On the gap between ess(f) and cnf_size(f)

On the Gap Bet w een ess ( f ) and c nf size ( f ) Lisa Hellerstein 1 , Dev orah Kletenik 1 Polyte chnic Institute of NY U, 6 Metr ote ch Center, Br o oklyn, N.Y., 112 01 Abstract Giv en a Bo olean function f , the quan tit y ess ( f ) denotes the largest s et of assignmen ts that falsify f , no t w o of whic h falsify a common implicate o f f . Although ess ( f ) is clearly a low er b ound on cnf size ( f ) (the minim um n um b er of clauses in a CNF fo rm ula for f ), ˘ Cep ek et al. sho we d it is not, in general, a tig h t low er b ound [1 ]. They ga ve examples of functions f for whic h there is a small gap b et w een ess ( f ) and cnf size ( f ). W e demonstrate significan tly la rger gaps. W e sho w t hat the gap can b e exp onential in n for arbitrary Bo olean functions, and Θ( √ n ) for Horn functions, where n is the n um b er of v aria bles of f . W e also introduce a natural extension of the quan tit y ess ( f ), whic h w e call ess k ( f ), whic h is the largest set of assignmen ts, no k of whic h falsify a common implicate of f . Keywor ds: DNF, CNF, ess ( f ), Horn functions, formula size 1. Introduction Determining the smallest CNF form ula for a giv en Bo olean function f is a d ifficult pro blem that h as b een studied for man y y ears. (See [2] for an o v erview of relev a n t literature.) Recen tly , ˘ Cep ek et a l. in tro duced a com- binatorial quantit y , es s ( f ), whic h lo w er b ounds cnf size ( f ), the minimum n um b er of claus es in a CNF fo rm ula represe nting f [1]. The quan t ity ess ( f ) is equal to the size of the larg est set of falsep oints of f , no tw o of whic h falsify the same implicate of f . 1 Email addr esses: hs tein@p oly.e du (Lisa Hellerstein), dkle tenik@ cis.p oly.edu (Dev o rah Kletenik) 1 This definitio n immediately follows from Coro llary 3.2 o f ˘ Cep e k et al. [1]. 1 F or ce rta in sub classes of Bo olean functions, suc h as the monotone (i.e., p ositiv e) functions, ess ( f ) is equal to cn f size ( f ). Ho w ev er, ˘ Cep ek et al. demonstrated that there can b e a ga p b etw een ess ( f ) and cnf size ( f ). They constructed a Bo olean function f on n v ariables suc h that there is a m ulti- plicativ e ga p of size Θ(log n ) b et w een cnf size ( f ) and ess ( f ). 2 Their con- structed function f is a Hor n function. Their res ults lea ve op en the possibilit y that ess ( f ) could b e a close approx imatio n to cnf size ( f ). W e sho w that this is no t the case. W e construct a Bo o lean function f on n v a riables suc h that there is a m ultiplicativ e gap o f size 2 Θ( n ) b et w een cnf size ( f ) and ess ( f ). Note that suc h a gap could not b e larger tha n 2 n − 1 , since cnf size ( f ) ≤ 2 n − 1 for all functions f . W e also construct a Horn f unction f suc h that there is a multiplicativ e gap of size Θ( √ n ) b etw een cnf size ( f ) and ess ( f ). W e sho w that no gap larger than Θ( n ) is possible. If one expresses the gaps as a function of cnf s i z e ( f ), rather t ha n as a function o f the n um b er of v ariables n , then the g a p we obtain with b o t h the constructed non-Horn and Ho r n functions f is cnf size ( f ) 1 / 3 . Clearly , no gap larger than cnf size ( f ) is possible. W e briefly explore a na t ural generalization of t he quantit y ess ( f ), whic h w e call ess k ( f ), whic h is the largest set of falsep oin ts, no k of whic h falsify a common implicate of f . The quantit y ess ( f ) / ( k − 1) is a low er b ound on CNF-size, for an y k ≥ 2. The a b o ve results concern the size o f CNF for mulas. Analogous results hold for DNF form ulas b y duality . 2. Preliminaries 2.1. D efinitions A Bo o lean function f ( x 1 , . . . , x n ) is a mapping { 0 , 1 } n → { 0 , 1 } . (Where it do es not cause confusion, we often use the word “function” to refer to a Bo olean function.) A v aria ble x i and its negation ¬ x i are li ter als (p ositive and negative respective ly). A clause is a disjunction ( ∨ ) of literals. A term is a conjunction ( ∧ ) of literals. A CNF (conjunctiv e normal form) formula is a formula of the fo r m c 0 ∧ c 1 ∧ . . . c k , where eac h c i is a clause. A DNF 2 Their fun ctio n is actually defined in terms of tw o pa rameters n 1 and n 2 . Setting them to maximize the multiplicativ e gap betw een ess ( f ) a nd cnf size ( f ), as a function of the nu mber of v ar ia bles n , yields a g ap of size Θ (log n ). 