Stability of Boolean function classes with respect to clones of linear functions
aa r X i v : . [ m a t h . R A ] J a n STABILITY OF BOOLEAN FUNCTION CLASSES WITHRESPECT TO CLONES OF LINEAR FUNCTIONS
MIGUEL COUCEIRO AND ERKKO LEHTONEN
This paper is dedicated to Maurice Pouzet to whom we are deeply thankful for his guidance,friendship, knowledgeable support, and for being always a source of great motivation andinspiration.
Abstract.
We consider classes of Boolean functions stable under composi-tions both from the right and from the left with clones. Motivated by thequestion how many properties of Boolean functions can be defined by meansof linear equations, we focus on stability under compositions with the clone oflinear idempotent functions. It follows from a result by Sparks that there arecountably many such linearly definable classes of Boolean functions. In thispaper, we refine this result by completely describing these classes. This work istightly related with the theory of function minors, stable classes, clonoids, andhereditary classes, topics that have been widely investigated in recent years byseveral authors including Maurice Pouzet and his coauthors. Introduction
This paper is a study of classes of functions of several arguments from a set A to a set B that are closed under composition from the right with a clone C on A and under composition from the left with a clone C on B , in brief, ( C , C ) -stable classes of functions. Special instances of the notion of ( C , C )-stability appear inthe literature. For example, if both C and C are clones of projections on therespective sets, then we get minor-closed classes or minions or equational classes (see Pippenger [14], Ekin et al. [8]). If C the clone of projections and C is theclone of an algebra B , then we get clonoids with source set A and target algebra B (see Aichinger and Mayr [1]).If both C and C are equal to the clone L c of idempotent linear functions on { , } , then the ( C , C )-stable classes are exactly the classes of Boolean functionsdefinable by linear equations (see [4]). It was already observed in [4] that there areinfinitely many such linearly definable classes, but it remained an open questionwhether there are countably or uncountably many such classes and exactly whatthese classes are.More generally, we would like to describe ( C , C )-stable classes. This problemseems unfeasible in full generality, since there are uncountably many clones onsets with at least three elements (see Yanov and Muchnik [20]). This fact ledus to considering ( C , C )-stability for clones C and C on the two-element set.Motivated by linear definability, we focus on clones containing the clone L c .We show that that there are a countably infinite number of ( L c , L c )-stable classes(in brief, L c -stable classes), and we provide an explicit description thereof. Moreprecisely, the paper is organized as follows. (M. Couceiro) Universit´e de Lorraine, CNRS, Inria, LORIA, F-54000 Nancy, France (E. Lehtonen)
Centro de Matem´atica e Aplicac¸˜oes, Faculdade de Ciˆencias e Tecnolo-gia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal
Date : January 19, 2021. • Section 2: We provide the basic definitions and preliminary results that areneeded in the sequel. • Section 3: We establish some auxiliary tools for studying ( C , C )-stability. • Section 4: We make a little diversion to clones on arbitrary finite fields,and we describe the L -stable classes, where L denotes the clone of all linearfunctions on F q . • Section 5: We define various properties of Boolean functions that are neededfor describing the L c -stable classes. • Section 6: We present our main result: an explicit description of the L c -stable classes of Boolean functions. The proof has two parts. First we showthat the listed classes are L c -stable; this is straightforward verification. Themore difficult part of the proof is to show that there are no further L c -stableclasses. • Section 7: With the help of the result on L c -stable classes, we obtain withlittle effort also a description of ( C , C )-stable classes for clones C and C , where C is arbitrary and L c ⊆ C • Section 8: We make some concluding remarks and indicate directions forfuture research.The main results of this paper were presented without proofs in the 1st Interna-tional Conference on Algebras, Graphs and Ordered Sets (ALGOS 2020) [7]. Thereader should be cautious about the fact that some notation and terminology havebeen slightly changed from the conference paper.2.
Preliminaries
The symbols N and N + denote the set of all nonnegative integers and the set ofall positive integers, respectively. For any n ∈ N , the symbol [ n ] denotes the set { i ∈ N | ≤ i ≤ n } . Definition 2.1.
Let A and B be sets. A mapping of the form f : A n → B forsome n ∈ N + is called a function of several arguments from A to B (or simply a function ). The number n is called the arity of f and denoted by ar( f ). If A = B ,then such a function is called an operation on A . We denote by F AB and O A theset of all functions of several arguments from A to B and the set of all operationson A , respectively. For any n ∈ N + , we denote by F ( n ) AB the set of all n -ary functionsin F AB , and for any C ⊆ F AB , we let C ( n ) := C ∩ F ( n ) AB and call it the n -ary part of C . Definition 2.2.
For b ∈ B and n ∈ N , the n -ary constant function c ( n ) b : A n → B isgiven by the rule ( a , . . . , a n ) b for all a , . . . , a n ∈ A . In the case when A = B ,for n ∈ N and i ∈ [ n ], the i -th n -ary projection pr ( n ) i : A n → A is given by the rule( a , . . . , a n ) a i for all a , . . . , a n ∈ A . Definition 2.3.
Let f : A n → B and i ∈ [ n ]. The i -th argument is essential in f if there exist a , . . . , a n , a ′ i ∈ A such that f ( a , . . . , a n ) = f ( a , . . . , a i − , a ′ i , a i +1 , . . . , a n ) . An argument that is not essential is fictitious.
The essential arity of f is the numberof its essential arguments. Definition 2.4.
Let f : B n → C and g , . . . , g n : A m → B . The composition of f with g , . . . , g n is the function f ( g , . . . , g n ) : A m → C given by the rule f ( g , . . . , g n )( a ) := f ( g ( a ) , . . . , g n ( a ))for all a ∈ A m . The function f is called the outer function and g , . . . , g n are calledthe inner functions of the composition. TABILITY OF BOOLEAN FUNCTION CLASSES 3
Definition 2.5.
Let f : A n → B and σ : [ n ] → [ m ]. Define the function f σ : A m → B by the rule f σ ( a , . . . , a m ) = f ( a σ (1) , . . . , a σ ( n ) ) , for all a , . . . , a m ∈ A . Such a function f σ is called a minor of f , formed via the minor formation map σ . Intuitively, minors of f are all those functions that canbe obtained from f by manipulation of its arguments: permutation of arguments,introduction of fictitious arguments, identification of arguments. It is clear fromthe definition that the minor f σ can be obtained as a composition of f with m -aryprojections on A : f σ = f (pr ( m ) σ (1) , . . . , pr ( m ) σ ( n ) ) . An important special case of minors is the identification of a pair of arguments.This is obtained with minor formation maps of the following form: for i, j ∈ [ n ]with i < j , let σ ij : [ n ] → [ n −
1] be given by σ ij ( m ) = m, if m < j , i, if m = j , m − , if m > j .We call such a map σ ij an identification map, and we write f ij for f σ ij . Moreexplicitly, f ij ( a , . . . , a n − ) = f ( a , . . . , a i , . . . , a j − , a i , a j , . . . , a n − ) . We write f ≤ g if f is a minor of g . The minor relation ≤ is a quasiorder (areflexive and transitive relation) on F AB , and it induces an equivalence relation ≡ on F AB and a partial order on the quotient F AB / ≡ in the usual way: f ≡ g if f ≤ g and g ≤ f , and f / ≡ ≤ g/ ≡ if f ≤ g .The effect of successive formations of minors is captured by the composition ofminor-forming maps. Lemma 2.6.
Let f : A n → B , σ : [ n ] → [ m ] , and τ : [ m ] → [ ℓ ] . Then ( f σ ) τ = f τ ◦ σ .Proof. For all a , . . . , a ℓ ∈ A , we have( f σ ) τ ( a , . . . , a ℓ ) = ( f σ )( a τ (1) , . . . , a τ ( m ) ) = f ( a τ ( σ (1)) , . . . , a τ ( σ ( n )) )= f ( a ( τ ◦ σ )(1) , . . . , a ( τ ◦ σ )( n ) ) = f τ ◦ σ ( a , . . . , a ℓ ) . (cid:3) Remark 2.7.
It is well known that any function can be decomposed into a surjec-tion and an injection. This obviously holds for minor formation maps σ : [ n ] → [ m ];we obtain σ = ρ ◦ τ where τ : [ n ] → [ ℓ ] is surjective and ρ : [ ℓ ] → [ m ] is injective.Moreover, as explained in [12, Section 2.2], we can choose the surjective map τ sothat it is a composition of a number of identification maps: τ = σ i k j k ◦ · · · ◦ σ i j (we regard the empty composition as the identity map on [ ℓ ]).Intuitively, this means that any minor of a function f : A n → B can be formedby first successively identifying pairs of arguments, and then introducing fictitiousarguments and permuting arguments.Composition of functions satisfies the so-called superassociative law. Conse-quently, formation of minors commutes with composition. Lemma 2.8.
Let f : C n → D , g , . . . , g n : B m → C , h , . . . , h m ∈ A ℓ → B .Then ( f ( g , . . . , g n ))( h , . . . , h m ) = f ( g ( h , . . . , h m ) , . . . , g n ( h , . . . , h m )) . Conse-quently, for any σ : [ ℓ ] → [ m ] , we have ( f ( g , . . . , g n )) σ = f (( g ) σ , . . . , ( g n ) σ )) . STABILITY OF BOOLEAN FUNCTION CLASSES
Proof.
For any a ∈ A ℓ , we have( f ( g , . . . , g n ))( h , . . . , h m )( a ) = ( f ( g , . . . , g n ))( h ( a ) , . . . , h m ( a ))= f ( g ( h ( a ) , . . . , h m ( a )) , . . . , g n ( h ( a ) , . . . , h m ( a )))= f ( g ( h , . . . , h m )( a ) , . . . , g n ( h , . . . , h m )( a ))= f ( g ( h , . . . , h m ) , . . . , g n ( h , . . . , h m ))( a ) . The statement about minors follows by taking h i := pr ( ℓ ) σ ( i ) , 1 ≤ i ≤ m . (cid:3) The notion of functional composition extends naturally to classes of functions.
Definition 2.9.
Let C ⊆ F BC and K ⊆ F AB . The composition of C with K isdefined as CK := { f ( g , . . . , g n ) | f ∈ C ( n ) , g , . . . , g n ∈ K ( m ) , n, m ∈ N + } . Remark 2.10.
It follows immediately from the definition of function class com-position that if
C, C ′ ⊆ F BC and K, K ′ ⊆ F AB satisfy C ⊆ C ′ and K ⊆ K ′ , then CK ⊆ C ′ K ′ . Lemma 2.11.
For any C , C ⊆ F BC , K ⊆ F AB , it holds that ( C ∩ C ) K ⊆ C K ∩ C K and ( C ∪ C ) K = C K ∪ C K .Proof. We clearly have ( C ∩ C ) K = ( C ∩ C ) K ∩ ( C ∩ C ) K ⊆ C K ∩ C K and C K ∪ C K ⊆ ( C ∪ C ) K ∪ ( C ∪ C ) K = ( C ∪ C ) K . In order to prove the inclusion( C ∪ C ) K ⊆ C K ∪ C K , let h ∈ ( C ∪ C ) K . Then h = f ( g , . . . , g n ) for some f ∈ C ∪ C and g , . . . , g ∈ K . Since f ∈ C or f ∈ C , we have that f ( g , . . . , g n )belongs to C K or C K ; therefore h = f ( g , . . . , g n ) ∈ C K ∪ C K . (cid:3) Remark 2.12.
The inclusion C K ∩ C K ⊆ ( C ∩ C ) K does not hold in general.For a counterexample, let C := { π (1)1 } , C := { c (1)0 } , K := { c (1)0 } , subsets of O { , } ,where c (1)0 denotes the unary constant function taking value 0. Then C K = C K = { c (1)0 } , so C K ∩ C K = { c (1)0 } , but ( C ∩ C ) K = ∅ because C ∩ C = ∅ . Definition 2.13.
A class C ⊆ O A is called a clone on A if CC ⊆ C and C containsall projections. The set of all clones on A is a closure system in which the greatestand least elements are the clone O A of all operations on A and the clone of allprojections on A , respectively. For any K ⊆ O A , we denote by h K i the clonegenerated by K , i.e., the smallest clone on A containing K . Definition 2.14.
Let K ⊆ F AB , C ⊆ O A , and C ⊆ O B . We say that K is stable under right composition with C if KC ⊆ K , and that K is stable under leftcomposition with C is C K ⊆ K . If both KC ⊆ K and C K ⊆ K hold, we saythat K is ( C , C ) -stable. If K, C ⊆ O A and K is ( C, C )-stable, we say that K is C -stable. The set of all ( C , C )-stable subsets of F AB constitutes a closure system, andfor any K ⊆ F AB , we denote by h K i ( C ,C ) the ( C , C ) -closure of K , i.e., thesmallest ( C , C )-stable class containing K . We also write h K i C for h K i ( C,C ) andcall it the C -closure of K . Remark 2.15.
A set K ⊆ F AB is minor-closed if and only if it is stable underright composition with the set of all projections on A . Every clone is minor-closed.A clone C is ( C, C )-stable.
Lemma 2.16.
Let C and C ′ be clones on A and C and C ′ clones on B suchthat C ⊆ C ′ and C ⊆ C ′ . Then for every K ⊆ F AB , it holds that if K is ( C ′ , C ′ ) -stable then K is ( C , C ) -stable.Proof. Assume that K is ( C ′ , C ′ )-stable. Then, in view of Remark 2.10, we have KC ⊆ KC ′ ⊆ K and C K ⊆ C ′ K ⊆ K , i.e., K is ( C , C )-stable. (cid:3) TABILITY OF BOOLEAN FUNCTION CLASSES 5 Stability and generators
The task of verifying whether a function class is stable under right or left compo-sitions with certain clones may appear complicated because the defining conditionsinvolve compositions with arbitrary members of each clone. We now develop helpfultools that simplify this task.For right stability, it is enough to consider closure under minors and certain sim-ple compositions involving only generators of the clone. In order to formalize this,let us consider the elementary superposition operations ζ (cyclic shift of arguments), τ (transposition of the first two arguments), ∆ (identification of arguments or di-agonalization), ∇ (introduction of a fictitious argument or cylindrification), and ∗ (composition) defined by Mal’cev [13] (see also [11, Section II.1.2]). The algebra( O A ; ζ, τ, ∆ , ∇ , ∗ ) is called the iterative function algebra on A , and its subuniversesare called closed classes. Closed classes containing all projections are precisely theclones on A .Let F ⊆ O A and f ∈ O A . We say that f is a superposition of F if f can beobtained from the members F by a finite number of applications of the operations ζ , τ , ∆, ∇ , ∗ . Lemma 3.1.
For any f ∈ O ( n ) A and g , . . . , g n ∈ O ( m ) A , the composition f ( g , . . . , g n ) is a superposition of { f, g , . . . , g n } .Proof. Let f := ( ζf ) ∗ g n , and For i = 1 , . . . , n −
1, let f i := ( ζf i − ) ∗ g n − i . Then f ( x , . . . , x n + m − )= ( ζf )( g n ( x , . . . , x m ) , x m +1 , . . . , x n + m − )= f ( x m +1 , . . . , x n + m − , g n ( x , . . . , x m )) ,f ( x , . . . , x n +2 m − )= ( ζf )( g n − ( x , . . . , x m ) , x m +1 , . . . , x n +2 m − )= f ( x m +1 , . . . , x n +2 m − , g n − ( x , . . . , x m ))= f ( x m +1 , . . . , x n +2 m − , g n − ( x , . . . , x m ) , g n ( x m +1 , . . . , x m )) ,f ( x , . . . , x n +3 m − )= ( ζf )( g n − ( x , . . . , x m ) , x m +1 , . . . , x n +3 m − )= f ( x m +1 , . . . , x n +3 m − , g n − ( x , . . . , x m ))= f ( x m +1 , . . . , x n +3 m − ,g n − ( x , . . . , x m ) , g n − ( x m +1 , . . . , x m ) , g n ( x m +1 , . . . , x m )) , ... f n ( x , . . . , x nm )= f ( g ( x , . . . , x m ) , g ( x m +1 , . . . , x m ) , . . . , g n ( x ( n − m +1 , . . . , x nm )) . Let θ be the composition of elementary operations that identifies arguments x i and x j if and only if i ≡ j (mod m ). Then θf n ( x , . . . , x m ) = f ( g ( x , . . . , x m ) , g ( x , . . . , x m ) , . . . , g n ( x , . . . , x m ))= f ( g , . . . , g n )( x , . . . , x m ) . By construction, the functions f , . . . , f n and θf n = f ( g , . . . , g n ) are superpositionsof { f, g , . . . , g n } . (cid:3) Lemma 3.2.
Let F ⊆ O A . Let C be a clone on A , and let G be a generating setof C . Then the following conditions are equivalent. (i) F C ⊆ F STABILITY OF BOOLEAN FUNCTION CLASSES (ii) F is minor-closed and f ∗ g ∈ F whenever f ∈ F and g ∈ C . (iii) F is minor-closed and f ∗ g ∈ F whenever f ∈ F and g ∈ G .Proof. (i) = ⇒ (iii): For any f ∈ F , any minor of f is of the form f (pr ( m ) i , . . . , pr ( m ) i m ), for some m ∈ N and i , . . . , i m ∈ [ m ]. Since all projections are members ofthe clone C , we have f (pr ( m ) i , . . . , pr ( m ) i m ) ∈ F C ⊆ F . Thus F is minor-closed.Let g ∈ G and define g ′ := g (pr ( m + n − , . . . , pr ( m + n − m ). Then g ′ ∈ C , and wehave f ∗ g = f ( g ′ , pr ( m + n − m +1 , . . . , pr ( m + n − m + n − ) ∈ F C ⊆ F .(iii) = ⇒ (ii): Let g ∈ C . If g is a projection, then for every f ∈ F , the function f ∗ g is a minor of f , obtained by introducing fictitious arguments, so f ∗ g ∈ F because F is minor-closed. If g is not a projection, then g is a superposition of G , that is, there is a term t , say ℓ -ary, in the language of iterative algebras and h , . . . , h ℓ ∈ G such that t O A ( h , . . . , h ℓ ) = g . We prove by induction on thestructure of the term t that for every f ∈ F it holds that f ∗ g ∈ F . If t = x i ,then t O A ( h , . . . , h ℓ ) = h i ∈ G , and we have f ∗ h i ∈ F by assumption. Considerthen the case that t = ϕu , where ϕ ∈ { ζ, τ, ∆ , ∇} and u is a term, and assumethat f ∗ u O A ( h , . . . , h ℓ ) ∈ F for every f ∈ F . Then also f ∗ t O A ( h , . . . , h ℓ ) = f ∗ ϕu O A ( h , . . . , h ℓ ) ∈ F for every f ∈ F , because F is minor-closed and thefollowing identities hold for any functions f and h (say h is n -ary): f ∗ ζh = π (1 2 ··· n ) ( f ∗ h ) ,f ∗ τ h = τ ( f ∗ h ) ,f ∗ ∆ h = ∆( f ∗ h ) ,f ∗ ∇ h = ∇ ( f ∗ h ) . Finally, consider the case that t = u ∗ v , and assume that f ∗ u O A ( h , . . . , h ℓ ) ∈ F and f ∗ v O A ( h , . . . , h ℓ ) ∈ F for every f ∈ F . Then also f ∗ t O A ( h , . . . , h ℓ ) = f ∗ ( u O A ( h , . . . , h ℓ ) ∗ v O A ( h , . . . , h ℓ )) = ( f ∗ u O A ( h , . . . , h ℓ )) ∗ v O A ( h , . . . , h ℓ ) ∈ F for every f ∈ F .(ii) = ⇒ (i): Let f ∈ F ( n ) and g , . . . , g n ∈ C ( m ) . A simple inductive argumentshows that, in the construction of f ( g , . . . , g n ) as a superposition of { f, g , . . . , g n } given in the proof of Lemma 3.1, the functions f i are in F , because F is minor-closed and each f i is of the form ζϕ ∗ γ for some ϕ ∈ F and γ ∈ G . Finally, f ( g , . . . , g n ) = θf n ∈ F , because F is minor-closed. (cid:3) For left stability, it is enough to consider compositions with generators of theclone.
Lemma 3.3.
Let F ⊆ O A . Let C be a clone on A , and let G be a generating setof C . Then the following conditions are equivalent. (i) CF ⊆ F (ii) g ( f , . . . , f n ) ∈ F whenever g ∈ C ( n ) and f , . . . , f n ∈ F ( m ) for some n, m ∈ N . (iii) g ( f , . . . , f n ) ∈ F whenever g ∈ G ( n ) and f , . . . , f n ∈ F ( m ) for some n, m ∈ N .Proof. (i) ⇐⇒ (ii): Holds by the definition of function class composition.(ii) = ⇒ (iii): Obvious.(iii) = ⇒ (ii): Let g ∈ C . Then there is a term t of the language of the algebra A = ( A ; G ) such that g = t A . We prove the claim by induction on the structureof the term t . Let f , . . . , f n ∈ F ( m ) . The inductive basis holds, because if t = x i ,then t A = pr ( n ) i , and we have pr ( n ) i ( f , . . . , f n ) = f i ∈ F . Consider now the casewhen t = h ( t , . . . , t ℓ ) for some h ∈ G and terms t , . . . , t ℓ , and assume that for TABILITY OF BOOLEAN FUNCTION CLASSES 7 i ∈ { , . . . , ℓ } , we have already shown that t A i ( f , . . . , f n ) ∈ F . It then follows fromsuperassociativity and our assumptions that t A ( f , . . . , f n ) = h A ( t A , . . . , t A ℓ )( f , . . . , f n )= h A ( t A ( f , . . . , f n ) , . . . , t A ℓ ( f , . . . , f n )) ∈ F. (cid:3) Let us record here a simple yet useful observation on the C -stable class generatedby a projection. Lemma 3.4.
For any clone C , h pr (1)1 i C = C .Proof. Since pr (1)1 ∈ C and C is C -stable, we clearly have h pr (1)1 i ⊆ C . ByLemma 3.2(ii), we also have f = pr (1)1 ∗ f ∈ h pr (1)1 i C for every f ∈ C , so C ⊆h pr (1)1 i C . (cid:3) Linear stability over finite fields
In this section we consider classes of operations on an arbitrary finite field andtheir right and left stability under clones of linear functions. Assume that A =GF( q ), a finite field of order q = p m , with p prime. Definition 4.1.
It is well known that every n -ary operation on A is representedby a unique polynomial over GF( q ) in n variables wherein no variable appears withan exponent greater than q −
1. We call such polynomials reduced polynomials.
Areduced polynomial can be written as(1) X ( a ,...,a n ) ∈{ ,...,q − } n α ( a ,...,a n ) Y i ∈{ ,...,n } x a i i , where each coefficient α ( a ,...,a n ) is an element of GF( q ). We will use the shorthand α a x a to designate the monomial α ( a ,...,a n ) Q i ∈{ ,...,n } x a i i with a = ( a , . . . , a n ).A monomial with coefficient 1 is called monic. The degree of a monomial α a x a is P ni =1 a i . The degree of a polynomial p , denoted deg( p ), is the maximum of thedegrees of its monomials with a nonzero coefficient; we agree that deg(0) := 0. Ingeneral, when we speak of the monomials of a polynomial, we mean the monomialswith a nonzero coefficient. As is usual when writing polynomials, we may omitcoefficients equal to 1, and we may omit monomials with coefficient 0. Withoutany risk of confusion, we will denote functions by reduced polynomials.The degree of an operation f , denoted deg( f ), is the degree of the unique reducedpolynomial representing f . For k ∈ N , denote by D k the set of all operations on A of degree at most k . Clearly, these sets constitute an infinite ascending chain D ⊂ D ⊂ D ⊂ · · · whose union is the set O A of all operations on A . In particular, D is the set of all constant operations, and D is the set of all linear operations. We shall also use the symbol L to denote the set D . The set L is a clone on A ; infact, it is a maximal clone according to Rosenberg’s classification [17]. Proposition 4.2.
