Monotone Circuit Lower Bounds from Robust Sunflowers
aa r X i v : . [ c s . CC ] D ec Monotone Circuit Lower Bounds from Robust Sunflowers
Bruno Pasqualotto Cavalar , Mrinal Kumar , and Benjamin Rossman Institute of Mathematics and Statistics, University of S˜ao Paulo [email protected] IIT Bombay [email protected] University of Toronto [email protected]
Abstract.
Robust sunflowers are a generalization of combinatorial sunflowers that have applicationsin monotone circuit complexity [22], DNF sparsification [10], randomness extractors [15], and recentadvances on the Erd˝os-Rado sunflower conjecture [3,16,19]. The recent breakthrough of Alweiss, Lovett,Wu and Zhang [3] gives an improved bound on the maximum size of a w -set system that excludes arobust sunflower. In this paper, we use this result to obtain an exp( n / − o (1) ) lower bound on themonotone circuit size of an explicit n -variate monotone function, improving the previous best knownexp( n / − o (1) ) due to Andreev [5] and Harnik and Raz [11]. We also show an exp( Ω ( n )) lower boundon the monotone arithmetic circuit size of a related polynomial via a very simple proof. Finally, weintroduce a notion of robust clique-sunflowers and use this to prove an n Ω ( k ) lower bound on themonotone circuit size of the CLIQUE function for all k n / − o (1) , strengthening the bound of Alonand Boppana [1]. A monotone Boolean circuit is a Boolean circuit with
AND and OR gates but no negations ( NOT gates).Although a restricted model of computation, monotone Boolean circuits seem a very natural model towork with when computing monotone
Boolean functions, i.e., Boolean functions f : { , } n → { , } suchthat for all pairs of inputs ( a , a , . . . , a n ) , ( b , b , . . . , b n ) ∈ { , } n where a i b i for every i , we have f ( a , a , . . . , a n ) f ( b , b , . . . , b n ). Many natural and well-studied Boolean functions such as Clique and
Majority are monotone.Monotone Boolean circuits have been very well studied in Computational Complexity over the years, andcontinue to be one of the few seemingly largest natural sub-classes of Boolean circuits for which we haveexponential lower bounds. This line of work started with an influential paper of Razborov [21] from 1985which proved an n Ω ( k ) lower bound on the size of monotone circuits computing the Clique k,n function on n -vertex graphs for k log n ; this bound is super-polynomial for k = log n . Prior to Razborov’s result, super-linear lower bounds for monotone circuits were unknown, with the best bound being a lower bound of 4 n dueto Tiekenheinrich [25]. Further progress in this line of work included the results of Andreev [4] who provedan exponential lower bound for another explicit function. Alon and Boppana [1] extended Razborov’s resultby proving an n Ω ( √ k ) lower bound for Clique k,n for all k n / − o (1) . A second paper of Andreev [5] from thesame time period proved an 2 Ω ( n / / log n ) lower bound for an explicit n -variate monotone function. Usinga different technique, Harnik and Raz [11] proved a lower bound of 2 Ω (( n/ log n ) / ) for a family of explicit n -variate functions defined using a small probability space of random variables with bounded independence.However, modulo improvements to the polylog factor in this exponent, the state of art monotone circuitlower bounds have been stuck at 2 n / − o (1) since 1987. To this day, the question of proving truly exponentiallower bounds for monotone circuits (of the form 2 Ω ( n ) ) for an explicit n -variate function) remains open!(Truly exponential lower bounds for monotone formulas were obtained only recently [18].)In the present paper, we are able to improve the best known lower bound for monotone circuits byproving an 2 Ω ( n / / (log n ) / ) lower bound for an explicit n -variate monotone Boolean function (Section 2).The function is based on the same construction first considered by Harnik and Raz, but our argument employs Stasys Jukna (personal communication) observed that Andreev’s bound [5] can be improved to 2 Ω (( n/ √ log n ) / ) using the lower bound criterion of [14]. he approximation method of Razborov with recent improvements on robust sunflower bounds [3,19]. Byapplying the same technique with a variant of robust sunflowers that we call robust clique-sunflowers, we areable to prove an n Ω ( k ) lower bound for the Clique k,n function when k n / − o (1) , thus improving the resultof Alon and Boppana when k is in this range (Appendix B). Finally, we are able to prove truly exponentiallower bounds in the monotone arithmetic setting to a fairly general family of polynomials, which shares somesimilarities to the functions considered by Andreev and Harnik and Raz (Section 3). The original lower bound for
Clique k,n due to Razborov employed a technique which came to be known as the approximation method . Given a monotone circuit C of “small size”, it consists into constructing gate-by-gate,in a bottom-up fashion, another circuit e C that approximates C on most inputs of interest. One then exploitsthe structure of this approximator circuit to prove that it differs from Clique k,n on most inputs of interest,thus implying that no “small” circuit can compute this function. This technique was leveraged to obtainlower bounds for a host of other monotone problems [1].A crucial step in Razborov’s proof involved the sunflower lemma due to Erd˝os and Rado. A family F ofsubsets of [ n ] is called a sunflower if there exists a set Y such that F ∩ F = Y for every F , F ∈ F . Thesets of F are called petals and the set Y = T F is called the core . We say that the family F is ℓ -uniform ifevery set in the family has size ℓ . Theorem 1 (Erd˝os and Rado [7]).