2 (disjunctiv e normal form) fo rm ula is a form ula of the form t 0 ∨ t 1 ∨ . . . t k , where eac h t i is a term. A clause c con taining v a r ia bles from X n = { x 1 , . . . , x n } is an implic ate of f if for all x ∈ { 0 , 1 } n , if c is fa lsified b y x then f ( x ) = 0. A term t con taining v a r ia bles from X n is an imp lic ant of function f ( x 1 , . . . , x n ) if for all x ∈ { 0 , 1 } n , if t is satisfie d b y x then f ( x ) = 1. W e de fine the size of a CNF formula t o b e the n um b er of its clauses , and the size of a DNF f o rm ula to b e the n umber o f its terms. Giv en a Bo olean function f , cnf size ( f ) is t he size of the smallest CNF form ula represen ting f . Analog o usly , dnf size ( f ) is the size of the smallest DNF form ula represen ting f . An assignmen t x ∈ { 0 , 1 } n is a fals ep oint of f if f ( x ) = 0, and is a truep oint o f f if f ( x ) = 1. W e sa y that a clause c c overs a f alsep oin t x of f if x falsifies c . A term t c overs a truep oin t x of f if x satisfies t . A CNF form ula φ represen ting a function f forms a c over of the falsep oin ts of f , in that eac h falsep oin t of f m ust b e co v ered b y at least o ne clause of φ . F urther, if x is a truep o int of f , t hen no clause of φ cov ers x . Similarly , a DNF fo rm ula φ represen ting a function f fo rms a c ov e r of the truep oints of f , in that each truep oint of f mu st b e co v ered by at least one term of φ . F urther, if x is a falsepoint of f , then no term of φ cov ers x . Giv en tw o assignmen ts x, y ∈ { 0 , 1 } n , w e write x ≤ y if ∀ i, x i ≤ y i . An assignmen t r sep ar ates t w o assignmen ts p and q if ∀ i , p i = r i or q i = r i . A p artial function f maps { 0 , 1 } n to { 0 , 1 , ∗} , where ∗ indicates that the v alue of f is not defined on the assignmen t. A Bo olean formula φ is c on s istent with a partial function f if φ ( a ) = f ( a ) for all a ∈ { 0 , 1 } n where f ( a ) 6 = ∗ . If f is a partial Bo olean function, then c n f size ( f ) and dnf s i ze ( f ) are the size of the smallest CNF and DNF f orm ulas consisten t with the f , respective ly . A Bo olean function f ( x 1 , . . . , x n ) is mono tone if f or all x, y ∈ { 0 , 1 } n , if x ≤ y then f ( x ) ≤ f ( y ). A Bo o lean function is anti-mon o tone if for all x, y ∈ { 0 , 1 } n , if x ≥ y then f ( x ) ≤ f ( y ). A DNF or CNF form ula is mono tone if it con tains no negations; it is an ti-monot o ne if all v ariables in it a r e negated. A CNF formula is a Horn- CNF if each clause contains at most one v a r iable without a negation. If eac h clause contains exactly one v ariable without a negation it is a pur e Horn- CNF. A Horn function is a Bo olean f unction that can b e represen ted b y a Horn-CNF. It is a pur e Horn function if it can b e represen ted b y a pure Horn-CNF. Horn functions are a generalization of an ti-mono tone f unctions, and ha v e applications in artficial in telligence [3]. 3 W e sa y that t wo falsep oints, x and y , of a function f are indep endent if no implic ate of f cov ers b oth x and y . Similarly , w e say that t w o truep oin ts x and y o f a function f are indep endent if no implican t of f cov ers b oth x and y . W e sa y that a set S of falsep oints (truepo in ts) of f is indep endent if all pairs of falsepoints (truep oin ts) in S are independen t. The set c overing pr o blem is as follo ws: Given a gro und set A = { e 1 , . . . , e m } of ele ments , a set S = { S 1 , . . . , S n } of subsets of A , and a p ositiv e in teger k , do es there exis t S ′ ⊆ S suc h that S S i ∈S ′ = S and |S ′ | ≤ k ? Each set S i ∈ S is said to co ve r the elemen ts it con tains. Th us the set co v ering problem asks whether A has a “co v er” of size at most k . A set co v ering instance is r - uniform, for some r > 0, if all subsets S i ∈ S ha v e size r . Giv en an instance of the set co ve ring problem, w e sa y that a subset A ′ of ground set A is ind e p ende n t if no tw o elemen ts of A ′ are con tained in a common subset S i of S . 3. The quan tity ess ( f ) W e b egin b y restating the definition of ess ( f ) in terms of indep enden t falsep oin ts. W e also introduce an analog o us quan tity f or truep oints. (The notation ess d refers to the fact that this is a dual definition.) Definition 1. Let f b e a Bo olean function. The quan tity e ss ( f ) denotes the size of the la rgest indep enden t set o f falsep oin ts of f . The quantit y ess d ( f ) denotes the largest independen t set of truep oints of f . As w as stated ab ov e, ˘ Cep ek et al. introduced the quan tit y ess ( f ) as a low er b ound on cn f size ( f ). The fact that ess ( f ) ≤ cnf size ( f ) fo llo ws easily from the ab ov e definitions, a nd from the following facts: (1) if φ is a CNF formula represen ting f , then ev ery falsep oint of f mus t b e co v ered b y some clause of φ , and (2) each clause of φ m ust b e an implicate of f . Let f ′ denote the function that is t he complemen t of f , i.e. f ′ ( a ) = ¬ f ( a ) for all assignmen ts a . Since, b y dualit y , ess ( f ′ ) = ess d ( f ) and cnf size ( f ′ ) = dnf size ( f ), it follo ws that ess ( f ′ ) ≤ dnf s i z e ( f ). Prop erty 1. [1] Tw o falsep oints of f , x and y , a re indep enden t iff there exists a truepoint a of f that separates x and y . 