For every k ∈ N , the set D k is L -stable.Proof. Noting that the clone L is generated by { x + x } ∪ { cx | c ∈ A } ∪ { c | c ∈ A } , we apply Lemmata 3.3 and 3.2. The stability under left composition with L follows from the fact that for any f, g ∈ D k and any c ∈ A we have c ( f ) = c ∈ D ⊆ D k , cx ( f ) = c · f ∈ D k , and ( x + x )( f, g ) = f + g ∈ D k . As for the right stability,note that D k is minor-closed because the formation of minors does not increase thedegree of functions, and that for any f ∈ D k and for any c ∈ A , it holds that f ∗ c , f ∗ cx , and f ∗ ( x + x ) are members of D k . (cid:3) Strictly speaking, operations of degree at most 1 are affine in the sense of linear algebra. Wego along with the term linear that is common in the context of clone theory and especially in thetheory of Boolean functions.
STABILITY OF BOOLEAN FUNCTION CLASSES
Proposition 4.3.
The empty set ∅ and the set O A of all operations on A are L -stable.Proof. Trivial. (cid:3)
Lemma 4.4.
Every nonempty L -stable class contains all constant functions.Proof. Let K be a nonempty L -stable class. Since L contains all projections ofany arity, K L contains functions of any arity, and so does K because K L ⊆ K .Note that for any g , . . . , g n ∈ O ( m ) A , it holds that c ( n ) b ( g , . . . , g n ) = c ( m ) b . Since allconstant functions are members of L and K contains functions of any arity, it followsthat L K contains all constant functions, and so does K because L K ⊆ K . (cid:3) Lemma 4.5.
For any k ∈ N , h x x . . . x k i L = D k .Proof. Clearly x x . . . x k ∈ D k and D k is L -stable by Proposition 4.2, so we have h x x . . . x k i L ⊆ D k . By identification of variables, permutation of variables, andsubstitution of constant 1 for variables, we obtain every monic monomial of degreeat most k . By taking the sum of monic monomials of degree at most k , with suitablecoefficients, we can obtain any polynomial of degree at most k , in other words, bycomposing a suitable linear function with functions represented by monic monomialsof degree at most k , we obtain any function of degree at most k . Therefore, D k ⊆h x x . . . x k i L . (cid:3) Lemma 4.6.
If the reduced polynomial of f : A n → A has degree k , then h f i L = D k .Proof. Let p be the reduced polynomial representing f as in (1). Let u = ( u , . . . ,u n ) ∈ { , . . . , q − } n be such that α u x u has degree k and α u = 0. We may assumethat α u = 1, because by composing f from the left by α − u x , which belongs to L ,we obtain a function in h f i L that has the same monomials as f but with coefficientsmultiplied by α − u .Let U := { i ∈ [ n ] | u i = 0 } . By substituting 0 for the variables x i with i ∈ [ n ] \ U , we obtain a function f ′ in h f i L with reduced polynomial p ′ such that p ′ has degree k and contains only variables x i with i ∈ U , and α u x u is a monomial ofdegree k in p ′ . We may consider the function f ′ in place of f and assume, withoutloss of generality, that U = [ n ].Let { B , . . . , B n } be a partition of [ k ] in n parts such that | B j | = u j for all j ∈ [ n ]. For j ∈ [ n ], let g j = P i ∈ B j x i . Note that g j ∈ L . Consider the function h := f ( g , . . . , g n ), which is in h f i L . For every a ∈ { , . . . , q − } n with P ni =1 a i ≤ k ,the expansion of the product Q ni =1 g a i i results in a polynomial of degree at most k inwhich no monomial contains all variables x , . . . , x k , with the exception of a = u , forwhich the expansion yields a polynomial in which one of the monomials is x . . . x k and the other monomials do not contain all variables x , . . . , x k . Consequently, h = x . . . x k + h ′ where h ′ is a polynomial in variables x , . . . , x k in which nomonomial contains all variables x , . . . , x k .Now, let us define a sequence of functions h , . . . , h k recursively as follows: h := h . For i = 1 , . . . , k , let h i := h i − − h i − ( x , . . . , x i − , , x i +1 , . . . , x k ). We have h i ∈ h h i − i L . It is easy to see that the polynomial of h i can be obtained fromthe polynomial of h i − by removing all monomials in which x i does not occur.Consequently, x . . . x k = h k ∈ h h k − i L ⊆ h h k − i L ⊆ · · · ⊆ h h i L ⊆ h f i L . Now itfollows from Lemma 4.5 that D k = h x . . . x k i L ⊆ h f i L ⊆ D k . (cid:3) Lemma 4.7.
Let K ⊆ O A , K = ∅ . If the set { deg( f ) | f ∈ K } has a maximum m , then h K i L = D m . Otherwise h K i L = O A . TABILITY OF BOOLEAN FUNCTION CLASSES 9
Proof.
If said maximum m exists, we have K ⊆ D m and there exists a g ∈ K withdeg( g ) = m . Since D m is L -stable by Lemma 4.2, we have h K i L ⊆ D m . Lemma 4.6implies D m = D deg( g ) ⊆ [ f ∈ K D deg( f ) = [ f ∈ K h f i L ⊆ h K i L ⊆ D m . Otherwise there is no finite upper bound on the degrees of the members of K .Then for every i ∈ N , there exists an f i ∈ K with deg( f i ) ≥ i . Now we have O A = [ i ∈ N D i ⊆ [ i ∈ N D deg( f i ) = [ i ∈ N h f i i L ⊆ h K i L ⊆ O A . (cid:3) Theorem 4.8.
The L -stable classes are O A , D k , and ∅ , for k ∈ N .Proof. The classes mentioned in the statement are L -stable by Propositions 4.2 and4.3. Lemma 4.7 implies that there are no further L -stable classes. (cid:3) Boolean functions
Definition 5.1.
Operations on { , } are called Boolean functions.
The class ofall Boolean functions is denoted by Ω . Definition 5.2.
By particularizing Definition 4.1 to the two-element field, we ob-tain that every Boolean function is represented by a unique multilinear polynomial over the two-element field, i.e., a polynomial with coefficients in GF(2) in whichno variable appears with an exponent greater than 1. Since the coefficients comefrom the set { , } , every monomial with a nonzero coefficient is monic. The uniquemultilinear polynomial representing a Boolean function f is known as the Zhegalkinpolynomial of f , and it can be written as(2) X S ∈ M f x S , where x S is a shorthand for Q i ∈ S x i and M f ⊆ P ([ n ]). Note that x ∅ = 1 and P S ∈∅ x S = 0. The terms x S with S = ∅ are called monomials. If ∅ ∈ M f , then wesay that f has constant term
1; otherwise f has constant term
0. Without any riskof confusion, we will denote Boolean functions by their Zhegalkin polynomials, andwe refer to the set M f as the set of monomials of f . Definition 5.3.
Some well-known Boolean functions are defined in Table 1: mod-ulo-2 addition +, conjunction ∧ , disjunction ∨ , triple sum ⊕ , median µ . TheirZhegalkin polynomials are the following: x + x ,x ∧ x = x x , ⊕ ( x , x , x ) = x + x + x ,x ∨ x = x x + x + x , µ ( x , x , x ) = x x + x x + x x . Definition 5.4.
For a, b ∈ { , } , let Ω a ∗ := { f ∈ Ω | f (0 , . . . ,
0) = a } , Ω ∗ b := { f ∈ Ω | f (1 , . . . ,
1) = b } , and let Ω ab := Ω a ∗ ∩ Ω ∗ b . Furthermore, define Ω = := { f ∈ Ω | f (0 , . . . ,
0) = f (1 , . . . , } , Ω = := { f ∈ Ω | f (0 , . . . , = f (1 , . . . , } , that is, Ω = = Ω ∪ Ω and Ω = = Ω ∪ Ω .Clearly Ω ∗ ∩ Ω ∗ = ∅ and Ω ∗ ∪ Ω ∗ = Ω ; similarly, Ω ∗ ∩ Ω ∗ = ∅ and Ω ∗ ∪ Ω ∗ = Ω , and Ω = ∩ Ω = = ∅ and Ω = ∪ Ω = = Ω . It is easy to see that Ω a ∗ is the class of allBoolean functions with constant term a . x y x + y x ∧ y x ∨ y x y z ⊕ ( x, y, z ) µ ( x, y, z )0 0 0 0 00 0 1 1 00 1 0 1 00 1 1 0 11 0 0 1 01 0 1 0 11 1 0 0 11 1 1 1 1 Table 1.
Well-known Boolean functions
Definition 5.5.
For a ∈ { , } , a Boolean function f is a -preserving if f ( a, . . . , a ) = a . A function is constant-preserving if it is both 0- and 1-preserving. We denotethe classes of all 0-preserving, of all 1-preserving, and of all constant-preservingfunctions by T , T , and T c , respectively. Note that T c = T ∩ T . It follows fromthe definitions that T = Ω ∗ , T = Ω ∗ , and T c = Ω . Remark 5.6.
The reason why we have introduced multiple notation for the classes T = Ω ∗ and T = Ω ∗ is to facilitate writing certain statements in a parameterizedform and to make reference, as the case may be, to either the classes Ω a ∗ ( a ∈{ , } ), Ω ∗ b ( b ∈ { , } ), or T a ( a ∈ { , } ). Definition 5.7.
The parity of a Boolean function f , denoted par( f ), is a number,either 0 or 1, which is given bypar( f ) := | M f \ {∅}| mod 2 . We call a function even or odd if its parity is 0 or 1, respectively. Note that Ω = and Ω = are precisely the classes of even and odd functions, respectively. Definition 5.8.
The set { , } is endowed with the natural order ≤ , with 0 < ≤ , on the Cartesian power { , } n : for ( a , . . . , a n ) , ( b , . . . , b n ) ∈ { , } n , ( a , . . . , a n ) ≤ ( b , . . . , b n ) if andonly if a i ≤ b i for all i ∈ [ n ].A Boolean function f : { , } n → { , } is monotone if f ( a ) ≤ f ( b ) whenever a ≤ b . We denote by M the class of all monotone Boolean functions. Definition 5.9.
For a ∈ { , } , let a denote the negation of a , that is, a := 1 − a .For any f ∈ Ω ( n ) , denote by f the negation of f , that is, the function f : { , } n →{ , } with f ( a ) = f ( a ) for all a ∈ { , } n . For C ⊆ Ω , let C := { f | f ∈ C } .A function f is self-dual if f ( a , . . . , a n ) = f ( a , . . . , a n ) for all a , . . . , a n ∈{ , } . A function f is reflexive (or self-anti-dual ) if f ( a , . . . , a n ) = f ( a , . . . , a n )for all a , . . . , a n ∈ { , } . We denote by S the class of all self-dual functions. Let S c := S ∩ T c and SM := S ∩ M , the classes of constant-preserving self-dual functionsand monotone self-dual functions, respectively. Definition 5.10.
By particularizing the definition of degree (see Definition 4.1) tomonomials and polynomials over GF(2), we obtain that the degree of a monomial x S is just | S | , and the degree of a Boolean function f is the size of the largestmonomial in the Zhegalkin polynomial of f , i.e., deg( f ) := max S ∈ M f | S | for f = 0,and we agree that deg(0) := 0. As before, for k ∈ N , we denote by D k the class ofall Boolean functions of degree at most k . Clearly D k ( D k +1 for all k ∈ N . TABILITY OF BOOLEAN FUNCTION CLASSES 11
A Boolean function f is linear if deg( f ) ≤
1. We denote by L the class of alllinear functions. Thus L = D . We also let L := L ∩ T = L ∩ Ω ∗ , L := L ∩ T = L ∩ Ω ∗ , LS := L ∩ S = L ∩ Ω = , L c := L ∩ T c = L ∩ Ω . The equalities in the above definitions are clear by Remark 5.6, except for theequality LS = L ∩ Ω = which is easy to verify and also follows from Lemma 5.12below. Definition 5.11.
Let f be an n -ary Boolean function. The characteristic of a set S ⊆ [ n ] in f is a number, either 0 or 1, which is given bych( S, f ) := |{ A ∈ M f | S ( A }| mod 2 . The characteristic rank of f , denoted by χ ( f ), is the smallest integer m such thatch( S, f ) = 0 for all subsets S ⊆ [ n ] with | S | ≥ m . Clearly χ ( f ) ≤ n becausech([ n ] , f ) = 0.For k ∈ N , denote by X k the class of all Boolean functions of characteristic rankat most k . For any k ∈ N , we have X k ( X k +1 . The inclusion is proper, as witnessedby the function x . . . x k +1 ∈ X k +1 \ X k . Moreover, for any k ∈ N , we have D k ⊆ X k .Reflexive and self-dual functions have a beautiful characterization in terms ofthe characteristic rank. Lemma 5.12 (Selezneva, Bukhman [18, Lemmata 3.1, 3.5]) . (i) A Boolean function f is reflexive if and only if χ ( f ) = 0 . (ii) A Boolean function f is self-dual if and only if f + x is reflexive. (iii) A Boolean function f is self-dual if and only if f is odd and χ ( f ) = 1 . In other words, X = X ∩ Ω = is the class of all reflexive functions, X ∩ Ω = isthe class of all self-dual functions, and X is the class of all self-dual or reflexivefunctions. Definition 5.13.
Let Λ c and V c denote the classes of all conjunctions of argumentsand of all disjunctions of arguments, respectively, that is, Λ c := { f ∈ Ω ( n ) | n ∈ N + , ∅ 6 = { i , . . . , i r } ⊆ [ n ] , f ( a , . . . , a n ) = a i ∧ · · · ∧ a i r } , V c := { f ∈ Ω ( n ) | n ∈ N + , ∅ 6 = { i , . . . , i r } ⊆ [ n ] , f ( a , . . . , a n ) = a i ∨ · · · ∨ a i r } . Let I c , I , I , and I ∗ denote the class of all projections, the class of all projectionsand constant 0 functions, the class of all projections and constant 1 functions, andthe class of all projections and negated projections, respectively, that is, I c := { pr ( n ) i | i, n ∈ N + , ≤ i ≤ n } , I := I c ∪ { c ( n )0 | n ∈ N + } , I := I c ∪ { c ( n )1 | n ∈ N + } , I ∗ := I c ∪ I c . It was shown by Post [15] that there are a countably infinite number of clones ofBoolean functions. In this paper, we will only need a handful of them, namely theclones Ω , T , T , T c , M , S , S c , SM , L , L , L , LS , L c , Λ c , V c , I ∗ , I , I , and I c thatwere defined above. The lattice of clones of Boolean functions, the so-called Post’slattice, is shown in Figure 1, and the above-mentioned clones are indicated in thediagram. In what follows, we will often make use of the following generating sets I c I ∗ I I L c LS L L LSMS c SM Λ c V c T c T T Ω Figure 1.
Post’s lattice.for some of these clones. Ω = h x x + 1 i , S = h µ, x + 1 i , SM = h µ i , L = h x + x , i , LS = h⊕ , x + 1 i , L c = h⊕ i , Λ c = h∧i , V c = h∨i , I ∗ = h x + 1 i , I = h i , I = h i , I c = h∅i . Let us conclude this introductory section with a couple of lemmata that help usexpress sums and minors of Boolean functions in terms of their sets of monomials.
Lemma 5.14.
Let f : { , } n → { , } and f : { , } n → { , } . Then M f + g = M f △ M g .Proof. By adding the polynomials of f and g and by cancelling equal monomials(because we do addition modulo 2), we obtain f + g = X S ∈ M f x S + X S ∈ M g x S = X S ∈ M f △ M g x S . Consequently, M f + g = M f △ M g by the uniqueness of Zhegalkin polynomials. (cid:3) Lemma 5.15.
Let f : { , } n → { , } and σ : [ n ] → [ m ] . Then M f σ = (cid:8) S ⊆ [ m ] (cid:12)(cid:12) |{ T ∈ M f | σ ( T ) = S }| ≡ (cid:9) . Proof.
A straightforward calculation using the definitions of minor and M f (Defi-nitions 2.5 and 5.2) shows that for all a , . . . , a m ∈ { , } , f σ ( a , . . . , a m ) = f ( a σ (1) , . . . , a σ ( n ) ) = X T ∈ M f Y i ∈ T a σ ( i ) = X T ∈ M f Y i ∈ σ ( T ) a i TABILITY OF BOOLEAN FUNCTION CLASSES 13 ΩΩ = Ω ∗ Ω ∗ Ω ∗ Ω ∗ Ω = Ω Ω Ω Ω Figure 2.
A block of eleven L c -stable classes.By cancelling pairs of summands corresponding to indices T, T ′ ∈ M f such that σ ( T ) = σ ( T ′ ), which are equal for any a , . . . , a m , we get X T ∈ M f Y i ∈ σ ( T ) a i = X S ∈ M ′ Y i ∈ S a i , where M ′ = (cid:8) S ⊆ [ m ] (cid:12)(cid:12) |{ T ∈ M f | σ ( T ) = S }| ≡ (cid:9) . Consequently, M f σ = M ′ by the uniqueness of Zhegalkin polynomials. (cid:3) L c -stable classes We are now ready to state the main result of this paper, a complete descriptionof the L c -stable classes of Boolean functions. Theorem 6.1.
The L c -stable classes or, equivalently, the ( I c , L c ) -stable classes are Ω , Ω a ∗ , Ω ∗ b , Ω ≈ , Ω ab , D k , D k ∩ Ω a ∗ , D k ∩ Ω ∗ b , D k ∩ Ω ≈ , D k ∩ Ω ab , X k , X k ∩ Ω a ∗ , X k ∩ Ω ∗ b , X k ∩ Ω ≈ , X k ∩ Ω ab , D i ∩ X j , D i ∩ X j ∩ Ω a ∗ , D i ∩ X j ∩ Ω ∗ b , D i ∩ X j ∩ Ω ≈ , D i ∩ X j ∩ Ω ab , D , D ∩ Ω a ∗ , ∅ , for a, b ∈ { , } , ≈ ∈ { = , = } , and i, j, k ∈ N + with i > j ≥ . Several L c -stable classes were known previously: the clones Ω , S = X ∩ Ω = , L = D , T = Ω ∗ , T = Ω ∗ , T c = Ω , S c = X ∩ Ω , L = D ∩ Ω ∗ , L = D ∩ Ω ∗ , LS = D ∩ X ∩ Ω = , L c = D ∩ Ω , as well as the classes D k for any k ∈ N andthe class X of reflexive (self-anti-dual) functions [4, pp. 29, 33]. The classes D k for k ∈ N were also known to be L -stable [6, Example 1, p. 111].In order to describe the structure of the lattice of L c -stable classes, it is helpfulto first look at the poset comprising the eleven classes Ω , Ω = , Ω = , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω , Ω , Ω , Ω that is shown in Figure 2. It is noteworthy that the fourminimal classes of this poset are pairwise disjoint, and that the six lower covers of Ω are precisely the unions of the six different pairs of minimal classes.The lattice of all L c -stable classes is shown in Figure 3. It has rather regularstructure; it is isomorphic to the direct product of the 11-element poset of Figure 2and the set { ( i, j ) ∈ ( N + ∪ {∞} ) | i ≥ j ≥ } with the componentwise order, anda few additional elements near the bottom of the lattice. In order to avoid clutter,we have used some shorthand notation in Figure 3. The diagram includes multiplecopies of the 11-element poset of Figure 2 (the shaded blocks) connected by thick triple lines. Each thick triple line between a pair of blocks represents eleven edges,each connecting a vertex of one poset to its corresponding vertex in the other poset.We have labeled in the diagram the meet-irreducible classes, as well as a few otherclasses of interest; the remaining classes are intersections of the meet-irreducibleones.The remainder of this section is devoted to the proof of Theorem 6.1. The proofhas two parts. First we observe that the classes listed in Theorem 6.1 are L c -stable.Secondly, we need to show that there are no other L c -stable classes.To this end, we start with verifying that the classes of Theorem 6.1 are L c -stable. Since intersections of L c -stable classes are L c -stable, it suffices to verify thisfor the meet-irreducible classes. With the help of the following lemma, we canfurther simplify the task of checking the stability under left and right compositionwith clones containing the triple sum. In fact L c -stability is equivalent to ( I c , L c )-stability. Lemma 6.2. (i)
For any f ∈ Ω , we have f ∗ ⊕ = ⊕ ( f σ , f σ , f σ ) , where, for i ∈ [3] , σ i : [ n ] → [ n + 2] , i , j j + 2 for ≤ j ≤ n . (ii) Let G ⊆ Ω , let C := h G ∪ {⊕ }i , C ′ := h G i , and let C be a clonecontaining ⊕ . Then a class F ⊆ Ω is ( C , C ) -stable if and only if it is ( C ′ , C ) -stable. (iii) The following are equivalent for a class F ⊆ Ω . (a) F is L c -stable. (b) F is ( I c , L c ) -stable. (c) F is minor-closed and f + g + h ∈ F whenever f, g, h ∈ F .Proof. (i) Let A i := σ i ( { S ∈ M f | ∈ S } ) for i ∈ [3], B := σ ( { S ∈ M f | / ∈ S } ) = σ ( { S ∈ M f | / ∈ S } ) = σ ( { S ∈ M f | / ∈ S } ) . Since the sets A , A , A , B are pairwise disjoint, their union equals their symmetricdifference. Using the commutativity and associativity of the symmetric differenceand the fact that X △ X = ∅ and X △ ∅ = X for any set X , we obtain M f ∗⊕ = A ∪ A ∪ A ∪ B = A △ A △ A △ B = A △ A △ A △ B △ B △ B = ( A △ B ) △ ( A △ B ) △ ( A △ B ) = ( A ∪ B ) △ ( A ∪ B ) △ ( A ∪ B )= M f σ △ M f σ △ M f σ = M f σ + f σ + f σ = M ⊕ ( f σ ,f σ ,f σ ) , that is, f ∗ ⊕ = ⊕ ( f σ , f σ , f σ ).(ii) Since C ′ ⊆ C , stability under right composition with C implies stabil-ity under right composition with C ′ . Assume now that F is ( C ′ , C )-stable. ByLemma 3.2, F is minor-closed and f ∗ g ∈ F whenever f ∈ F and g ∈ G . More-over, f ∗ ⊕ = ⊕ ( f σ , f σ , f σ ), where f σ , f σ , f σ are the minors of f specifiedin part (i). Since F is minor-closed, we have f σ , f σ , f σ ∈ F . By our assump-tion, ⊕ ∈ C , and since F is stable under left composition with C , it followsthat ⊕ ( f σ , f σ , f σ ) ∈ F . It follows from Lemma 3.2 that F is stable under rightcomposition with C .(iii) Since L c = h⊕ i , this is a consequence of part (ii) and Lemma 3.3. (cid:3) In view of Lemma 6.2(iii), our task is reduced to verifying that each one of themeet-irreducible classes shown in Figure 3, namely Ω , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω = , Ω = , D k , and X k for k ∈ N , is minor-closed and closed under triple sums of its members. Lemma 6.3. Ω is minor-closed and closed under triple sums of its members.Proof. Trivial. (cid:3)
TABILITY OF BOOLEAN FUNCTION CLASSES 15 D D ∩ X D D ∩ X D ∩ X D D ∩ X D ∩ X D ∩ X D X X X X ΩΩ = Ω ∗ Ω ∗ Ω ∗ Ω ∗ Ω = T c X SS c L L LSL c D ∅ Figure 3. L c -stable classes. Lemma 6.4.