Let F be a ℓ -uniform family of subsets of [ n ] . If |F| > ℓ !( r − ℓ , then F contains a sunflower of r petals. Informally, the sunflower lemma allows one to prove that a monotone function can be approximated byone with fewer minterms by means of the “plucking” procedure: if the function has too many (more than ℓ !( r − ℓ ) minterms of size ℓ , then it contains a sunflower with r petals; remove all the petals, replacingthem with the core. One can then prove that this procedure does not introduce many errors.The notion of robust sunflowers was introduced by the third author in [22], to achieve better boundsvia the approximation method on the monotone circuit size of Clique k,n when the negative instances areErd˝os-R´enyi random graphs G n,p below the k -clique threshold. A family
F ⊆ [ n ] is called a ( p, ε )- robustsunflower if P W ⊆ p [ n ] [ ∃ F ∈ F : F ⊆ W ∪ Y ] > − ε, where Y := T F and W is a p -random subset of [ n ]. Henceforth, we consistently write random objects usingboldface symbols (such as W , G n,p , etc).As remarked in [22], every ℓ -uniform sunflower of r petals is a ( p, e − rp ℓ )-robust sunflower. Moreover, asobserved in [16], every (1 /r, /r )-robust sunflower contains a sunflower of r petals. A corresponding boundfor the appearance of robust sunflowers in large families was also proved in [22]. Theorem 2 ([22]).
Let F be a ℓ -uniform family such that |F| > ℓ !(2 log(1 /ε ) /p ) ℓ . Then F contains a ( p, ε ) -robust sunflower. For many choice of parameters p and ε , this bound is better than the one by Erd˝os and Rado, thus leadingto better approximation bounds. In a recent breakthrough, this result was significantly improved in [3]. Theorem 3 (Theorem 2.5 of [3]).
Let F be a ℓ -uniform family such that |F| > (log ℓ ) ℓ · (log log ℓ · log(1 /ε ) /p ) O ( ℓ ) . Then F contains a ( p, ε ) -robust sunflower. Because of the connection between robust sunflowers and sunflowers explained above, this result was usedby the authors to significantly improve the standard sunflower bounds of Erd˝os and Rado. Soon afterwards,Rao [19] provided an alternative proof which slightly improved the bound. It is this bound we are going touse, which we introduce in the next section. Robust sunflowers were called quasi-sunflowers in [22,10,15,16] and approximate sunflowers in [17]. FollowingAlweiss et al [3], we adopt the new name robust sunflower . Crucially for our application, the O ( ℓ ) exponent in the bound of Theorem 3 is only 2 ℓ when ε = 2 − Ω ( ℓ ) . To getany improvement over the Harnik-Raz bound, we require ℓ + o ( ℓ ), which is given by the result of Rao [19]. .2 Slice sunflowers In what follows, let m be a positive integer such that m < n . Definition 1.
Let F be a family of subsets of [ n ] and let Y := T F . Let also W ⊆ [ n ] be a set of size m chosen uniformly at random. The family F is called a ( m, ε ) - slice-sunflower if P W [ ∃ F ∈ F : F ⊆ W ∪ Y ] > − ε. Theorem 4 ([19]).
There exists an universal constant
B > such that the following holds. Let p ∈ (0 , and let F ⊆ (cid:0) [ n ] ℓ (cid:1) be such that |F| > ( Bx log x ) ℓ , where x = log( ℓ/ε ) /p . Then F contains a ( m, ε ) -slice-sunflower, where m = ⌊ np ⌋ . The theorem above is implicit in Rao [19]. For this reason, we include most of its proof in Appendix A,closely following the argument and notation of [19].
