New Generalizations of Shannon Decomposition

Pawel KerntopfInstitute of Computer Science, Department of Electronics and Information TechnologyWarsaw University of Technology, ul. Nowowiejska 15/19, 00-665 Warsaw, Poland

e-mail:pke@ii.pw.edu.pl

ABSTRACT: New types of functional decompositions are presented which we call function-drivendecompositions. Namely, it is shown in the paper that in the well-known Shannon formula it is possible toreplace a variable by an arbitrary function having the property of self-duality with respect to a variable ofthe decomposed function. In this way a generalization of recently studied linear decompositions (used forconstructing Linearly Transformed Binary Decision Diagrams) is obtained. Further extensions of function-driven decompositions are also mentioned. These extensions can be defined by combining the function-driven decompositions with other known decompositions, e.g. Reed-Muller (Davio) decompositions. It isalso shown how to extend function-driven decompositions to multiple-valued functions.

1. Introduction

Binary Decision Diagrams (BDDs) [5] is a popular data structure for representation andmanipulation of Boolean functions. They are based on the so-called Shannon decomposition of aBoolean function f:

f = x' f 0 + x f 1,where x is a variable of f, x' is the negation of x, f 0 and f 1 are cofactors of f obtained by replacingx by constants, respectively 0 and 1. During the last decade many variants of BDDs wereproposed in search for a better representation of Boolean functions (e.g., see [10, 29, 37]). Thekey idea behind proposing these variants is relaxing the limitations that have been imposed onBDDs in their definition.

Two well-known decompositions used for defining efficient variants of decision diagrams arebased on the following modifications of Shannon decomposition:

f = f 0 ⊕ x f 2 - positive Reed-Muller (Davio) decompositionf = f 1 ⊕ x’ f 2 - negative Reed-Muller (Davio) decomposition,

where f 2 = f 0 ⊕ f 1. On the basis of these decompositions Functional Decision Diagrams,Kronecker Functional Decision Diagrams and Pseudo Kronecker Functional Decision Diagramshave been introduced [20, 11, 1, 2]. On the basis of subsequent modifications of Reed-Muller(Davio) decompositions many variants of decision diagrams have been developed, especially forefficient verification of arithmetic devices [10, 29, 37], e.g., Edge-Valued Decision Diagrams(EVBDDs), Multi-Terminal Binary Decision Diagrams (MTBDDs), Binary Moment Diagrams(BMDs), Multiplicative Binary Moment Diagrams (*BMDs), Kronecker Multiplicative BinaryMoment Diagrams (K*BMDs), Hybrid Decision Diagrams (HDDs), Linearly-IndependentDecision Diagrams (LIDDs) [30] as well as their word-level versions. Recently many researchershave studied properties and succinctness of linearly transformed BDDs (LTBDDs) [12, 14-16, 26,28, 32] based on linear decompositions:

f = g'G + gH,where g is a linear function of some variables of f.

In [21] the author has proposed new generalizations of Shannon decomposition. They areobtained by extending linear decompositions to the case where f may be a nonlinear function.Namely, for an arbitrary function f there exists an expansion

f = X'G + XH,where X, G and H are Boolean functions whose supports are subsets of the support of f, such thatX uniquely determines G and H in the above decomposition. The formula f = g’ f 0g + g f 1g (1)was studied at the end of 1980s in the context of research on verification of sequential machines[7-9, 35], however, it was observed only that for arbitrary f and g the equation (1) may have morethan one solution with respect to the unknowns f 0g and f 1g.

In this paper we give, for the first time, a precise characterization of the class of functions g forwhich the equation (1) has the unique solution for arbitrary function f. Due to the uniqueness ofsolutions for f 0

g and f 1g it is possible to use this decomposition as a basis for defining new types

of decision diagrams (we call them function-driven decision diagrams) in the same manner asBDDs. These diagrams are also canonical representations of Boolean functions and standardBDD packages may be used for manipulating them. The present paper is an extended version of[21] as well as it contains new notions and results.

The paper is organized as follows. In Section 2 three classes of Boolean functions needed forfurther presentation are defined. Section 3 presents the new generalizations of Shannondecomposition obtained on the basis of using additional functions. Then a completecharacterization of these additional functions is given and function-driven decision diagramsbased on the new decomposition are defined. Finally, conclusions are formulated.

2. Basic notions

We begin with definitions of some classes of Boolean functions.

Definition 1 A Boolean function in n variables f (x1, x2, ... , xn) is linear iff it can be defined byan expression a0 ⊕ a1x1 ⊕ ... ⊕ anxn, where ⊕ is an exclusive or operation (EXOR) and thecoefficients ai can take the values 0 or 1. Otherwise, f is nonlinear.