Let a, b ∈ { , } . (i) Ω a ∗ is minor-closed and closed under triple sums of its members. (ii) Ω ∗ b is minor-closed and closed under triple sums of its members.Proof. (i) Let f ∈ Ω ( n ) a ∗ , and let σ : [ n ] → [ m ]. We have f σ (0 , . . . ,
0) = f (0 , . . . ,
0) = a , so f σ ∈ Ω a ∗ ; thus Ω a ∗ is minor-closed. Let now f, g, h ∈ Ω ( n ) a ∗ . We have ( f + g + h )(0 , . . . ,
0) = f (0 , . . . ,
0) + g (0 , . . . ,
0) + h (0 , . . . ,
0) = a + a + a = a ; thus f + g + h ∈ Ω a ∗ .(ii) The proof is similar to that of part (i). (cid:3) Lemma 6.5.
For ≈ ∈ { = , = } , Ω ≈ is minor-closed and closed under triple sums ofits members.Proof. We show first that Ω ≈ is minor-closed. Let f : { , } n → { , } and σ : [ n ] → [ m ]. For each T ⊆ [ m ], let Z T := { S ∈ M f | σ ( T ) = S } ; clearly the sets Z T arepairwise disjoint and their union is M f . By Lemma 5.15, | Z T | is odd if and onlyif T ∈ M f σ ; thus for each T ⊆ [ m ], there exists a number k T ∈ N such that | Z T | = 2 k T + 1 if T ∈ M f σ and | Z T | = 2 k T if T / ∈ M f σ . Consequently, | M f | = X T ⊆ [ m ] | Z T | = X T ⊆ [ m ] T ∈ M fσ (2 k Z + 1) + X T ⊆ [ m ] T / ∈ M fσ k Z ≡ X T ⊆ [ m ] T ∈ M fσ X T ⊆ [ m ] T / ∈ M fσ | M f σ | (mod 2) . Note also that Z ∅ = {∅} ; thus ∅ ∈ M f σ if and only if ∅ ∈ M f . It follows thatpar( f σ ) = par( f ). Therefore f σ ∈ Ω ≈ if and only if f ∈ Ω ≈ , that is, Ω ≈ is minor-closed.We now show that Ω ≈ is closed under triple sums of its members. Let f, g, h ∈ Ω ( n ) ≈ . By definition, it holds that par( f ) = | M f \ {∅}| mod 2 = a , par( g ) = | M g \ {∅}| mod 2 = a , par( h ) = | M h \ {∅}| mod 2 = a , where a = 0 if ≈ is =and a = 1 otherwise. Then M f + g + h = M f △ M g △ M h by Lemma 5.14, so M f + g + h \{∅} = ( M f \{∅} ) △ ( M g \{∅} ) △ ( M h \{∅} ), which implies par( f + g + h ) = | M f + g + h \ {∅}| mod 2 = a + a + a mod 2 = a . Therefore f + g + h ∈ Ω ≈ . (cid:3) Lemma 6.6.
For k ∈ N , D k is minor-closed and closed under sums of its members.Proof. Let f ∈ D ( n ) k , and let σ : [ n ] → [ m ]. Let T ∈ M f σ be such that | T | = deg f σ .By Lemma 5.15, there exists U ∈ M f such that σ ( U ) = T . We must have | T | ≤ | U | ,so deg f σ = | T | ≤ | U | ≤ deg f ≤ k ; therefore f σ ∈ D k , so D k is minor-closed.Let now f, g ∈ D ( n ) k . Since M f + g = M f △ M g by Lemma 5.14, we have deg( f + g ) = max S ∈ M f + g | S | ≤ max(deg( f ) , deg( g )) ≤ k , so f + g ∈ D k . (cid:3) Lemma 6.7.
For k ∈ N , X k is minor-closed.Proof. In view of Remark 2.7, it is sufficient to consider closure under minors formedvia injective maps or identification maps (see Definition 2.5). Let f ∈ X ( n ) k .Consider first an injective minor formation map σ : [ n ] → [ m ]. It is easy toverify that M f σ = { σ ( A ) | A ∈ M f } . Let S ⊆ [ m ] with | S | ≥ k . If S * Im σ , thenthere is clearly no A ∈ M f σ such that S * A ; hence ch( S, f σ ) = 0. If S ⊆ Im σ ,then ch( S, f σ ) = ch( σ − ( S ) , f ) = 0 because | σ − ( S ) | = | S | ≥ k and f ∈ X k . Weconclude that f σ ∈ X k .Consider now an identification map σ ij : [ n ] → [ n −
1] for some i, j ∈ [ n ] with i < j . Let S ⊆ [ n −
1] with | S | ≥ k . Let H := { A ∈ M f | S ( σ ij ( A ) } . We claimthat | H | is even. TABILITY OF BOOLEAN FUNCTION CLASSES 17 If i / ∈ S , then the only subset of [ n ] mapped onto S by σ ij is σ − ij ( S ), andany proper superset of σ − ij ( S ) is mapped to a proper superset of S . Therefore H = { A ∈ M f | σ − ij ( S ) ( A } . Since | σ − ( S ) | = | S | ≥ k and f ∈ X k , it followsthat | H | is even.If i ∈ S , then there are three subsets of [ n ] that are mapped onto S by σ ij : T := σ − ij ( S ), T := σ − ij ( S ) \ { i } , and T := σ − ij ( S ) \ { j } . The proper supersets of T , T , and T are mapped to proper supersets of S , with the exception of the set T = σ − ij ( S ), which is a proper superset of both T and T but σ ij ( T ) = S . Since | σ − ij ( S ) | = | S | + 1 ≥ k + 1 and f ∈ X k , it follows that the sets U i := { A ∈ M f | T i ( A } , 1 ≤ i ≤
3, have even cardinality. Let U ′ i := U i \ { T } , i ∈ { , } , andobserve that U = U ′ ∩ U ′ . We have H = U ′ ∪ U ′ , so | H | = | U ′ | + | U ′ | − | U | = ( | U | + | U | − | U | , if T / ∈ M f , | U | + | U | − | U | − , if T ∈ M f .In either case, | H | is even.For any subset A ⊆ [ n − Z A := { T ⊆ M f | σ ij ( T ) = A } . By definition,we have H = S S ( A ⊆ [ n − Z A . The sets Z A are clearly pairwise disjoint, so | H | = P S ( A ⊆ [ n − | Z A | . Since | H | is even, there must be an even number of sets A with S ( A ⊆ [ n −
1] such that | Z A | is odd. It follows from Lemma 5.15 thatch( S, f σ ij ) = |{ A ∈ M f σij | S ( A }| mod 2 = 0, and we conclude that f σ ij ∈ X k . (cid:3) Lemma 6.8.
Let k ∈ N . For any f, g ∈ X ( n ) k , we have f + g ∈ X k .Proof. Write h := f + g . We have M h = M f △ M g by Lemma 5.14, and for any S ⊆ [ n ], it holds that { A ∈ M h | S ( A } = { A ∈ M f △ M g | S ( A } = { A ∈ M f | S ( A } △ { A ∈ M g | S ( A } . By our assumption, for any S ⊆ [ n ] with | S | ≥ k , we have |{ A ∈ M f | S ( A }| mod 2 = ch( S, f ) = 0 , |{ A ∈ M g | S ( A }| mod 2 = ch( S, g ) = 0 . Since the symmetric difference of sets of even cardinality is again of even cardinality,it follows that ch(
S, h ) = |{ A ∈ M h | S ( A }| mod 2 = 0 for any S ⊆ [ n ] with | S | ≥ k . Therefore h ∈ X k . (cid:3) Proposition 6.9.
The classes listed in Theorem 6.1 are L c -stable.Proof. According to Lemmata 6.3, 6.4, 6.5, 6.6, 6.7, and 6.8, each of the classes Ω , Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω = , Ω = , D k , and X k for k ∈ N is minor-closed and closedunder triple sums of its members, so by Lemma 6.2(iii), each is L c -stable. It followsthat the remaining classes listed in Theorem 6.1, being intersections of the aboveclasses, are also L c -stable. (cid:3) It remains to show that the classes listed in Theorem 6.1 are the only L c -stableclasses. To this end, we are going to verify that any set of Boolean functionsgenerates exactly what is suggested by Figure 3. More precisely, we prove thateach class K is generated by any subset of K that is not contained in any propersubclass fo K , i.e., the subset contains for each proper subclass C of K an elementin K \ C . If each proper subclass is contained in a lower cover of K , then it sufficesto consider the lower covers of K . We begin with some helpful lemmata. Lemma 6.10.
For any F ⊆ Ω , we have f ∈ h F i L c if and only if f is the sum ofan odd number of minors of members of F , i.e., f = P k +1 i =1 ( g i ) σ i for some k ∈ N , g i ∈ F , σ i : [ n i ] → [ n ] , where n i := ar( g i ) and n := ar( f ) (1 ≤ i ≤ k + 1) .Proof. “ ⇐ ”: Clear because h F i L c is closed under minors and triple sums and henceunder any odd sums of its members by Lemma 6.2(iii).“ ⇒ ”: By Lemma 6.2(iii), h F i L c is the set obtained by a finite number of thefollowing construction steps:(1) Every f ∈ F is a member of h F i L c .(2) If f ∈ h F i L c , ar( f ) = n , and σ : [ n ] → [ m ] for some m ∈ N + , then f σ ∈h F i L c .(3) If f, g, h ∈ h F i L c , all of arity n ∈ N + , then f + g + h ∈ h F i L c .We will show by induction on the construction that every f ∈ h F i L c is an odd sumof minors of members of F . This obviously holds for every f ∈ F : f = P i =1 f id .Assume f = P k +1 i =1 ( g i ) σ i for some g i ∈ F and σ i : [ n i ] → [ n ] (1 ≤ i ≤ k + 1).Then for any τ : [ n ] → [ m ], we have f τ = (cid:0) k +1 X i =1 ( g i ) σ i (cid:1) τ = k +1 X i =1 (( g i ) σ i ) τ = k +1 X i =1 ( g i ) τ ◦ σ i , where the second and the third equalities hold by Lemmata 2.8 and 2.6, respectively.Finally, assume that f = P k +1 i =1 ( f i ) σ i , g = P ℓ +1 i =1 ( g i ) τ i , h = P m +1 i =1 ( h i ) ρ i for some f i , g i , h i ∈ F , σ i : [ar( f i )] → [ n ], τ i : [ar( g i )] → [ n ], ρ i : [ar( h i )] → [ n ]. Then f + g + h = k +1 X i =1 ( f i ) σ i + ℓ +1 X i =1 ( g i ) τ i + m +1 X i =1 ( h i ) ρ i , which is an odd sum of minors of members of F . (cid:3) Lemma 6.11.
Assume that C is an L c -stable class and h F i L c = C . Then C is L c -stable and h F i L c = C .Proof. Assume that h F i L c = C . Then C is L c -stable because for all n -ary f +1 , g + 1 , h + 1 ∈ C , we have f, g, h ∈ C and hence ( f + 1) + ( g + 1) + ( h + 1) =( f + g + h ) + 1 ∈ C , and for any σ : [ n ] → [ m ], we have, by Lemma 2.8, ( f + 1) σ = f σ + 1 σ = f σ + 1 ∈ C .In order to show that C is generated by F , let f + 1 ∈ C . Then f ∈ C , and byLemma 6.10, f = P k +1 i =1 ( g i ) σ i for some g i ∈ F and some minor formation map σ i (1 ≤ i ≤ k + 1). Consequently, f + 1 = P k +1 i =1 (( g i ) σ i + 1) = P k +1 i =1 (( g i ) σ i + 1 σ i ) = P k +1 i =1 ( g i + 1) σ i by Lemma 2.8. Since each g i + 1 is in F , Lemma 6.10 implies that f ∈ h F i L c . (cid:3) Proposition 6.12. (i) h∅i L c = ∅ . (ii) For any f ∈ D ∩ Ω ∗ , we have h f i L c = D ∩ Ω ∗ . (iii) For any f ∈ D ∩ Ω ∗ , we have h f i L c = D ∩ Ω ∗ . (iv) For any f, g ∈ D such that f / ∈ Ω ∗ , g / ∈ Ω ∗ , we have h f, g i L c = D .Proof. (i) Obvious.(ii) The function f is a constant 0 function of some arity. We obtain any constant0 function by identifying arguments or introducing fictitious arguments. Therefore D ∩ Ω ∗ ⊆ h f i L c ⊆ D ∩ Ω ∗ .(iii) Follows from part (ii) by Lemma 6.11 because D ∩ Ω ∗ = D ∩ Ω ∗ . TABILITY OF BOOLEAN FUNCTION CLASSES 19 (iv) Since Ω ∗ and Ω ∗ partition Ω , it follows that f ∈ D ∩ Ω ∗ and g ∈ D ∩ Ω ∗ .By parts (ii) and (iii), D = ( D ∩ Ω ∗ ) ∪ ( D ∩ Ω ∗ ) = h g i L c ∪ h f i L c ⊆ h f, g i L c ⊆ D . (cid:3) Lemma 6.13.
Let f ∈ Ω with n := ar( f ) . Let k ∈ N . (i) If n > deg( f ) , then f has a minor of degree deg( f ) and arity deg( f ) + 1 . (ii) If f ∈ X k and deg( f ) > k , then n > deg( f ) . (iii) If f ∈ X k and n − f ) > k , then M f contains all subsets of [ n ] ofcardinality n − . (iv) If f ∈ X k \ X k − , then f has a k -ary minor g such that [ k ] ∈ M g and g ∈ D k \ X k − . (v) If f ∈ X k and there is an S ∈ M f with ℓ := | S | > k , then f has an ( ℓ + 1) -ary minor g such that M g contains all subsets of [ ℓ + 1] of cardinality ℓ but [ ℓ + 1] / ∈ M g . Moreover, if ℓ > k + 1 , then M g contains also a subset ofcardinality ℓ − . (vi) If deg f = n , then f has a minor of arity n − and degree n − .Proof. (i) Let m := deg( f ). There exists an S ∈ M f with | S | = m . Let us identifyall arguments not in S , i.e., we form the minor f σ with a minor formation map σ : [ n ] → [ m + 1] that maps S onto [ m ] and every element of [ n ] \ S to m + 1.Then f σ has arity m + 1. Clearly every monomial of f σ has degree at most m , and[ m ] ∈ M f σ ; hence deg( f σ ) = m .(ii) Clearly n = ar( f ) ≥ deg( f ). Assume that n > k , and suppose, to the con-trary, that n = deg( f ). But then ch([ n − , f ) = 1 and | [ n − | ≥ k , contradicting f ∈ X k .(iii) Assume that n − f ) > k . Then there exists an S ∈ M f with | S | = n −
1. Let A ⊆ S with | A | = n −
2. Since n − ≥ k and f ∈ X k , there must bean even number of proper supersets of A in M f . We already have S ∈ M f , so theremust be another one. In fact there is only one other possibility, namely A ∪ { i } ,where i is the unique element of [ n ] \ S . By letting A range over all ( n − S , we conclude that M f indeed contains all subsets of [ n ] of cardinality n − f / ∈ X k − , there exists a subset A ⊆ [ n ] with | A | = k − A, f ) = 1. Let us identify all arguments not in A , i.e., we form the minor f σ witha minor formation map σ : [ n ] → [ k ] that maps A onto [ k −
1] and every element of[ n ] \ A to k . Then f σ has arity k . Since those subsets of [ n ] whose image under σ equals [ k ] are precisely all proper supersets of A , and since ch( A, f ) = 1, there arean odd number of sets T ∈ M f such that σ ( T ) = [ k ]. By Lemma 5.15, [ k ] ∈ M f σ .Then clearly f σ ∈ D k \ X k − .(v) By part (ii), we must have n > deg( f ) ≥ ℓ . By identifying all arguments thatare not in S , we obtain a minor g of f that has arity ℓ + 1 and contains a monomialof degree ℓ . Since X k is minor-closed, g ∈ X k , so by part (iii), [ ℓ + 1] / ∈ M g ; hencedeg( g ) = ℓ . By part (iii), M g contains all subsets of [ ℓ + 1] of cardinality ℓ . If ℓ > k + 1, then M g must also contain a subset of cardinality ℓ −
1. For, considera subset A ⊆ [ ℓ + 1] with | A | = ℓ −
2. Since ℓ − ≥ k and g ∈ X k , we havech( A, g ) = 0, so there must be an even number of sets S ∈ M g with A ( S .There are exactly three such sets S of cardinality ℓ , namely [ ℓ + 1] \ { i } for each i ∈ [ ℓ + 1] \ A ; therefore there must also be a set of cardinality ℓ − M g .(vi) If f has no monomial of degree n −
1, then for any i, j ∈ [ n ] with i < j , the( n − f ij has degree n −
1. If f has exactly one monomial of degree n −
1, say S ∈ M f , | S | = n −
1, then for any i, j ∈ S with i < j , the minor f ij has degree n −
1. If f has at least two monomials of degree n −
1, say
S, T ∈ M f , S = T , | S | = | T | = n −
1, then for { i, j } := S △ T with i < j , the minor f ij hasdegree n − (cid:3) In what follows, we are going to make use of a family of special Boolean functions W k that was inspired by the “unitrades” and the proof methods presented byPotapov [16, Section 4]. There is a minor difference in the definition, though. WhilePotapov’s unitrade W k is composed of all subsets of cardinality k , we neverthelessinclude all nonempty proper subsets of [ k + 1] in the set of monomials of W k , asthis will serve better our needs. Definition 6.14.
For k ∈ N , let W k : { , } k +1 → { , } be the function with M W k = { S ⊆ [ k + 1] | < | S | < k + 1 } . Equivalently, W k ( a ) = 1 if and only if a / ∈ { (0 , . . . , , (1 , . . . , } . For n ≥ k and B ⊆ [ n ] with | B | = k , denote by W Bk the minor ( W k ) σ where σ : [ k ] → [ n ] is an injective map with range B (since W k istotally symmetric, any such map σ produces the same minor). In other words, W Bk is obtained from W k by introducing n − k fictitious arguments and then permutingarguments so that the essential arguments are the ones indexed by the elements of B . While the arity of W Bk is not explicit in the notation, it will be clear from thecontext. Lemma 6.15. (i)
For any k ∈ N , we have deg( W k ) = k , par( W k ) = 0 , and χ ( W k ) = 0 ; hence W k ∈ D k ∩ X ∩ Ω . (ii) For any k, ℓ ∈ N with k ≤ ℓ , W k is a minor of W ℓ .Proof. (i) It is clear from the definition that deg( W k ) = k . Since M W k comprises allsubsets of [ k + 1] except ∅ and [ k + 1], we have | M W k \ {∅}| = 2 k +1 − k − W k ) = 0. As for the characteristic rank, for any S ⊆ [ k + 1], we have { A ∈ M W k | S ( A } = { A ⊆ [ k + 1] | S ( A ( [ k + 1] } . Thisset has 2 k +1 −| S | − k −| S | −
1) elements if A = [ k + 1] and no element if A = [ k + 1]. Therefore ch( S, f ) = 0 for every S ⊆ [ k + 1], and we conclude that χ ( f ) = 0. The constant term of W k is clearly 0, and we conclude that W k ∈ D k ∩ X ∩ Ω = ∩ Ω ∗ = D k ∩ X ∩ Ω .(ii) By the transitivity of the minor relation, it suffices to show that W k is a minorof W k +1 for any k ∈ N . By identifying the ( k + 1)-st and ( k + 2)-nd arguments,i.e., by taking σ to be the identification map σ k +1 ,k +2 , we obtain, by Lemma 5.15, M ( W k +1 ) σ = (cid:8) S ⊆ [ k + 1] (cid:12)(cid:12) |{ T ∈ M W k +1 | σ ( T ) = S }| ≡ (cid:9) =: M. We now determine which subsets of [ k + 1] belong to the set M on the right sideof the above equality. Recall that M W k +1 = { T ⊆ [ k + 2] | < | T | < k + 2 } . Forany S ⊆ [ k ], the only subset S ′ of [ k + 2] such that σ ( S ) = σ ( S ′ ) is S itself; hence S ∈ M for all ∅ 6 = S ⊆ [ k ]. For any set of the form S ∪ { k + 1 } with S ⊆ [ k ], thereare exactly three subsets S ′ of [ n + 2] such that σ ( S ′ ) = S ∪{ k +1 } , namely the sets S ∪ { k + 1 } , S ∪ { k + 2 } , and S ∪ { k + 1 , k + 2 } . If S = [ k ], then all three sets belongto M W k +1 . If S = [ k ], then only the first two belong to M W k +1 . Hence S ∪ { k } ∈ M for all S ( [ k ]. We conclude that M = { S ⊆ [ k + 1] | < | S | < k + 1 } = M W k ,that is ( W k +1 ) σ = W k . (cid:3) Here is another functional construction that we will use in what follows.
Definition 6.16.
For any function f : { , } n → { , } and any i ∈ [ n ], let f ′ i : { , } n → { , } be the function with M f ′ i := { S \ { i } | S ∈ M f , i ∈ S } .The effect of negating an argument in a function f can be expressed in a conve-nient way with the help of f ′ i . TABILITY OF BOOLEAN FUNCTION CLASSES 21
Lemma 6.17.
Let f : { , } n → { , } , i ∈ [ n ] , and let g := f ( x , . . . , x i − , x i +1 , x i +1 , . . . , x n ) . Then g = f + f ′ i .Proof. Given f = P S ∈ M f x S , we have g = f ( x , . . . , x i − , x i + 1 , x i +1 , . . . , x n ) = X S ∈ M f i/ ∈ S x S + X S ∈ M f i ∈ S ( x i + 1) x S \{ i } = X S ∈ M f i/ ∈ S x S + X S ∈ M f i ∈ S ( x S + x S \{ i } ) = X S ∈ M f x S + X S ∈ M f i ∈ S x S \{ i } = f + f ′ i . (cid:3) Lemma 6.18.
Let f : { , } n → { , } , and assume that f depends on the i -thargument. (i) If f ∈ X k for some k > , then f ′ i ∈ X k − . (ii) deg( f ′ i ) < deg( f ) .Proof. (i) Let S ⊆ [ n ] with | S | ≥ k −
1. If i ∈ S , then clearly ch( S, f ′ i ) = 0 becauseno set in M f ′ i contains the element i . If i / ∈ S , then a set A with S ( A ⊆ [ n ]belongs to M f ′ i if and only if i / ∈ A and A ∪ { i } ∈ M f . Hence there is a one-to-onecorrespondence between the sets { A ∈ M f ′ i | S ( A } and { B ∈ M f | S ∪{ i } ⊆ B } ,so it follows that ch( S, f ′ i ) = ch( S ∪ { i } , f ) = 0, where the second equality holdsbecause | S ∪ { i }| ≥ k and f ∈ X k . We conclude that f ′ i ∈ X k − .(ii) Obvious from the construction of f ′ i . (cid:3) Lemma 6.19.