4 Consider the follow ing decision problem, whic h w e will call ESS : “Give n a CNF fo r mula represen ting a Bo olean function f , and a n um b er k , is ess ( f ) ≤ k ?” Using Prop erty 1, this problem is easily sho wn to b e in co-NP [1]. W e can com bine the fact that ESS is in co-NP with results o n the hardness of appro ximating CNF-minimization, to get the following preliminary result, based on a complex ity-theoretic assumption. Prop osition 1. If c o - NP 6 = Σ P 2 , then for some γ > 0 , ther e exists an infinite set of B o ole an functions f such that e ss ( f ) n γ < cnf size ( f ) , wher e n is the numb er of variables of f . Pr o of. Consider the Min-CNF pro blem (decision v ersion): Giv en a CNF form ula represen ting a Bo olean function f , and a n um b er k , is cnf size ( f ) ≤ k ? Umans prov ed that it is Σ P 2 -complete to approx imate this problem to within a f a ctor of n γ , f or some γ > 0, where n is the nu mber of v ariables o f f [4]. (Appro ximating this problem to within some fa ctor q means answe ring “y es” whenev er cnf size ( f ) ≤ k , and answ ering “no” whenev er cnf size ( f ) > k q . If k < cnf size ( f ) ≤ k q , either answ er is acceptable.) Supp ose ess ( f ) n γ ≥ cnf siz e ( f ) for all Bo olean functions f . Then one can approx imate Min-CNF to within a factor of n γ in co-NP by simply us- ing the co-NP algo rithm for ESS to determine whether e ss ( f ) ≤ k . Ev en if ess ( f ) n γ ≥ cnf size ( f ) for a finite set S of functions, one can still ap- pro ximate Min-CNF to within a factor of n γ in co-NP , by simply handling the finite n um b er of functions in S explicitly as special cases. Since appro x- imating Min-CNF to within this factor is Σ P 2 -complete, Σ P 2 ⊆ co-NP . By definition, co-NP ⊆ Σ P 2 , so Σ P 2 = co-NP . The no n-approx imability r esult of Umans for Min-CNF , used in the ab o v e pro of, is expressed in terms of the n um b er of v ariables n of the function. Umans also sho w ed [5] that it is Σ P 2 complete to appro ximate Min-CNF to within a factor of m γ , for some γ ≥ 0, where m = cnf size ( f ). Thus we can also prov e that, if NP 6 = Σ P 2 , then for some γ > 0, there is an infinite set of functions f suc h tha t ess ( f ) < cnf size ( f ) 1 − γ . The assumption that Σ P 2 6 = co-NP is not unreasonable, so we ha v e grounds to b eliev e that there is an infinite set of f unctions for which the gap betw een ess ( f ) and cnf size ( f ) is greater than n γ (or cnf size ( f ) γ ) for some γ . Below, w e will explicitly construct suc h sets with larger gaps than that of Prop osition 1, and with no complexit y theoretic assumptions. 5 W e can also pro ve a pro p osition similar to Prop o sition 1 for Horn func- tions, using a differen t complexit y theoretic assumption. (Since the statemen t of the prop o sition includes a complexit y class parameterized b y the standard input-size pa rameter n , w e use N instead o f n to denote the n umber of inputs to a Boo lean function.) Prop osition 2. If NP 6⊆ c o-NTIME( n poly l og ( n ) ) , then for some ǫ such that 0 < ǫ < 1 , ther e exists an infinite set of Horn functions f such that cnf size ( f ) ess ( f ) ≥ 2 log 1 − ǫ N , wher e N is the numb e r of input variables of f . Pr o of. Consider the f ollo wing Min-Horn-CNF problem (decision v ersion): Giv en a Horn-CNF φ represen ting a Horn function f , and an integer k ≥ 0, is cnf size ( f ) ≤ k ? Bhattac harya et al. [6] sho w ed that there exists a deterministic, man y-one reduction (i.e. a Karp reduction), running in time O ( n poly l og ( n ) ) (where n is the size of the input), from an NP-complete problem to the problem of approximating Min-Horn-CNF to within a f a ctor of 2 log 1 − ǫ N , where N is the n um b er of input v ariables of f . Supp ose that cnf size ( f ) ess ( f ) is at most 2 log 1 − ǫ N for all Bo olean f unctions f . It is w ell kno wn that giv en a Horn-CNF f , the size o f the smallest (functionally) equiv alen t Horn-CNF is precisely cnf size ( f ). Th us giv en a Horn-CNF φ on N v ariables, and a n um b er k , if there do es not exist a Horn- CNF equiv a lent to φ of size less than 2 log 1 − ǫ N × k , this can b e verifie d non-deterministically in p olynomial time (b y verifyin g tha t ess ( f ) ≥ k ). Thus the complemen t of Min-Horn-CNF is approx imable to within a factor of 2 log 1 − ǫ N , in determin- istic time n poly l og ( n ) (where n is the size in bits of the input Hor n- CNF, and N is the n umber of v ariables in the input Horn-CNF). Combinin g this fact with the reduction of Bhattachary a et al. implies that the complemen t of an NP-complete problem can b e solv ed in non-deterministic time n poly l og ( n ) . Th us NP is con tained in co-NTIME( n poly l og ( n ) ). The same holds if cnf size ( f ) ess ( f ) is at most 2 log 1 − ǫ n for all but a finite set of Bo olean functions f . 4. Constructions of functions with large gaps b etw een es s ( f ) and cnf size ( f ) W e will b egin b y constructing a function f , such that cnf size ( f ) ess ( f ) = Θ( n ). This is already a larger g ap tha n the multiplicativ e gap of log( n ) ac hiev ed b y the construction o f ˘ Cep ek et al. [1], and the gap of n γ in Prop osition 1. 