The strongest lower bound known for monotone circuits computing an explicit n -variate monotone Booleanfunction is exp (cid:0) Ω (cid:0) ( n/ log n ) / (cid:1)(cid:1) , and it was obtained by Harnik and Raz [11]. In this section, we will prove alower bound of exp( Ω ( n / / (log n ) / )) for the same Boolean function they considered. We apply the methodof approximations [21] and the new robust sunflower bound [3,19]. We do not expect that a lower boundbetter than exp( n / − o (1) ) can be obtained by this technique, even with better sunflower bounds.We start by giving a high level outline of the proof. We define the Harnik-Raz function f HR : { , } n →{ , } and find two distributions Y and N with support in { , } n satisfying the following properties: – f HR outputs 1 on Y with high probability (Lemma 1); – f HR outputs 0 on N with high probability (Lemma 2).Because of these properties, the distribution Y is called the positive test distribution , and N is called the negative test distribution . We also define a set of monotone Boolean functions called approximators , and weshow that: – every approximator commits many mistakes on either Y or N with high probability (Lemma 8); – every Boolean function computed by a “small” monotone circuit agrees with an approximator on both Y and N with high probability (Lemma 9).Together these suffice for proving that “small” circuits cannot compute f HR . The crucial part where therobust sunflower result comes into play is in the second item. For A ⊆ [ n ], let x A ∈ { , } n be the binary vector with support in A . For a set A ∈ [ n ] , let ⌈ A ⌉ be theindicator function satisfying ⌈ A ⌉ ( x ) = 1 ⇐⇒ x A x. Define also { , } n = m := (cid:8) x A : A ∈ (cid:0) nm (cid:1)(cid:9) . For a monotone Boolean function f : { , } n → { , } , let M ( f )denote the set of minterms of f , and let M ℓ ( f ) := M ( f ) ∩{ , } n = ℓ . Elements of M ℓ ( f ) are called ℓ -mintermsof f . In what follows, we will mostly ignore ceilings and floors for the sake of convenience, since these do notmake any substantial difference in the final calculations.3 .2 The function We now describe the construction of the function f HR : { , } n → { , } considered by Harnik and Raz [11].First observe that, for every n -bit monotone Boolean function f , there exists a family S ⊆ [ n ] such that f ( x , . . . , x n ) = f S ( x , . . . , x n ) := _ S ∈S ^ j ∈ S x j . Indeed, S can be chosen to be the family of the coordinate-sets of minterms of f . Now, in order to constructthe Harnik-Raz function, we will suppose n is a prime power and let F n be the field of n elements. Moreover,we fix two positive integers c and k with c < k . For a polynomial P ∈ F n [ x ], we let S P be the set of thevaluations of P in each element of { , , . . . , k } (in other words, S P = { P (1) , . . . , P ( k ) } ). Observe that it isnot necessarily the case that | S P | = k , since it may happen that P ( i ) = P ( j ) for some i, j such that i = j .Finally, we consider the family S HR defined as S HR := { S P : P ∈ F n [ x ] , P has degree at most c − | S P | > k/ } . We thus define f HR as f HR := f S HR .We now explain the choice of S HR . First, the choice for valuations of polynomials with degree at most c − P ∈ F n [ x ] with degree c − P (1) , . . . , P ( k ) are c -wise independent, and are eachuniform in [ n ]. This allows us to define a distribution on the inputs (the positive test distribution) thathas high agreement with f HR and is easy to analyze. Observe further that, since |S HR | n c , the monotonecomplexity of f HR is at most 2 c log n . Later we will chose c to be roughly n / , and prove that the monotonecomplexity of f HR is 2 Ω ( c ) .Finally, the restriction | S P | > k/ f HR is very small.Otherwise, if f HR had small minterms, it might have been a function that almost always outputs 1. Suchfunctions have very few maxterms and are therefore computed by a small CNF. Since we desire f HR to havehigh complexity, this is an undesirable property. The fact that f HR doesn’t have small minterms is importantin the proof that f HR almost surely outputs 0 in the negative test distribution (Lemma 2).We now define the positive and negative test distributions. Let Y ∈ { , } n be the random variable whichchooses a polynomial P ∈ F n [ x ] with degree at most c − x S P ∈ { , } n . Let p := n − c/k and m := ⌊ np ⌋ . Let also N be the distribution which chooses an input from { , } n = m uniformly at random. For a Booleanfunction f and a probability distribution µ on the inputs on f , we write f ( µ ) to denote the random variablewhich evaluates f on a random instance of µ . Harnik and Raz proved that f HR outputs 1 on Y with highprobability. Lemma 1 (Claim 4.2 in [11]).