Definition 2 A Boolean function f (x1, x2, ... , xn) = f(X) is balanced iff it has an equal number oftrue [f (X) = 1] and false [f (X) = 0] vectors. Otherwise, f is unbalanced.

Balanced Boolean functions have been considered in many papers on switching theory, logicdesign (in these fields they have also been called “neutral functions” [17, 36]), cryptography andcomplexity theory. They also play a substantial role in the areas of reversible computing (e.g.,see [22, 23]) and transformations of decision diagrams [3, 27, 24, 25].

There are altogether 70 balanced functions of 3 variables belonging to 10 pn-equivalence classes(under permutation of variables and/or negation of functions). Representatives of these classesare shown in Table 1 (for convenience, in positive polarity Reed-Muller forms). Note that alllinear functions are balanced (classes C1, C2, C4).

TABLE 1PN-EQUIVALENCE CLASSES OF 3-VARIABLE BALANCED FUNCTIONS

Class Number offunctions

Representative

C1 6 xC2 6 x ⊕ yC3 6 x ⊕ yzC4 2 x ⊕ y ⊕ zC5 12 x ⊕ y ⊕ xzC6 12 x ⊕ xy ⊕ yzC7 2 xy ⊕ xz ⊕ yzC8 6 x ⊕ y ⊕ z ⊕ xyC9 12 x ⊕ y ⊕ xy ⊕ xz

C10 6 x ⊕ y ⊕ xy ⊕ xz ⊕ yz

By analogy to the notion of self-dual Boolean functions [31] which satisfy the equationf (x1, ... , xi-1, xi, xi+1, ... , xn) = f’ (x1’, ... , xi-1’, xi’, xi+1’, ... , xn’)

we define the following class of functions:

Definition 3 A Boolean function f (x1, x2, ... , xn) is called self-dual with respect to a variable xi

(in short xi-SD), i = 1,2,...,n, iff f (x1, ... , xi-1, xi, xi+1, ... , xn) = f’ (x1, ... , xi-1, xi’, xi+1, ... , xn), i.e.negating the variable xi is equivalent to negating the function itself.

An xi-SD function f (x1, ... , xi-1, xi, xi+1, ... , xn) is also called linear with respect to a variable xi. Itmay be written as follows

f (x1, ... , xi-1, xi, xi+1, ... , xn) = xi ⊕ h (x1, ... , xi-1, xi+1, ... , xn),where the function h does not depend on the variable xI (the Boolean difference f 2

i of f withrespect to xi is equal to 1). It is easy to observe that xi-SD functions are always balanced.

Example 1 Six out of ten classes listed in Table 1 (including all classes of linear functions)consist of x-SD functions:C1: x, 1 ⊕ x (linear)C2: x ⊕ y, x ⊕ z, 1 ⊕ x ⊕ y, 1 ⊕ x ⊕ z (linear)C3: x ⊕ yz, 1 ⊕ x ⊕ yz (nonlinear)C4: x ⊕ y ⊕ z, 1 ⊕ x ⊕ y ⊕ z (linear)C5: x ⊕ y ⊕ yz, x ⊕ z ⊕ yz, 1 ⊕ x ⊕ y ⊕ yz, 1 ⊕ x ⊕ z ⊕ yz (nonlinear)C8: x ⊕ y ⊕ z ⊕ yz, 1 ⊕ x ⊕ y ⊕ z ⊕ yz (nonlinear)

(End of Example 1)

Self-dual functions differ from xi-SD functions. For example, the functions in the class C7 areself-dual but not xi-SD for all i. The number of all xi-SD functions for any i = 1, 2, ... ,n, growsexponentially with the number of variables n. Namely, it follows from Definition 3 that if we fixthe values of a function f (x1, ... , xi-1, xi, xi+1, ... , xn) for all variable assignments with a specifiedvalue of xI then f will have the xi-SD property iff the other values will be determined according tothe formula f (x1, ... , xi-1, 0, xi+1, ... , xn) = f’ (x1, ... , xi-1, 1, xi+1, ... , xn). Thus there are as many xi-SD functions as there are ways of choosing values of 2n-1 bits.

3. New generalizations of Shannon Decomposition

In this section we introduce the notion of a function-driven generalized cofactor with respect toan additional function and next prove theorems concerning existence and properties of function-driven generalizations of Shannon decomposition.