6 W e describe the construction of f , pro ve bo unds on cnf size ( f ) and ess ( f ), and then pro v e that the ratio cnf size ( f ) ess ( f ) = Θ( n ). W e will then sho w ho w t o mo dify this construction to give a function f suc h that cnf size ( f ) ess ( f ) = 2 Θ( n ) , th us increasing the gap to b e exp onen tial in n . A t the end of this section, w e will explore ess k ( f ), our generalization of ess ( f ). 4.1. C onstructing a function with a line a r gap Theorem 1. Ther e exists a function f ( x 1 , . . . , x n ) such that cnf size ( f ) ess ( f ) = Θ( n ) . Pr o of. W e construct a function f suc h that dnf size ( f ) ess d ( f ) = Θ( n ). Theorem 4.1 then follo ws immediately by duality . Our construction relies hea vily on a reduction of Gimp el from the 196 0’s [7 ], whic h reduces a generic instance of the set co v ering problem to a DNF- minimization problem. (See Czort [8] or Allender et al. [9] for more recen t discussions of this reduction.) Gimp el’s reduction is as follows. Let A = { e 1 , . . . , e m } be the ground set of the set cov ering instance, a nd let S b e the set of subsets A from whic h the co v er m ust b e formed. With eac h elemen t e i in A , asso ciat e a Bo olean input v ariable x i . F or each S ∈ S , let x S denote the assignmen t in { 0 , 1 } m where x i = 0 iff e i ∈ S . Define the partial function f ( x 1 , . . . , x m ) as follo ws: f ( x ) =    1 if x con tains exactly m − 1 ones ∗ if x ≥ x S for some S ∈ S 0 otherwise There is a DNF fo rm ula of size at most k that is consisten t with this partial function if a nd only if the elemen t s e i of the set cov ering instance A can b e co v ered using at most k subsets in S (cf. [8]). W e apply t his reduction to the simple, 2 - uniform, set co vering instance o v er m elemen ts where S consists of all subsets con taining exactly t w o o f those m elemen t s. The smallest set co v er f o r this instance is clearly ⌈ m/ 2 ⌉ . The largest indep enden t set of elemen ts is only of size 1, since ev ery pair of elemen ts is contained in a common subset of S . Note that this gives a ratio of minimal set co v er to largest indep enden t set of Θ( m ) . Applying Gimp el’s reduction to this sim ple set co v ering instance, w e g et the follo wing partial function ˆ f : 7 ˆ f ( x ) =        1 if x con tains exactly m − 1 ones ∗ if x con ta ins exactly m − 2 ones ∗ if x con ta ins exactly m ones 0 otherwise Since the smallest set co v er for the instance has size ⌈ m/ 2 ⌉ , dnf size ( ˆ f ) = ⌈ m/ 2 ⌉ . Allender et al. extended the reduction of G imp el b y con v erting the partial function f to a total function g . The con v ersion is as follo ws: Let t = m + 1 and let s be the n um b er of ∗ ’s in f ( x ). Let y 1 and y 2 b e t w o additio na l Bo olean v aria bles, and let z = z 1 . . . z t b e a v ector of t more Bo olean v a r ia bles. Let S ⊆ { 0 , 1 } t b e a collection of s v ectors, each con taining an o dd num b er of 1’s (since s ≤ 2 m , suc h a collection exists). Let χ be the function suc h that χ ( x ) = 0 if the parit y of x is ev en and χ ( x ) = 1 otherwise. The total function g is defined as follo ws: g ( x, y 1 , y 2 , z ) =        1 if f ( x ) = 1 and y 1 = y 2 = 1 and z ∈ S 1 if f ( x ) = ∗ and y 1 = y 2 = 1 1 if f ( x ) = ∗ , y 1 = χ ( x ) , and y 2 = ¬ χ ( x ) 0 otherwise Allender et al. pro v ed that this total function g obeys the f o llo wing prop erty : dnf size ( g ) = s ( dnf size ( f ) + 1) . Let ˆ g b e the tot a l function obtained by setting f = ˆ f in the ab ov e defi- nition of g . W e can no w compute dnf size ( ˆ g ). Le t n b e the n um b er of input v ar i- ables of ˆ f . The total f unction ˆ g is defined on n = 2 m + 3 v a riables. Since dnf size ( ˆ f ) = ⌈ m/ 2 ⌉ , w e hav e dnf size ( ˆ g ) = s  ⌈ m 2 ⌉ + 1  ≥ s  n − 3 4 + 1  where s is the n umber o f assignmen ts x for whic h ˆ f ( x ) = ∗ . 8 W e will upp er b o und ess d ( ˆ g ) b y dividing the truep oin ts of ˆ g in to t w o disjoin t sets and upp er-b o unding the size of a maxim um indep endent set of truep oin ts in each. (Recall that t w o truep oin ts of ˆ g ar e indep enden t if they do not satisfy a common implican t of ˆ g .) Set 1: The set of all truep oin ts o f ˆ g whose x comp onent has the prop erty f ( x ) = ∗ . Let a 1 b e a maxim um indep enden t set of truep oints of ˆ g consisting only of p oints in this set. Consider t w o truep o ints p and q in this set that ha v e the same x v alue. It follo ws that they share the same v alues for y 1 and y 2 . Let t b e the term con taining all v ariables x i , and exactly one of the t wo y j v ariables, suc h that eac h x i app ears without negation if it set to 1 b y p and q , and with negation otherwise, and y j is set to 1 b y b oth p a nd q . Clearly , t is a n implicant of ˆ g b y definiton of ˆ g , and clearly t cov ers b oth p