We have P [ f HR ( Y ) = 1] > − k/n. We now claim that f HR also outputs 0 on N with high probability. Lemma 2.
We have P [ f HR ( N ) = 0] > − n − c . Proof.
Let x A be an input sampled from N . Observe that f HR ( x A ) = 1 only if there exists a minterm x of f HR such that x x A . Since all minterms of f HR have Hamming weight at least k/ f HR has at most n c minterms, we have P [ f HR ( N ) = 1] n c · (cid:0) n − k/ m − k/ (cid:1)(cid:0) nm (cid:1) n c · (cid:16) mn (cid:17) k/ n − c . As a consequence of Lemmas 1 and 2, we obtain the following result.
Lemma 3.
For large enough n , we have P [ f HR ( Y ) = 1] + P [ f HR ( N ) = 0] > / . .3 A closure operator In this section, we describe a closure operator in the lattice of monotone Boolean functions. We provethat the closure of a monotone Boolean function f is a good approximation for f on the negative testdistribution (Lemma 4), and we give a bound on the size of the set of minterms of closed monotone functions.This bound makes use of the robust sunflower lemma (Theorem 4), and is crucial to bounding errors ofapproximation (Lemma 7). Throughout this section, we let ε := n − c . Definition 2.
We say that a monotone function f : { , } n → { , } is ε -closed if, for every A ∈ (cid:0) [ n ] c (cid:1) , wehave P [ f ( N ∨ x A ) = 1 ] > − ε = ⇒ f ( x A ) = 1 . This means that for, an ε -closed function, we always have P [ f ( N ∨ x A ) = 1] / ∈ [1 − ε,
1) when | A | c . Notemorever that if f, g are both ε -closed monotone Boolean functions, then so is f ∧ g . Therefore, there existsa unique minimum closed function cl( f ) satisfying f cl( f ). We call cl( f ) the closure of f . We now give abound on the error of approximating f by cl( f ) under the distribution N . Lemma 4.
For every monotone f : { , } n → { , } , we have P [ f ( N ) = 0 and cl( f )( N ) = 1] n − c . Proof.
We first prove that there exists a positive integer t and sets A , . . . , A t and monotone functions h , h , . . . , h t : { , } n → { , } such that1. h = f ,2. h i = h i − ∨ ⌈ A i ⌉ ,3. P [ h i − ( N ∪ x A i ) = 1] > − ε ,4. h t = cl( f ).Indeed, if h i − is not closed, there exists A i ∈ (cid:0) [ n ] c (cid:1) such that P [ h i − ( N ∪ x A i ) = 1] > − ε but h i − ( x A i ) = 0.We let h i := h i − ∨ ⌈ A i ⌉ . Clearly, we have that h t is closed, and that the value of t is at most the numberof subsets of [ n ] of size at most c . Therefore, we get t P cj =0 (cid:0) nj (cid:1) . Moreover, by induction we obtain that h i cl( f ) for every i ∈ [ t ]. It follows that h t = cl( f ). Now, observe that P [ f ( N ) = 0 and cl( f )( N ) = 1] t X i =1 P [ h i − ( N ) = 0 and h i ( N ) = 1]= t X i =1 P [ h i − ( N ) = 0 and x A i ⊆ N ] t X i =1 P [ h i − ( N ∪ x A i ) = 0] ε c X j =0 (cid:18) nj (cid:19) n − c . We now bound the size of the set of ℓ -minterms of an ε -closed function. This bound is dependent on therobust sunflower theorem (Theorem 4). Lemma 5.