Definition 4 Let f (x1, x2, ... , xn) be a Boolean function, g (x1, ... , xi-1, xi, xi+1 , ... , xn) be an xi-SDBoolean function and fi,g (x1, ... , xi-1, xi, xi+1, ... , xn) = f (x1, ... , xi-1, g, xi+1, ... , xn).The functions

f 0i,g = fi,g (x1, ... , xi-1, 0, xi+1, ... , xn)f 1i,g = fi,g (x1, ... , xi-1, 1, xi+1, ... , xn)

are called the negative and positive function-driven generalized cofactors of f with respect to thevariable xi and the function g, respectively.

Note that both f 0i,g and f 1

i,g (similarly to f 0 and f 1) do not depend on the variable xI. Thisproperty is important for using these decompositions as a basis for defining decision diagrams.When dealing with generalized cofactors it is sufficient to consider only one xi-SD function fromeach pair g, g’ because taking the other xi-SD function results only in the exchanging (flipping) ofcofactors.

Example 2 The well-known hidden bit function of n variables (HWBn) was introduced by R.E.Bryant who showed that the size of BDDs representing it grows exponentially in n [6]. Sincethen sizes of variants of BDDs for HWBn have been studied by many authors to overcome thisinherent difficulty (for a survey see [4, 37]). The hidden bit function is defined as follows:

HWBn(x1, x2, x3, x4) = xsum,where sum=x1+x2+...+xn (+ means arithmetic addition) and x0=0, i.e. if the sum of ones in thevariable assignment is equal to sum>0 then the function takes the value of xsum, and if it is equalto 0 then the function takes the value 0.The positive polarity Reed-Muller form of f = HWB4 is as follows:

f(x1, x2, x3, x4) = x1 ⊕ x1x3 ⊕ x1x4 ⊕ x2x3 ⊕ x2x4 ⊕ x1x2x4 ⊕ x2x3 x4

We will determine function-driven generalized cofactors of f with respect to x2 andg= x2 ⊕ x3 ⊕ x1x3:

First the function f2,g has to be determined:f2,g(x1, x2, x3, x4) = f(x1, g, x3, x4) = x1 ⊕ x1x3 ⊕ x1x4 ⊕ (x2 ⊕ x3 ⊕ x1x3)x3 ⊕ (x2 ⊕ x3 ⊕ x1x3)x4 ⊕x1(x2 ⊕ x3 ⊕ x1x3)x4 ⊕ (x2 ⊕ x3 ⊕ x1x3)x3x4 = x1 ⊕ x1x3 ⊕ x1x4 ⊕ x2x3 ⊕ x3 ⊕ x1x3 ⊕ x2x4 ⊕ x3x4

⊕ x1x3x4 ⊕ x1x2 x4 ⊕ x1x3x4 ⊕ x1x3x4 ⊕ x2x3x4 ⊕ x3x4 ⊕ x1x3x4 = x1 ⊕ x3 ⊕ x1x4 ⊕ x2x3 ⊕ x2x4 ⊕x1x2 x4 ⊕ x2x3x4

Thus f 02,g = x1 ⊕ x3 ⊕ x1x4 and f 12,g = x1 ⊕ x4 ⊕ x3x4. (End of Example 2)

The theorem below presents new generalizations of Shannon decomposition.

Theorem 1 The following decompositions hold true:f = g’ f 0i,g+ g f 1i,g

where g (x1, ... , xi-1, xi, xi+1 , ... , xn) is an arbitrary xi-SD function for i = 1,2, ... ,n. They will becalled function-driven generalizations of Shannon decompositions. (or simply function-driven S-decompositions).

ProofLet i be fixed. As g is an xi-SD function there exists a function h (x1 , ... , xi-1 , xi+1 , ... , xn) suchthat

g (x1 , ... , xi-1 , xi, xi+1 , ... , xn) = xi ⊕ h (x1 , ... , xi-1 , xi+1 , ... , xn)Then

g(x1 , ... , xi-1 , 0 , xi+1 , ... , xn)= h(x1 , ... , xi-1 , xi+1 , ... , xn)g(x1 , ... , xi-1 , 1 , xi+1 , ... , xn)= 1 ⊕ h(x1 , ... , xi-1 , xi+1 , ... , xn)