and q . It follows that p and q are not indep enden t. Because any t w o truep o ints in this set with the same x v alue are not indep enden t, | a 1 | cannot excee d the n umber of differen t x assignmen ts. There are s assignmen ts suc h that ˆ f ( x ) = ∗ , so | a 1 | ≤ s . Set 2: The set of all truep oin ts o f ˆ g whose x comp onent has the prop erty ˆ f ( x ) = 1. Let a 2 b e a maxim um indep enden t set consisting only of p oin ts in this set. Consider any tw o truep oints p and q in this set that con tain the same assignmen t for z . W e can construct a term t of the form w y 1 y 2 e z suc h that w con tains exactly m − 2 x i ’s that are set to 1 by b oth p and q , and all z i s that ar e set to 1 b y p and q app ear in e z without negatio n, and a ll other z i s app ear with negatio n. It is clear that t is an implican t of ˆ g and that t co v ers b oth p and q . Once again, it follows that p and q are not indep enden t truep oin ts of g . Because any tw o truep oints in this set with the same z v alue are not indep enden t, | a 2 | cannot exceed the n um b er of differen t z assignmen ts. There are s assignmen ts to z such that z ∈ S , so | a 2 | ≤ s . Since a ma ximum indep enden t set of truep oints of ˆ g can b e pa rtitioned in to an indep enden t set of p oin ts from the first set, and an indep enden t set 9 of p oin ts from the second set, it immediately follows that 3 ess d ( ˆ g ) ≤ | a 1 | + | a 2 | ≤ s + s = 2 s. Hence, the ratio b et w een the DNF size and ess ( g ) size is: s ( n − 3 4 + 1) 2 s ≥ n + 1 8 = Θ( n ) Note that the ab o v e function giv es a class of functions satisfying the conditions of Prop o sition 1, for γ = 1. Corollary 1. Ther e exists a function f such that cnf size ( f ) ess ( f ) ≥ cnf size ( f ) ǫ for an ǫ ≥ 0. Pr o of. In the previous construction, ˆ f ( x ) = ∗ for exactly  m 2  + 1 p oints , yielding s = Θ( n 2 ). Hence, the DNF size is Θ( m 3 ), making the ratio b et w een dnf size ( ˆ g ) a nd ess d ( ˆ g ) at least Θ( dn f size ( ˆ g ) 1 3 ). The CNF result follow s b y dualit y . 4.2. C onstructing a function with an exp onential gap Theorem 2. Ther e exists a function f on n variables such that cnf size ( f ) ess ( f ) ≥ 2 Θ( n ) . Pr o of. As b efore, we will reduce a set cov ering instance to a DNF -minimization problem in v olving a partial Bo olean function f . How ev er, here w e will rely on a more general version of Gimp el’s reduction, due to Allender et al., de- scrib ed in the follo wing lemma. Lemma 1. [9] L et S = { S 1 , . . . , S p } b e a set of subsets of gr ound set A = { e 1 , . . . , e m } . L et t > 0 and let V = { v i : i ∈ { 1 , . . . , m }} and W = { w j : j ∈ { 1 , . . . , p }} b e sets of ve ctors fr om { 0 , 1 } t such that for al l j ∈ { 1 , . . . , p } and i ∈ { 1 , . . . , m } , e i ∈ S j iff v i ≥ w j 3 It can ac tually be prov ed that in fact, ess d ( ˆ g ) = 2 s , but details of this pro of ar e omitted. 10 L et f : { 0 , 1 } t → { 0 , 1 , ∗ } b e the p artial function such that f ( x ) =    1 if x ∈ V ∗ if x ≥ w for some w ∈ W and x / ∈ V 0 otherwise Then S h a s a minimum c over of size k iff dnf size ( f ) = k . (Note tha t the construction in the ab ov e lemma is equiv a lent to Gimp el’s if w e tak e t = m , V = { v ∈ { 0 , 1 } m | v contains exactly m − 1 1’s } , and W = { x S | S ∈ S } , where x S denotes the assignmen t in { 0 , 1 } m where x i = 0 iff e i ∈ S .) As b efore, w e use the simple 2-uniform set co v ering instance o ve r m el- emen ts where S consists o f all subsets of t w o of those elemen ts. The next step is to construct sets V and W satisfying the pro p erties in the ab ov e lemma for t his set co v ering instance. T o do this, w e use a randomized con- struction of Allender et a l. tha t generates sets V and W fro m an r -uniform set-co v ering instance, for an y r > 0. This randomized construction a pp ears in the app endix of [9], and is described in the follo wing lemma. Lemma 2. L et r > 0 and let S = { S 1 , . . . , S p } b e a set of subsets of { e 1 , . . . , e m } , wh e r e e a ch S i c ontains exactly r elements. L et t ≥ 3 r (1 + ln( pm )) . L et V = { v 1 , . . . , v m } b e a set of m ve ctors of length t , whe r e e ach v i ∈ V is pr o duc e d by r andomly and indep endently setting e ach bit of v i to 0 w ith pr ob ability 1 /r . L et W = { w 1 , . . . , w p } , wher e e ach w j = the bitwise AND of al l v i such that e i ∈ S j . Then, the fol l o wing holds with pr o b abil- ity g r e ater than 1/2: F o r al l j ∈ { 1 , . . . , p } and i ∈ { 1 , . . . , m } , e i ∈ S j iff v i ≥ w j . By Lemma 2, there exist sets V a nd W , each consisting of v ectors o f length 6(1 + ln( m 2 ( m − 2) / 2)) = O (log m ), satisfying the conditio ns o f Lemma 1 for our simple 2- uniform se t co v ering instance. Let ˜ f b e the partial f unction on O (log m ) v ariables o btained b y using these V a nd W in the definition o f f in Lemma 1, The DNF- size o f ˜ f is the size of the smallest set cov er, whic h is ⌈ m/ 2 ⌉ , and the n umber of v aria bles n = Θ(log m ); hence the D NF size is 2 Θ( n ) . W e can con v ert the partial function ˜ f ( x ) to a total function ˜ g ( x ) just as done in the previous section. The arguments regarding DNF-size a nd ess d ( ˜ g ) remain the same. Hence, the DNF-size is now s  2 Θ( n ) + 1  , a nd ess d ( ˜ g ) is again at most 2 s . 