Let
B > be as in Theorem 4. If a monotone function f : { , } n → { , } is ε -closed, then, forall ℓ ∈ [ c ] , we have |M ℓ ( f ) | (cid:18) B log( ℓ/ε ) p log (cid:18) log( ℓ/ε ) p (cid:19)(cid:19) ℓ . roof. Fix ℓ ∈ [ c ]. Suppose we have |M ℓ ( f ) | > ( C log( ℓ/ε ) /p log (log( ℓ/ε ) /p )) ℓ . Consider also the family F := n A ∈ (cid:0) [ n ] ℓ (cid:1) : x A ∈ M ℓ ( f ) o . Observe that |F| = |M ℓ ( f ) | . By Theorem 4, there exists a ( m, ε )-slice-sunflower F ′ ⊆ F . Let Y := T F ′ and let W ∈ (cid:0) [ n ] m (cid:1) be chosen uniformly at random. We have P [ f ( N ∨ x Y ) = 1] > P [ ∃ x ∈ M ℓ ( f ) : x N ∨ x Y ]= P [ ∃ F ∈ F : F ⊆ W ∪ Y ] > P [ ∃ F ∈ F ′ : F ⊆ W ∪ Y ] > − ε. Therefore, since f is ε -closed, we get that f ( x Y ) = 1. However, since Y = T F ′ , there exists F ∈ F ′ suchthat Y ( F . This is a contradiction, because x F is a minterm of f . In this section, we define a trimming operation for Boolean functions. We will bound the probability that a trimmed function gives the correct output on the distribution Y , and we will give a bound on the error ofapproximating a Boolean function f by the trimming of f on that same distribution. Definition 3.
We say that a monotone function f ∈ { , } n → { , } is trimmed if all the minterms of f have size at most c/ . We define the trimming operation trim( f ) as follows: trim( f ) := c/ _ ℓ =1 _ A ∈M ℓ ( f ) ⌈ A ⌉ . That is, the trim operation takes out from f all the minterms of size larger than c/
2, yielding a trimmedfunction. We will first prove the following claim.
Claim.
For every monotone function f : { , } n → { , } and ℓ c , we have P [ ∃ x ∈ M ℓ ( f ) : x Y ] ( k/n ) ℓ |M ℓ ( f ) | . Proof.
Recall (Section 2.2) that the distribution Y takes a polynomial P ∈ F n [ x ] with degree at most c − x { P (1) , P (2) ,..., P ( k ) } ∈ { , } n . Let A ∈ (cid:0) [ n ] ℓ (cid:1) for ℓ c .Observe that x A Y if and only if A ⊆ { P (1) , P (2) , . . . , P ( k ) } . Therefore, if x A Y , then there existsindices { j , . . . , j ℓ } such that { P ( j ) , P ( j ) , . . . , P ( j ℓ ) } = A . Since ℓ c , we get by the c -wise independenceof P (1) , . . . , P ( k ) that the random variables P ( j ) , P ( j ) , . . . , P ( j ℓ ) are independent. It follows that P [ { P ( j ) , P ( j ) , . . . , P ( j ℓ ) } = A ] = ℓ ! n ℓ . Therefore, we have P [ x A Y ] = P [ A ⊆ { P (1) , P (2) , . . . , P ( k ) } ] (cid:18) kℓ (cid:19) ℓ ! n ℓ (cid:18) kn (cid:19) ℓ . The claim now follows by an union bound.
Lemma 6.
If a monotone function f ∈ { , } n → { , } is trimmed and f = (i.e., f is not identically ),then P [ f ( Y ) = 1] c/ X ℓ =1 (cid:18) kn (cid:19) ℓ |M ℓ ( f ) | . Proof.
It suffices to see that, since f is trimmed, if f ( Y ) = 1 and f = then there exists a minterm x of f with Hamming weight between 1 and c/ x Y . The result follows by the claim above.6 emma 7. Let f ∈ { , } n → { , } be a monotone function, all of whose minterms have Hamming weightat most c . We have P [ f ( Y ) = 1 and trim( f )( Y ) = 0] c X ℓ = c/ (cid:18) kn (cid:19) ℓ |M ℓ ( f ) | . Proof.