Let us transform the right side of the equation in Theorem 1:g’ f 0i,g+ g f 1i,g = g’ f 0i,g ⊕ g f 1i,g = (1 ⊕ xi ⊕ h) f 0i,g ⊕ (xi ⊕ h) f 1i,g = = (1 ⊕ xi ⊕ h) fi,g (x1, ... , xi-1, 0, xi+1, ... , xn) ⊕ (xi ⊕ h) fi,g (x1, ... , xi-1 , 1, xi+1, ... , xn) = = (1 ⊕ xi ⊕ h) f (x1, ... , xi-1 , g(x1 , ... , xi-1 , 0, xi+1 , ... , xn) , xi+1 , ... , xn) ⊕ ⊕ (xi ⊕ h) f (x1, ... , xi-1 , g(x1 , ... , xi-1 , 1, xi+1 , ... , xn) , xi+1 , ... , xn) = = (1 ⊕ xi ⊕ h) f (x1, ... , xi-1 , h , xi+1, ... , xn) ⊕ (xi ⊕ h) f (x1, ... , xi-1, xi ⊕ h , xi+1 , ... , xn)The last expression is equal to xi’ f 0

i,g ⊕ xi f 1i,g = f for both h=0 and h=1. Thus the above

decompositions holds for each variable xi, i = 1,2, ... ,n. (End of Proof)

Thus for each variable xI there are as many function-driven S-decompositions as xi-SD functions.Similarly to Shannon decomposition also function-driven S-decompositions correspond to amultiplexer operation but with a specified function g as the controlling variable instead of avariable. Figures 1 to 3 illustrate differences between the three decompositions: Shannon, positiveReed-Muller (Davio) and function-driven S-decomposition (for simplicity, the figures correspondto the case of x1-SD function g). In Shannon decomposition the cofactors are formed by splittingthe function vector into the halves f 0 and f 1. In the positive Reed-Muller (Davio) decompositionone cofactor (f 0) is obtained in the same way as in the Shannon decomposition but the other oneis obtained by exoring f 0 and f 1 (in negative decomposition of this kind one cofactor is equal to f1, and the other is obtained by exoring f 0 and f 1). In function-driven S-decompositionpropagation of each pair of corresponding bits

f (a1, ... , ai-1, 0 , ai+1, ... , an) and f (a1, ... , ai-1, 1 , ai+1, ... , an)is controlled by the value g (a1, ... , ai-1, 0 , ai+1, ... , an) = g’ (a1, ... , ai-1, 1 , ai+1, ... , an). Wheng=0 the bits of the pair go to the same positions in the cofactors as in Shannon decomposition,while for g = 1 the bits of the pair are flipped. It is easy to observe that in both Shannon andfunction-driven S-decompositions the sum of weights of the function vectors of cofactors of f isalways equal to the weight of the function vector of f while it may not be true for Reed-Muller(Davio) decompositions.

In the context of using BDDs for verification of digital circuits the notion of generalized cofactorshas been introduced by analogy to Shannon decomposition (e.g., see [35, 29]). Let

f = g’ f 0g + g f 1g

where g is an arbitrary Boolean function and f 0g and f 1

g are called generalized cofactors of f. Ingeneral, these cofactors are not uniquely determined. This property was used in the papers [7-9]for introducing operators called constrain and restrict for implicit computations using BDDs. Inthis context the following theorem presents an interesting result. It also shows for whichfunctions g it is possible to define canonical function-driven decision diagrams in a mannersimilar to defining BDDs.

Theorem 2 Let f (x1, x2, ... , xn) be an arbitrary Boolean function. All xi-SD functions (i=1,2,...,n)form the maximal set of functions such that there exists a unique solution of the equation inTheorem 1 with respect to the unknowns f 0i,g and f

1i,g..

ProofLet us assume that a non-xi-SD function g is satisfying the equation in Theorem 1. Then thereexists a binary vector (a1, ... , ai-1, ai+1, ... , an) such that

g (a1, ... , ai-1, 0 , ai+1, ... , an) = g (a1, ... , ai-1, 1 , ai+1, ... , an) = b,where a1, ... , ai-1, ai+1, ... , an,, b ∈ {0, 1}.Thusf 0i,g = fi,g(a1, ... , ai-1, 0, ai+1, ... , an) = f (a1, ... , ai-1 , g(a1 , ... , ai-1 , 0, ai+1 , ... , an) , ai+1 , ... , an)= f (a1, ... , ai-1 , b , ai+1 , ... , an)andf 1i,g = fi,g(a1, ... , ai-1, 1, ai+1, ... , an) = f (a1, ... , ai-1 , g(a1 , ... , ai-1 , 1, ai+1 , ... , an) , ai+1 , ... , an)= f (a1, ... , ai-1 , b , ai+1 , ... , an).From Theorem 1 we havef (a1, ... , ai-1, 0 , ai+1, ... , an) = g’(a1, ... , ai-1 0, ai+1, ... , an) f

0i,g+ g(a1, ... , ai-1, 0, ai+1, ... , an) f

1i,g

= b’ f 0i,g + b f 1i,g = b’ f 0i,g + b f 0i,g = f 0i,g

andf(a1, ... , ai-1, 1 , ai+1, ... , an) = g’(a1, ... , ai-1 1, ai+1, ... , an) f

