Locally monotone Boolean and pseudo-Boolean functions

LOCALL Y MONOTONE BOOLEAN AND PSEUDO-BOOLEAN FUNCTIONS MIGUEL COUCEIRO, JEAN-LUC MARICHAL, AND T AM ´ AS W ALDHAUSER Abstract. W e propose lo cal versions of monotonicit y for Boolean and pseudo- Boolean functions: sa y that a p seudo-Boolean (Boolean) function i s p -locally monotone if none of i ts partial deri v ativ es change s in sign on tuples which differ in less than p p ositions. As it turns out, this parameterized not i on provides a hierarch y of mono tonicities for pseudo-Boolean (Boolean) functions. Local monotonicities are shown to b e tigh tly related to lattice counter- parts of classi cal partial deriv atives via the notion of p ermutable de r iv ative s. More precisely , p - locally monot one functions are sho wn to ha ve p -p ermut able lattice deriv ativ es and, in the ca s e of symmetric f unctions, these t wo notions coincide. W e pro vide f urther results relating t hese t wo notions, and present a classification of p -lo cally monotone f unctions, as w el l as of functions having p - permutable deri v ativ es, in terms of certain forbidden “sections”, i .e., functions which can be obtained b y substituting constan ts f or v ariables. This description is made expli cit in the sp ecial case whe n p = 2. 1. Intr oduction Throughout this paper, let [ n ] = { 1 , . . . , n } and B = { 0 , 1 } . W e ar e in ter ested in the so-c a lled Bo o lean functions f ∶ B n → B and pseudo- Bo olean functions f ∶ B n → R , where n denotes the ar ity of f . The p oint wis e order ing of functions is denoted by ≤ , i.e., f ≤ g mea ns that f ( x ) ≤ g ( x ) f or all x ∈ B n . The nega tion of x ∈ B is defined b y x = x ⊕ 1, where ⊕ stands for addition mo dulo 2 . F o r x, y ∈ B , w e set x ∧ y = min ( x, y ) and x ∨ y = max ( x, y ) . F o r k ∈ [ n ] , x ∈ B n , and a ∈ B , let x a k be the tuple in B n whose i -th comp onent is a , if i = k , and x i , otherwise. W e use the shorthand nota tio n x ab j k for ( x a j ) b k = ( x b k ) a j . More generally , for S ⊆ [ n ] , a ∈ B n , and x ∈ B S , let a x S be the tuple in B n whose i -th comp onent is x i , if i ∈ S , a nd a i , otherwise. Let i ∈ [ n ] and f ∶ B n → R . A v ariable x i is s a id to be essential in f , or tha t f dep ends on x i , if there exists a ∈ B n such th at f ( a 0 i ) ≠ f ( a 1 i ) . Otherwise, x i is said to b e inessential in f . Let S ⊆ [ n ] and f ∶ B n → R . W e say that g ∶ B S → R is an S -se ction of f if there exists a ∈ B n such that g ( x ) = f ( a x S ) for all x ∈ B S . By a se ction of f we mean a n S -section of f for so me S ⊆ [ n ] , i.e., any function which can b e obtained fro m f by replacing some of its v aria bles b y cons tant s . The (discr ete) p artial derivative of f ∶ B n → R with resp ect to its k -th v ariable is the function ∆ k f ∶ B n → R defined by ∆ k f ( x ) = f ( x 1 k ) − f ( x 0 k ) ; see [8 , 12]. Note that Date : March 27, 2012. 2010 Mathematics Subject Classific ation. 06E30, 94C10. Key wor ds and phr ases. Bo olean function, pseudo-Bo olean function, lo cal monotonicit y , dis- crete partial deriv ative, join and meet de r iv ativ es. 1 2 MIGUEL COUCEIRO, JEAN-LUC MARICHAL, AND T AM ´ AS W ALDHA USE R ∆ k f doe s not dep end on its k -th v ariable, hence it could b e regarded as a function of arity n − 1, but for no tational conv enience we define it as an n - ary function. A pseudo-Bo olea n function f ∶ B n → R ca n always b e repr e s ent e d by a multilinear po lynomial of degree at most n (see [13]), that is, (1) f ( x ) =  S ⊆[ n ] a S  i ∈ S x i , where a S ∈ R . F or instance, the m ultilinear expression for a binar y pseudo-Boo lean function is given b y (2) a 0 + a 1 x 1 + a 2 x 2 + a 12 x 1 x 2 . This representation is very c o nv enient for computing the pa rtial deriv a tives of f . Indeed, ∆ k f can b e obtained by a pplying the cor resp onding for mal deriv ative to the multilinear representation of f . Thus, from (1), we immediately obtain (3) ∆ k f ( x ) =  S ∋ k a S  i ∈ S ∖{ k } x i . W e sa y that f is isotone (resp. antitone ) in its k -th variable if ∆ k f ( x ) ≥ 0 (resp. ∆ k f ( x ) ≤ 0) for a ll x ∈ B n . If f is either isotone or antitone in its k -th v a r iable, then we sa y that f is monotone in its k -th variable . If f is iso to ne (resp. antitone, monotone) in all o f its v ar iables, then f is an isotone (resp. antitone , monotone ) function . 1 It is clear that any section o f an is otone (res p. antitone, monotone) function is also isotone (resp. antitone, monotone). Thus defined, a function f ∶ B n → R is monotone if and only if none of its partial deriv atives changes in sign on B n . Noteworth y e x amples of monotone functions include the so -called pseudo-p olyno- mial functions [2, 3] which play an imp ortant role, for instance, in the qualita- tive appro a ch to decision making; for gener al background see, e.g., [1, 6]. In the current setting, pseudo-p olynomial functions can b e tho ught of as comp ositions p ○ ( ϕ 1 , . . . , ϕ n ) o f (lattice) polynomial functions p ∶ [ a, b ] n → [ a, b ] , a < b , with unary functions ϕ i ∶ B → [ a, b ] , i ∈ [ n ] . Interestingly , pseudo-p olynomial functions f ∶ B n → R coincide exactly with those pseudo-Bo olea n functions that are mono to ne. Theorem 1. A pseudo - Bo ole an funct ion is m onotone if and only if it is a pseudo- p olynomial function. Pr o of. Clear ly , every pseudo -p olynomial function is mono tone. F o r the conv erse, suppo se that f ∶ B n → R is monoto ne and let a ∈ R b e the minimum and b ∈ R the maximum of f . Constant functions are obviously pseudo-p olyno mia l functions, therefore we assume a < b . Define ϕ i ∶ B → { a, b } b y ϕ i ( 0 ) = a a nd ϕ i ( 1 ) = b if f is isotone in its i - th v ariable and ϕ i ( 0 ) = b and ϕ i ( 1 ) = a otherwise. Let p ∶ { a, b } n → [ a, b ] b e given by p = f ○ ( ϕ − 1 1 , . . . , ϕ − 1 n ) . Thu s defined, p is isotone (i.e., o rder-pres erving) in eac h v aria ble and hence, b y Theorem D in Goo dstein [1 0, p. 237], there exists a p olynomial function p ′ ∶ [ a, b ] n → [ a, b ] such that p ′  { a,b } n = p . Therefore f is the pseudo -p o lynomial function p ′ ○ ( ϕ 1 , . . . , ϕ n ) .  