If we have f ( Y ) = 1 and trim( f )( Y ) = 0, then there was a minterm x of f with Hamming weightlarger than c/ | x | c by assumption, the resultfollows by the claim. Let A := { trim(cl( f )) : f : { , } n → { , } is monotone } . Functions in A will be called approximators . Wedefine the approximating operations ⊔ , ⊓ : A × A → A as follows: for f, g ∈ A , let f ⊔ g := trim(cl( f ∨ g )) ,f ⊓ g := trim(cl( f ∧ g )) . Observe that every input function x i is an approximator. Therefore, we can replace each gate of amonotone {∨ , ∧} -circuit C by its corresponding approximating gate, thus obtaining a {⊔ , ⊓} -circuit C A computing an approximator.The rationale for choosing this set of approximators is as follows. By letting approximators be thetrimming of a closed function, we are able to plug the bound on the set of ℓ -minterms given by the robustsunflower lemma (Lemma 5) on Lemmas 6 and 7, since the trimming operation can only reduce the set ofminterms. Moreover, since trimmings can only help to get a negative answer on the negative test distribution,we can safely apply Lemma 4 when bounding the errors of approximation. In this section, we will prove that the function f HR requires monotone circuits of size 2 Ω ( c ) . By properlychoosing c and k , this will imply the promised exp( Ω ( n / − o (1) )) lower bound for the Harnik-Raz function.First, we fix some parameters. Choose B as in Lemma 5. We also let c := 16 Be /B (cid:18) n (log n ) (cid:19) / , k := (cid:18) n log n (cid:19) / . For simplicity, we assume these values are integers. We clearly have c < k . Moreover, observe that, becauseof this choice of parameters, we have p = Ω (1). Indeed, we have p = n − c/k = n − / (3 Be /B log n ) = e − / (3 Be /B ) > e − /B . We will now show that, when f is an approximator, the bound of Lemma 6 can be replaced by 1 /
2, and alsothat, when f is an ε -closed function, the bound of Lemma 7 can be replaced by 2 − Ω ( c ) . We will first need tobound the sequence s ℓ , defined as follows. For every 1 ℓ c , let s ℓ := (cid:18) kn (cid:19) ℓ · (cid:18) B log( c/ε ) p log (cid:18) log( c/ε ) p (cid:19)(cid:19) ℓ . Note that, when f is a n -bit ε -closed monotone function, we get by Lemma 5 that (cid:0) kn (cid:1) ℓ |M ℓ ( f ) | s ℓ . Inother words, the summands of Lemma 6 and Lemma 7 can be replaced by s ℓ in some applications. Observemoreover that s ℓ = ( s ) ℓ . Now we are going to show that, for n sufficiently large, we have s /
3, whichimplies s ℓ − ℓ . First, observe thatlog( c/ε ) /p = log( n c c ) /p log( n c ) /p = 4 cp log n. c/ε ) /p ) = log (cid:18) cp log n (cid:19) = 12 log n −
12 log log n + O (1)
12 log n, for n sufficiently large. From the previous two inequalities, we obtain for n sufficiently large that s = Bk log( c/ε ) log(log( c/ε ) /p ) / ( pn ) Bck (log n ) / ( pn ) / , as desired. Lemma 8 (Approximators make many errors).
For every approximator f ∈ A , we have P [ f ( Y ) =1] + P [ f ( N ) = 0] / . Proof.
Let f ∈ A . By definition, there exists an ε -closed function h such that f = trim( h ). Observe that M ℓ ( f ) ⊆ M ℓ ( h ) for every ℓ ∈ [ c ]. Hence, applying Lemma 6 and the bounds for s ℓ , we obtain that, if f = ,we have P [ f ( Y ) = 1] c/ X ℓ =1 (cid:18) kn (cid:19) ℓ |M ℓ ( h ) | c/ X ℓ =1 s ℓ c/ X ℓ =1 − ℓ / . Therefore, for every f ∈ A we have P [ f ( Y ) = 1] + P [ f ( N ) = 0] / / . Lemma 9 ( C is well-approximated by C A ). Let C be a monotone circuit. We have P [ C ( Y ) = 1 and C A ( Y ) = 0] + P [ C ( N ) = 0 and C A ( N ) = 1] size( C ) · − Ω ( c ) . Proof.