0i,g+ g(a1, ... , ai-1, 1, ai+1, ... , an) f

1i,g

= b’ f 0i,g + b f 1i,g = b’ f 0i,g + b f 0i,g = f 0i,g,,so f (a1, ... , ai-1, 0 , ai+1, ... , an) = f (a1, ... , ai-1, 1 , ai+1, ... , an).However, this is in contradiction with the assumption that f is an arbitrary Boolean function of nvariables. Thus the Theorem 2 holds. (End of Proof)

On the basis of function-driven S-decompositions new modifications of the notion of binarydecision diagram can be created. The details can be found in the papers [24, 25] where we alsogive some experimental results.

Function-driven decompositions can be combined with other decompositions (e.g. thosementioned in Section 1). Below we present an example.

Theorem 3 The following decompositions hold true:f = f 0ig, ⊕ g f 2i,g - function-driven positive Reed-Muller (Davio) decompositions

f = f 1ig, ⊕ g’ f 2i,g - function-driven negative Reed-Muller (Davio) decompositions,where g(x1,... , xi-1, xi, xi+1 , ... , xn) is an arbitrary xi-SD function, i=1,2,...,n,and f 2i,g = f 0ig, ⊕ f 1i,g.

We omit the proof of Theorem 3 as it is similar to the proof of Theorem 1.

Function-driven decompositions can also be extended to multiple-valued functions f: Mn → Mover a finite set M. We assume that M={0, 1, ... , m-1}, m ∈ N. First we have to generalizeDefinitions 2-4.

Definition 5 A multiple-valued function in n variables f (x1, x2, ... , xn) = f(X) is balanced iff thenumber of vectors X for which f(X)=j is the same for j=0, 1, ..., m-1. Otherwise, f is unbalanced.

Definition 6 A multiple-valued function g(x1, x2, ... , xn) is called self-dual with respect to avariable xi (in short xi-SD), i=1,2,...,n, iff for a fixed vector (a1, ... , ai-1, ai+1, ... , an) ∈ Mn-1

{g(a1, ... , ai-1, j, ai+1, ... , an): j ∈ M } = M.

Definition 7 Let f (x1, x2, ... , xn) be a multiple-valued function, g(x1, ... , xi-1, xi, xi+1 , ... , xn) be anxi-SD multiple-valued function, and fi,g (x1, ... , xi-1, xi, xi+1, ... , xn) = f (x1, ... , xi-1, g, xi+1, ... , xn).The functions

f 0i,g = fi,g (x1, ... , xi-1, 0, xi+1, ... , xn)f 1i,g = fi,g (x1, ... , xi-1, 1, xi+1, ... , xn)

.

.

.f m-1

i,g = fi,g (x1, ... , xi-1, m-1, xi+1, ... , xn)

are called generalized cofactors of f with respect to the variable xi and the multiple-valuedfunction g.

Now function-driven multiple-valued decompositions can be easily introduced in analogousmanner to the binary case and used to define function-driven Shannon (or mixed-type) multi-valued decision diagrams in a manner analogous to defining multi-valued decision diagrams(MDD) [19, 33, 34].

4. Conclusions

We have presented new generalizations of the well-known Shannon decomposition starting withthe notion of the class of functions self-dual with respect to a variable of the function. We callthem function-driven Shannon decompositions because additional Boolean functions form a basisfor defining these decompositions. The number of such generalizations grows exponentially withthe number of variables. This notion of function-driven decompositions is independent from thepreviously developed decompositions so further extensions are possible by combining them withsome other decompositions (as we have shown in Theorem 3). Moreover, we have also provedthat the maximal set of functions which can form a basis for defining a function-driven Shannondecomposition is equal to the set of all xi-SD functions.

The results presented in the paper are closely related to the problem of minimization of decisiondiagrams. However, although function-driven decision diagrams can provide more compactrepresentations for a specified function than other decision diagrams, finding a minimal function-driven diagram is very hard as the search space is much larger in this case. We have alreadystarted to work in this direction and in [24, 25] present some promising experimental resultsbased on even more general approach. In this approach arbitrary balanced functions (including xi-SD functions) may be used for defining new variants of decision diagrams. It corresponds to thenotion of BDD transformations introduced in [3]. Unfortunately, no heuristic algorithms havebeen developed up to now for finding efficient algorithms leading to minimal transformeddiagrams. Nevertheless we plan to continue studying decision diagrams based on function-drivenS-decompositions.

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