For any k ∈ N , every function in D k ∩ X ∩ Ω is a sum of minorsof W k . Consequently, h W k i L c = D k ∩ X ∩ Ω .Proof. We follow the proof technique of Potapov [16, Proposition 11]. Note that Ω ⊆ Ω = , so every function in D k ∩ X ∩ Ω is even. We proceed by inductionon k . The claim is obvious for k = 0, since D ∩ X ∩ Ω = D ∩ Ω ∗ , and everyconstant 0 function (of any arity) can be obtained from W , the unary constant0 function, by introducing fictitious arguments. The claim is also clear for k = 1,since D ∩ X ∩ Ω = D ∩ Ω = ∩ Ω ∗ , and any even function of degree 1 with constantterm 0 can be obtained by adding together suitable minors of W = x + x obtainedby introducing fictitious arguments and permuting arguments.Assume now that the claim holds for k = ℓ for some ℓ ≥
1. Every functionin D ℓ +1 ∩ X ∩ Ω of degree less than ℓ + 1 is a sum of minors of W ℓ by theinduction hypothesis and is therefore a sum of minors of W ℓ +1 because W ℓ ≤ W ℓ +1 by Lemma 6.15(ii). We only need to consider functions of degree exactly ℓ + 1. Weproceed by induction on the arity of functions. By Lemma 6.13(ii), for any f ∈ X with deg( f ) = ℓ +1, we must have ar( f ) ≥ ℓ +2. Therefore, in order to establish thebasis of induction, we need to consider an arbitrary function f ∈ D ℓ +1 ∩ X ∩ Ω withar( f ) = ℓ + 2. By Lemma 6.13(iii), M f contains all subsets of [ ℓ + 2] of cardinality ℓ + 1. Then g := f + W ℓ +1 = f + W ℓ +1 + 0 ∈ D ℓ ∩ X ∩ Ω because f , W ℓ +1 , and 0belong to X ∩ Ω , which is L c -stable by Proposition 6.9, and deg( g ) ≤ ℓ becauseall monomials of degree ℓ + 1 are cancelled in the sum f + W ℓ +1 . By the inductivehypothesis, g is a sum of minors of W ℓ ; hence f = g + W ℓ +1 is a sum of minors of W ℓ +1 .For the inductive step, assume that every m -ary function in D ℓ +1 ∩ X ∩ Ω ofdegree ℓ + 1 is a sum of minors of W ℓ +1 . Let f ∈ D ℓ +1 ∩ X ∩ Ω be ( m + 1)-aryand of degree ℓ + 1. If f does not depend on the ( m + 1)-st argument, then f isobtained from an m -ary function f ∗ ∈ D ℓ +1 ∩ X ∩ Ω by introducing a fictitiousargument; then f ∗ is a sum of minors of W ℓ +1 , and by introducing a fictitiousargument to the summands we obtain f as a sum of minors of W ℓ +1 . From now on, assume that f depends on the ( m + 1)-st argument. Let g := f ′ m +1 , and let c bethe constant term (0 or 1) of g . By Lemma 6.18 we have g ∈ D ℓ ∩ X ; furthermore, g + c ∈ D ℓ ∩ X ∩ Ω ∗ = D ℓ ∩ X ∩ Ω . By the inductive hypothesis, g + c is asum of minors of W ℓ , say g + c = P pi =1 W S i k i , with k i ≤ ℓ for each i . Now let h := P pi =1 W S i ∪{ m +1 } k i +1 + c ∗ , where c ∗ := 0 if c = 0 and c ∗ := W { m,m +1 } if c = 1,and let f ∗ := f + h . We have f ∗ ∈ D ℓ +1 ∩ X ∩ Ω because f , h , and 0 belong to D ℓ +1 ∩ X ∩ Ω , which is L c -stable. Moreover, f ∗ does not depend on the ( m + 1)-st argument because none of its monomials contains x m +1 . Let f ∗∗ be the m -aryfunction obtained from f ∗ by removing the fictitious ( m + 1)-st argument; then f ∗ and f ∗∗ are minors of each other. By the induction hypothesis, f ∗∗ is a sum ofminors of W ℓ +1 , and consequently so is f ∗ and hence also f ∗ + h = f .As for the last claim about h W k i L c , since 0 = W is a minor of W k , it followsthat every sum of minors of W k (not just every odd sum) is in h W k i L c . Therefore,by what we have shown above, D k ∩ X ∩ Ω ⊆ h W k i L c ⊆ D k ∩ X ∩ Ω . (cid:3) Lemma 6.20.
A Boolean function f belongs to X k if and only if f = g + h forsome g ∈ X and h ∈ D k .Proof. “ ⇐ ”: Clear because X ⊆ X k , D k ⊆ X k , and X k is closed under sums byLemma 6.8.“ ⇒ ”: We proceed by induction on k . For k = 0, the claim is obvious: if f ∈ X ,then f = f + 0, where f ∈ X and 0 ∈ D . For k = 1, this follows from Lemma 5.12:if f ∈ X ∩ X , then we are done by the above; if f ∈ X \ X , then f + x ∈ X , sowe have the decomposition f = ( f + x ) + x , where f + x ∈ X and x ∈ D .Assume now that the claim holds for k = ℓ for some ℓ ≥
0. Let f ∈ X ℓ +1 . Weproceed by induction on the degree of f . If deg( f ) ≤ ℓ + 1, then we clearly have f = 0 + f with 0 ∈ X and f ∈ D ℓ +1 . Assume that the claim holds for functions ofdegree at most m ≥ ℓ +1. Consider now the case when deg( f ) = m +1. We proceedby induction on the arity n of f . By Lemma 6.13(ii), n ≥ m + 2. If n = m + 2, then M f contains all subsets of [ n ] of cardinality m + 1 = n − f ∗ := f + W m +1 . Then deg( f ∗ ) ≤ m , so by the induction hypothesis f ∗ = g ∗ + h ∗ forsome g ∗ ∈ X and h ∗ ∈ D ℓ +1 ; therefore f = ( g ∗ + W m +1 )+ h ∗ , where g ∗ + W m +1 ∈ X by Lemma 6.8 and h ∗ ∈ D ℓ +1 . Assume that the claim holds for functions of arity p , and consider the case when ar( f ) = p + 1; we may assume that f depends onthe ( p + 1)-st argument. By Lemma 6.18, we have f ′ p +1 ∈ X ℓ , so by the inductionhypothesis f ′ p +1 = g ∗ + h ∗ for some g ∗ ∈ X and h ∗ ∈ D ℓ ; by changing the constantterms in g ∗ and h ∗ if necessary, we may assume that the constant term of g ∗ is 0.By Lemma 6.19, we can write g ∗ as g ∗ = P si =1 W S i k i . Let g + := P si =1 W S i ∪{ p +1 } k i +1 ,and let h + be the function with M h + = { S ∪ { p + 1 } | S ∈ M h ∗ } ; then clearly g + ∈ X and h + ∈ D ℓ +1 . Let ϕ := f + g + + h + . Clearly ϕ ∈ X ℓ +1 and ϕ doesnot depend on the ( p + 1)-st argument, so by the induction hypothesis ϕ = γ + η with γ ∈ X and η ∈ D ℓ +1 . Then f = ϕ + g + + h + = ( γ + g + ) + ( η + h + ), where γ + g + ∈ X and η + h + ∈ D ℓ +1 , which gives us the desired decomposition. (cid:3) Lemma 6.21.
For any k ≥ , h x . . . x k + x i L c = D k ∩ Ω .Proof. Let f := x . . . x k + x . We have f ∈ D k ∩ Ω , so h f i L c ⊆ D k ∩ Ω .By permuting arguments we get g := x x k +1 x k +2 . . . x k − + x ∈ h f i L c , and byidentifying all arguments we get 0 ∈ h f i L c ; hence also h := f + g + 0 = x . . . x k + x x k +1 x k +2 . . . x k − ∈ h f i L c . Again by permuting the arguments of h we get h ′ := x x k +1 x k +2 . . . x k − + x k − x k . . . x k − ∈ h f i L c ; hence also h ′′ := h + h ′ + 0 = x . . . x k + x k − x k . . . x k − ∈ h f i L c . It is clear that any even sum of monomialsof degree at most k can be obtained by adding (an odd number of) minors of h ′′ .Therefore D k ∩ Ω = D k ∩ Ω = ∩ Ω ∗ ⊆ h h ′′ i L c ⊆ h f i L c . (cid:3) TABILITY OF BOOLEAN FUNCTION CLASSES 23
Proposition 6.22.
Let a ∈ { , } . (i) Let k ∈ N + . For any f ∈ ( D k ∩ X ∩ Ω aa ) \ D k − , we have h f i L c = D k ∩ X ∩ Ω aa . (ii) Let k ∈ N + with k ≥ . For any g ∈ ( D k ∩ Ω aa ) \ X k − , we have h g i L c = D k ∩ Ω aa . (iii) Let i, j ∈ N with i > j ≥ . For any f, g ∈ D i ∩ X j ∩ Ω aa such that f / ∈ D i − and g / ∈ X j − , we have h f, g i L c = D i ∩ X j ∩ Ω aa .Proof. It suffices to prove the statements for a = 0. The statements for a = 1follow by Lemma 6.11 because D i ∩ X j ∩ Ω = D i ∩ X j ∩ Ω , D i − = D i − , and X j − = X j − . Note that Ω ⊆ Ω = .(i) We proceed by induction on k . For k = 1, let f ∈ ( D ∩ X ∩ Ω ) \ D = ( D ∩ Ω = ∩ Ω ∗ ) \ D . The function f is a sum of an even nonzero number of arguments,so by identification of arguments we get 0 , x + x ∈ h f i L c , and with these we cangenerate every even sum: D ∩ X ∩ Ω = D ∩ Ω = ∩ Ω ∗ ⊆ h f i L c ⊆ D ∩ X ∩ Ω .Assume that the claim holds for k = ℓ for some ℓ ≥
1. Let f ∈ ( D ℓ +1 ∩ X ∩ Ω ) \ D ℓ . Since X ∩ Ω ⊆ X ∩ Ω = = X , Lemma 6.13(v) implies that f has an ( ℓ +2)-aryminor ϕ such that deg( ϕ ) = ℓ + 1 and M ϕ contains all ( ℓ + 1)-element subsets of[ ℓ + 2] and a subset S of cardinality ℓ . By identifying the two arguments not in S ,we obtain a minor ϕ ′ of ϕ such that ϕ ′ ∈ X , ar( ϕ ′ ) = ℓ + 1, and deg( ϕ ′ ) ≥ ℓ > ϕ ′ ) = ℓ . Since ϕ ′ ∈ ( D ℓ ∩ X ∩ Ω ) \ D ℓ − ,it holds that h ϕ ′ i L c = D ℓ ∩ X ∩ Ω by the induction hypothesis. All monomials ofdegree ℓ +1 are cancelled in the sum ϕ + W ℓ +1 , so we have ϕ + W ℓ +1 ∈ D ℓ ∩ X ∩ Ω = h ϕ ′ i L c ⊆ h f i L c . Since also ϕ, ∈ h f i L c , we get W ℓ +1 = ( ϕ + W ℓ +1 ) + ϕ + 0 ∈ h f i L c .By Lemma 6.19, D ℓ +1 ∩ X ∩ Ω = h W ℓ +1 i L c ⊆ h f i L c ⊆ D ℓ +1 ∩ X ∩ Ω .(ii) We proceed by induction on k . For k = 2, let g ∈ ( D ∩ Ω ) \ X . Since D ⊆ X , g has a binary minor γ with [2] ∈ M γ by Lemma 6.13(iv). Since γ ∈ Ω ⊆ Ω = , we have γ ≡ x x + x . It follows from Lemma 6.21 that D ∩ Ω = h x x + x i L c ⊆ h g i L c ⊆ D ∩ Ω .Assume that the claim holds for k = ℓ for some ℓ ≥
2. Let g ∈ ( D ℓ +1 ∩ Ω ) \ X ℓ . Since D ℓ +1 ⊆ X ℓ +1 , g has an ( ℓ + 1)-ary minor γ with [ ℓ + 1] ∈ M γ byLemma 6.13(iv). By Lemma 6.13(vi), γ has an ℓ -ary minor γ ij ∈ ( D ℓ ∩ Ω ) \ X ℓ − .By the inductive hypothesis, D ℓ ∩ Ω = h γ ij i L c ⊆ h g i L c . The functions γ ′ := γ +( x . . . x ℓ +1 + x ) and 0 are members of D ℓ ∩ Ω ⊆ h g i L c , so also x . . . x ℓ +1 + x = γ ′ + γ + 0 ∈ h g i L c . By Lemma 6.21, D ℓ +1 ∩ Ω = h x . . . x ℓ +1 + x i L c ⊆ h g i L c ⊆ D ℓ +1 ∩ Ω .(iii) Let f, g ∈ D i ∩ X j ∩ Ω such that f / ∈ D i − and g / ∈ X j − . By Lemma 6.13(iv), g has a j -ary minor g ′ ∈ D j \ X j − . Since D i ∩ X j ∩ Ω is minor-closed, we have g ′ ∈ ( D j ∩ Ω ) \ X j − , and by part (ii), h g ′ i L c = D j ∩ Ω .By Lemma 6.20, f = f + f for some f ∈ X and f ∈ D j . Since X ⊆ Ω = and f ∈ Ω ⊆ Ω = , we must also have f ∈ Ω = . Since f ∈ Ω ∗ , it is clear thatby changing the constant terms in f and f if necessary, we can assume that both f and f are in Ω = ∩ Ω ∗ = Ω . Thus f ∈ D j ∩ Ω = h g ′ i L c ⊆ h g i L c , so f = f + f + 0 ∈ h f, g i L c . Since f ∈ ( D i ∩ X ∩ Ω ∗ ) \ D i − = ( D i ∩ X ∩ Ω ) \ D i − ,we have h f i L c = D i ∩ X ∩ Ω by part (i). It follows from Lemma 6.20 that D i ∩ X j ∩ Ω = { α + β | α ∈ D i ∩ X ∩ Ω , β ∈ D j ∩ Ω } = { α + β + 0 | α ∈ D i ∩ X ∩ Ω , β ∈ D j ∩ Ω }⊆ h f , g ′ i L c ⊆ h f, g i L c ⊆ D i ∩ X j ∩ Ω . (cid:3) Lemma 6.23.
For any k ∈ N + , h x . . . x k i L c = D k ∩ Ω . Proof.
It is clear that any monomial of degree at most k can be obtained as a minorof x . . . x k . Any function in D k ∩ Ω = D k ∩ Ω = ∩ Ω ∗ is an odd sum of monomialsof degree at most k . Therefore D k ∩ Ω ⊆ h x . . . x k i L c ⊆ D k ∩ Ω . (cid:3) Proposition 6.24.
Let a ∈ { , } . (i) For any f ∈ D ∩ Ω aa , we have h f i L c = D ∩ Ω aa . (ii) Let k ∈ N + with k ≥ . For any f ∈ ( D k ∩ X ∩ Ω aa ) \ D k − , we have h f i L c = D k ∩ X ∩ Ω aa . (iii) Let k ∈ N + with k ≥ . For any g ∈ ( D k ∩ Ω aa ) \ X k − , we have h g i L c = D k ∩ Ω aa . (iv) Let i, j ∈ N with i > j ≥ . For any f, g ∈ D i ∩ X j ∩ Ω aa such that f / ∈ D i − and g / ∈ X j − , we have h f, g i L c = D i ∩ X j ∩ Ω aa .Proof. It suffices to prove the statements for a = 0. The statements for a = 1follow by Lemma 6.11 because D i ∩ X j ∩ Ω = D i ∩ X j ∩ Ω , D i − = D i − , and X j − = X j − .(i) If f ∈ D ∩ Ω , then by identifying all arguments we obtain x ∈ h f i L c . ByLemma 6.23, we have D ∩ Ω = L c = h x i L c ⊆ h f i L c ⊆ D ∩ Ω .(ii) We show by induction on k that the claim holds for any k ≥ k ≥ k = 1, is, in fact, statement(i) that we have already established; note that ( D ∩ X ∩ Ω ) \ D = D ∩ Ω .For the induction step, assume that the claim holds for k = ℓ for some ℓ ≥ f ∈ ( D ℓ +1 ∩ X ∩ Ω ) \ D ℓ . By Lemma 6.13(v), f has an ( ℓ + 2)-ary minor ϕ ∈ ( D ℓ +1 ∩ X ∩ Ω ) \ D ℓ such that M ϕ contains all subsets of [ ℓ + 2] of cardinality ℓ + 1. If ℓ ≥
2, then M ϕ furthermore contains a subset of cardinality ℓ ; then,again by Lemma 6.13(v), ϕ has an ( ℓ + 1)-ary minor ϕ ′ ∈ ( D ℓ ∩ X ∩ Ω ) \ D ℓ − ,and by the inductive hypothesis, D ℓ ∩ X ∩ Ω = h ϕ ′ i L c ⊆ h f i L c . If ℓ = 1, then D ℓ ∩ X ∩ Ω = L c = h x i L c ⊆ h f i L c because x is a minor of f (identify allarguments). In either case, let λ := W ℓ +1 + ϕ . We have W ℓ +1 ∈ D ℓ +1 ∩ X ∩ Ω by Lemma 6.15 and ϕ ∈ D ℓ +1 ∩ X ∩ Ω . Consequently λ ∈ D ℓ ∩ X ∩ Ω ⊆ h f i L c because all monomials of degree ℓ + 1 are cancelled in the sum W ℓ +1 + ϕ , X isclosed under sums by Lemma 6.8, and λ (0 , . . . ,
0) = W ℓ +1 (0 , . . . ,
0) + ϕ (0 , . . . ,
0) =0 + 0 = 0, λ (1 , . . . ,
1) = W ℓ +1 (1 , . . . ,
1) + ϕ (1 , . . . ,
1) = 0 + 1 = 1.Let now h ∈ ( D ℓ +1 ∩ X ∩ Ω ) \ D ℓ be arbitrary. Then h + x ∈ D ℓ +1 ∩ X ∩ Ω , soby Lemma 6.19, h + x ∈ h W ℓ +1 i L c , that is, h + x = P m +1 i =1 W S i k i with k i ≤ ℓ +1. Wecan write W S i k i = ( W ℓ +1 ) σ i for a suitable minor formation map σ i . Consequently, h = (cid:0) m +1 X i =1 W S i k i (cid:1) + x = (cid:0) m +1 X i =1 ( W ℓ +1 ) σ i (cid:1) + x = (cid:0) m +1 X i =1 ( W ℓ +1 + ϕ + ϕ ) σ i (cid:1) + x = (cid:0) m +1 X i =1 (cid:0) ( W ℓ +1 + ϕ ) σ i | {z } ∈h f i L c + ϕ σ i |{z} ∈h f i L c (cid:1)(cid:1) + x , |{z} ∈h f i L c where the last equality holds by Lemma 2.8. Since the last expression is an odd sumof elements of h f i L c , it follows that h ∈ h f i L c . We conclude that D ℓ +1 ∩ X ∩ Ω ⊆h f i L c ⊆ D ℓ +1 ∩ X ∩ Ω .(iii) We show by induction on k that the claim holds for any k ≥
1. The basisof the induction, the case when k = 1, is, in fact, statement (i) that we havealready established, because ( D ∩ Ω ) \ X = D ∩ Ω . For the induction step,assume that the claim holds for k = ℓ for some ℓ ≥
1. Let g ∈ ( D ℓ +1 ∩ Ω ) \ X ℓ . ByLemma 6.13(iv), g has an ( ℓ +1)-ary minor γ such that [ ℓ + 1] ∈ M γ and γ ∈ ( D ℓ +1 ∩ Ω ) \ X ℓ . By Lemma 6.13(vi), γ has an ℓ -ary minor γ ij ∈ ( D ℓ ∩ Ω ) \ X ℓ − . By theinductive hypothesis, D ℓ ∩ Ω = h γ ij i L c ⊆ h g i L c . We have γ ′ := γ + x . . . x ℓ +1 + x ∈ TABILITY OF BOOLEAN FUNCTION CLASSES 25 D ℓ ∩ Ω ⊆ h g i L c and clearly x ∈ h g i L c , so also x . . . x ℓ +1 = γ ′ + γ + x ∈ h g i L c .By Lemma 6.23, we have D ℓ +1 ∩ Ω = h x . . . x ℓ +1 i L c ⊆ h g i L c ⊆ D ℓ +1 ∩ Ω .(iv) Let f, g ∈ D i ∩ X j ∩ Ω such that f / ∈ D i − and g / ∈ X j − . By Lemma 6.13(iv), g has a j -ary minor g ′ ∈ D j \ X j − . Since D i ∩ X j ∩ Ω is minor-closed, we have g ′ ∈ ( D j ∩ Ω ) \ X j − , and by part (iii), h g ′ i L c = D j ∩ Ω .By Lemma 6.20, f = f + f for some f ∈ X and f ∈ D j . Since X ⊆ Ω = and f ∈ Ω = , we must also have f ∈ Ω = . Since f ∈ Ω ∗ , it is clear that by changing theconstant terms if necessary, we may assume that both f and f are in Ω ∗ . Thus f ∈ D j ∩ Ω = ∩ Ω ∗ = D j ∩ Ω = h g ′ i L c ⊆ h g i L c , so f + x = f + f + x ∈ h f, g i L c .Since f ∈ ( D i ∩ X ∩ Ω ∗ ) \ D i − = ( D i ∩ X ∩ Ω = ∩ Ω ∗ ) \ D i − , we have f + x ∈ ( D i ∩ X ∩ Ω = ∩ Ω ∗ ) \ D i − = ( D i ∩ X ∩ Ω ) \ D i − , so h f + x i L c = D i ∩ X ∩ Ω by part (ii).Now, with the help of Lemma 6.20, we can see that for any h ∈ D i ∩ X j ∩ Ω ,we have h = h + h for some h ∈ D i ∩ X ∩ Ω ∗ = D i ∩ X ∩ Ω = ∩ Ω ∗ and h ∈ D j ∩ Ω = ∩ Ω ∗ = D j ∩ Ω , and hence h + x ∈ D i ∩ X ∩ Ω ⊆ h f, g i L c and h ∈ h g i L c . Since x ∈ h f i L c as well, we have h = ( h + x ) + h + x ∈ h f, g i L c .We conclude that D i ∩ X j ∩ Ω ⊆ h f, g i L c ⊆ D i ∩ X j ∩ Ω . (cid:3) Proposition 6.25.
Let a, b ∈ { , } . (i) For any f i ∈ ( X ∩ Ω ab ) \ D i ( i ∈ N + ) , we have h{ f i | i ∈ N + }i L c = X ∩ Ω ab . (ii) Let k ∈ N + with k ≥ . For any f i ∈ ( X k ∩ Ω ab ) \ D i ( i ∈ N + ) and g ∈ ( X k ∩ Ω ab ) \ X k − , we have h{ f i | i ∈ N + } ∪ { g }i L c = X k ∩ Ω ab . (iii) For any g i ∈ ( Ω ab ) \ X i ( i ∈ N + ) , we have h{ g i | i ∈ N + }i L c = Ω ab .Proof. (i) For i ∈ N + , let f i ∈ ( X ∩ Ω ab ) \ D i , and let n i := deg( f i ); we have n i > i .Then f i ∈ ( D n i ∩ X ∩ Ω ab ) \ D n i − , so by Proposition 6.22(i) and Proposition 6.24(ii), h f i i L c = D n i ∩ X ∩ Ω ab . Therefore X ∩ Ω ab = [ i ∈ N + ( D i ∩ X ∩ Ω ab ) ⊆ [ i ∈ N + ( D n i ∩ X ∩ Ω ab )= [ i ∈ N + h f i i L c ⊆ h{ f i | i ∈ N + }i L c ⊆ X ∩ Ω ab . (ii) For i ∈ N + , let f i ∈ ( X k ∩ Ω ab ) \ D i , and let g ∈ ( X k ∩ Ω ab ) \ X k − , and let n i := deg( f i ); we have n i > i . By Lemma 6.13(iv), g has a k -ary minor γ of degree k such that γ ∈ ( D k ∩ Ω ab ) \ X k − . By Proposition 6.22(iii) and Proposition 6.24(iv),it holds for i ≥ k that h f i , g i L c = D n i ∩ X k ∩ Ω ab . Therefore X k ∩ Ω ab = [ i ∈ N + ( D i ∩ X k ∩ Ω ab ) = [ i ≥ k ( D i ∩ X k ∩ Ω ab ) ⊆ [ i ≥ k ( D n i ∩ X k ∩ Ω ab ) = [ i ≥ k h f i , g i L c ⊆ h{ f i | i ∈ N + } ∪ { g }i L c ⊆ X k ∩ Ω ab . (iii) For i ∈ N + , let g i ∈ ( Ω ab ) \ X i , and let k i := χ ( g i ). Then g i ∈ ( X k i ∩ Ω ab ) \ X k i − . By Lemma 6.13(iv), g i has a k i -ary minor γ i of degree k i suchthat γ i ∈ ( D k i ∩ Ω ab ) \ X k i − . By Proposition 6.22(ii) and Proposition 6.24(iii), h γ i i L c = D k i ∩ Ω ab . Therefore Ω ab = [ i ∈ N + ( D i ∩ Ω ab ) ⊆ [ i ∈ N + ( D k i ∩ Ω ab ) = [ i ∈ N + h γ i i L c ⊆ [ i ∈ N + h g i i L c ⊆ h{ g i | i ∈ N + }i L c ⊆ Ω ab . (cid:3) Proposition 6.26.
