View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Open Repository and Bibliography - Luxembourg Pivotal decompositions of functions Jean-Luc Marichal, Bruno Teheux Mathematics Research Unit, FSTC, University of Luxembourg, 6, rue Coudenhove-Kalergi, L-1359 Luxembourg, Luxembourg. Abstract We extend the well-known Shannon decomposition of Boolean functions to more general classes of functions. Such decompositions, which we call piv- otal decompositions, express the fact that every unary section of a function only depends upon its values at two given elements. Pivotal decompositions appear to hold for various function classes, such as the class of lattice poly- nomial functions or the class of multilinear polynomial functions. We also define function classes characterized by pivotal decompositions and function classes characterized by their unary members and investigate links between these two concepts. Keywords: Shannon decomposition, pivotal decomposition, Boolean function, pseudo-Boolean function, switching theory, aggregation function. 2010 MSC: 94C10 1. Introduction A remarkable (though immediate) property of Boolean functions is the so-called Shannon decomposition, or Shannon expansion (see [20]), also called pivotal decomposition [2]. This property states that, for every Boolean func- tion f∶ {0; 1}n → {0; 1} and every k ∈ [n] = {1; : : : ; n}, the following decom- position formula holds: ( ) = ( 1) + ( 0) = ( ) ∈ { }n f x xk f xk xk f xk ; x x1; : : : ; xn 0; 1 ; (1) Email addresses: jean-luc.marichal[at]uni.lu (Jean-Luc Marichal), bruno.teheux[at]uni.lu (Bruno Teheux) Preprint submitted to Discrete Applied Mathematics April 28, 2014 = − a = where xk 1 xk and xk is the n-tuple whose i-th coordinate is a, if i k, and xi, otherwise. Here the `+' sign represents the classical addition for real numbers. Decomposition formula (1) means that we can precompute the function values for xk = 0 and xk = 1 and then select the appropriate value depending on the value of xk. By analogy with the cofactor expansion formula for ( 1) ( 0) determinants, here f xk (resp. f xk ) is called the cofactor of xk (resp. xk) for f and is derived by setting xk = 1 (resp. xk = 0) in f. Clearly, the addition operation in (1) can be replaced with the maximum operation ∨, thus yielding the following alternative formulation of (1): ( ) = ( ( 1)) ∨ ( ( 0)) ∈ { }n ∈ [ ] f x xk f xk xk f xk ; x 0; 1 ; k n : Equivalently, (1) can also be put in the form ( ) = ( ∨ ( 0)) ( ∨ ( 1)) ∈ { }n ∈ [ ] f x xk f xk xk f xk ; x 0; 1 ; k n : (2) As it is well known, repeated applications of (1) show that any n-ary Boolean function can always be expressed as the multilinear polynomial func- tion n f(x) = ∑ f(1S) ∏ xi ∏ xi ; x ∈ {0; 1} ; (3) S⊆[n] i∈S i∈[n]∖S n where 1S is the characteristic vector of S in {0; 1} , that is, the n-tuple whose i-th coordinate is 1, if i ∈ S, and 0, otherwise. ↦ ( z ) ∈ If f is nondecreasing (i.e., the map z f xk is isotone for every x {0; 1}n and every k ∈ [n]), then by expanding (2) we see that the decompo- sition formula reduces to ( ) = ( ( 1)) ∨ ( 0) ∈ { }n ∈ [ ] f x xk f xk f xk ; x 0; 1 ; k n ; (4) or, equivalently, ( ) = ( ( 1) ( 0)) ∈ { }n ∈ [ ] f x med xk; f xk ; f xk ; x 0; 1 ; k n ; (5) where med is the ternary median operation defined by med(x; y; z) = (x ∧ y) ∨ (y ∧ z) ∨ (z ∧ x) and ∧ is the minimum operation. 2 Interestingly, the following decomposition formula also holds for nonde- creasing n-ary Boolean functions: ( ) = ( ( 1)∨ ( 0))+ ( ( 1)∧ ( 0)) ∈ { }n ∈ [ ] f x xk f xk f xk xk f xk f xk ; x 0; 1 ; k n : (6) Actually, any of the decomposition formulas (4){(6) exactly expresses the fact that f should be nondecreasing and hence characterizes the subclass of nondecreasing n-ary Boolean functions. We state this result as follows. Proposition 1.1. A Boolean function f∶ {0; 1}n → {0; 1} is nondecreasing if and only if it satisfies any of the decomposition formulas (4){(6). Decomposition property (1) also holds for functions f∶ {0; 1}n → R, called n-ary pseudo-Boolean functions. As a consequence, these functions also have the representation given in (3). Moreover, formula (6) clearly characterizes the subclass of nondecreasing n-ary pseudo-Boolean functions. The multilinear extension of a pseudo-Boolean function f∶ {0; 1}n → R is the function f^∶ [0; 1]n → R defined by (see Owen [16, 17]) ^ n f(x) = ∑ f(1S) ∏ xi ∏ (1 − xi) ; x ∈ [0; 1] : (7) S⊆[n] i∈S i∈[n]∖S Actually, a function