In the sp ecial case of Bo o le a n functions, mono tone functions are most frequent among functions of small (essential) arity . F o r insta nce, among binary functions 1 Note that the terms “p ositive” an d “nondecreasing” (resp. “negative ” and “non increasing”) are often used i nstead of isotone (resp. an titone), and it is al so customary to use the word “mono- tone” only for isotone func tions. LOCALL Y MONOTONE BOOLEAN AND PSEUDO-BOOLEAN FUNCTIONS 3 f ∶ B 2 → B , there are exactly t wo no n-monotone functions, namely the Bo olean sum x 1 ⊕ x 2 and its nega tion x 1 ⊕ x 2 ⊕ 1. E ach o f these functions is in fact highly non-monotone in the sense that any of its partial deriv atives c hang es in sign when negating its unique essen tial v ariable; this is not the case, e.g., with f ( x 1 , x 2 , x 3 ) = x 1 − x 1 x 2 + x 2 x 3 which is non-mo notone but none of its partial deriv atives c hanges in sign when nega ting any of its v ar ia bles (see Example 6 b elow). This fac t motiv a tes the study of these “skew” functions, i.e., these highly non- monotone functions. T o formalize this proble m we pr op ose the following par am- eterized r elaxations o f mono to nicit y : a function f ∶ B n → R is p - lo cally mono tone if no ne of its partial der iv a tiv e s changes in s ig n when nega ting less than p of its v a r iables, or eq uiv alently , on tuples which differ in les s than p p ositions. With this terminolo gy , our problem reduces to ask ing whic h Bo olean functions are not 2- lo cally mo notone. As w e will see (Corollary 10), these are precisely those functions that hav e the Bo o lean sum or its negatio n as a binary se c tio n. In this paper we extend this study to pseudo- Bo olean functions and show that these par ameterized relaxatio ns of mo no tonicity are tig h tly related to th e f o llowing lattice versions of partial deriv a tives. F or f ∶ B n → R a nd k ∈ [ n ] , let ∧ k f ∶ B n → R and ∨ k f ∶ B n → R be the p artial latt ic e derivatives defined by ∧ k f ( x ) = f ( x 0 k ) ∧ f ( x 1 k ) and ∨ k f ( x ) = f ( x 0 k ) ∨ f ( x 1 k ) . The latter, kno wn a s the k -th j oin deriva t ive o f f , w as pro po sed by F adini [9] while the former , kno w n as the k - th me et derivative of f , was introduced by Thayse [16]. In [17] these lattice der iv atives were shown to b e rela ted to so- called prime implicants a nd implica tes of Boo lean functions whic h pla y an impor tant r o le in the consensus metho d for Bo olea n and pseudo-Bo o lean functions. F o r further ba ck- ground and applica tions see, e.g., [4, 5, 7 , 15, 18]. Observe that, just like in the ca se of the partia l deriv ative ∆ k f , the k -th v aria ble of each of the lattice deriv atives ∧ k f and ∨ k f is inessential. The following pro p os ition assem bles some ba sic prop e rties of lattice deriv atives. Prop ositio n 2. F or any pseudo-Bo ole an functions f , g ∶ B n → R and j, k ∈ [ n ] , j ≠ k , the fol lowing hold: ( i ) ∧ k ∧ k f = ∧ k f and ∨ k ∨ k f = ∨ k f ; ( ii ) if f ≤ g , then ∧ k f ≤ ∧ k g and ∨ k f ≤ ∨ k g ; ( iii ) ∧ j ∧ k f = ∧ k ∧ j f and ∨ j ∨ k f = ∨ k ∨ j f ; ( iv ) ∨ k ∧ j f ≤ ∧ j ∨ k f . F r om equa tions (1) and (3 ) it follows that every function is (up to an additive constant) uniquely determined b y its partial deriv atives. As it turns out, this do es not hold when lattice der iv atives ar e considered. How ever, as we shall see (Theorem 22), there a re only tw o types o f such exceptions. Now, if a n n -ar y pseudo-Bo olea n function is 2-lo cally mo no tone, then for every j, k ∈ [ n ] , j ≠ k , we hav e ∨ k ∧ j f = ∧ j ∨ k f (see Lemma 11 b elow). This motiv ates the no tio n o f p ermutable lattice deriv atives. As it turns out, p -lo cal monotonicity of f implies per mutabilit y of p of its la ttice deriv atives (see Theorem 2 1). How ever the conv er se does not ho ld (see Exa mple 24). The structure of this pap er go es as follo ws. In Section 2 w e formalize the notion of p -lo cal monotonicity a nd show that it g ives r ise to a hierar chy of mo notonici- ties whose lar gest mem ber is the cla ss of all n -ary pseudo-Bo olea n functions (this 4 MIGUEL COUCEIRO, JEAN-LUC MARICHAL, AND T AM ´ AS W ALDHAUSER is the cas e whe n p = 1) and who s e smallest member is the class of n -ar y mono- tone functions (this is the case when p = n ). W e also pr ovide a characterization of p -lo cally mo notone functions in terms o f “for bidden” se ctions; as mentioned, this characterization is made explicit in the sp ecial c a se when p = 2. In Sectio n 3 we int r o duce the notion of p ermutable lattice deriv atives. Similarly to lo ca l mono- tonicity , the notion of p ermutable lattice deriv atives gives ris e to nested classes , each of whic h is also describ ed in terms of its sections. In the Bo olea n c ase and for p = 2 , these t wo par a meterized notions a re shown to coincide; this does not hold for pseudo-Bo olea n functions even