Let a ∈ { , } . (i) Let k ∈ N + . For any f, h, h ′ ∈ D k ∩ X ∩ Ω a ∗ with f / ∈ D k − , h / ∈ Ω ∗ a , h ′ / ∈ Ω ∗ a , we have h f, h, h ′ i L c = D k ∩ X ∩ Ω a ∗ . (ii) Let k ∈ N + with k ≥ . For any g, h, h ′ ∈ D k ∩ Ω a ∗ with g / ∈ X k − , h / ∈ Ω ∗ a , h ′ / ∈ Ω ∗ a , we have h g, h, h ′ i L c = D k ∩ Ω a ∗ . (iii) Let i, j ∈ N with i > j ≥ . For any f, g, h, h ′ ∈ D i ∩ X j ∩ Ω a ∗ such that f / ∈ D i − , g / ∈ X j − , h / ∈ Ω ∗ a , h ′ / ∈ Ω ∗ a , we have h f, g, h, h ′ i L c = D i ∩ X j ∩ Ω a ∗ .Proof. It suffices to prove the statements for a = 0. The statements for a = 1 followby Lemma 6.11 because D i ∩ X j ∩ Ω ∗ = D i ∩ X j ∩ Ω ∗ , D i − = D i − , X j − = X j − , Ω ∗ = Ω ∗ , and Ω ∗ = Ω ∗ . We consider only statement (iii). The proofs ofstatements (i) and (ii) are analogous; we just need to omit the parts of the proofthat deal with the function f or g , as the case may be, that does not appear in thestatement.Since { Ω ∗ , Ω ∗ } is a partition of Ω , we have that h ∈ Ω ∗ and h ′ ∈ Ω ∗ . Byidentifying all arguments, we get x ∈ h h i L c and 0 ∈ h h ′ i L c , so we have f + x = f + x + 0 ∈ h f, h, h ′ i L c and g + x = g + x + 0 ∈ h g, h, h ′ i L c . One of f and f + x belongs to ( D i ∩ X j ∩ Ω ) \ D i − and the other to ( D i ∩ X j ∩ Ω ) \ D i − , and,similarly, one of g and g + x belongs to ( D i ∩ X j ∩ Ω ) \ X j − and the other to( D i ∩ X j ∩ Ω ) \ X j − . Propositions 6.22(iii) and 6.24(iv) imply that h f, g, h, h ′ i L c contains a generating set for both D i ∩ X j ∩ Ω and D i ∩ X j ∩ Ω . Therefore D i ∩ X j ∩ Ω ∗ = ( D i ∩ X j ∩ Ω ) ∪ ( D i ∩ X j ∩ Ω ) ⊆ h f, g, h, h ′ i L c ⊆ D i ∩ X j ∩ Ω ∗ . (cid:3) Proposition 6.27.
Let a ∈ { , } . (i) Let k ∈ N + . For any f, h, h ′ ∈ D k ∩ X ∩ Ω ∗ a with f / ∈ D k − , h / ∈ Ω a ∗ , h ′ / ∈ Ω a ∗ , we have h f, h, h ′ i L c = D k ∩ X ∩ Ω ∗ a . (ii) Let k ∈ N + with k ≥ . For any g, h, h ′ ∈ D k ∩ Ω ∗ a with g / ∈ X k − , h / ∈ Ω a ∗ , h ′ / ∈ Ω a ∗ , we have h g, h, h ′ i L c = D k ∩ Ω ∗ a . (iii) Let i, j ∈ N with i > j ≥ . For any f, g, h, h ′ ∈ D i ∩ X j ∩ Ω ∗ a such that f / ∈ D i − , g / ∈ X j − , h / ∈ Ω a ∗ , h ′ / ∈ Ω a ∗ , we have h f, g, h, h ′ i L c = D i ∩ X j ∩ Ω ∗ a .Proof. It suffices to prove the statements for a = 1. The statements for a = 0 followby Lemma 6.11 because D i ∩ X j ∩ Ω ∗ = D i ∩ X j ∩ Ω ∗ , D i − = D i − , X j − = X j − , Ω ∗ = Ω ∗ , and Ω ∗ = Ω ∗ . We consider only statement (iii). The proofs ofstatements (i) and (ii) are analogous; we just need to omit the parts of the proofthat deal with the function f or g , as the case may be, that does not appear in thestatement.Since { Ω ∗ , Ω ∗ } is a partition of Ω , we have that h ∈ Ω ∗ and h ′ ∈ Ω ∗ . Byidentifying all arguments, we get x ∈ h h i L c and 1 ∈ h h ′ i L c , so we have f + x + 1 ∈h f, h, h ′ i L c and g + x + 1 ∈ h g, h, h ′ i L c . One of f and f + x + 1 belongs to( D i ∩ X j ∩ Ω ) \ D i − and the other to ( D i ∩ X j ∩ Ω ) \ D i − , and, similarly, one of g and g + x +1 belongs to ( D i ∩ X j ∩ Ω ) \ X j − and the other to ( D i ∩ X j ∩ Ω ) \ X j − .Propositions 6.22(iii) and 6.24(iv) imply that h f, g, h, h ′ i L c contains a generating setfor both D i ∩ X j ∩ Ω and D i ∩ X j ∩ Ω . Therefore D i ∩ X j ∩ Ω ∗ = ( D i ∩ X j ∩ Ω ) ∪ ( D i ∩ X j ∩ Ω ) ⊆ h f, g, h, h ′ i L c ⊆ D i ∩ X j ∩ Ω ∗ . (cid:3) Proposition 6.28.
Let ≈ ∈ { = , = } . (i) Let k ∈ N + . For any f, h, h ′ ∈ D k ∩ X ∩ Ω ≈ with f / ∈ D k − , h / ∈ Ω ∗ , h ′ / ∈ Ω ∗ , we have h f, h, h ′ i L c = D k ∩ X ∩ Ω ≈ . (ii) Let k ∈ N + with k ≥ . For any g, h, h ′ ∈ D k ∩ Ω ≈ with g / ∈ X k − , h / ∈ Ω ∗ , h ′ / ∈ Ω ∗ , we have h g, h, h ′ i L c = D k ∩ Ω ≈ . TABILITY OF BOOLEAN FUNCTION CLASSES 27 (iii)
Let i, j ∈ N with i > j ≥ . For any f, g, h, h ′ ∈ D i ∩ X j ∩ Ω ≈ such that f / ∈ D i − , g / ∈ X j − , h / ∈ Ω ∗ , h ′ / ∈ Ω ∗ , we have h f, g, h, h ′ i L c = D i ∩ X j ∩ Ω ≈ .Proof. We consider only statement (iii). The proofs of statements (i) and (ii) areanalogous; we just need to omit the parts of the proof that deal with the function f or g , as the case may be, that does not appear in the statement.Since { Ω ∗ , Ω ∗ } is a partition of Ω , we have that h ∈ Ω ∗ and h ′ ∈ Ω ∗ . Byidentifying all arguments, we get 1 ∈ h h i L c and 0 ∈ h h ′ i L c if ≈ is =; or x +1 ∈ h h i L c and x ∈ h h ′ i L c if ≈ is =. With the triple sum and these two minors of h and h ′ we are able to negate functions ( ϕ + 1 = ϕ + 1 + 0 and ϕ + 1 = ϕ + ( x + 1) + x );hence f + 1 ∈ h f, h, h ′ i L c and g + 1 ∈ h g, h, h ′ i L c . One of f and f + 1 belongs to( D i ∩ X j ∩ Ω ∗ ∩ Ω ≈ ) \ D i − and the other to ( D i ∩ X j ∩ Ω ∗ ∩ Ω ≈ ) \ D i − , and,similarly, one of g and g + 1 belongs to ( D i ∩ X j ∩ Ω ∗ ∩ Ω ≈ ) \ X j − and the other to( D i ∩ X j ∩ Ω ∗ ∩ Ω ≈ ) \ X j − , Propositions 6.22(iii) and 6.24(iv) imply that h f, g, h, h ′ i L c contains a generating set for both D i ∩ X j ∩ Ω ∗ ∩ Ω ≈ and D i ∩ X j ∩ Ω ∗ ∩ Ω ≈ . Therefore D i ∩ X j ∩ Ω ≈ = ( D i ∩ X j ∩ Ω ∗ ∩ Ω ≈ ) ∪ ( D i ∩ X j ∩ Ω ∗ ∩ Ω ≈ ) ⊆ h f, g, h, h ′ i L c ⊆ D i ∩ X j ∩ Ω ≈ . (cid:3) Proposition 6.29.
Let C ∈ { Ω ∗ , Ω ∗ , Ω ∗ , Ω ∗ , Ω = , Ω = } , and let ( K , K ) := ( ( Ω ∗ , Ω ∗ ) if C ∈ { Ω ∗ , Ω ∗ } , ( Ω ∗ , Ω ∗ ) , if C ∈ { Ω ∗ , Ω ∗ , Ω = , Ω = } . (i) For any f i ∈ ( X ∩ C ) \ D i ( i ∈ N + ) , h ∈ ( X ∩ C ) \ K , h ∈ ( X ∩ C ) \ K ,we have h{ f i | i ∈ N + } ∪ { h , h }i L c = X ∩ C . (ii) Let k ∈ N + with k ≥ . For any f i ∈ ( X k ∩ C ) \ D i ( i ∈ N + ) , g ∈ ( X k ∩ C ) \ X k − , h ∈ ( X ∩ C ) \ K , h ∈ ( X ∩ C ) \ K , we have h{ f i | i ∈ N + } ∪ { g, h , h }i L c = X k ∩ C . (iii) For any g i ∈ C \ X i ( i ∈ N + ) , h ∈ C \ K , h ∈ C \ K , we have h{ g i | i ∈ N + } ∪ { h , h }i L c = C .Proof. (i) For i ∈ N + , let f i ∈ ( X ∩ C ) \ D i , h ∈ ( X ∩ C ) \ K , h ∈ ( X ∩ C ) \ K and let n i := deg( f i ); we have n i > i . By identifying all arguments of h and h , we get minors η ∈ ( D ∩ X ∩ C ) \ K , η ∈ ( D ∩ X ∩ C ) \ K . Since f i ∈ ( D n i ∩ X ∩ C ) \ D n i − , it follows from Propositions 6.26(i), 6.27(i), and 6.28(i)that h f i , η , η i L c = D n i ∩ X ∩ C for any i ∈ N + . Therefore X ∩ C = [ i ∈ N + ( D i ∩ X ∩ C ) ⊆ [ i ∈ N + ( D n i ∩ X ∩ C )= [ i ∈ N + h f i , η , η i L c ⊆ h{ f i | i ∈ N + } ∪ { h , h }i L c ⊆ X ∩ C. (ii) For i ∈ N + , let f i ∈ ( X k ∩ C ) \ D i , g ∈ ( X k ∩ C ) \ X k − , h ∈ ( X ∩ C ) \ K , h ∈ ( X ∩ C ) \ K , and let n i := deg( f i ); we have n i > i . By Lemma 6.13(iv), g has a k -ary minor γ of degree k such that γ ∈ ( D k ∩ X k ∩ C ) \ X k − . Byidentifying all arguments of h and h , we get minors η ∈ ( D ∩ X k ∩ C ) \ K , η ∈ ( D ∩ X k ∩ C ) \ K . By Propositions 6.26(iii), 6.27(iii), and 6.28(iii) it holdsthat h f i , g, η , η i L c = D n i ∩ X k ∩ C whenever n i ≥ k (this certainly holds whenever i ≥ k ). Therefore X k ∩ C = [ i ∈ N + ( D i ∩ X k ∩ C ) = [ i ≥ k ( D i ∩ X k ∩ C ) ⊆ [ i ≥ k ( D n i ∩ X k ∩ C ) = [ i ≥ k h f i , g, η , η i L c ⊆ h{ f i | i ∈ N + } ∪ { g, h , h }i L c ⊆ X k ∩ C. (iii) For i ∈ N + , let g i ∈ C \ X i , h ∈ C \ K , h ∈ C \ K , and let k i := χ ( g i ).Then g i ∈ ( X k i ∩ C ) \ X k i − . By Lemma 6.13(iv), g i has a k i -ary minor γ i of degree k i such that γ i ∈ ( D k i ∩ C ) \ X k i − . By identifying all arguments of h and h ,we get minors η ∈ ( D ∩ C ) \ K , η ∈ ( D ∩ C ) \ K . By Propositions 6.26(ii),6.27(ii), and 6.28(ii) it holds that h γ i , η , η i L c = D k i ∩ C for any i ∈ N + . Therefore C = [ i ∈ N + ( D i ∩ C ) ⊆ [ i ∈ N + ( D k i ∩ C ) = [ i ∈ N + h γ i , η , η i L c ⊆ [ i ∈ N + h g i , h , h i L c ⊆ h{ g i | i ∈ N + } ∪ { h , h }i L c ⊆ C. (cid:3) Lemma 6.30.
For any h ∈ Ω ∗ , h ∈ Ω ∗ , h ∈ Ω ∗ , h ∈ Ω ∗ , h ∈ Ω = , h ∈ Ω = , we have D ⊆ h h , h , h , h , h , h i L c .Proof. By identifying all arguments, we see that(3) either 0 or x is in h h i L c , either 1 or x + 1 is in h h i L c , either 0 or x + 1 is in h h i L c , either 1 or x is in h h i L c , either 0 or 1 is in h h i L c , either x or x + 1 is in h h i L c . Let G := { , , x , x + 1 } . Clearly h G i L c = D . Any three-element subset of G also generates D because each element of G is the sum of the other three elements.Any choice of functions from the six pairs in (3) includes at least three differentelements of G , so we conclude that D ⊆ h h , h , h , h , h , h i L c . (cid:3) Proposition 6.31. (i)
Let k ∈ N + . For any f, h , h , h , h , h , h ∈ D k ∩ X with f / ∈ D k − , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω = , h / ∈ Ω = , we have h f, g , g , g , g , g , g i L c = D k ∩ X . (ii) Let k ∈ N + with k ≥ . For any g, h , h , h , h , h , h ∈ D k with g / ∈ X k − , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω = , h / ∈ Ω = , we have h g, h , h , h , h , h , h i L c = D k . (iii) Let i, j ∈ N with i > j ≥ . For any f, g, h , h , h , h , h , h ∈ D i ∩ X j with f / ∈ D i − , g / ∈ X j − , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω = , h / ∈ Ω = , we have h f, g, h , h , h , h , h , h i L c = D i ∩ X j .Proof. We consider only statement (iii). The proofs of statements (i) and (ii) areanalogous; we just need to omit the parts of the proof that deal with the function f or g , as the case may be, that does not appear in the statement.Since { Ω = , Ω = } , { Ω ∗ , Ω ∗ } , and { Ω ∗ , Ω ∗ } are partitions of Ω , we have that h ∈ Ω ∗ , h ∈ Ω ∗ , h ∈ Ω ∗ , h ∈ Ω ∗ , h ∈ Ω = , and h ∈ Ω = . By Lemma 6.30,we have D ⊆ h h , h , h , h , h , h i L c . Hence f + x + 1 ∈ h f, h , h , h , h , h , h i L c and g + x + 1 ∈ h g, h , h , h , h , h , h i L c . One of f and f + x + 1 belongs to( D i ∩ X j ∩ Ω = ) \ D i − and the other to ( D i ∩ X j ∩ Ω = ) \ D i − , and, similarly, one of g and g + x +1 belongs to ( D i ∩ X j ∩ Ω = ) \ X j − and the other to ( D i ∩ X j ∩ Ω = ) \ X j − .Proposition 6.28(iii) implies that h f, g, h , h , h , h , h , h i L c contains a generatingset for both D i ∩ X j ∩ Ω = and D i ∩ X j ∩ Ω = . Therefore D i ∩ X j = ( D i ∩ X j ∩ Ω = ) ∪ ( D i ∩ X j ∩ Ω = ) ⊆ h f, g, h , h , h , h , h , h i L c ⊆ D i ∩ X j . (cid:3) Proposition 6.32.
TABILITY OF BOOLEAN FUNCTION CLASSES 29 (i)
For any f i ∈ X \ D i ( i ∈ N + ) and h , h , h , h , h , h ∈ X such that h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω = , h / ∈ Ω = , we have h{ f i | i ∈ N + } ∪ { h , h , h , h , h , h }i L c = X . (ii) Let k ∈ N + with k ≥ . For any f i ∈ X k \ D i ( i ∈ N + ) , g ∈ X k \ X k − , and h , h , h , h , h , h ∈ X k such that h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω = , h / ∈ Ω = , we have h{ f i | i ∈ N + } ∪ { g, h , h , h , h , h , h }i L c = X k . (iii) For any g i ∈ Ω \ X i ( i ∈ N + ) and h , h , h , h , h , h ∈ Ω such that h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω = , h / ∈ Ω = , we have h{ g i | i ∈ N + } ∪ { h , h , h , h , h , h }i L c = Ω .Proof. Observe first that 0 , , x ∈ D ⊆ X k for any k ∈ N + and 1 / ∈ Ω ∗ , 0 / ∈ Ω ∗ ,1 / ∈ Ω ∗ , 0 / ∈ Ω ∗ , x / ∈ Ω = , 0 / ∈ Ω = .(i) For i ∈ N + , let f i ∈ X \ D i and h , h , h , h , h , h ∈ X be such that h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω = , h / ∈ Ω = , and let n i :=deg( f i ); we have n i > i . Since f i ∈ ( D n i ∩ X ) \ D n i − , Proposition 6.31(i)implies h f i , , , x i L c = D n i ∩ X for any i ∈ N + . We have { , , x } ⊆ D ⊆h h , h , h , h , h , h i L c by Lemma 6.30. Therefore X = [ i ∈ N + ( D i ∩ X ) ⊆ [ i ∈ N + ( D n i ∩ X ) = [ i ∈ N + h f i , , , x i L c ⊆ h{ f i | i ∈ N + } ∪ { h , h , h , h , h , h }i L c ⊆ X . (ii) For i ∈ N + , let f i ∈ X k \ D i , g ∈ X k \ X k − , and h , h , h , h , h , h ∈ X k such that h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω = , h / ∈ Ω = , and let n i := deg( f i ); we have n i > i . By Lemma 6.13(iv), g has a k -ary minor γ of degree k such that γ ∈ X k \ X k − ; hence γ ∈ D k \ X k − . By Proposition 6.31(iii), it holdsthat h f i , γ, , , x i L c = D n i ∩ X k whenever n i ≥ k . We have { , , x } ⊆ D ⊆h h , h , h , h , h , h i L c by Lemma 6.30. Therefore X k = [ i ∈ N + ( D i ∩ X k ) = [ i ≥ k ( D i ∩ X k ) ⊆ [ i ≥ k ( D n i ∩ X k ) = [ i ≥ k h f i , γ, , , x i L c ⊆ h{ f i | i ∈ N + } ∪ { g, h , h , h , h , h , h }i L c ⊆ X k . (iii) For i ∈ N + , let g i ∈ Ω \ X i , and h , h , h , h , h , h ∈ X k such that h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω ∗ , h / ∈ Ω = , h / ∈ Ω = , and let k i := χ ( g i ). Then g i ∈ X k i \ X k i − . By Lemma 6.13(iv), g i has a k i -ary minor γ i of degree k i suchthat γ i ∈ D k i \ X k i − . By Proposition 6.31(ii), it holds that h γ i , , , x i L c = D k i for any i ∈ N + . We have { , , x } ⊆ D ⊆ h h , h , h , h , h , h i L c by Lemma 6.30.Therefore Ω = [ i ∈ N + D i ⊆ [ i ∈ N + D k i = [ i ∈ N + h γ i , , , x i L c ⊆ h{ g i | i ∈ N + } ∪ { h , h , h , h , h , h }i L c ⊆ Ω . (cid:3) Proof of Theorem 6.1.
By Lemma 6.2(iii), L c -stability is equivalent to ( I c , L c )-sta-bility. The given classes are L c -stable by Proposition 6.9. The fact that there areno further L c -stable classes distinct from these follows from Propositions 6.12, 6.22,6.24, 6.25, 6.26, 6.27, 6.28, 6.29, 6.31, 6.32, in which we have shown that any setof Boolean functions generates one of the classes listed in the statement – moreprecisely, that for each class C and for any set F ⊆ C that is not included in anyproper subclass of C it holds that h F i L c = C . (cid:3)
7. ( C , C ) -stable classes for L c ⊆ C Theorem 6.1 allows us to describe also all ( C , C )-stable classes of Booleanfunctions for clones C and C such that C is arbitrary and L c ⊆ C . Namely, by Lemma 6.2, L c -stability is equivalent to ( I c , L c )-stability. Since ( C , C )-stabilityimplies ( I c , L c )-stability whenever L c ⊆ C , it suffices to search for ( C , C )-stableclasses among the L c -stable ones. To this end, we determine, for each ( I c , L c )-stableclass K , the clones C and C for which it holds that KC ⊆ K and C K ⊆ K .The results are summarized in the following theorem which refers to Table 2. Theorem 7.1.
For each L c -stable class K , as determined in Theorem 6.1, thereexist clones C K and C K , as prescribed in Table 2, such that for every clone C , itholds that KC ⊆ K if and only if C ⊆ C K , and CK ⊆ K if and only if C ⊆ C K . The proof of Theorem 7.1 will be developed in the remainder of this section.The following two lemmata will be useful. The first one (Lemma 7.2) providessufficient conditions for right and left stability for classes that are intersections ofclasses for which we already know sufficient conditions for right and left stability.The second one (Lemma 7.3) provides necessary conditions. These will be appliedin the subsequent propositions in which necessary and sufficient stability conditionsare established for each L c -stable class. Lemma 7.2.
Let K , K , C , C ⊆ Ω . Then the following statements hold. (i) Assume that K C ⊆ K whenever C ⊆ C and K C ⊆ K whenever C ⊆ C . Then ( K ∩ K ) C ⊆ K ∩ K whenever C ⊆ C ∩ C . (ii) Assume that CK ⊆ K whenever C ⊆ C and CK ⊆ K whenever C ⊆ C . Then C ( K ∩ K ) ⊆ K ∩ K whenever C ⊆ C ∩ C .Proof. (i) If C ⊆ C ∩ C , then ( K ∩ K ) C ⊆ K C ⊆ K and ( K ∩ K ) C ⊆ K C ⊆ K by the monotonicity of function class composition and the stabilityof K and K under right composition with C and C , respectively. Therefore( K ∩ K ) C ⊆ K ∩ K .(ii) The proof is analogous to that of part (i). (cid:3) Lemma 7.3.