11 The ratio b et w een the D NF-size and e ss d ( ˜ g ) is therefore at least 2 Θ( n ) . Once again, the CNF result follo ws. 4.3. T he quantity ess k ( f ) W e sa y that a set S of falsep oints (truep oints) of f is a “ k -indep enden t set” if no k of the falsep oints (truep oints ) of f can b e cov ered by the same implicate (implicate) of f . W e de fine ess k ( f ) to b e the size of the largest k - indep enden t set of false- p oin ts of f , and ess d k ( f ) to b e the size o f the largest k -indep endent set of truep oin ts of f . If S is a k -indep enden t set of falsep oints of f , then eac h implicate of f can co v er at most k − 1 falsep oints in S . W e th us hav e the fo llo wing low er-b o und on CNF-size: cnf size ( f ) ≥ ess k ( f ) k − 1 . Lik e ess ( f ), this lo w er b ound is not tight. Theorem 3. F or any arbitr ary 2 ≤ k ≤ h ( n ) , wher e h ( n ) = Θ( n ) , ther e exists a function f on n variable s, such that the gap b etwe en cnf size ( f ) and ess k ( f ) k − 1 is at le ast 2 Θ( n k ) . Pr o of. Consider t he k -unifo rm set co v er instance consisting of all subsets of { e 1 , . . . , e m } of size k . Construct V and W randomly using the construction from the app endix of [9] describ ed in Lemma 2, and define a corresp onding partial function ˜ f , a s in Lemma 1. Note that according t o t he definition o f ˜ f , there can be no k v i for any k v alues of i ∈ { 1 , . . . , m } , suc h that all v i ≥ w j for some j ∈ { 1 , . . . , p } . The maxim um size k - indep enden t set of t ruep oin ts of ˜ f consists of k − 1 truep o in ts. W e can conv ert the partia l function ˜ f to a total function ˜ g according to the construction detailed in Section 4.1. O nce again, w e in tro duce s new truep oin ts such t ha t ˜ f ( x ) = ∗ , yielding a maximum of s pairwise indep enden t truep oin ts. The definiton of k - indep endence, ho wev er, allow s k − 1 “copies” of these truep oints that differ in the assignmen ts to z for each of the s p oin ts. Hence, the largest k - indep enden t set of these p oin ts can contain a maxim um of s ( k − 1) p oints. W e hav e previously men tioned that there exist k − 1 k -indep endent ground elemen ts (i.e., ˜ f ( x ) = 1 truep oints). Once again, when w e consider the s ˜ z p ortion of t he term, where no t wo ˜ z p ortions can b e cov ered b y the prime implicate, w e can include a total o f s ( k − 1) of these truep oints. Hence, the largest indep enden t set fo r p oin ts of this type is of size is of size no 12 greater than s ( k − 1). Since these tw o ty p es of truep o in ts are indep enden t, ess d k ( ˜ g ) ≤ 2 s ( k − 1). The lo w er b o und on DNF size, ess d k ( f ) k − 1 , is, fo r this ˜ g , ≤ 2 s ( k − 1) k − 1 ≤ 2 s . The ratio b et w een tha t and the actual DNF size is s (2 Θ( n k ) + 1) 2 s ≥ 2 Θ( n k ) . The CNF result clearly follo ws. 5. Size of the gap for Horn F unctions Because Horn-CNFs con tain at most o ne unnegated v ariable p er clause, they can b e expressed as implications; eg. ¯ a ∨ b is equiv alen t to a → b . Moreo v er, a conjunction of sev eral clauses that hav e the same an teceden t can b e represen ted as a single meta-cla use , where the an teceden t is the anteced ent common to all the clauses and the conse quen t is comprised of a conjunction of all the consequen ts, eg. ( a → b ) ∧ ( a → c ) can b e represen ted as a → ( b ∧ c ). 5.1. B ounds on the r atio b etwe en cnf size ( f ) and ess ( f ) Angluin, F razier and Pitt [10] presen ted an algorithm (henceforth: the AFP algorithm) to learn Horn-CNFs, where the output is a series of meta- clauses. It can b e prov en [1 1, 12] that the output of the algorithm is of minim um implication size (henceforth: min imp ( f )) – that is, it contains the few est n um b er of meta-clauses needed to represen t function f . Eac h meta- clause can b e a conjunction of at most n clauses; hence, eac h implication is equiv alen t to the conjunction o f at most n claus es. Therefore, cnf size ( f ) ≤ n × min imp ( f ) . The learning algorithm maintains a list of negativ e and po sitiv e examples (falsep oin ts and truep oints of the Horn function, resp ectiv ely), con taining at most min imp ( f ) examples of eac h. Lemma 3. The set of ne gative examples ma intaine d by the AFP algo ri thm is an inde p enden t set. Pr o of. The pro of for t his lemma relies hea vily on [11]; see there for f ur t her details. Let us cons ider an y tw o negativ e ex amples, n i and n j , main ta ined by the algorithm. There are tw o p ossibilities: 13 1. n i ≤ n j or n j ≤ n i . (These t w o examples are comparable p oin ts; one is b elo w the other on the Bo olean lattice.) 