when p = 2 (see E xample 13). (At the end of Sec- tion 3 we also pro v ide some g a me-theoretic int erpretations of p -lo ca l monotonicit y and p -p ermutabilit y o f lattice deriv a tives.) How ever, in Section 4, we show that a symmetric function is p -lo cally monotone if and only if it ha s p -p ermutable lattice deriv atives. In the last se ction we discuss directions for future resear ch. 2. Local monotonicities The following definition formulates a lo cal version of monoto nicit y given in ter ms of Hamming distance b etw een tuples. In what follows we assume that p ∈ [ n ] . Definition 3. W e say that f ∶ B n → R is p -lo c al ly monotone if, for every k ∈ [ n ] and every x , y ∈ B n , we ha ve  i ∈[ n ]∖{ k }  x i − y i  < p ⇒ ∆ k f ( x ) ∆ k f ( y ) ≥ 0 . An y p -lo ca lly mo notone pseudo-Bo olean function is also p ′ -lo cally monotone f o r every p ′ ≤ p . Every function f ∶ B n → R is 1-lo ca lly monotone, and f is n -lo cally monotone if and only if it is monotone. Thus p -lo cal monotonicity is a r elaxation of monotonicity , and the nested classes of p -lo cally monotone functions for p = 1 , . . . , n provide a hierar ch y of monotonicities fo r n -ary pseudo-B o olean functions. The weak est nontrivial condition is 2 -lo cal mo notonicity , therefore we will simply say that f is lo c al ly monotone whenever f is 2-lo cally mono to ne. 2 If f is p -lo ca lly monotone for so me p < n but not ( p + 1 ) -lo cally mono tone, then we say that f is exactly p -lo c al ly monotone , or that the de gr e e of lo c al monotonicity of f is p . If f ∶ B n → B is a Bo olean function, then ∆ k f ( x ) ∈ { − 1 , 0 , 1 } for all x ∈ B n , hence the condition ∆ k f ( x ) ∆ k f ( y ) ≥ 0 in the definition of p -local monotonicity is equiv a lent to (4)  ∆ k f ( x ) − ∆ k f ( y ) ≤ 1 . F r om this it follows that a B o o lean function f ∶ B n → B is lo cally monotone if and only if (5)  ∆ k f ( x ) − ∆ k f ( y )  ≤  i ∈[ n ]∖{ k }  x i − y i  . (see [14, Lemma 5.1 ] for a pr o of of (5) in a s lightly more general framework). In a sense, the latter identit y means that ∆ k f is “ 1-Lipschitz con tinuous”. The following pro po sition is just a r eformulation of the definition of p - lo cal mono- tonicity . 2 In [11], lo cal monotonicity is used to refer to Boolean functions which are monotone (i.e., isotone or ant i tone in each v ariable). LOCALL Y MONOTONE BOOLEAN AND PSEUDO-BOOLEAN FUNCTIONS 5 Prop ositio n 4. A function f ∶ B n → R is p - lo c al ly monotone if and only if, for every k ∈ [ n ] , S ⊆ [ n ] ∖ { k } , with  S  = p − 1 , and every a ∈ B n , x , y ∈ B S , we have (6) ∆ k f ( a x S ) ∆ k f ( a y S ) ≥ 0 . Equivalently, a pseudo-Bo ole an function is p - lo c al ly monotone if and only if n one of its p artial derivatives changes in sign when ne gating less than p of its variables. As a sp ecial cas e, w e hav e tha t f ∶ B n → R is lo cally mono tone if a nd only if, for every j, k ∈ [ n ] , j ≠ k , and every x ∈ B n , we ha ve (7) ∆ k f ( x 0 j ) ∆ k f ( x 1 j ) ≥ 0 . Equiv alently , a pseudo-Bo o lean function is lo cally mono tone if and only if none of its partial deriv atives changes in sign when negating any of its v a riables. By (4) we see that, for Bo o lean functions f ∶ B n → B , inequality (7) can be replaced with  ∆ j k f ( x ) ≤ 1 , where ∆ j k f ( x ) = ∆ j ∆ k f ( x ) = ∆ k ∆ j f ( x ) . Example 5. As observed, the binary Bo olea n sum f 1 ( x 1 , x 2 ) = x 1 ⊕ x 2 = x 1 + x 2 − 2 x 1 x 2 and the binary B o olean equiv alence f 2 ( x 1 , x 2 ) = f 1 ( x 1 , x 2 ) = x 1 ⊕ x 2 ⊕ 1 = 1 − x 1 − x 2 + 2 x 1 x 2 are not lo cally mo notone. Indeed, we ha ve  ∆ 12 f 1 ( x 1 , x 2 ) =  ∆ 12 f 2 ( x 1 , x 2 ) = 2. Example 6. Consider the ternary Bo ole a n function f ∶ B 3 → B given b y f ( x 1 , x 2 , x 3 ) = x 1 − x 1 x 2 + x 2 x 3 . Since ∆ 2 f may c hange in sign (∆ 2 f ( x ) = x 3 − x 1 ), the function f is not monotone. How ever, f is lo cally monotone since  ∆ 12 f ( x ) = 1 ,  ∆ 13 f ( x ) = 0, and  ∆ 23 f ( x ) = 1. Thus f is e x actly 2-lo cally monotone. Ex ample 26 in Section 4 provides, for each p ≥ 2, examples of exa c tly p -lo cally monotone functions. F act 7. A fun ction f ∶ B n → R is p -lo c al ly monotone if and only if so is αf + β for every α, β ∈ R , with α ≠ 0 . The same holds for any function obtaine d fr om f by ne gating some of its variables. The next theorem gives a characterization of p -lo cally monotone functions in terms of their se ctions. Theorem 8. A function f ∶ B n → R is p -lo c al ly monotone if and only if every p -ary se ction of f is monotone. Pr o of. W e just need to observe that the inequa lit y (6) is equiv a lent to ∆ k g ( x ) ∆ k g ( y ) ≥ 0, whe r e g is the p -ary section of f defined by g ( x ) = f ( a x S ∪{ k } ) , where S is a ( p − 1 ) - subset of [ n ] ∖ { k } . Thus f is p -lo cally monotone if and only if ∆ k g ( x ) ∆ k g ( y ) ≥ 0 holds for every x , y ∈ B S ∪{ k } , and for every S ∪ { k } -section g of f .  By combining (7) with Theorem 8, we can eas ily verify the following corollary . Corollary 9. A fun ction f ∶ B n → R is lo c al ly monotone if and only if every binary se ction (2) of f satisfies a 1 ( a 1 + a 12 ) ≥ 0 and a 2 ( a 2 + a 12 ) ≥ 0 . Since every binary Boolean function is monotone except for x ⊕ y and x ⊕ y ⊕ 1, we also obtain the following corollary . Corollary 10. A Bo ole an function f ∶ B n → B is lo c al ly m onotone if and only if neither x ⊕ y nor x ⊕ y ⊕ 1 is a se ction of f . 