Let a, b ∈ { , } , ≈ ∈ { = , = } , i, j ∈ N + with i ≥ j ≥ . (i) For any ∅ 6 = K ⊆ Ω , the following statements hold. (a) I a K * Ω a ∗ ∪ Ω ∗ a . (b) I a K * Ω = . (c) If a = b , then I K * Ω ab , I K * Ω ab . (ii) For K := D i ∩ X j ∩ Ω ab , the following statements hold. (d) I ∗ K * Ω a ∗ ∪ Ω ∗ b . (e) Λ c K * D i , V c K * D i . If j ≥ or a = b , then Λ c K * X j , V c K * X j . (f) SM K * D i . If j ≥ , then SM K * X j . (g) K I * Ω ∗ b , K I * Ω a ∗ , K I ∗ * Ω a ∗ ∪ Ω ∗ b . (h) If i > j , then K I * X j , K I * X j . (i) K Λ c * D i ∪ X j , K V c * D i ∪ X j , (j) K SM * D i . If j ≥ , then K SM * X j . (iii) (k) SM ( X ∩ Ω a ∗ ) * X , SM ( X ∩ Ω ∗ a ) * X . (iv) For K := D i ∩ X j ∩ Ω ≈ , the following statements hold. (l) K I * Ω ≈ , K I * Ω ≈ . (m) If j ≥ , then K I ∗ * Ω ≈ . (n) If ≈ = = , then Λ c K * Ω ≈ , V c K * Ω ≈ ,Proof. Throughout the proof, we will use Lemmata 3.3 and 3.2 together with thefact that I = h i , I = h i , I ∗ = h x + 1 i , Λ c = h∧i , V c = h∨i , SM = h µ i .(i) (a) For any ϕ ∈ Ω , we have a ( ϕ ) = a / ∈ Ω a ∗ ∪ Ω ∗ a . Therefore I a K * Ω a ∗ ∪ Ω ∗ a .(b) For any ϕ ∈ Ω , we have a ( ϕ ) = a / ∈ Ω = . Therefore I a K * Ω = .(c) If a = b , then, by (a), we have I K * Ω ∗ ∪ Ω ∗ and I K * Ω ∗ ∪ Ω ∗ . Since Ω ∗ a b is a subset of both Ω ∗ ∪ Ω ∗ and Ω ∗ ∪ Ω ∗ , it follows that I i K * Ω ab for i ∈ { , } . TABILITY OF BOOLEAN FUNCTION CLASSES 31 KC ⊆ K CK ⊆ KK if and only if if and only if result C ⊆ . . . C ⊆ . . . Ω Ω Ω
Proposition 7.4 Ω a ∗ T T a Proposition 7.5 Ω ∗ a T T a Proposition 7.5 Ω = T c Ω Proposition 7.7 Ω = T c S Proposition 7.7 Ω ab T c T a ∩ T b Proposition 7.8 X k k ≥ LS L
Proposition 7.10 k = 1 S LX k ∩ Ω a ∗ k ≥ L c L a Proposition 7.11 k = 1 S c L a X k ∩ Ω ∗ a k ≥ L c L a Proposition 7.12 k = 1 S c L a X k ∩ Ω = k ≥ L c L Proposition 7.13 k = 1 S ΩX k ∩ Ω = k ≥ L c LS Proposition 7.14 k = 1 S SX k ∩ Ω ab k ≥ L c L a ∩ L b Proposition 7.15 k = 1, a = b S c T a k = 1, a = b S c S c D k L L
Proposition 7.16 D k ∩ Ω a ∗ L L a Proposition 7.18 D k ∩ Ω ∗ a L L a Proposition 7.18 D k ∩ Ω = k ≥ L c L Proposition 7.19 k = 1 LS LD k ∩ Ω = k ≥ L c LS Proposition 7.20 k = 1 LS LSD k ∩ Ω ab L c L a ∩ L b Proposition 7.21 D i ∩ X j LS L
Proposition 7.22 D i ∩ X j ∩ Ω a ∗ L c L a Proposition 7.23 D i ∩ X j ∩ Ω ∗ a L c L a Proposition 7.23 D i ∩ X j ∩ Ω = j ≥ L c L Proposition 7.24 j = 1 LS LD i ∩ X j ∩ Ω = j ≥ L c LS Proposition 7.25 j = 1 LS LSD i ∩ X j ∩ Ω ab L c L a ∩ L b Proposition 7.26 D Ω Ω
Proposition 7.17 D ∩ Ω a ∗ Ω T a Proposition 7.17 ∅ Ω Ω
Proposition 7.4
Table 2.
The L c -stable classes K and their stability under rightand left compositions with clones C . Parameters: a, b ∈ { , } , i, j, k ∈ N with k ≥ i > j ≥ (ii) Let f := x + x + a, g := W i + a, h := x . . . x j + x j +1 + a,f := x + a, g := W i + x i +1 + a, h := x . . . x j + a, and note that f , g , h ∈ D i ∩ X j ∩ Ω aa and f , g , h ∈ D i ∩ X j ∩ Ω aa .(d) Clearly for any a, b ∈ { , } and for any f ∈ Ω a ∗ ∪ Ω ∗ b we have ( x + 1)( f ) = f + 1 / ∈ Ω a ∗ ∪ Ω ∗ b . For any a, b ∈ { , } there exists a function in D i ∩ X j ∩ Ω ab ;consider the functions f and f defined above. It follows that I ∗ K * Ω a ∗ ∪ Ω ∗ b .(e) The reduced polynomial of each of the functions ∧ ( W i + a, x i +1 + x i +2 + a ) , ∧ ( W i + x i +1 + a, x i +1 + a ) , ∨ ( W i + a, x i +1 + x i +2 + a ) , ∨ ( W i + x i +1 + a, x i +1 + a ) , contains the monomial x x . . . x i +1 and hence has degree at least i + 1; thereforenone of them is an element of D i . Note that the inner functions of the two compo-sitions on the left (right, resp.) are minors of f and g ( f and g , resp.) and hencebelong to K if a = b (if a = b , resp.). This shows that Λ c K * D i , V c K * D i .If j ≥
2, then ∧ ( h , x j +1 + x j +2 + a ) = x . . . x j x j +1 + x . . . x j x j +2 + . . . , ∨ ( h , x j +1 + x j +2 + a ) = x . . . x j x j +1 + x . . . x j x j +2 + . . . , ∧ ( h , x j +1 + a ) = x . . . x j x j +1 + . . . , ∨ ( h , x j +1 + a ) = x . . . x j x j +1 + . . . , where the terms that have not been written out have degree at most j . The j -element set { , . . . , j + 1 } has characteristic 1 in each, so these functions are notin X j . Note that the inner functions of the first (last, resp.) two compositions areminors of h and f ( h and f , resp.) and hence belong to K if a = b (if a = b ,resp.). This shows that Λ c K * X j , V c K * X j if j ≥ j = 1 and a = b , then ∧ ( x + a, x + a ) = x x + ax + ax + a / ∈ X , ∨ ( x + a, x + a ) = x x + ( a + 1) x + ( a + 1) x + a / ∈ X . Note that the inner functions are minors of f and hence belong to K . This showsthat Λ c K * X j , V c K * X j also in this case.(f) The reduced polynomial of each of the functions µ ( W i + a, x i +1 + x i +2 + a, a ) , µ ( W i + x i +1 + a, x i +1 + a, x i +2 + a )contains the monomial x x . . . x i +1 and hence has degree at least i + 1; thereforenone of them is an element of D i . Note that the inner functions of the two com-positions on the left (right, resp.) are minors of f and g ( f and g , resp.) andhence belong to K if a = b (if a = b , resp.). This shows that SM K * D i .If j ≥
2, then µ ( h , x j +1 + x j +2 + a, a ) = x . . . x j x j +1 + x . . . x j x j +2 + x j +1 + x j +1 x j +2 + a,µ ( h , x j +1 + a, x j +2 + a ) = x . . . x j x j +1 + x . . . x j x j +2 + x j +1 x j +2 + a. Neither of these functions is in X j , which can be seen by considering the character-istic of the j -element set { , . . . , j + 1 } . Note that the inner functions of the first(second, resp.) composition are minors of h and f ( h and f , resp.) and hencebelong to K if a = b (if a = b , resp.). This shows that SM K * X j if j ≥ a = b , then f ( x ,
0) = x + a / ∈ Ω ∗ b , f ( x ,
1) = x + a + 1 / ∈ Ω a ∗ ,f ( x + 1 , x ) = x + x + a + 1 / ∈ Ω a ∗ ∪ Ω ∗ b . TABILITY OF BOOLEAN FUNCTION CLASSES 33 If a = b , then f (0) = a / ∈ Ω ∗ b , f (1) = a + 1 / ∈ Ω a ∗ , f ( x + 1) = x + a + 1 / ∈ Ω a ∗ ∪ Ω ∗ b . These calculations show the non-inclusions K I * Ω ∗ b , K I * Ω a ∗ , and K I ∗ * Ω a ∗ ∪ Ω ∗ b .(h) Assume that i > j . Observe that each one of the functions g ( x , . . . , x i , g ( x , . . . , x i , g ( x , . . . , x i , , g ( x , . . . , x i , ,
1) contains the monomial x . . . x i , and it is the only monomial of degree i . Therefore none of them is amember of X j , which can be seen by considering the characteristic of the set [ i − j . We conclude that K I * X j and K I * X j .(i) For i ∈ { , } , the reduced polynomial of each of the functions g i ∗ ∧ , g i ∗ ∨ contains the monomial x x . . . x i +1 and hence has degree at least (in fact, exactly) i + 1; therefore none of them is an element of D i . Therefore K Λ c * D i , K V c * D i .For i ∈ { , } , the reduced polynomial of each of h i ∗ ∧ , h i ∗ ∨ contains themonomial x . . . x j +1 , and this is the only monomial of degree j + 1. We see thatthe characteristic of the j -element set [ j ] is 1 in each, so none is an element of X j ;therefore K Λ c * X j , K V c * X j .(j) For i ∈ { , } , the reduced polynomial of g i ∗ µ contains the monomial x x . . . x i +1 and hence has degree at least (in fact, exactly) i + 1; therefore g i ∗ µ / ∈ D i . Therefore K SM * D i ,If j ≥
2, then h ∗ µ = x x x . . . x j +2 + x x x . . . x j +2 + x x x . . . x j +2 + x j +3 + a,h ∗ µ = x x x . . . x j +2 + x x x . . . x j +2 + x x x . . . x j +2 + a, so the characteristic of the j -element set { , . . . , j + 1 } \ { } is 1. Therefore, for i ∈ { , } , h i ∗ µ / ∈ X j , which shows that K SM * X j .(iii) (k) The following calculations show that SM ( X ∩ Ω a ∗ ) * X (the first line)and SM ( X ∩ Ω ∗ a ) * X (the second line) for a ∈ { , } : µ ( x , x ,
0) = x x , µ ( x + 1 , x + 1 ,
1) = x x + 1 ,µ ( x , x ,
1) = x x + x + x , µ ( x + 1 , x + 1 ,
0) = x x + x + x + 1 . (iv) (l) We have f := x + x ∈ D i ∩ X j ∩ Ω = and f ′ := x ∈ D i ∩ X j ∩ Ω = , but f ( x ,
0) = x / ∈ Ω = , f ′ (0) = 0 / ∈ Ω = , f ( x ,
1) = x + 1 / ∈ Ω = , f ′ (1) = 1 / ∈ Ω = ,which shows that K I * Ω ≈ and K I * Ω ≈ .(m) Assume that j ≥
2. We have g := x x + x ∈ D i ∩ X j ∩ Ω = and g ′ := x x ∈ D i ∩ X j ∩ Ω = , but g ( x , x + 1) = x x + x + x / ∈ Ω = and g ′ ( x , x + 1) = x x + x / ∈ Ω = ; therefore K I ∗ * Ω ≈ .(n) We have x , x + 1 ∈ D i ∩ X j ∩ Ω = , but ∧ ( x , x + 1) = x · ( x + 1) = x + x = 0 / ∈ Ω = , ∨ ( x , x + 1) = x · ( x + 1) + x + ( x + 1) = 1 / ∈ Ω = ; therefore Λ c K * Ω = and V c K * Ω = . (cid:3) Proposition 7.4.
For every clone C , we have Ω C ⊆ Ω , C Ω ⊆ Ω , ∅ C ⊆ ∅ , C ∅ ⊆ ∅ .Proof. Trivial. (cid:3)
Proposition 7.5.
Let a ∈ { , } , and let C be a clone. (i) Ω a ∗ C ⊆ Ω a ∗ if and only if C ⊆ T . (ii) C Ω a ∗ ⊆ Ω a ∗ if and only if C ⊆ T a . (iii) Ω ∗ a C ⊆ Ω ∗ a if and only if C ⊆ T . (iv) C Ω ∗ a ⊆ Ω ∗ a if and only if C ⊆ T a .Proof. (i) Assume first that C ⊆ T . For any f ∈ Ω ( n ) a ∗ and g , . . . , g n ∈ C ( m ) , wehave f ( g , . . . , g n )(0 , . . . ,
0) = f ( g (0 , . . . , , . . . , g n (0 , . . . , f (0 , . . . ,
0) = a, so f ( g , . . . , g n ) ∈ Ω a ∗ . We conclude that Ω a ∗ C ⊆ Ω a ∗ . Conversely, if Ω a ∗ C ⊆ Ω a ∗ ,then C includes neither I nor I ∗ by Lemma 7.3(g), so C ⊆ T .(ii) Assume first that C ⊆ T a . For any f ∈ C ( n ) and g , . . . , g n ∈ Ω ( m ) a ∗ , we have f ( g , . . . , g n )(0 , . . . ,
0) = f ( g (0 , . . . , , . . . , g n (0 , . . . , f ( a, . . . a ) = a, so f ( g , . . . g n ) ∈ Ω a ∗ . We conclude that C Ω a ∗ ⊆ Ω a ∗ . Conversely, if C Ω a ∗ ⊆ Ω a ∗ ,then C includes neither I a nor I ∗ by Lemma 7.3(a), (d), so C ⊆ T a .(iii) Assume first that C ⊆ T . For any f ∈ Ω ( n ) ∗ a and g , . . . , g n ∈ C ( m ) , we have f ( g , . . . , g n )(1 , . . . ,
1) = f ( g (1 , . . . , , . . . , g n (1 , . . . , f (1 , . . . ,
1) = a, so f ( g , . . . , g n ) ∈ Ω ∗ a . We conclude that Ω ∗ a C ⊆ Ω ∗ a . Conversely, if Ω ∗ a C ⊆ Ω ∗ a ,then C includes neither I nor I ∗ by Lemma 7.3(g), so C ⊆ T .(iv) Assume first that C ⊆ T a . For any f ∈ C ( n ) and g , . . . , g n ∈ Ω ( m ) ∗ a , we have f ( g , . . . , g n )(1 , . . . ,
1) = f ( g (1 , . . . , , . . . , g n (1 , . . . , f ( a, . . . a ) = a, so f ( g , . . . g n ) ∈ Ω ∗ a . We conclude that C Ω ∗ a ⊆ Ω ∗ a . Conversely, if C Ω ∗ a ⊆ Ω ∗ a ,then C includes neither I a nor I ∗ by Lemma 7.3(a), (d), so C ⊆ T a . (cid:3) Lemma 7.6. (i)
For any f, g ∈ Ω = , we have f · g ∈ Ω = . (ii) For any f, g ∈ Ω = , we have f · g ∈ Ω = if and only if both f and g haveequal constant terms ( i.e, f, g ∈ Ω ∗ or f, g ∈ Ω ∗ ) .Proof. (i) Let α, β ∈ Ω = ∩ Ω ∗ . Then both α and β are sums of an even number ofmonomials. We have α · β ∈ Ω = because the expansion of the product of the twoeven sums of monomials yields a sum of an even number of monomials. We clearlyalso have that ( α + 1) · β = α · β + β , α · ( β + 1) = α · β + α , and ( α + 1) · ( β + 1) = α · β + α + β + 1 belong to Ω = because they are sums of polynomials with an evennumber of monomials plus a possible constant term. The claim now follows becauseany f ∈ Ω = is of the form α or α + 1 for some α ∈ Ω = ∩ Ω ∗ .(ii) Let α, β ∈ Ω = ∩ Ω ∗ . Then both α and β are sums of an odd number ofmonomials. We have α · β ∈ Ω = because the expansion of the product of the two oddsums of monomials yields a sum of an odd number of monomials. Consequently,( α + 1) · β = α · β + β ∈ Ω = , α · ( β + 1) = α · β + α ∈ Ω = , and ( α + 1) · ( β + 1) = α · β + α + β + 1 ∈ Ω = . (cid:3) Proposition 7.7.
Let C be a clone. (i) Ω = C ⊆ Ω = if and only if C ⊆ T c . (ii) C Ω = ⊆ Ω = for any clone C . (iii) Ω = C ⊆ Ω = if and only if C ⊆ T c . (iv) C Ω = ⊆ Ω = if and only if C ⊆ S .Proof. Recall that T c = Ω ∗ ∩ Ω = .(i) Assume first that C ⊆ T c . Let f ∈ Ω ( n )= and g , . . . , g n ∈ C ( m ) . Observingthat T c = Ω = Ω = ∩ Ω ∗ , we have f ( g , . . . , g n ) = X S ∈ M f Y i ∈ S g i ∈ Ω = , because each summand Q i ∈ S g i is odd by Lemma 7.6, and there are an even numberof such summands since f is even. We conclude that Ω = C ⊆ Ω = . Conversely, if Ω = C ⊆ Ω = , then C includes neither I , I , nor I ∗ by Lemma 7.3(l), (m), so C ⊆ T c .(ii) It is enough to prove the claim for C = Ω . Using the fact that Ω = h x x + 1 i ,we will apply Lemma 3.3. Let g , g ∈ Ω = . With the help of Lemma 7.6, we seethat ( x x + 1)( g , g ) = g g + 1 ∈ Ω = . Now it follows from Lemma 3.3 that ΩΩ = ⊆ Ω = . TABILITY OF BOOLEAN FUNCTION CLASSES 35 (iii) Assume first that C ⊆ T c . Let f ∈ Ω ( n ) = and g , . . . , g n ∈ C ( m ) . If f ∈ Ω ∗ ,then f ∈ Ω ∗ ∩ Ω = = T c , and it follows immediately from the fact that T c is aclone that f ( g , . . . , g n ) ∈ T c = Ω ∗ ∩ Ω = ⊆ Ω = . If f ∈ Ω ∗ , then f ′ := f + 1 ∈ Ω ∗ ∩ Ω = = T c . It follows from Lemma 2.8 that f ( g , . . . , g n ) = ( f ′ + 1)( g , . . . , g n ) = f ′ ( g , . . . , g n ) + 1( g , . . . , g n )= f ′ ( g , . . . , g n ) | {z } ∈ T c = Ω ∗ ∩ Ω = +1 ∈ Ω ∗ ∩ Ω = ⊆ Ω = . We conclude that Ω = C ⊆ Ω = . Conversely, if Ω = C ⊆ Ω = , then C includes neither I , I , nor I ∗ by Lemma 7.3(l), (m), so C ⊆ T c .(iv) For sufficiency, it is enough to prove the claim for C = S . Using the factthat S = h µ, x + 1 i , we will apply Lemma 3.3. Let g , g , g ∈ Ω = . We clearlyhave ( x + 1)( g ) = g + 1 ∈ Ω = . Applying Lemma 7.6, we see that µ ( g , g , g ) = g g + g g + g g ∈ Ω = ; for if g , g , g have the same constant term, then the threesummands g g , g g , g g belong to Ω = ; if they do not all have the same constantterm, then it is easy to see that exactly one of the summands belongs to Ω = andthe other two belong to Ω = . Now it follows from Lemma 3.3 that SΩ = ⊆ Ω = .For necessity, assume that C Ω = ⊆ Ω = . Then C includes neither I , I , Λ c , nor V c by Lemma 7.3(b), (n), so C ⊆ S . (cid:3) Proposition 7.8.
Let a, b ∈ { , } , and let C be a clone. (i) ( Ω ab ) C ⊆ ( Ω ab ) if and only if C ⊆ T c . (ii) C ( Ω ab ) ⊆ ( Ω ab ) if and only if C ⊆ T a ∩ T b .Proof. (i) Lemma 7.2 and Proposition 7.5(i), (iii) imply that ( Ω ab ) C ⊆ Ω ab when-ever C ⊆ T ∩ T = T c . Conversely, if ( Ω ab ) C ⊆ Ω ab , then C includes neither I , I , nor I ∗ by Lemma 7.3(g), so C ⊆ T c .(ii) Lemma 7.2 and Proposition 7.5(ii), (iv) imply that C ( Ω ab ) ⊆ Ω ab whenever C ⊆ T a ∩ T b .Assume now that C ( Ω ab ) ⊆ Ω ab . If a = b , then C includes neither I a nor I ∗ byLemma 7.3(a), (d), so C ⊆ T a = T a ∩ T b . If a = b , then C includes neither I , I ,nor I ∗ by Lemma 7.3(c), (d), so C ⊆ T c = T a ∩ T b . (cid:3) Lemma 7.9. (i) X S ⊆ X . (ii) ΩX ⊆ X .Proof. (i) Let f ∈ X ( n )0 and g , . . . , g n ∈ S ( m ) . Since X is the class of all reflexivefunctions and S is the class of all self-dual functions, we have, for any a ∈ { , } m that f ( g , . . . , g n )( a ) = f ( g ( a ) , . . . , g n ( a )) = f ( g ( a ) , . . . , g n ( a ))= f ( g ( a ) , . . . , g n ( a )) = f ( g , . . . , g n )( a ) , so f ( g , . . . , g n ) ∈ X .(ii) Let f ∈ Ω ( n ) , g , . . . , g n ∈ X ( m )0 . We have, for any a ∈ { , } m , f ( g , . . . , g n )( a ) = f ( g ( a ) , . . . , g n ( a )) = f ( g ( a ) , . . . , g n ( a )) = f ( g , . . . , g n )( a ) , so f ( g , . . . , g n ) ∈ X . (cid:3) Proposition 7.10.
Let k ∈ N + , and let C be a clone. (i) For k ≥ , X k C ⊆ X k if and only if C ⊆ LS . (ii) X C ⊆ X if and only if C ⊆ S . (iii) C X k ⊆ X k if and only if C ⊆ L . Proof. (i) For sufficiency, it is enough to prove the claim for C = LS . Using thefact that LS = h⊕ , x + 1 i , we apply Lemma 3.2. Let f ∈ X k . We have f ∗ ⊕ = ⊕ ( f σ , f σ , f σ ), where the σ i are as in Lemma 6.2. Since X k is closed underminors and sums by Lemmata 6.7 and 6.8, we have ⊕ ( f σ , f σ , f σ ) ∈ X k . As for f ∗ ( x + 1), note that f ∗ ( x + 1) = f + f ′ by Lemma 6.17. Since f ′ ∈ X k − ⊆ X k by Lemma 6.18, we have f ∗ ( x + 1) = f + f ′ ∈ X k by Lemma 6.8. It follows fromLemma 3.2 that X k LS ⊆ X k .For necessity, assume that X k C ⊆ X k . Then C includes neither I , I , Λ c , V c ,nor SM by Lemma 7.3(h), (i), (j), so C * LS .(ii) Assume first that C ⊆ S . Since X = ( X ∩ Ω = ) ∪ ( X ∩ Ω = ) = X ∪ S and S is a clone, it follows from Lemmata 2.11 and 7.9(i) that X S ⊆ ( X ∪ S ) S = X S ∪ SS ⊆ X ∪ S = X . Conversely, if X C ⊆ X , then C includes neither I , I , Λ c , nor V c by Lemma 7.3(h),(i), so C ⊆ S .(iii) For sufficiency, it is enough to prove the claim for C = L . Using the factthat L = h x + x , i , we apply Lemma 3.3. For any g , g ∈ X ( n ) k , we clearly have1( g ) = 1 ∈ X k and ( x + x )( g , g ) = g + g ∈ X k by Lemma 6.8. It follows fromLemma 3.3 that LX k ⊆ X k .For necessity, assume that C X k ⊆ X k . Then C includes neither Λ c , V c , nor SM by Lemma 7.3(e), (f), (k) so C ⊆ L . (cid:3) Proposition 7.11.