2. n i and n j are incomparable p oints (Neither is b elow the other on the lattice). Let us consider the first ty p e o f p oin ts: Without loss of generalit y , assume that n i ≤ n j . Arias et al. define a p ositiv e example n ∗ i for eac h negativ e example n i . This example n ∗ i has sev eral unique pro p erties; among st them, that n i < n ∗ i for all negative examples n i (Section 3 in [11 ]). They further pro v e (L emma 6 in [11]) that if n i ≤ n j , then n ∗ i ≤ n j as we ll. Henc e, any attempt to falsify b ot h falsep oints , n i and n j , with a common implicate o f the Horn function would falsify the p ositiv e example ( n ∗ i ) that lies b etw een them as w ell. Therefore, these tw o p o in ts are indep enden t. No w let us a ssume that n i and n j are incomparable. An y implicate that falsifies b oth p o ints is compo sed of v ariables on whic h the t w o p oints a gree. Clearly , this implicate w ould lik ewise cov er a p oin t that is the comp o nen t wise in tersection of n i and n j . Ho w ev er, Arias et al. pro v e (Lemma 7 in [11]) t ha t n i ∧ n j is a p ositiv e p oin t if n i and n j are incomparable. Hence, any implicate that falsifies both n i and n j w ould lik ewise falsify the truep oin t n i ∧ n j that lies b et w een them. Therefore, these tw o p oin ts cannot b e falsified by the same implicate and they are independen t. Theorem 4. F or an y Horn function f , cnf size ( f ) ess ( f ) ≤ n Pr o of. F o r any Horn f unction f , there exists a set of negativ e examples of size at most min imp ( f ), and these examples are all indep enden t. Hence, ess ( f ) ≥ min imp ( f ). W e hav e a lr eady stated that min imp ( f ) is at most a factor of n times larger than the minim um CNF size for this function. Hence, cnf size ( f ) ≤ n × ess ( f ). Moreo v er, since Lemma 3 holds for general Horn functions in addition to pure Horn [12], this bo und holds for all Horn functions. 5.2. C onstructing a Horn function with a lar ge g ap b etwe en ess ( f ) and cn f size ( f ) Theorem 5. Th er e ex i s ts a defi n ite Horn function f on n v ariables such that cnf size ( f ) ess ( f ) ≥ Θ( √ n ) . Pr o of. Consider the 2-uniform set co v ering instance o ver k elemen ts consist- ing of all subs ets of t w o elemen ts. W e can construct a definite Horn form ula ϕ corresp onding to this set co v ering according to the construction in [13], with mo difications based on [6]. 14 The form ula ϕ will con tain 3 ty p es of v ariables: • Elemen t v ariables: There is a v ar ia ble x for eac h of the k elemen ts. • Set v ariables: There is a v ariable s for eac h of the  k 2  subsets. • Amplification v ariables: There are t v ar iables z 1 . . . z t . The clauses in ϕ fall in to the follo wing 3 groups: • Witness clauses: There is a clause s j → x i for eac h subset and for each elemen t that the subset co vers . There are 2  k 2  suc h clauses. • F eedbac k clauses : There is a clause x 1 . . . x k → s j for eac h subset. There are  k 2  suc h clauses. • Amplification clauses: There is a clause z h → s j for ev ery h ∈ { 1 . . . t } and for ev ery subset. There are t  k 2  suc h clauses. It follows from [1 3] that an y minim um CNF for this f unction m ust con tain all witness and feedbac k clauses, along with tc amplification clauses, where c is the siz e of the smallest set co v er. This particular function f has a minim um set co ver of size k / 2; hence, cnf size ( f ) = 2  k 2  +  k 2  + t ( k / 2) . W e will upp er b ound ess ( f ) b y dividing the falsep o ints of f in to three disjoin t sets and finding the maxim um indep enden t set for eac h. Set 1: The set of all falsep oints of f that contain at least one x i = 0 for i ∈ { 1 , . . . , k } and some s j = 1 for a sub set s j that co v ers x i . Let a 1 b e the la rgest indep enden t set o f f consisting of p oin ts in this set. These p o ints can b e co v ered by an implicates of the form s j → x i , o f whic h there are 2  k 2  . W e will define t he function f ′ whose falsep oints are just the T yp e 1 p oints. Since these p o in ts a re cov ered b y the s j → x i implicates, cnf size ( f ′ ) is no more than the num b er o f s j → x i implicates. W e hav e earlier said that ess ( f ′ ) ≤ cnf size ( f ′ ), hence it follo ws that e ss ( f ′ ) ≤ 2  k 2  . ess ( f ′ ) is precisely the size of a 1 ; hence, a 1 can con tain no more than 2  n 2  p oin ts. 15 Set 2: The set of all falsep o ints that are not in t he first set, ha ve x i = 1 for all i ∈ { 1 , . . . , k } , and at least one s j = 0 for s ome j ∈ { 1 , . . . ,  k 2  } . Let a 2 of f b e the larg est indep enden t set consisting of p oints in this set. These p oints can b e co v ered by implicates of the form x 1 . . . x k → s j . There are  k 2  suc h implicates . Hence, b y the same argumen t as ab o v e, a 2 can con tain