6 MIGUEL COUCEIRO, JEAN-LUC MARICHAL, AND T AM ´ AS W ALDHAUSER 3. Permut abl e l a ttice deriv a tives The a im of this section is to relate commutation of lattice deriv atives to p - lo cal mono tonicit y . The starting p oint is the characteriza tio n of lo cally monotone Bo olean functions given in Theorem 12 b elow. Lemma 1 1 . If f ∶ B n → R is lo c al ly monotone, the n ∨ k ∧ j f = ∧ j ∨ k f for al l j, k ∈ [ n ] , j ≠ k . Pr o of. Let f ∶ B n → R b e a lo c a lly monotone function, and let j, k ∈ [ n ] , j ≠ k . Setting a = f ( x 00 j k ) , b = f ( x 01 j k ) , c = f ( x 10 j k ) , and d = f ( x 11 j k ) , the desired equality ∨ k ∧ j f ( x ) = ∧ j ∨ k f ( x ) takes the form (8) ( a ∧ c ) ∨ ( b ∧ d ) = ( a ∨ b ) ∧ ( c ∨ d ) . Since f is 2 -lo cally monoto ne , the bina r y sectio n g ( u , v ) = f ( x uv j k ) is mono tone, according to Theo r em 8. If g is iso tone in u , then a ≤ c and b ≤ d , while if g is antitone in u , then a ≥ c and b ≥ d . Similarly , w e hav e either a ≤ b and c ≤ d or a ≥ b and c ≥ d , dep e nding o n whether g is isotone o r antitone in v . Thu s we need to consider four cases , and in each one o f them it is str aightforw ar d to verify (8).  Theorem 12. A Bo ole an function f ∶ B n → B is lo c al ly monotone if and only if ∨ k ∧ j f = ∧ j ∨ k f holds for al l j, k ∈ [ n ] , j ≠ k . Pr o of. If f is lo cally monotone , then ∨ k ∧ j f = ∧ j ∨ k f by L e mma 11. If f is not lo cally monotone, then Cor o llary 10 implies that there exists a ∈ B n and j, k ∈ [ n ] , j ≠ k , suc h that the binary section g ( u, v ) = f ( a uv j k ) is of the form g ( u , v ) = u ⊕ v or g ( u, v ) = u ⊕ v ⊕ 1. The n we ha ve ∨ k ∧ j f ( a ) = ( g ( 0 , 0 ) ∧ g ( 1 , 0 )) ∨ ( g ( 0 , 1 ) ∧ g ( 1 , 1 )) = 0 , ∧ j ∨ k f ( a ) = ( g ( 0 , 0 ) ∨ g ( 0 , 1 )) ∧ ( g ( 1 , 0 ) ∨ g ( 1 , 1 )) = 1 , showing t hat ∨ k ∧ j f ≠ ∧ j ∨ k f .  As the next example shows, Theor em 12 is no t v alid for pseudo-Bo o lean func- tions. Example 13 . Let f b e the bina ry pseudo-Bo olean function defined by f ( 0 , 0 ) = 1, f ( 0 , 1 ) = 4, f ( 1 , 0 ) = 2 and f ( 1 , 1 ) = 3 . Then we hav e ∨ 2 ∧ 1 f = ∧ 1 ∨ 2 f = 3 and ∨ 1 ∧ 2 f = ∧ 2 ∨ 1 f = 2. Howev er, f is not lo cally monotone since ∆ 1 f ( x 0 2 ) ∆ 1 f ( x 1 2 ) = − 1. Lemma 11 and Theor em 1 2 motiv ate the following notion of p ermutabilit y o f lattice deriv atives, and its relation to lo ca l monotonicities. Definition 14. W e say tha t a ps eudo-Bo olean function f ∶ B n → R has p -p ermutable lattic e derivatives if, for every p -subset { k 1 , . . . , k p } ⊆ [ n ] , every choice of the op- erators O k i ∈ { ∧ k i , ∨ k i } ( i = 1 , . . . , p ) , and every p er m utatio n π ∈ S p , the fo llowing ident ity holds: O k 1 ⋯ O k p f = O k π ( 1 ) ⋯ O k π ( p ) f . If f ∶ B n → R has n -per m utable lattice der iv atives, then w e s imply say that f has p ermu table lattic e derivatives . Every function f ∶ B n → R has 1-p ermutable lattice deriv atives. W e will see in Theorem 23 that if a function f ∶ B n → R has p -p ermutable lattice deriv atives, then it also has p ′ -p ermutable lattice deriv a tives for e very p ′ ≤ p . LOCALL Y MONOTONE BOOLEAN AND PSEUDO-BOOLEAN FUNCTIONS 7 F act 15. A function f ∶ B n → R has p -p ermu table lattic e deriv atives if and only if so has αf + β for every α, β ∈ R , with α ≠ 0 . The same ho lds for any function obtaine d fr om f by ne gating some of its variables. F act 16. A funct ion f ∶ B n → R has p -p erm u table lattic e derivatives if and only if every p -ary se ction of f has p ermutable lattic e derivatives. In the particular case when p = 2, we ha ve the f o llowing descr iptio n of functions having 2-p ermutable lattice deriv atives. The pro of is a s tr aightforw a r d verification of cases. Prop ositio n 17 . A function f ∶ B n → R has 2 -p ermutable latt ic e derivatives if and only if every binary se ct ion (2) of f satisfies a 1 a 12 ≥ 0 or a 2 a 12 ≥ 0 or  a 12  ≤  a 1  ∨  a 2  . Lemma 11 sho ws that the class o f 2-lo cally monotone pseudo-Bo olean fun ctions is a sub class of that of pseudo-Bo ole a n functions which hav e 2-p ermutable la ttice deriv atives. E xample 13 then shows that this inclusio n is strict. On the other hand, accor ding to Theor em 12, a Bo olean function is 2-lo cally monotone if and only if it has 2 -p e r mu table lattice deriv atives. E xample 24 b elow shows tha t the analogo us equiv alence do es not hold for p > 2. How ever, p -loca l monotonicity implies p -p ermutabilit y of lattice der iv atives of any pseudo-Bo o lean function (see Theorem 21 b elow). T o this extent, let us firs t study how the deg ree of lo ca l monotonicity is affected by taking lattice deriv atives. Lemma 18. If f ∶ B n → R is monotone, then ∧ j f and ∨ j f ar e also monotone for al l j ∈ [ n ] . Pr o of. Clear ly , if f is monotone, then so a re f 0 j ( x ) = f ( x 0 j ) and f 1 j ( x ) = f ( x 1 j ) , for all j ∈ [ n ] . Mor eov er, if f is isotone (resp. an titone) in x k , then b oth f 0 j and f 1 j are also isotone (res p. antitone) in x k . Since ∧ and ∨ are isotone functions, we hav e that fo r every j ∈ [ n ] , both ∧ j f ( x ) = f 0 j ( x ) ∧ f 1 j ( x ) a nd ∨ j f ( x ) = f 0 j ( x ) ∨ f 1 j ( x ) a r e monotone.  