Let k ∈ N + , a ∈ { , } , and let C be a clone. (i) For k ≥ , ( X k ∩ Ω a ∗ ) C ⊆ X k ∩ Ω a ∗ if and only if C ⊆ L c . (ii) ( X ∩ Ω a ∗ ) C ⊆ X ∩ Ω a ∗ if and only if C ⊆ S c . (iii) C ( X k ∩ Ω a ∗ ) ⊆ X k ∩ Ω a ∗ if and only if C ⊆ L a .Proof. (i) Lemma 7.2 and Propositions 7.5(i) and 7.10(i) imply that ( X k ∩ Ω a ∗ ) C ⊆ X k ∩ Ω a ∗ whenever C ⊆ T ∩ LS = L c . Conversely, if ( X k ∩ Ω a ∗ ) C ⊆ X k ∩ Ω a ∗ ,then C includes neither I , I , I ∗ , Λ c , V c , nor SM by Lemma 7.3(g), (h), (i), (j), so C ⊆ L c .(ii) Lemma 7.2 and Propositions 7.5(i) and 7.10(i) imply that ( X ∩ Ω a ∗ ) C ⊆ X ∩ Ω a ∗ whenever C ⊆ T ∩ S = S c . Conversely, if ( X ∩ Ω a ∗ ) C ⊆ X ∩ Ω a ∗ , then C includes neither I , I , I ∗ , Λ c , nor V c by Lemma 7.3(g), (h), (i), so C ⊆ S c .(iii) Lemma 7.2 and Propositions 7.5(ii) and 7.10(iii) imply that C ( X k ∩ Ω a ∗ ) ⊆ X k ∩ Ω a ∗ whenever C ⊆ T a ∩ L = L a . Conversely, if C ( X k ∩ Ω a ∗ ) ⊆ X k ∩ Ω a ∗ , then C includes neither I a , I ∗ , Λ c , V c , nor SM by Lemma 7.3(a), (d), (e), (f), (k), so C ⊆ L a . (cid:3) Proposition 7.12.
Let k ∈ N + , a ∈ { , } , and let C be a clone. (i) For k ≥ , ( X k ∩ Ω ∗ a ) C ⊆ X k ∩ Ω ∗ a if and only if C ⊆ L c . (ii) ( X ∩ Ω ∗ a ) C ⊆ X ∩ Ω ∗ a if and only if C ⊆ S c . (iii) C ( X k ∩ Ω ∗ a ) ⊆ X k ∩ Ω ∗ a if and only if C ⊆ L a .Proof. (i) Lemma 7.2 and Propositions 7.5(iii) and 7.10(i) imply that ( X k ∩ Ω ∗ a ) C ⊆ X k ∩ Ω ∗ a whenever C ⊆ T ∩ LS = L c . Conversely, if ( X k ∩ Ω ∗ a ) C ⊆ X k ∩ Ω ∗ a ,then C includes neither I , I , I ∗ , Λ c , V c , nor SM by Lemma 7.3(g), (h), (i), (j), so C ⊆ L c .(ii) Lemma 7.2 and Propositions 7.5(iii) and 7.10(i) imply that ( X ∩ Ω ∗ a ) C ⊆ X ∩ Ω ∗ a whenever C ⊆ T ∩ S = S c . Conversely, if ( X ∩ Ω ∗ a ) C ⊆ X ∩ Ω ∗ a , then C includes neither I , I , I ∗ , Λ c , nor V c by Lemma 7.3(g), (h), (i), so C ⊆ S c .(iii) Lemma 7.2 and Propositions 7.5(iv) and 7.10(iii) imply that C ( X k ∩ Ω a ∗ ) ⊆ X k ∩ Ω a ∗ whenever C ⊆ T a ∩ L = L a . Conversely, if C ( X k ∩ Ω ∗ a ) ⊆ X k ∩ Ω ∗ a , then TABILITY OF BOOLEAN FUNCTION CLASSES 37 C includes neither I a , I ∗ , Λ c , V c , nor SM by Lemma 7.3(a), (d), (e), (f), (k), so C ⊆ L a . (cid:3) Proposition 7.13.
Let k ∈ N + , and let C be a clone. (i) For k ≥ , ( X k ∩ Ω = ) C ⊆ X k ∩ Ω = if and only if C ⊆ L c . (ii) ( X ∩ Ω = ) C ⊆ X ∩ Ω = if and only if C ⊆ S . (iii) For k ≥ , C ( X k ∩ Ω = ) ⊆ X k ∩ Ω = if and only if C ⊆ L . (iv) C ( X ∩ Ω = ) ⊆ X ∩ Ω = for any clone C .Proof. (i) Lemma 7.2 and Propositions 7.7(i) and 7.10(i) imply that ( X k ∩ Ω = ) C ⊆ X k ∩ Ω = whenever C ⊆ L c ∩ T c = L c . Conversely, if ( X k ∩ Ω = ) C ⊆ X k ∩ Ω = , then C includes neither I , I , I ∗ , Λ c , V c , nor SM by Lemma 7.3(i), (j), (l), (m), so C ⊆ L c .(ii) Assume first that C ⊆ S . Since X ∩ Ω = = X , it follows from Lemma 7.9(i)that ( X ∩ Ω = ) C ⊆ X S ⊆ X = X ∩ Ω = . Conversely, if ( X ∩ Ω = ) C ⊆ X ∩ Ω = ,then C includes neither I , I , Λ c , nor V c by Lemma 7.3(i), (l), so C ⊆ S .(iii) Lemma 7.2 and Propositions 7.7(ii) and 7.10(iii) imply that C ( X k ∩ Ω = ) ⊆ X k ∩ Ω = whenever C ⊆ L ∩ Ω = L . Conversely, if C ( X k ∩ Ω = ) ⊆ X k ∩ Ω = , then C includes neither Λ c , V c , nor SM by Lemma 7.3(e), (f) (note that Ω ⊆ Ω = ), so C ⊆ L .(iv) Observing that X ∩ Ω = = X , this follows immediately from Lemma 7.9(ii). (cid:3) Proposition 7.14.
Let k ∈ N + , and let C be a clone. (i) For k ≥ , ( X k ∩ Ω = ) C ⊆ X k ∩ Ω = if and only if C ⊆ L c . (ii) ( X ∩ Ω = ) C ⊆ X ∩ Ω = if and only if C ⊆ S . (iii) For k ≥ , C ( X k ∩ Ω = ) ⊆ X k ∩ Ω = if and only if C ⊆ LS . (iv) C ( X ∩ Ω = ) ⊆ X ∩ Ω = if and only if C ⊆ S .Proof. (i) Lemma 7.2 and Propositions 7.7(iii) and 7.10(i) imply that ( X k ∩ Ω = ) C ⊆ X k ∩ Ω = whenever C ⊆ LS ∩ T c = L c . Conversely, if ( X k ∩ Ω = ) C ⊆ X k ∩ Ω = , then C includes neither I , I , I ∗ , Λ c , V c , nor SM by Lemma 7.3(i), (j), (l), (m), so C ⊆ L c .(ii) Assume first that C ⊆ S . Since X ∩ Ω = = S and S is a clone, it is immediatelyobvious that ( X ∩ Ω = ) C ⊆ SS ⊆ S = X ∩ Ω = . Conversely, if ( X ∩ Ω = ) C ⊆ X ∩ Ω = ,then C includes neither I , I , Λ c , nor V c by Lemma 7.3(i), (l), so C ⊆ S .(iii) Lemma 7.2 and Propositions 7.7(iv) and 7.10(iii) imply that C ( X k ∩ Ω = ) ⊆ X k ∩ Ω = whenever C ⊆ L ∩ S = LS . Conversely, if C ( X k ∩ Ω = ) ⊆ X k ∩ Ω = , then C includes neither I , I , Λ c , V c , nor SM by Lemma 7.3(b), (e), (f), so C ⊆ LS .(iv) Assume first that C ⊆ S . Since X ∩ Ω = = S and S is a clone, it is immediatelyobvious that C ( X ∩ Ω = ) ⊆ SS ⊆ S = X ∩ Ω = . Conversely, if C ( X ∩ Ω = ) ⊆ X ∩ Ω = ,then C includes neither I , I , Λ c , nor V c by Lemma 7.3(b), (e), so C ⊆ S . (cid:3) Proposition 7.15.
Let k ∈ N + , a, b ∈ { , } , and let C be a clone. (i) For k ≥ , ( X k ∩ Ω ab ) C ⊆ X k ∩ Ω ab if and only if C ⊆ L c . (ii) ( X ∩ Ω ab ) C ⊆ X ∩ Ω ab if and only if C ⊆ S c . (iii) If k ≥ , then C ( X k ∩ Ω ab ) ⊆ X k ∩ Ω ab if and only if C ⊆ L a ∩ L b . (iv) If a = b , then C ( X ∩ Ω ab ) ⊆ X ∩ Ω ab if and only if C ⊆ T a . (v) If a = b , then C ( X ∩ Ω ab ) ⊆ X ∩ Ω ab if and only if C ⊆ S c .Proof. (i) Lemma 7.2 and Propositions 7.8(i) and 7.10(i) imply that ( X k ∩ Ω ab ) C ⊆ X k ∩ Ω ab whenever C ⊆ LS ∩ T c = L c . Conversely, if ( X k ∩ Ω ab ) C ⊆ X k ∩ Ω ab , then C includes neither I , I , I ∗ , Λ c , V c , nor SM by Lemma 7.3(g), (i), (j), so C ⊆ L c .(ii) Lemma 7.2 and Propositions 7.8(i) and 7.10(ii) imply that ( X ∩ Ω ab ) C ⊆ X ∩ Ω ab whenever C ⊆ S ∩ T c = S c . Conversely, if ( X ∩ Ω ab ) C ⊆ X ∩ Ω ab , then C includes neither I , I , I ∗ , Λ c , nor V c by Lemma 7.3(g), (i), so C ⊆ S c . (iii) Lemma 7.2 and Propositions 7.8(ii) and 7.10(iii) imply that C ( X k ∩ Ω ab ) ⊆ X k ∩ Ω ab whenever C ⊆ L ∩ T a ∩ T b = L a ∩ L b .Assume now that C ( X k ∩ Ω ab ) ⊆ X k ∩ Ω ab . Then C includes neither I ∗ , Λ c , V c ,nor SM by Lemma 7.3(d), (e), (f). If a = b , then C includes neither I nor I byLemma 7.3(c), so C ⊆ L c = L a ∩ L b . If a = b , then C does not include I a byLemma 7.3(a), so C ⊆ L a = L a ∩ L b .(iv) Assume first that C ⊆ T a . We have C ( X ∩ Ω aa ) ⊆ T a ( X ∩ Ω a ∗ ) ⊆ X ∩ Ω a ∗ = X ∩ Ω aa , where the second inclusion holds because T a ( X ∩ Ω a ∗ ) ⊆ ΩX ⊆ X byLemma 7.9(ii) and T a ( X ∩ Ω a ∗ ) ⊆ T a Ω a ∗ ⊆ Ω a ∗ as can be easily seen.Assume now that C ( X ∩ Ω aa ) ⊆ X ∩ Ω aa . Then C includes neither I a nor I ∗ byLemma 7.3(a), (d), so C ⊆ T a .(v) Assume first that C ⊆ S c . Since X ∩ Ω = X ∩ Ω = ∩ Ω ∗ = S ∩ Ω ∗ = S c ,we have C ( X ∩ Ω ) ⊆ S c S c ⊆ S c = X ∩ Ω . Note also that X ∩ Ω = X ∩ Ω = ∩ Ω ∗ = S ∩ Ω ∗ = S \ S c . For any f ∈ S ( n )c , g , . . . , g n ∈ ( S \ S c ) ( m ) , it holds that f ( g , . . . , g ) ∈ S and f ( g , . . . , g n )(0 , . . . ,
0) = f ( g (0 , . . . , , . . . , g n (0 , . . . , f (1 , . . . ,
1) = 1 , so f ( g , . . . , g n ) / ∈ S c , that is, f ( g , . . . , g n ) ∈ S \ S c . Consequently, S c ( S \ S c ) ⊆ S \ S c ,and it follows that C ( X ∩ Ω ) ⊆ S c ( S \ S c ) ⊆ S \ S c = X ∩ Ω . Assume now that C ( X ∩ Ω ab ) ⊆ X ∩ Ω ab . Then C includes neither I , I , I ∗ , Λ c ,nor V c by Lemma 7.3(c), (d), (e), so C ⊆ S c . (cid:3) Proposition 7.16.
Let k ∈ N + , and let C be a clone. (i) D k C ⊆ D k if and only if C ⊆ L . (ii) C D k ⊆ D k if and only if C ⊆ L .Proof. (i) For sufficiency, it is enough to prove the claim for C = L . Using the factthat L = h x + x , i , we apply Lemma 3.2. It is easy to see that for any function f ∈ D k , we have deg( f ∗ ( x + x )) ≤ deg( f ) ≤ k and deg( f ∗ ≤ deg( f ) ≤ k , so f ∗ ( x + x ) , f ∗ ∈ D k . It follows from Lemma 3.2 that D k L ⊆ D k .For necessity, assume that D k C ⊆ D k . Then C includes neither Λ c , V c , nor SM by Lemma 7.3(i), (j), so C ⊆ L .(ii) For sufficiency, it is enough to prove the claim for C = L . Using the fact that L = h x + x , i , we apply Lemma 3.3. It is clear that for any g , g ∈ D ( m ) k , thefunctions ( x + x )( g , g ) = g + g and 1( g ) = 1 have degree at most k , and aretherefore members of D k . It follows from Lemma 3.3 that LD k ⊆ D k .For necessity, assume that C D k ⊆ D k . Then C includes neither Λ c , V c , nor SM by Lemma 7.3(e), (f), so C ⊆ L . (cid:3) Proposition 7.17.
Let a ∈ { , } , and let C be a clone. (i) D C ⊆ D and C D ⊆ D for any clone C . (ii) ( D ∩ Ω a ∗ ) C ⊆ D ∩ Ω a ∗ for any clone C . (iii) C ( D ∩ Ω a ∗ ) ⊆ D ∩ Ω a ∗ if and only if C ⊆ T a .Proof. (i) Clear, as any composition in which either all inner functions are constantor the outer function is constant is a constant function.(ii) Clear, as for any m -ary g , . . . , g n ∈ Ω we have c ( n ) a ( g , . . . , g n ) = c ( m ) a ∈ D ∩ Ω a ∗ .(iii) Lemma 7.2, Proposition 7.5(ii), and part (i) imply that C ( D ∩ Ω a ∗ ) ⊆ D ∩ Ω a ∗ whenever C ⊆ Ω ∩ T a = T a . TABILITY OF BOOLEAN FUNCTION CLASSES 39
Assume now that C * T a . Then there exists a g ∈ C that does not preserve a ,and we have g (c ( n ) a , . . . , c ( n ) a ) = c ( n )1 − a / ∈ Ω a ∗ . Therefore C ( D ∩ Ω a ∗ ) * D ∩ Ω a ∗ . (cid:3) Proposition 7.18.
Let k ∈ N + , a ∈ { , } , and let C be a clone. (i) ( D k ∩ Ω a ∗ ) C ⊆ ( D k ∩ Ω a ∗ ) if and only if C ⊆ L . (ii) C ( D k ∩ Ω a ∗ ) ⊆ ( D k ∩ Ω a ∗ ) if and only if C ⊆ L a . (iii) ( D k ∩ Ω ∗ a ) C ⊆ ( D k ∩ Ω ∗ a ) if and only if C ⊆ L . (iv) C ( D k ∩ Ω ∗ a ) ⊆ ( D k ∩ Ω ∗ a ) if and only if C ⊆ L a .Proof. (i) Lemma 7.2 and Propositions 7.5(i) and 7.16(i) imply that ( D k ∩ Ω a ∗ ) C ⊆ D k ∩ Ω a ∗ whenever C ⊆ L ∩ T = L . Conversely, if ( D k ∩ Ω a ∗ ) C ⊆ D k ∩ Ω a ∗ , then C includes neither I , I ∗ , Λ c , V c , nor SM by Lemma 7.3(g), (i), (j), so C ⊆ L .(ii) Lemma 7.2 and Propositions 7.5(ii) and 7.16(ii) imply that C ( D k ∩ Ω a ∗ ) ⊆ D k ∩ Ω a ∗ whenever C ⊆ L ∩ T a = L a . Conversely, if C ( D k ∩ Ω a ∗ ) ⊆ D k ∩ Ω a ∗ , then C includes neither I a , I ∗ , Λ c , V c , nor SM by Lemma 7.3(a), (d), (e), (f), so C ⊆ L a .(iii) Lemma 7.2 and Propositions 7.5(iii) and 7.16(i) imply that ( D k ∩ Ω ∗ a ) C ⊆ D k ∩ Ω ∗ a whenever C ⊆ L ∩ T = L . Conversely, if ( D k ∩ Ω ∗ a ) C ⊆ D k ∩ Ω ∗ a , then C includes neither I , I ∗ , Λ c , V c , nor SM by Lemma 7.3(g), (i), (j), so C ⊆ L .(iv) Lemma 7.2 and Propositions 7.5(iv) and 7.16(ii) imply that C ( D k ∩ Ω ∗ a ) ⊆ D k ∩ Ω ∗ a whenever C ⊆ L ∩ T a = L a . Conversely, if C ( D k ∩ Ω ∗ a ) ⊆ D k ∩ Ω ∗ a ,then C includes neither I a , I ∗ , Λ c , V c , nor SM by Lemma 7.3(a), (d), (e), (f), so C ⊆ L a . (cid:3) Proposition 7.19.
Let k ∈ N + , and let C be a clone. (i) For k ≥ , ( D k ∩ Ω = ) C ⊆ D k ∩ Ω = if and only if C ⊆ L c . (ii) ( D ∩ Ω = ) C ⊆ D ∩ Ω = if and only if C ⊆ LS . (iii) C ( D k ∩ Ω = ) ⊆ D k ∩ Ω = if and only if C ⊆ L .Proof. (i) Lemma 7.2 and Propositions 7.7(i) and 7.16(i) imply that ( D k ∩ Ω = ) C ⊆ D k ∩ Ω = whenever C ⊆ L ∩ T c = L c . Conversely, if ( D k ∩ Ω = ) C ⊆ D k ∩ Ω = , then C includes neither I , I , I ∗ , Λ c , V c , nor SM by Lemma 7.3(i), (j), (l), (m) so C ⊆ L c .(ii) Assume first that C ⊆ LS . Note that LS = D ∩ Ω = and L = D , and let f ∈ ( D ∩ Ω = ) ( n ) and g , . . . , g n ∈ ( D ∩ Ω = ) ( m ) . The composition f ( g , . . . , g n ) is amember of L because the outer and inner functions all belong to D = L . Moreover,it is a sum of an even number of odd polynomials, that is, an even polynomial, so f ( g , . . . , g n ) ∈ Ω = . We conclude that ( D ∩ Ω = ) C ⊆ D ∩ Ω = .Assume now that ( D ∩ Ω = ) C ⊆ D ∩ Ω = . Then C includes neither I , I , Λ c , V c , nor SM by Lemma 7.3(i), (j), (l) so C ⊆ LS .(iii) Lemma 7.2 and Propositions 7.7(ii) and 7.16(ii) imply that C ( D k ∩ Ω = ) ⊆ D k ∩ Ω = whenever C ⊆ L ∩ Ω = L . Conversely, if C ( D k ∩ Ω = ) ⊆ D k ∩ Ω = , then C includes neither Λ c , V c , nor SM by Lemma 7.3(e), (f), so C ⊆ L . (cid:3) Proposition 7.20.
Let k ∈ N + , and let C be a clone. (i) For k ≥ , ( D k ∩ Ω = ) C ⊆ D k ∩ Ω = if and only if C ⊆ L c . (ii) ( D ∩ Ω = ) C ⊆ D ∩ Ω = if and only if C ⊆ LS . (iii) C ( D k ∩ Ω = ) ⊆ D k ∩ Ω = if and only if C ⊆ LS .Proof. (i) Lemma 7.2 and Propositions 7.7(iii) and 7.16(i) imply that ( D k ∩ Ω = ) C ⊆ D k ∩ Ω = whenever C ⊆ L ∩ T c = L c . Conversely, if ( D k ∩ Ω = ) C ⊆ D k ∩ Ω = , then C includes neither I , I , I ∗ , Λ c , V c , nor SM by Lemma 7.3(i), (j), (l), (m), so C ⊆ L c .(ii) If C ⊆ LS , then, since D ∩ Ω = = LS and LS is a clone, it clearly holds that( D ∩ Ω = ) C ⊆ LS LS ⊆ LS = D ∩ Ω = . Conversely, if ( D ∩ Ω = ) C ⊆ D ∩ Ω = , then C includes neither I , I , Λ c , V c , nor SM by Lemma 7.3(i), (j), (l), so C ⊆ LS . (iii) Lemma 7.2 and Propositions 7.7(iv) and 7.16(ii) imply that C ( D k ∩ Ω = ) ⊆ D k ∩ Ω = whenever C ⊆ L ∩ S = LS . Conversely, if C ( D k ∩ Ω = ) ⊆ D k ∩ Ω = , then C includes neither I , I , Λ c , V c , nor SM by Lemma 7.3(b), (e), (f), so C ⊆ LS . (cid:3) Proposition 7.21.
Let k ∈ N + , a, b ∈ { , } , and let C be a clone. (i) ( D k ∩ Ω ab ) C ⊆ D k ∩ Ω ab if and only if C ⊆ L c . (ii) C ( D k ∩ Ω ab ) ⊆ D k ∩ Ω ab if and only if C ⊆ L a ∩ L b .Proof. (i) Lemma 7.2 and Propositions 7.8(i) and 7.16(i) imply that ( D k ∩ Ω ab ) C ⊆ D k ∩ Ω ab whenever C ⊆ L ∩ T c = L c . Conversely, if ( D k ∩ Ω ab ) C ⊆ D k ∩ Ω ab , then C includes neither I , I , I ∗ , Λ c , V c , nor SM by Lemma 7.3(g), (i), (j), so C ⊆ L c .(ii) Lemma 7.2 and Propositions 7.8(ii) and 7.16(ii) imply that C ( D k ∩ Ω ab ) ⊆ D k ∩ Ω ab whenever C ⊆ L ∩ T a ∩ T b = L a ∩ L b .Assume now that C ( D k ∩ Ω ab ) ⊆ D k ∩ Ω ab . Then C includes neither I ∗ , Λ c , V c , nor SM by Lemma 7.3(d), (e), (f). If a = b , then C does not include I a byLemma 7.3(a), so C ⊆ L a = L a ∩ L b . If a = b , then C includes neither I nor I byLemma 7.3(c), so C ⊆ L c = L a ∩ L b . (cid:3) Proposition 7.22.