no more than  k 2  p oin ts. Set 3: The set of all fa lsep o ints that are not in the first tw o sets, and therefore ha v e z h = 1 for some h ∈ { 1 , . . . , t } , x i = 0 for some i ∈ { 1 , . . . , k } , and y j = 0 for all subsets y j co v ering x i . Let a 3 b e the largest indep enden t set of f consisting of p oin ts in this set. Let us fix h = 1. Consider a falsep oint p in this set where x i = 0 for at least one i ∈ { 1 , . . . , k } . If p con tained a y j = 1 suc h that the subset y j co v ers x i , tha t p oint w ould be a p oin t in the first se t. Hence, the only p o in ts of this form in this set ha ve y j = 0 for al l k − 1 subs ets y j that co v er x i . No w consid er another fa lsep oin t q in this set, where x a = 0 for at least one a ∈ { 1 , . . . , k } . Once aga in, the only p oin ts in this set m ust set y b = 0 for all k − 1 subsets y b that co v er x a . Because t he set co v ering problem included a set fo r eac h pair of x i p oin ts, there exists some y j that cov ers b ot h x i and x a . By the previous argumen t, that y j is set to 0 in all assignmen ts t ha t set x i or x a = 0. F o r a fixed k , all of these p oin ts can b e cov ered b y the implicate z k → y j . Hence, p oin ts p and q are not indep enden t. In fact, any tw o falsep oints ch osen t ha t a r e not in the first set and con- tain z h = 1 for the same h and at least o ne x i = 0 are not independen t. Because there are t v alues o f h , a 3 therefore has size t . The largest indep enden t set for all fa lsepo in ts cannot exceed the sum of the indep endent sets for these three dis jo int sets, hence ess ( f ) ≤ | a 1 | + | a 2 | + | a 3 | ≤ 2  k 2  +  k 2  + t. The gap b et w een cnf size ( f ) and ess ( f ) = cnf size ( f ) ess ( f ) ≥ 3  k 2  + t ( k / 2) 3  k 2  + t . 16 Let us set t = 3  k 2  . The difference is no w: cnf size ( f ) ess ( f ) ≥ t (1 + k / 2) 2 t ≥ Θ( k ) . W e hav e k elemen t v ariables,  k 2  set v a riables, and 3  k 2  amplification v ari- ables, yielding n = Θ( k 2 ) v ariables in total. The difference b et w een cn f size ( f ) and ess ( f ) is therefore ≥ Θ( √ n ). W e earlier p osited that if Σ 2 p 6 = c o-NP , there exists an infinite set of functions for whic h cnf size ( f ) ess ( f ) ≥ cnf size ( f ) γ for some γ > 0 . W e can no w pro v e a stronger theorem: Theorem 6. Ther e exists an infini te set of Horn functions f for which cnf size ( f ) ess ( f ) ≥ cnf size ( f ) γ . Pr o of. See construction ab o v e. Because cnf size ( f ) = Θ( k 3 ), cnf size ( f ) ess ( f ) = Θ( cnf size ( f ) 1 / 3 ) . 6. Ackno wledgemen ts This work w as partially supp orted b y the US D epart ment of Education GAANN gran t P200A0901 5 7, and b y NSF Gr a n t CCF-0917153. [1] O. ˘ Cep ek, P . Ku˘ cera, P . Sa vic k´ y, Bo olean F unctions with a simple cer- tificate for CNF complexit y , T echnic al Rep o rt, Rutgers Cente r for Op- erations Researc h, 2 010. [2] O. Coudert, Tw o- lev el logic minimization: an ov erview, In tegr a tion, the VLSI Journal (1994). [3] S. J. Russell, P . Norvig, Artificial In telligence: A Mo dern Approa c h, P earson Education, 2003. [4] C. Umans, Hardness o f a pproximating Σ p 2 minimization problems, in: Pro c. IEEE Sy mp osium on F oundations of Computer Science, pp. 465– 474. [5] C. Umans, The minim um equiv alen t D NF problem and shortest impli- can ts, in: IEEE Symp osium on F oundations of Computer Science, pp. 556–563. 17 [6] A. Bhatta chary a , B. DasGupta , D . Muba yi, G. T ur´ an, On appro ximate horn for mula minimization, in: S. Abramsky , C. Gav oille, C. Kirc h- ner, F. Mey er auf der Heide, P . Spirakis (Eds.), Automat a , La nguages and Programming, v olume 6 1 98 of L e ctur e Notes in Computer Scienc e , Springer Berlin / Heidelberg , 2010, pp. 438–4 50. [7] J. Gimp el, Metho d of pro ducing a b o olean function having an arbitrarily presrib ed prime impliant table, IEEE T rans. Computers (1965). [8] S. L. A. Czort, The Complexit y of Minimizing Disjunctiv e Normal F orm F ormulas, Master’s thesis, Univ ersit y of Aarh us, Aar h us, Denmark, 1999. [9] E. Allender, L. Hellerstein, P . McCab e, T. Pitassi, M. E. Saks, Mini- mizing disjunctiv e normal form formu las and AC 0 circuits giv en a truth table, SIAM J. Comput. 38 (2008) 63–84. [10] D. Angluin, M. F r a zier, L. Pitt, Learning conjunctions of horn clauses, Mac hine Learning 9 (19 92) 147–164. [11] M. Arias, J. L. Balc´ azar, Query learning and certificates in lattices, in: Y . F reund, L. Gy¨ orfi, G . T ur´ an, T. Zeugmann (Eds.), Alg o rithmic Learning Theory , volume 5 2 54 of L e ctur e Notes in Computer Scienc e , Springer Berlin / Heidelberg , 2008, pp. 303–3 15. [12] M. Arias, J. Balc´ azar, Construction a nd learnability of canonical horn form ulas, Mac hine Learning (201 1) 1–25. 10.1007/s10994-0 11-5248 -5. [13] Y. Crama, P . L. Hammer (Eds.), Bo olean F unctions: Theory , Algo- rithms, and Applications, Cam bridge Unive rsity Press, 2011. 18