Theorem 19. If f ∶ B n → R is p -lo c al ly monotone, then ∧ j f and ∨ j f ar e ( p − 1 ) - lo c al ly monotone for al l j ∈ [ n ] . Pr o of. Supp ose that f ∶ B n → R is p -lo ca lly monotone. By Theorem 8, it suffices to show that all ( p − 1 ) -ary se ctions of ∧ j f and ∨ j f are mono tone. W e consider only ∨ j f , the other case c an be dealt with in a similar way . Let h b e a ( p − 1 ) -ary section of ∨ j f defined by h ( x ) = ∨ j f ( a x S ) for a ll x ∈ B S , where a ∈ B n and S ⊆ [ n ] is a ( p − 1 ) - subset. Let T = S ∪ { j } , and let us define g ∶ B T → R b y g ( y ) = f ( a y T ) for a ll y ∈ B T . Clear ly , g is a section of f , and the arity of g is e ither p − 1 o r p , depending on whether j be lo ngs to S or not. A simple calculation shows that h ( y  S ) = ∨ j g ( y ) for a ll y ∈ B T , where y  S stands for the restriction of y to S . This mea ns that if j ∉ S , then h ca n be o btained fro m ∨ j g by de le ting its inessential j -th v ar ia ble, and h = ∨ j g if j ∈ S . Since f is p - lo c ally monotone, g is mo notone by Theor e m 8, thus we can conclude with the help of Lemma 18 that h is monotone as well.  Corollary 20. L et 0 ≤ ℓ < p ≤ n . If f ∶ B n → R is p -lo c al ly monotone, then, for every ℓ -subset { k 1 , . . . , k ℓ } ⊆ [ n ] and every choic e of the op er ators O k i ∈ { ∨ k i , ∧ k i } ( i = 1 , . . . , ℓ ) , the function O k 1 ⋯ O k ℓ f is ( p − ℓ ) - lo c al ly monotone. In p articular, if ℓ ≤ p − 2 , then O k 1 ⋯ O k ℓ f is lo c al ly monotone. 8 MIGUEL COUCEIRO, JEAN-LUC MARICHAL, AND T AM ´ AS W ALDHAUSER R emark 1 . W e will see in E xample 26 o f Sectio n 4 that Theo rem 19 ca nnot b e sharp ened, i.e., the lattice deriv atives o f a p -lo cally mo notone function are not necessarily p -lo ca lly monotone, not even in the case of Bo o lean functions. With the help o f Co rollary 20 we can now prove the promised implication b et ween p -lo cal mono to nicit y and p - per m uta bilit y of lattice der iv atives, thus genera lizing Lemma 11. Theorem 21. If f ∶ B n → R is p -lo c al ly monotone, then it has p -p erm u table lattic e derivatives. Pr o of. Let f ∶ B n → R b e a p -lo cally monotone function, let { k 1 , . . . , k p } b e a p - subset of [ n ] , and let O k i ∈ { ∧ k i , ∨ k i } for i = 1 , . . . , p . W e need to show that for an y per mu tation π ∈ S p the following iden tity holds: O k 1 ⋯ O k p f = O k π ( 1 ) ⋯ O k π ( p ) f . Since S p is gener ated by trans p os itions of the form ( i i + 1 ) , it suffices to prov e that O k 1 ⋯ O k i − 1 O k i O k i + 1 O k i + 2 ⋯ O k p f = O k 1 ⋯ O k i − 1 O k i + 1 O k i O k i + 2 ⋯ O k p f , and for this it is sufficient to verify that (9) O k i O k i + 1 g = O k i + 1 O k i g , where g s ta nds for the function O k i + 2 ⋯ O k p f . F rom Coro llary 20 it follows that g is lo ca lly mo notone, a nd then Lemma 11 prov e s (9 ) if one of O k i , O k i + 1 is a meet and the other is a join deriv a tive. (If b oth are meet o r both a r e join, then (9) is trivial.)  A natural question regarding la ttice deriv atives is whether a function can be re- constructed f r om its deriv atives. As the next theorem sho ws, the answer is p ositive for almost all functions. Theorem 22. L et f , g ∶ B n → R b e pseudo-Bo ole an functions such that for al l k ∈ [ n ] we have ∨ k f = ∨ k g and ∧ k f = ∧ k g . Then either f = g or ther e exists a one-to-one function α ∶ B → R su ch that f ( x ) = α ( x 1 ⊕ ⋯ ⊕ x n ) and g ( x ) = α ( x 1 ⊕ ⋯ ⊕ x n ⊕ 1 ) for al l x ∈ B n . Pr o of. T o ma ke the pro of more vivid, we present it thro ugh the analysis of the following game. Alice picks a s e c ret function f ∶ B n → R , and Bob tries to identif y this function by as k ing the v alues o f its lattice deriv atives. If he c an do this, then he wins , otherwise Alice is the winner. W e show that B ob ha s a winning strategy unless f is a function o f the sp ecial for m in the statement of the theorem. Let us r egard B n as the s et o f vertices of the n - dimens io nal cub e, a nd let Alice write the v alues of f to the cor resp onding vertices. Now the possible winning strategy for Bob is based on the following f o ur basic o bserv ations. 1 . Bo b c an determine the un or der e d p air of numb ers written t o the endp oints of any e dge of the cub e. Indeed, the endp o int s of a n edge are of the for m x 0 k , x 1 k , and it is clear that { ∧ k f ( x ) , ∨ k f ( x )} = { f ( x 0 k ) , f ( x 1 k )} . 2 . If Bob c an find the value of f at one p oint, then he c an win. Accor ding to the pr e vious observ ation, knowing the v alue at one vertex o f the cube, Bob can figur e out the v alues w r itten to the neig hboring vertices. Since the graph of the c ub e is connected, he can determine all v a lues of f this way . LOCALL Y MONOTONE BOOLEAN AND PSEUDO-BOOLEAN FUNCTIONS 9 3 . If f ( x 0 k ) = f ( x 1 k ) for some x ∈ B n , k ∈ [ n ] , then Bo b c an win. This follo ws immediately from the first tw o o bserv ations. 4 . If the r ange of f c ontains at le ast thr e e elements, t hen Bob c an win. W e can supp ose that the previous obser v atio n do es not a pply , i.e., for every edge Bob detects a tw o-ele men t set. If f takes on a t least three different v a lue s , then, by the connectedness of the cube, there exis ts a vertex x and t wo edge s incident with this vertex suc h that the tw o-element sets E 1 and E 2 corres p onding to these edges are different. Then E 1 ∩ E 2 m ust b e a one-element s e t 3 containing the v alue of f ( x ) , and then Bo b ca n win as explained in the second observ atio n. F r om these observ ations we can conclude that Bob ha s a winning strateg y unless the rang e of f contains exactly tw o num b ers and f ( x 0 k ) ≠ f ( x 1 k ) , for all x ∈ B n , k ∈ [ n ] . This means that f is of the following form for some u ≠ v ∈ R : f ( x ) =  u , if  x  is ev e n ; v , if  x  is odd , where  x  = ∑ n i = 1 x i . In other words, f ( x ) = α ( x 1 ⊕ ⋯ ⊕ x n ) , wher e α ( 0 ) = u, α ( 1 ) = v . In this case Bob can determine f only up to interc hang ing u and v , i.e., he cannot distinguish f fr om g ( x ) = α ( x 1 ⊕ ⋯ ⊕ x n ⊕ 1 ) , so he has only 50 % chance to win. (Indeed, f and g hav e the sa me lattice deriv atives, namely their meet deriv atives are all cons tant u ∧ v , while their join deriv atives are all constant u ∨ v .)  