Let i, j ∈ N + with i > j ≥ , and let C be a clone. (i) ( D i ∩ X j ) C ⊆ D i ∩ X j if and only if C ⊆ LS . (ii) C ( D i ∩ X j ) ⊆ D i ∩ X j if and only if C ⊆ L .Proof. (i) Lemma 7.2 and Propositions 7.10(i), (ii) and 7.16(i) imply that ( D i ∩ X j ) C ⊆ D i ∩ X j whenever C ⊆ LS ∩ L = LS if k ≥ C ⊆ S ∩ L = LS if k = 1. Conversely, if ( D i ∩ X j ) C ⊆ D i ∩ X j , then C includes neither I , I , Λ c , V c ,nor SM by Lemma 7.3(h), (i), (j), so C ⊆ LS .(ii) Lemma 7.2 and Propositions 7.10(iii) and 7.16(ii) imply that C ( D i ∩ X j ) ⊆ D i ∩ X j whenever C ⊆ L ∩ L = L . Conversely, if C ( D i ∩ X j ) ⊆ D i ∩ X j , then C includes neither Λ c , V c , nor SM by Lemma 7.3(e), (f), so C ⊆ L . (cid:3) Proposition 7.23.
Let i, j ∈ N + with i > j ≥ , a ∈ { , } , and let C be a clone. (i) ( D i ∩ X j ∩ Ω a ∗ ) C ⊆ D i ∩ X j ∩ Ω a ∗ if and only if C ⊆ L c . (ii) C ( D i ∩ X j ∩ Ω a ∗ ) ⊆ D i ∩ X j ∩ Ω a ∗ if and only if C ⊆ L a . (iii) ( D i ∩ X j ∩ Ω ∗ a ) C ⊆ D i ∩ X j ∩ Ω ∗ a if and only if C ⊆ L c . (iv) C ( D i ∩ X j ∩ Ω ∗ a ) ⊆ D i ∩ X j ∩ Ω ∗ a if and only if C ⊆ L a .Proof. (i) Lemma 7.2 and Propositions 7.5(i) and 7.22(i) imply that ( D i ∩ X j ∩ Ω a ∗ ) C ⊆ D i ∩ X j ∩ Ω a ∗ whenever C ⊆ LS ∩ T = L c . Conversely, if ( D i ∩ X j ∩ Ω a ∗ ) C ⊆ D i ∩ X j ∩ Ω a ∗ , then C includes neither I , I , I ∗ , Λ c , V c , nor SM by Lemma 7.3(g),(h), (i), (j), so C ⊆ L c .(ii) Lemma 7.2 and Propositions 7.5(ii) and 7.22(ii) imply that C ( D i ∩ X j ∩ Ω a ∗ ) ⊆ D i ∩ X j ∩ Ω a ∗ whenever C ⊆ L ∩ T a = L a . Conversely, if C ( D i ∩ X j ∩ Ω a ∗ ) ⊆ D i ∩ X j ∩ Ω a ∗ , then C includes neither I a , I ∗ , Λ c , V c , nor SM by Lemma 7.3(a), (d),(e), (f), so C ⊆ L a .(iii) Lemma 7.2 and Propositions 7.5(iii) and 7.22(i) imply that ( D i ∩ X j ∩ Ω ∗ a ) C ⊆ D i ∩ X j ∩ Ω ∗ a whenever C ⊆ LS ∩ T = L c . Conversely, if ( D i ∩ X j ∩ Ω ∗ a ) C ⊆ D i ∩ X j ∩ Ω ∗ a , then C includes neither I , I , I ∗ , Λ c , V c , nor SM by Lemma 7.3(g),(h), (i), (j), so C ⊆ L c .(iv) Lemma 7.2 and Propositions 7.5(iv) and 7.22(ii) imply that C ( D i ∩ X j ∩ Ω ∗ a ) ⊆ D i ∩ X j ∩ Ω ∗ a whenever C ⊆ L ∩ T a = L a . Conversely, if C ( D i ∩ X j ∩ Ω ∗ a ) ⊆ D i ∩ X j ∩ Ω ∗ a , then C includes neither I a , I ∗ , Λ c , V c , nor SM by Lemma 7.3(a), (d),(e), (f), so C ⊆ L a . (cid:3) Proposition 7.24.
Let i, j ∈ N + with i > j ≥ , and let C be a clone. (i) For j ≥ , ( D i ∩ X j ∩ Ω = ) C ⊆ D i ∩ X j ∩ Ω = if and only if C ⊆ L c . TABILITY OF BOOLEAN FUNCTION CLASSES 41 (ii) ( D i ∩ X ∩ Ω = ) C ⊆ D i ∩ X ∩ Ω = if and only if C ⊆ LS . (iii) C ( D i ∩ X j ∩ Ω = ) ⊆ D i ∩ X j ∩ Ω = if and only if C ⊆ L .Proof. (i) Lemma 7.2 and Propositions 7.7(ii) and 7.22(i) imply that ( D i ∩ X j ∩ Ω = ) C ⊆ D i ∩ X j ∩ Ω = whenever C ⊆ LS ∩ T c = L c . Conversely, if ( D i ∩ X j ∩ Ω = ) C ⊆ D i ∩ X j ∩ Ω = , then C includes neither I , I , I ∗ , Λ c , V c , nor SM by Lemma 7.3(i),(j), (l), (m), so C ⊆ L c .(ii) Lemma 7.2 and Propositions 7.13(ii) and 7.16(i) imply that ( D i ∩ X ∩ Ω = ) C ⊆ D i ∩ X ∩ Ω = whenever C ⊆ S ∩ L = LS . Conversely, if ( D i ∩ X ∩ Ω = ) C ⊆ D i ∩ X ∩ Ω = ,then C includes neither I , I , Λ c , V c , nor SM by Lemma 7.3(i), (j), (l), so C ⊆ LS .(iii) Lemma 7.2 and Propositions 7.7(ii) and 7.22(ii) imply that C ( D i ∩ X j ∩ Ω = ) ⊆ D i ∩ X j ∩ Ω = whenever C ⊆ L ∩ Ω = L . Conversely, if C ( D i ∩ X j ∩ Ω = ) ⊆ D i ∩ X j ∩ Ω = ,then C includes neither Λ c , V c , nor SM by Lemma 7.3(e), (f), so C ⊆ L . (cid:3) Proposition 7.25.
Let i, j ∈ N + with i > j ≥ , and let C be a clone. (i) For j ≥ , ( D i ∩ X j ∩ Ω = ) C ⊆ D i ∩ X j ∩ Ω = if and only if C ⊆ L c . (ii) ( D i ∩ X ∩ Ω = ) C ⊆ D i ∩ X ∩ Ω = if and only if C ⊆ LS . (iii) C ( D i ∩ X j ∩ Ω = ) ⊆ D i ∩ X j ∩ Ω = if and only if C ⊆ LS .Proof. (i) Lemma 7.2 and Propositions 7.7(iii) and 7.22(i) imply that ( D i ∩ X j ∩ Ω = ) C ⊆ D i ∩ X j ∩ Ω = whenever C ⊆ LS ∩ T c = L c . Conversely, if ( D i ∩ X j ∩ Ω = ) C ⊆ D i ∩ X j ∩ Ω = , then C includes neither I , I , I ∗ , Λ c , V c , nor SM by Lemma 7.3(i),(j), (l), (m), so C ⊆ L c .(ii) Lemma 7.2 and Propositions 7.14(ii) and 7.16(i) imply that ( D i ∩ X ∩ Ω = ) C ⊆ D i ∩ X ∩ Ω = whenever C ⊆ S ∩ L = LS . Conversely, if ( D i ∩ X ∩ Ω = ) C ⊆ D i ∩ X ∩ Ω = ,then C includes neither I , I , Λ c , V c , nor SM by Lemma 7.3(i), (j), (l), so C ⊆ LS .(iii) Lemma 7.2 and Propositions 7.7(iv) and 7.22(ii) imply that C ( D i ∩ X j ∩ Ω = ) ⊆ D i ∩ X j ∩ Ω = whenever C ⊆ L ∩ S = LS . Conversely, if C ( D i ∩ X j ∩ Ω = ) ⊆ D i ∩ X j ∩ Ω = ,then C includes neither I , I , Λ c , V c , nor SM by Lemma 7.3(b), (e), (f), so C ⊆ LS . (cid:3) Proposition 7.26.
Let i, j ∈ N + with i > j ≥ , a, b ∈ { , } , and let C be aclone. (i) ( D i ∩ X j ∩ Ω ab ) C ⊆ D i ∩ X j ∩ Ω ab if and only if C ⊆ L c . (ii) C ( D i ∩ X j ∩ Ω ab ) ⊆ D i ∩ X j ∩ Ω ab if and only if C ⊆ L a ∩ L b .Proof. (i) Lemma 7.2 and Propositions 7.10(i), (ii) and 7.21(i) imply that ( D i ∩ X j ∩ Ω ab ) C ⊆ D i ∩ X j ∩ Ω ab whenever C ⊆ L c ∩ LS = L c if j ≥ C ⊆ L c ∩ S = L c if j = 1. Conversely, if ( D i ∩ X j ∩ Ω ab ) C ⊆ D i ∩ X j ∩ Ω ab , then C includes neither I , I , I ∗ , Λ c , V c , nor SM by Lemma 7.3(g), (i), (j), so C ⊆ L c .(ii) Lemma 7.2 and Propositions 7.10(iii) and 7.21(ii) imply that C ( D i ∩ X j ∩ Ω ab ) ⊆ D i ∩ X j ∩ Ω ab whenever C ⊆ L a ∩ L b ∩ L = L a ∩ L b .Assume now that C ( D i ∩ X j ∩ Ω ab ) ⊆ D i ∩ X j ∩ Ω ab . If a = b , then C includesneither I a , I ∗ , Λ c , V c , nor SM by Lemma 7.3(a), (d), (e), (f), so C ⊆ L a = L a ∩ L b .If a = b , then C includes neither I , I , I ∗ , Λ c , V c , nor SM by Lemma 7.3(c), (d),(e), (f), so C ⊆ L c = L a ∩ L b . (cid:3) Proof of Theorem 7.1.
The theorem puts together Propositions 7.4, 7.5, 7.7, 7.8,7.10, 7.11, 7.12, 7.13, 7.14, 7.15, 7.16, 7.17, 7.18, 7.19, 7.20, 7.21, 7.22, 7.23, 7.24,7.25, 7.26. (cid:3)
With the help of Post’s lattice (Figure 1) and by reading off from Table 2, wecan determine for any pair ( C , C ) of clones which L c -stable classes are ( C , C )-stable. If L c ⊆ C , then any ( C , C )-stable class is ( I c , L c )-stable by Lemma 2.16and hence also L c -stable by Lemma 6.2. Therefore, in the case when L c ⊆ C , the( C , C )-stable classes are among the L c -stable ones and they can be easily picked out from Table 2. In particular, we have an explicit description of ( I c , C )-stableclasses (“clonoids” of Aichinger and Mayr [1]) and C -stable classes for L c ⊆ C . The L -stable classes (see Corollary 7.28(iii)) were determined earlier by Kreinecker [10,Theorem 3.12]. Corollary 7.27. (i)
The ( I c , L c ) -stable classes are Ω , Ω a ∗ , Ω ∗ a , Ω ≈ , Ω ab , D k , D k ∩ Ω a ∗ , D k ∩ Ω ∗ a , D k ∩ Ω ≈ , D k ∩ Ω ab , X k , X k ∩ Ω a ∗ , X k ∩ Ω ∗ a , X k ∩ Ω ≈ , X k ∩ Ω ab , D i ∩ X j , D i ∩ X j ∩ Ω a ∗ , D i ∩ X j ∩ Ω ∗ a , D i ∩ X j ∩ Ω ≈ , D i ∩ X j ∩ Ω ab , D , D ∩ Ω a ∗ , ∅ , for a, b ∈ { , } , ≈ ∈ { = , = } , and i, j, k ∈ N + with i > j ≥ . (ii) The ( I c , LS ) -stable classes are Ω , Ω ≈ , X k , X k ∩ Ω ≈ , D k , D k ∩ Ω ≈ , D i ∩ X j , D i ∩ X j ∩ Ω ≈ , D , ∅ , for ≈ ∈ { = , = } , and i, j, k ∈ N + with i > j ≥ . (iii) The ( I c , L ) -stable classes are Ω , Ω ∗ , Ω ∗ , Ω = , Ω , X k , X k ∩ Ω ∗ , X k ∩ Ω ∗ , X k ∩ Ω = , X k ∩ Ω , D k , D k ∩ Ω ∗ , D k ∩ Ω ∗ , D k ∩ Ω = , D k ∩ Ω , D i ∩ X j , D i ∩ X j ∩ Ω ∗ , D i ∩ X j ∩ Ω ∗ , D i ∩ X j ∩ Ω = , D i ∩ X j ∩ Ω , D , D ∩ Ω ∗ , ∅ , for k ∈ N + . (iv) The ( I c , L ) -stable classes are Ω , Ω ∗ , Ω ∗ , Ω = , Ω , X k , X k ∩ Ω ∗ , X k ∩ Ω ∗ , X k ∩ Ω = , X k ∩ Ω , D k , D k ∩ Ω ∗ , D k ∩ Ω ∗ , D k ∩ Ω = , D k ∩ Ω , D i ∩ X j , D i ∩ X j ∩ Ω ∗ , D i ∩ X j ∩ Ω ∗ , D i ∩ X j ∩ Ω = , D i ∩ X j ∩ Ω , D , D ∩ Ω ∗ , ∅ , for k ∈ N + . (v) The ( I c , L ) -stable classes are Ω , Ω = , X k , X k ∩ Ω = , D k , D k ∩ Ω = , D i ∩ X j , D i ∩ X j ∩ Ω = , D , ∅ , for k ∈ N + . (vi) The ( I c , S c ) -stable classes are Ω , Ω a ∗ , Ω ∗ a , Ω ≈ , Ω ab , X ∩ Ω ≈ , X ∩ Ω ab , D , D ∩ Ω a ∗ , ∅ , for a, b ∈ { , } and ≈ ∈ { = , = } . (vii) The ( I c , S ) -stable classes are Ω , Ω ≈ , X ∩ Ω ≈ , D , ∅ , for ≈ ∈ { = , = } . (viii) The ( I c , T c ) -stable classes are Ω , Ω a ∗ , Ω ∗ a , Ω = , Ω ab , X ∩ Ω = , X ∩ Ω aa , D , D ∩ Ω a ∗ , ∅ , for a, b ∈ { , } (ix) The ( I c , T ) -stable classes are Ω , Ω ∗ , Ω ∗ , Ω = , Ω , X ∩ Ω = , X ∩ Ω , D , D ∩ Ω ∗ , ∅ . (x) The ( I c , T ) -stable classes are Ω , Ω ∗ , Ω ∗ , Ω = , Ω , X ∩ Ω = , X ∩ Ω , D , D ∩ Ω ∗ , ∅ . (xi) The ( I c , Ω ) -stable classes are Ω , Ω = , X ∩ Ω = , D , ∅ . Corollary 7.28. (i)
The L c -stable classes are Ω , Ω a ∗ , Ω ∗ a , Ω ≈ , Ω ab , D k , D k ∩ Ω a ∗ , D k ∩ Ω ∗ a , D k ∩ Ω ≈ , D k ∩ Ω ab , X k , X k ∩ Ω a ∗ , X k ∩ Ω ∗ a , X k ∩ Ω ≈ , X k ∩ Ω ab , D i ∩ X j , D i ∩ X j ∩ Ω a ∗ , D i ∩ X j ∩ Ω ∗ a , D i ∩ X j ∩ Ω ≈ , D i ∩ X j ∩ Ω ab , D , D ∩ Ω a ∗ , ∅ , for a, b ∈ { , } , ≈ ∈ { = , = } , and i, j, k ∈ N + with i > j ≥ . (ii) The LS -stable classes are Ω , X k , X ∩ Ω ≈ , D k , D ∩ Ω ≈ , D i ∩ X j , D i ∩ X ∩ Ω ≈ , D , ∅ , for ≈ ∈ { = , = } and i, j, k ∈ N + with i > j ≥ . (iii) The L -stable classes are Ω , Ω ∗ , D k , D k ∩ Ω ∗ , D , D ∩ Ω ∗ , ∅ , for k ∈ N + . (iv) The L -stable classes are Ω , Ω ∗ , D k , D k ∩ Ω ∗ , D , D ∩ Ω ∗ , ∅ , for k ∈ N + . (v) The L -stable classes are Ω , D k , D , ∅ , for k ∈ N + . (vi) The S c -stable classes are Ω , Ω a ∗ , Ω ∗ a , Ω ≈ , Ω ab , X ∩ Ω ≈ , X ∩ Ω ab , D , D ∩ Ω a ∗ , ∅ , for a, b ∈ { , } and ≈ ∈ { = , = } . (vii) The S -stable classes are Ω , X ∩ Ω ≈ , D , ∅ , for ≈ ∈ { = , = } . (viii) The T c -stable classes are Ω , Ω a ∗ , Ω ∗ a , Ω = , Ω ab , D , D ∩ Ω a ∗ , ∅ , for a, b ∈ { , } (ix) The T -stable classes are Ω , Ω ∗ , D , D ∩ Ω ∗ , ∅ . (x) The T -stable classes are Ω , Ω ∗ , D , D ∩ Ω ∗ , ∅ . (xi) The Ω -stable classes are Ω , D , ∅ . Recall from Lemma 6.2(iii) that ( I c , L c )-stability is equivalent to L c -stability.Therefore, as expected, the classes listed in Corollary 7.27(i) are the same as those TABILITY OF BOOLEAN FUNCTION CLASSES 43 in Corollary 7.28(i). By comparing Corollary 7.27(vi) with Corollary 7.28(vi), wesee also that ( I c , S c )-stability is equivalent to S c -stability. Whether the reason forthis is a relationship similar to Lemma 6.2 is beyond the scope of this paper. Corollary 7.29. S c -stability is equivalent to ( I c , S c ) -stability. Final remarks and perspectives
Looking into directions of future research, one may consider arbitrary pairs ofclones C and C on arbitrary sets A and B and describe the ( C , C )-stable setsin this case. However, this task is challenging. Firstly, there are uncountablymany clones on sets with at least three elements (see [20]), and not all of themare known. Secondly, for given clones C and C , there may be uncountably many( C , C )-stable classes, in which case an explicit description may be unattainable.For this reason, a natural next step would be to consider ( C , C )-stability forclones C and C on the two-element set { , } , which are well known (see Post [15]).Moreover, the cardinality of the closure system of ( I c , C )-stable classes of Booleanfunctions is known for every clone C on { , } , due to the following result bySparks [19]. However, this result does not provide an explicit description of the( I c , C )-stable classes, even for the cases where the number of ( I c , C )-stable classesis finite. Theorem 8.1 ([19, Theorem 1.3]) . Let A be a finite set with | A | > , and let B := { , } . Denote by J A the clone of projections on A , and let C be a clone on B . Then the following statements hold. (i) L ( J A ,C ) is finite if and only if C contains a near-unanimity operation. (ii) L ( J A ,C ) is countably infinite if and only if C contains a Mal’cev operationbut no majority operation. (iii) L ( J A ,C ) has the cardinality of the continuum if and only if C contains nei-ther a near-unanimity operation nor a Mal’cev operation. Recall that an n -ary operation f ∈ O B with n ≥ near-unanimityoperation if f ( x, . . . , x, y, x, . . . , x ) = x for all x, y ∈ B , where the single occurrenceof y can occur in any of the n argument positions. A ternary near-unanimityoperation is called a majority operation. A ternary operation f ∈ O B is called a Mal’cev operation if f ( y, y, x ) = f ( x, y, y ) = x for all x, y ∈ B .A clone C on { , } contains a Mal’cev operation but no majority operation(statement (ii)) if and only if L c ⊆ C ⊆ L ; this situation is completely describedin the current paper. In view of Theorem 8.1, explicit descriptions of the ( C , C )-stable classes of Boolean functions seem attainable in the case when C containsa near-unanimity function (statement (i)), but this may not be the case when C contains neither a near-unanimity operation nor a Mal’cev operation (statement(iii)). This suggests a feasible direction for future research. Acknowledgments
This work is funded by National Funds through the FCT – Funda¸c˜ao para aCiˆencia e a Tecnologia, I.P., under the scope of the project UIDB/00297/2020 (Cen-ter for Mathematics and Applications) and the project PTDC/MAT-PUR/31174/2017.
References [1]
E. Aichinger, P. Mayr,
Finitely generated equational classes,
J. Pure Appl. Algebra (2016) 2816–2827.[2]
E. Aichinger, B. Rossi,
A clonoid based approach to some finiteness results in universalalgebraic geometry,
Algebra Universalis (1) (2020), Paper No. 8, 7 pp. [3] M. Bouaziz, M. Couceiro, M. Pouzet,
Join-irreducible Boolean functions,
Order (2010)261–282.[4] M. Couceiro, S. Foldes,
Definability of Boolean function classes by linear equations overGF(2),
Discrete Appl. Math. (2004) 29–34.[5]
M. Couceiro, S. Foldes,
Functional equations, constraints, definability of function classes,and functions of Boolean variables,
Acta Cybernet. (2007) 61–75.[6] M. Couceiro, S. Foldes,
Function classes and relational constraints stable under composi-tions with clones,
Discuss. Math. Gen. Algebra Appl. (2009) 109–121.[7] M. Couceiro, E. Lehtonen,
Linearly definable classes of Boolean functions, In: M. Couceiro,P. Monnin, A. Napoli (Eds.), Proceedings of the 1st International Conference on Algebras,Graphs and Ordered Sets (ALGOS 2020), pp. 39–46.[8]
O. Ekin, S. Foldes, P. L. Hammer, L. Hellerstein,
Equational characterizations ofBoolean function classes,
Discrete Math. (2000) 27–51.[9]
S. Fioravanti,
Closed sets of finitary functions between finite fields of coprime order,
AlgebraUniversalis (4) (2020), Paper No. 52, 14 pp.[10] S. Kreinecker,
Closed function sets on groups of prime order,
J. Mult.-Valued Logic SoftComput. (2019) 51–74.[11] D. Lau,
Function Algebras on Finite Sets, A Basic Course on Many-Valued Logic and CloneTheory,
Springer-Verlag, Berlin, Heidelberg, 2006.[12]
E. Lehtonen,
Reconstruction of functions from minors, habilitation thesis, Technische Uni-versit¨at Dresden, Dresden, 2018. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-237864 [13]
A. I. Mal’cev,
Iterative algebras and Post varieties (Russian),
Algebra Logika (2) (1966)5–24.[14] N. Pippenger,
Galois theory for minors of finite functions,
Discrete Math. (2002) 405–419.[15]
E. L. Post,
The Two-Valued Iterative Systems of Mathematical Logic,
Annals of Mathemat-ics Studies, no. 5, Princeton University Press, Princeton, 1941.[16]
V. N. Potapov,
Splitting of hypercube into k -faces and DP-colorings of hypergraphs, arXiv:1905.04461v4.[17] I. Rosenberg, ¨Uber die funktionale Vollst¨andigkeit in den mehrwertigen Logiken (Struk-tur der Funktionen von mehreren Ver¨anderlichen auf endlichen Mengen),
RozpravyˇCeskoslovensk´e Akad. Vˇed ˇRada Mat. Pˇr´ırod. Vˇed (1970) 3–93.[18] S. N. Selezneva, A. V. Bukhman,
Polynomial-time algorithms for checking some propertiesof Boolean functions given by polynomials,
Theory Comput. Syst. (2016) 383–391.[19] A. Sparks,
On the number of clonoids,
Algebra Universalis (4) (2019), Paper No. 53, 10pp.[20] Yu. I. Yanov, A. A. Muchnik,
On the existence of k -valued closed classes that have nobases, Dokl. Akad. Nauk SSSR127