is the multilinear extension of a pseudo-Boolean func- tion if and only if it is a multilinear polynomial function, i.e., a polynomial function of degree ⩽ 1 in each variable. Thus defined, one can easily see that the class of multilinear polynomial functions can be characterized as follows. Proposition 1.2. A function f∶ [0; 1]n → R is a multilinear polynomial func- tion if and only if it satisfies ( ) = ( 1) + ( − ) ( 0) ∈ [ ]n ∈ [ ] f x xk f xk 1 xk f xk ; x 0; 1 ; k n : (8) Interestingly, Eq. (8) provides an immediate proof of the property @f(x) = ( 1) − ( 0) f xk f xk ; @xk which holds for every multilinear polynomial function f∶ [0; 1]n → R. As far as nondecreasing multilinear polynomial functions are concerned, we have the following characterization, which is a special case of Corollary 4.8. Recall first that a multilinear polynomial function is nondecreasing if and only if so is its restriction to {0; 1}n (i.e., its defining pseudo-Boolean func- tion). 3 Proposition 1.3. A function f∶ [0; 1]n → R is a nondecreasing multilinear polynomial function if and only if it satisfies ( ) = ( ( 1)∨ ( 0))+ ( ( 1)∧ ( 0)) ∈ [ ]n ∈ [ ] f x xk f xk f xk xk f xk f xk ; x 0; 1 ; k n : (9) The decomposition formulas considered in this introduction share an in- teresting common feature, namely the fact that any variable, here denoted xk and called pivot, can be pulled out of the function, reducing the evaluation of ( ) ( 1) ( 0) 1 f x to the evaluation of a function of xk, f xk , and f xk . This feature ( 1) ( 0) may be useful when for instance the values f xk and f xk are much easier to compute than that of f(x). In addition to this, such (pivotal) decompo- sitions may facilitate inductive proofs and may lead to canonical forms such as (3). In this paper we define a general concept of pivotal decomposition for various functions f∶ Xn → Y , where X and Y are nonempty sets (Section 2). We also introduce function classes that are characterized by pivotal de- compositions (Section 3) and function classes that are characterized by their unary members and investigate relationships between these concepts (Section 4). We also introduce a natural generalization of the concept of pivotal de- composition, namely componentwise pivotal decomposition (Section 5). We then end our paper by some concluding remarks (Section 6). 2. Pivotal decompositions The examples presented in the introduction motivate the following defini- tion. Let X and Y be nonempty sets and let 0 and 1 be two fixed elements of ∶ n → = {( ( 1) ( 0)) ∶ ∈ n ∈ X. For every function f X Y , define Rf f xk ; f xk x X ; k [n]}. Throughout we assume that n ⩾ 1. Definition 2.1. We say that a function f∶ Xn → Y is pivotally decomposable if there exist a subset D of X × Y 2 and a function Π∶ D → Y , called pivotal function, such that D ⊇ X × Rf and ( ) = ( ( 1) ( 0)) ∈ n ∈ [ ] f x Π xk; f xk ; f xk ; x X ; k n : (10) In this case, we say that f is Π-decomposable. 1In applications, such as cooperative game theory or aggregation function theory, this means that, in a sense, one can isolate the marginal contribution of a factor (attribute, criterion, etc.) from the others. 4 From Definition 2.1 we immediately obtain the following two results. Fact 2.2. Let f∶ Xn → Y be a Π-decomposable function for some pivotal func- tion Π. Then, for every (u; v) ∈ Rf , we have Π(1; u; v) = u and Π(0; u; v) = v. Proposition 2.3 (Uniqueness of the pivotal function). If f∶ Xn → Y is Π- and Π′-decomposable for some pivotal functions Π and Π′, then Π and Π′ coincide on X × Rf . n Proof. Let (p; u; v) ∈ X × Rf . By definition of Rf , there exist x ∈ X and ∈ [ ] ( ) = ( ( 1) ( 0)) k n such that u; v f xk ; f xk . We then have ′( ) = ′( ( 1) ( 0)) = ( p) = ( ( 1) ( 0)) = ( ) Π p; u; v Π p; f xk ; f xk f xk Π p; f xk ; f xk Π p; u; v ; which completes the proof. Example 2.4. Every Boolean function is Π-decomposable, where Π∶ {0; 1}3 → {0; 1} is the classical `if-then-else' connective defined by Π(p; u; v) = (p ∧ u) ∨ (p ∧ v). If f is nondecreasing, we can restrict Π to {0; 1} × {(u; v) ∈ {0; 1}2 ∶ u ⩾ v} or consider Π′(p; u; v) = (p ∧ (u ∨ v)) ∨ (p ∧ (u ∧ v)) on {0; 1}3. Example 2.5. Every multilinear polynomial function f∶ [0; 1]n → R is Π- decomposable, where Π∶ D → R is defined by D = [0; 1] × R2 and Π(p; u; v) = p u + (1 − p) v. If f is nondecreasing, we can restrict Π to [0; 1] × {(u; v) ∈ R2 ∶ u ⩾ v} or consider Π′(p; u; v) = p (u ∨ v) + (1 − p)(u ∧ v) on [0; 1]n.