The following theorem s hows that, a s in the case of lo cal monotonicity , the cla sses of functions having pe r m utable lattice deriv atives form a chain under inclusion. Theorem 23. If f ∶ B n → R has ( p + 1 ) -p ermu table lattic e derivatives, then f has p -p ermutable lattic e derivatives. Pr o of. Let f ∶ B n → R be a function that has ( p + 1 ) -p ermutable la ttice deriv atives. Using the same no ta tion as in the pr o of of Theorem 21, it suffices to pr ov e that O k 1 ⋯ O k i − 1 O k i O k i + 1 O k i + 2 ⋯ O k p f = O k 1 ⋯ O k i − 1 O k i + 1 O k i O k i + 2 ⋯ O k p f . Let g 1 and g 2 be the ( n − p ) -ary functions obtained from the left-hand side a nd from the righ t-ha nd side of this equality by deleting their inessential v aria bles x k 1 , . . . , x k p . If O k i = ∧ k i , O k i + 1 = ∧ k i + 1 or O k i = ∨ k i , O k i + 1 = ∨ k i + 1 , then g 1 = g 2 holds trivially . Let us now assume that O k i = ∨ k i , O k i + 1 = ∧ k i + 1 ; the remaining case O k i = ∧ k i , O k i + 1 = ∨ k i + 1 is similar . By P rop osition 2, we hav e g 1 ≤ g 2 . Since the tw o (t yp es of ) functions given in Theorem 22 are or der-incompar able, if g 1 ≠ g 2 , then the la ttice deriv a tives o f g 1 and g 2 cannot all coincide. Thus there exists j ∈ [ n ] ∖ { k 1 , . . . , k p } and O j ∈ { ∧ j , ∨ j } such that O j g 1 ≠ O j g 2 . T ak ing into a ccount the definition o f g 1 and g 2 , we can rewrite this inequality as O j O k 1 ⋯ O k i − 1 O k i O k i + 1 O k i + 2 ⋯ O k p f ≠ O j O k 1 ⋯ O k i − 1 O k i + 1 O k i O k i + 2 ⋯ O k p f , which con tradicts the fact that f has ( p + 1 ) -p ermutable la ttice deriv atives.  If f ∶ B n → B is a Bo olean function with p -permutable lattice deriv atives for some p ≥ 2, then f has 2 -p ermutable lattice deriv atives by Theor em 23, and then Theorem 12 implies that f is 2-lo ca lly monoto ne . Unfortunately , nothing more 3 If E 1 ∩ E 2 is empt y , then Alice is cheating! 10 MIGUEL COUCEIRO, JEAN-LUC MARICHAL, AND T AM ´ AS W ALDHAUSER can be said ab out the deg ree of lo ca l monotonic ity o f a Bo olean function with p - per mu table lattice deriv atives. Indeed, the next exa mple shows tha t there exist n -ary Bo olean functions with n -p ermutable lattice deriv atives that ar e exa ctly 2 - lo cally monotone. Example 24. Let f n ∶ B n → B b e the function that takes the v a lue 1 o n all tuples of the form x = ( m     1 , . . . , 1 , 0 , . . . , 0 ) with 0 ≤ m ≤ n, and takes the v alue 0 everywhere else. Using Coro llary 10, it is not difficult to verify that f n is 2-lo cally monotone. How ever, if n ≥ 3, then f n is not 3-lo ca lly monotone, since ∆ 2 f ( 0 , 0 , 0 , 0 , . . . , 0 ) = − 1 , ∆ 2 f ( 1 , 0 , 1 , 0 , . . . , 0 ) = 1 . Thu s f n is exac tly 2-lo cally monotone. W e will s how by induction o n n tha t f n has n -p ermutable la ttice der iv a tiv e s. First we compute the meet deriv atives ∧ k f n ( x ) =  1 , if x 1 = ⋯ = x k − 1 = 1 and x k + 1 = ⋯ = x n = 0 ; 0 , otherwise . Since ∧ k f takes the v alue 1 o nly a t one tuple, it is monoto ne. The join deriv ative ∨ k f n is essentially the s ame as the function f n − 1 (up to the inessential k -th v ariable of ∨ k f n ), that is, (10) ∨ k f n ( x ) = f n − 1 ( x 1 , . . . , x k − 1 , x k + 1 , . . . , x n ) . Now it follo ws tha t if { k 1 , . . . , k n } = [ n ] and O k i ∈ { ∧ k i , ∨ k i } ( i = 1 , . . . , n ) , then (11) O k 1 ⋯ O k n − 1 O k n f = O k π ( 1 ) ⋯ O k π ( n − 1 ) O k n f holds for every p ermutation π ∈ S n − 1 . (If O k n = ∧ k n , then we use Theor em 21 and the fact that ∧ k n f is mono to ne, and if O k n = ∨ k n , then we use (10) a nd the induction hypo thesis.) On the other hand, from the 2-lo c al monotonicity o f f we can conclude that (12) O k 1 ⋯ O k n − 2 O k n − 1 O k n f = O k 1 ⋯ O k n − 2 O k n O k n − 1 f with the help of Theor em 12. Since S n is generated by S n − 1 and the transp os ition ( n − 1 n ) , we see from (11) and (12) that f has n -p ermutable la ttice deriv atives. W e finish this s ection with game- theoretic interpretations o f the para meterized notions of lo cal mo notonicity and p ermutabilit y o f lattice deriv atives. Identifying B n with the p ower set o f [ n ] , we can r egard a pseudo- Bo olean function f ∶ B n → R as a co op erative game, where [ n ] is the set o f players a nd f ( C ) is the worth of coalition C ⊆ [ n ] . The partia l der iv ative ∆ k f ( C ) gives the (marginal) co nt r ibution of the k -th play er to coalition C . Note that the same player might hav e a p ositive contribution to some coalitions a nd a negative contribution to other coa litio ns. Suc h a setting can model situations where some players hav e conflicts, which prev ents them from co op erating. The lattice deriv ative ∨ k f ( C ) gives the outcome if the k -th play er acts bene volently and joins (or le aves) the coa lition C only if this increases the worth. Similarly , ∧ k f ( C ) re pr esents the outcome if the k -th play er acts male volently . LOCALL Y MONOTONE BOOLEAN AND PSEUDO-BOOLEAN FUNCTIONS 11 Games corr esp onding to lo c ally monotone functions hav e the prop erty that if t wo c oalitions are close to ea ch o ther , then any g iven player relates in t he same wa y to these coalitions. Mo re precisely , f is p -lo cally monotone if and only if whenever t wo coalitions differ in less than p play ers, then the contribution of any play er is either nonnegative to bo th coalitions or it is nonp ositive to b o th. Finally , let us in terpret per m utability of la ttice deriv a tives. Let P be a p -subset of [ n ] , and le t C ⊆ [ n ] ∖ P . Suppose that some play ers of P are b enevolen t and some o f them are malevolen t, and they ar e asked one b y one to join co alition C if they want to . W e obtain the lea st p ossible outcome if the ma levolen t players are asked fir st, and we get the greates t outcome if the b enevolen t play er s make their choices first. The function f ha s p -p ermutable la ttice der iv a tiv e s if and only if these extremal outcomes coincide, i.e., if the order in whic h the play ers make their choices is irrelev ant for every p -subset of [ n ] . 4. Symmetric functions In the pr evious sections we saw that b oth notions of lo cal monoto nicit y and of per mu table lattice deriv a tives lead to tw o hierar chies of pseudo - Bo olean functions which a re r elated by the fact that each p -lo cal mono tone class is con taine d in the corr esp onding class of functions having p -p ermutable lattice deriv atives. Now, in gener al this containmen t is strict. How ever, under certa in assumptions (see, e.g., Theorem 12), p -lo cal monoto nicit y is equiv alent to p -p e rmut a bilit y of la ttice deriv atives. Hence it is natural to as k for conditions under whic h these tw o notio ns are equiv a lent . In this se c tion we pro v ide a partial answer to this pro blem b y fo cusing on sym- metric pseudo-Boo lean functions, i.e., functions f ∶ B n → R that a re in v ariant under all p er m utatio ns of their v ariables. Quite sur prisingly , in this ca se the notio ns of p -lo cal monotonicity and p -per m utabilit y of lattice deriv atives b ecome equiv alent. Symmetric functions o f ar it y n are in a one-to-o ne corresp ondence with sequences of r eal num b ers of length n + 1 , wher e the function co rresp onding to the s equence α = α 0 , . . . , α n is given by f ( x ) = α ∣ x ∣ ( x ∈ B n ) . Clearly , f is isotone if and only if the corresp onding sequence is nondecreas ing, i.e., α 0 ≤ α 1 ≤ ⋯ ≤ α n . Similar ly , f is a n tito ne if and only if α 0 ≥ α 1 ≥ ⋯ ≥ α n , and f is monotone if and only if f is either isotone or antitone. 4 It is easy to see that if f is symmetric, then every section of f is also s ym- metric; mor eov er, if f cor resp onds to the sequence α = α 0 , . . . , α n , then the p -a ry sections o f f are precisely the symmetric functions corr e spo nding to the subse- quences 5 α i , α i + 1 , . . . , α i + p of α of length p + 1. This observ ation and Theor em 8 lead to the following description of p - lo c a lly monoto ne symmetric pseudo-Bo olea n functions. Prop ositio n 25. L et f ∶ B n → R b e a symmetric function c orr esp onding t o the se- quenc e α = α 0 , . . . , α n . Then f is p -lo c al ly mo notone i f and only if e ach subse qu enc e of lengt h p + 1 of α is either nonde cr e asing or nonincr e asing. Unlik e in the previo us se c tio ns, here it will be more convenien t to disca rd the inessential k -th v aria ble of the lattice deriv atives ∧ k f and ∨ k f , and r e g ard the 4 Since if f is isot one (resp. antitone ) in one v ar iable, t hen it is isoton e (resp. antitone) i n all v ariables. 5 Here by a subsequence we mean a sequence of consecutiv e en tri es of the or iginal sequence. 12 MIGUEL COUCEIRO, JEAN-LUC MARICHAL, AND T AM ´ AS W ALDHAUSER latter a s ( n − 1 ) -a ry functions. Clea rly , if f is symmetr ic , then so are its lattice deriv atives. Mo reov er, if f corre s po nds to the seq uence α = α 0 , . . . , α n , then ∧ k f and ∨ k f co rresp ond to the sequences α 0 ∧ α 1 , α 1 ∧ α 2 , . . . , α n − 1 ∧ α n and α 0 ∨ α 1 , α 1 ∨ α 2 , . . . , α n − 1 ∨ α n , resp ectively , for a ll k ∈ [ n ] . Since these sequences do not dep end on k , w e will write ∧ f and ∨ f instead of ∧ k f and ∨ k f , a nd w e will abbrevia te ∧⋯∧  ℓ f and ∨⋯∨  ℓ f by ∧ ℓ f and ∨ ℓ f , r esp ectively . The next e x ample shows t ha t Theorem 1 9 cannot b e shar pene d. Example 26. Le t f ∶ B n → B b e the symmetric function corresp onding to the sequence α = 0 , 0 , p     1 , . . . , 1 , p     0 , . . . , 0 , 1 , 1 where n = 2 p + 4 and p ≥ 2. It f ollows fro m Pro po sition 2 5 that f is exactly p -lo cally monotone. T o co mpute ∧ f , it is handy to construct a ta ble whos e first row contains the sequence α , and in the s e c ond row we write α i ∧ α i + 1 betw een α i and α i + 1 : f ∶ 0 0 1 1 ⋯ 1 1 0 0 ⋯ 0 0 1 1 ∧ f ∶ 0 0 1 1 ⋯ 1 1 0 0 0 ⋯ 0 0 0 1 Thu s ∧ f co rresp onds to the sequence 0 , 0 , p − 1     1 , . . . , 1 , p + 1     0 , . . . , 0 , 1 , and a similar ca lculation yields that ∨ f cor resp onds to the sequence 0 , p + 1     1 , . . . , 1 , p − 1     0 , . . . , 0 , 1 , 1 . Now Propo s ition 25 sho ws tha t ∧ f a nd ∨ f ar e e xactly ( p − 1 ) -lo ca lly monotone. R emark 2 . Example 26 shows that the degree of lo cal mono to nicit y can decrea se, when taking lattice deriv atives, and Theor e m 19 s tates that it can decr e a se by at most one. Other examples can b e found to illustra te the cases when this degree stays the same, or even increases. F o r instance, consider the function f ( x ) = x 1 ⊕ ⋯ ⊕ x n , which is not ev en 2 -lo cally monotone, but its lattice deriv atives are constant. W e co nclude this se c tio n by pro ving that for symmetric functions the no tio ns of p -lo cal monotonicity and p -per