Rg4 = [yo y1 y4 rl4IT = [0 0 1 -1IT. Decomposing around x4, we have Rg4' = [0 0IT corresponding to G,,, = 0. From TIRIB-Synth(G,), G5 = G4X4 and the cell G4 = x4 A G5 is extracted. Since R g s
= [ro.rllT = [l -1IT, G5 = X I . The same TRIB circuit as that in [l] is shown in Fig. 6.
Table 4: Zero polarity arithmetic spectrum of the four variable function G2
Formulae for composite arithmetic spectra of logical manipulation on switching functions have been pre- sented in terms of the original spectra for individual functions making up the composition. In addition, the operations used in Boolean classification [7] have been carried out directly in arithmetic spectral domain with- out reconverting to the binary domain. The results are further extended to allow a complete analysis and in depth treatment of logical operations involving both the input variables and the output. Although the the- ory is derived for basic AND, OR and XOR logical operators, other logical operators such as NOR, NAND and XNOR have very similar relationships and
355
their derivations can be easily obtained from the given expression as immediate corollaries by combining the de Morgan theorem and the properties of the comple- mented spectrum. Arithmetic spectral disjoint decom- position is also presented. By considering the values of the minterms as an arithmetic average of all possible assignments of don’t cares, the presented formulae for composite spectra are valid for incompletely specified functions as well. Certainly, this is an advantage com- pared with the treatment of the composite Walsh spec- tra presented in [X, 151 that did not consider such cases at all. Practical application of developed theory for synthesis of tributary switching networks through oper- ations on the arithmetic spectrum is also shown. The resulting tributary network is obtained through a smaller number of steps than the same network design through the Walsh spectrum [l]. A major benefit of the work presented here is that other approaches to logic synthesis and minimisation, for example cube calculus, set theory and spectral techniques using other trans- forms can be translated into arithmetic spectral meth- ods in lower computational complexity with the availability of such formulae. As more experiences are gained in spectral methods, the formulation of arithme- tic spectral domain operations is definitely necessary and their advantages in CAD situations will be increas- ingly explored.
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