m utabilit y of lattice deriv atives coincide. Theorem 27. If f ∶ B n → R is symmetr ic, then f is p -lo c al ly monotone if and only if f ha s p -p ermutable lattic e derivatives. Pr o of. By Theor em 21, it is enough to show that if a symmetric function f ∶ B n → R is not p -lo cally monotone, then it do e s not hav e p -p ermutable lattice der iv a tiv e s. So supp ose that f is a symmetric function which is not p -lo cally monoto ne, and which co rresp onds to the sequence α = α 0 , . . . , α n . Let α i , . . . , α i + ℓ be a shortest subsequence of α that is neither nondecr easing no r nonincre a sing. Pro po sition 25 LOCALL Y MONOTONE BOOLEAN AND PSEUDO-BOOLEAN FUNCTIONS 13 implies that there is indeed such a subsequence for ℓ + 1 ≤ p + 1. F r om the mini- mality of ℓ it follows t hat the subsequence α i , . . . , α i + ℓ − 1 is either nondecrea sing or nonincreasing . W e may a ssume without loss of g enerality th at the first cas e holds; the s econd case is the dual o f the first one. Then we must have α i + ℓ − 1 > α i + ℓ , since otherwis e the whole s ubs equence α i , . . . , α i + ℓ would b e nondecr easing. Th us we ha ve t he following inequalities: (13) α i ≤ α i + 1 ≤ ⋯ ≤ α i + ℓ − 1 > α i + ℓ . F r om the minimality of ℓ , we can also conclude that the subseq uence α i + 1 , . . . , α i + ℓ is either nondecrea sing o r nonincrea s ing. As α i + ℓ − 1 > α i + ℓ , the first ca se is imp oss ible, therefore α i + 1 , . . . , α i + ℓ is nonincrea sing, and we must hav e α i < α i + 1 since o therwise the whole subseque nce α i , . . . , α i + ℓ would be nonincr e asing: (14) α i < α i + 1 ≥ ⋯ ≥ α i + ℓ − 1 ≥ α i + ℓ . Comparing (13) a nd (14), we obtain α i < α i + 1 = ⋯ = α i + ℓ − 1 > α i + ℓ . T o simplify notation, we set β ∶= α i , γ ∶= α i + 1 , δ ∶= α i + ℓ . With this notation w e hav e that α co ntains the subsequence β , γ , . . . , γ , δ of length ℓ + 1 with β , δ < γ . In the following we will use this observ atio n to prov e that f do es no t hav e ℓ -p ermutable lattice deriv atives. Let us co mpute the sequence corr esp onding to ∨ ∧ ℓ − 1 f . W e ca n co ns truct a table as in Example 26, but this time the table has ℓ + 1 rows (in the last row µ stands for β ∨ δ ): f ∶ α 0 ⋯ β γ γ γ ⋯ γ γ γ δ ⋯ α n ∧ f ∶ ⋯ β γ γ ⋯ γ γ δ ⋯ ∧ 2 f ∶ ⋯ β γ ⋯ γ δ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ∧ ℓ − 2 f ∶ ⋯ β γ δ ⋯ ∧ ℓ − 1 f ∶ ⋯ β δ ⋯ ∨ ∧ ℓ − 1 f ∶ ⋯ µ ⋯ A similar table ca n be constructed for ∧ ℓ − 1 ∨ f : f ∶ α 0 ⋯ β γ γ γ ⋯ γ γ γ δ ⋯ α n ∨ f ∶ ⋯ γ γ γ ⋯ γ γ γ ⋯ ∧ ∨ f ∶ ⋯ γ γ ⋯ γ γ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ∧ ℓ − 3 ∨ f ∶ ⋯ γ γ γ ⋯ ∧ ℓ − 2 ∨ f ∶ ⋯ γ γ ⋯ ∧ ℓ − 1 ∨ f ∶ ⋯ γ ⋯ Since β , δ < γ , we hav e µ < γ , and this means that the sequences cor resp onding to ∨ ∧ ℓ − 1 f and ∧ ℓ − 1 ∨ f differ in at least one p osition, ther efore f does not ha ve ℓ -p ermutable la ttice de r iv a tiv e s . As ℓ ≤ p , this implies that f do es not hav e p - per mu table lattice deriv atives either, accor ding to Theorem 23.  R emark 3 . As a co nsequence of Theo r em 27, we can observe that any ex actly p -lo cally monotone sy mmetric function (for insta nc e , the functions considered in Example 26) has p -p ermutable but not ( p + 1 ) -p ermutable lattice deriv a tives. 14 MIGUEL COUCEIRO, JEAN-LUC MARICHAL, AND T AM ´ AS W ALDHAUSER 5. Open problems and concluding remarks W e prop osed relaxatio ns of monotonicity , namely p -lo ca l mono tonicity , and we presented characterizations of each in terms of “forbidden” sections. Also, for eac h p , we o bserved that p - lo c ally monoto ne functions hav e the prop erty that any p of their lattice deriv atives p ermute, a nd showed that the co nv erse a ls o holds in the sp ecial case of symmetric functions. The cla sses of 2-lo cally monotone functions, and of functions having 2- per m uta ble la ttice deriv atives w ere explicitly descr ibed. How ever, similar descriptions elude us fo r p ≥ 3 . Hence we are left with the following problems. Problem 1. F or p ≥ 3 , describ e the class of p -lo c al ly m onotone functions and that of fu n ctions having p -p ermutable lattic e derivatives. Problem 2. F or p ≥ 3 , determine ne c essary and su fficient c onditions o n func- tions for the e quivalenc e b etwe en p -lo c al m onotonicity and p -p ermutability of lattic e derivatives. Ackno wledgments Miguel Couceiro and Jean-Luc Marichal are supp orted by the int e r nal resear ch pro ject F1R-MTH-PUL-12 RDO2 of the Universit y of Luxembo urg. T am´ as W ald- hauser is supp orted by the T ´ AMOP-4.2.1 /B-09 / 1/KONV-2010-0005 pr o gram of National Development Agency of Hungary , b y the Hungarian Nationa l F ounda- tion for Scientific Resear ch under grant nos K 77409 and K 83219 , by the National Research F und of Luxe mbo urg, and by the Marie Curie Actions of the Euro pea n Commission (FP7-COFUND). References [1] D. Bouyssou, D. Dub ois, H . Pr ade, M. 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Ma thema tics Research Unit, FSTC, University of Luxemb ourg, 6, rue Coudenhove- Kalergi, L-1359 Luxembourg, Luxembourg E-mail ad dr ess : miguel.co uceiro[at]uni.lu Ma thema tics Research Unit, FSTC, University of Luxemb ourg, 6, rue Coudenhove- Kalergi, L-1359 Luxembourg, Luxembourg E-mail ad dr ess : jean-luc. marichal[at]uni.lu Ma thema tics Research Unit, FSTC, University of Luxemb ourg, 6, rue Coudenhove- Kalergi, L-1359 Luxemb ourg, Luxembourg and Bol y ai Institute, University of Szeged, Aradi v ´ er t an ´ uk tere 1, H-6720 Szeged, Hungar y E-mail ad dr ess : twaldha[a t]math.u-szeged.hu