We begin by bounding the approximation errors under the distribution Y . We will show that, fortwo approximators f, g ∈ A , if f ∨ g accepts an input from Y , then f ⊔ g rejects that input with probabilityat most 2 − Ω ( c ) , and that the same holds for the approximation f ⊓ g .First note that, if f, g ∈ A , then all the minterms of both f ∨ g and f ∧ g have Hamming weight at most c ,since f and g are trimmed. Let now h = cl( f ∨ g ). We have ( f ⊔ g )( x ) < ( f ∨ g )( x ) only if trim( h )( x ) < h ( x ).Since h is closed, we obtain the following inequality by Lemma 7 and the bounds on s ℓ : P [( f ∨ g )( Y ) = 1 and ( f ⊔ g )( Y ) = 0] c X ℓ = c/ (cid:18) kn (cid:19) ℓ |M ℓ ( h ) | c X ℓ = c/ s ℓ = 2 − Ω ( c ) . The same argument shows P [( f ∧ g )( Y ) = 1 and ( f ⊓ g )( Y ) = 0] = 2 − Ω ( c ) . Since there are size( C ) gates in C , this implies that P [ C ( Y ) = 1 and C A ( Y ) = 0] size( C ) · − Ω ( c ) . To bound the approximation errors under N , note that ( f ∨ g )( x ) = 0 and ( f ⊔ g )( x ) = 1 only ifcl( f ∨ g )( x ) = ( f ∨ g )( x ), since trimming a Boolean function cannot decrease the probability that it rejectsan input. Therefore, by Lemma 4 we obtain P [( f ∨ g )( N ) = 0 and ( f ⊔ g )( N ) = 1] n − c − Ω ( c ) . Once again, doing this approximation for every gate in C implies P [ C ( N ) = 0 and C A ( N ) = 1] size( C ) · − Ω ( c ) . This finishes the proof.
Theorem 5.
Any monotone circuit computing f HR has size Ω ( c ) = 2 Ω ( n / / (log n ) ) .Proof. Let C be a monotone circuit computing f HR . For large n , we have9 / P [ f HR ( Y ) = 1] + P [ f HR ( N ) = 0] P [ C ( Y ) = 1 and C A ( Y ) = 0] + P [ C A ( Y ) = 1]+ P [ C ( N ) = 0 and C A ( N ) = 1] + P [ C A ( N ) = 0]= 3 / C )2 − Ω ( c ) . This implies size( C ) = 2 Ω ( c ) . 8 .7 Discussion In this application, we chose the values of c and k to be roughly √ n . We expect that, if c were chosen to becloser to n , the implied Harnik-Raz function would still have 2 Ω ( c ) complexity, and thus one would be ableto improve our bound. However, we do not think that the present technique would work for any c > √ n , asit seems to require that ck n . Therefore, in order to obtain a stronger bound to the Harnik-Raz function,we think a different technique has to be considered. In this section, we give a short and simple proof of a truly exponential (exp( Ω ( n ))) lower bound for realmonotone algebraic circuits computing a multilinear n variate polynomial. As we shall see, the lower boundargument holds for a general family of multilinear polynomials constructed in a very natural way from errorcorrecting codes, and the similarities to the hard function used by Harnik and Raz in the Boolean settingis quite evident (see Section 2.2). In particular, our lower bound just depends on the rate and relativedistance of the underlying code. We note that exponential lower bounds for monotone algebraic circuits arenot new, and have been known since the 80’s with various quantitative bounds. More precisely, Jerrum andSnir proved an exp( Ω ( √ n )) lower bound for an n variate polynomial in [13]. This bound was subsequentlyimproved to a lower bound of exp( Ω ( n )) by Raz and Yehudayoff in [20], via an extremely clever argument,which relied on deep and beautiful results on character sums over finite fields. A similar lower bound ofexp( Ω ( n )) was shown by Srinivasan [23] using more elementary techniques building on a work of Yehudayoff[26]. In a recent personal communication Igor Sergeev pointed out to us that truly exponential lower boundsfor monotone arithmetic circuits had also been proved in the 1980’s in the erstwhile Soviet Union by severalauthors, including the works of Kasim-Zade, Kuznetsov and Gashkov. We refer the reader to [9] for a detaileddiscussion on this line of work.We show a similar lower bound of exp( Ω ( n )) via a simple and short argument, which holds in a some-what general setting. Our contribution is just the simplicity, the (lack of) length of the argument and theobservation that it holds for families of polynomials that can be constructed from any sufficiently good errorcorrecting codes. Definition 4 (From sets of vectors to polynomials).
Let C ⊆ F nq be an arbitrary subset of F nq . Then,the polynomial P C is a multilinear homogeneous polynomial of degree n on qn variables { x i,j : i ∈ [ q ] , j ∈ [ n ] } and is defined as follows: P C = X c ∈ C Y j ∈ [ n ] x j,c ( j ) . Here, c ( j ) is the j th coordinate of c which is an element of F q , which we bijectively identify with the set [ q ] . Here, we will be interested in the polynomial P C when the set C is a good code, i.e it has high rate andhigh relative distance. The following observation summarizes the properties of P C and relations between theproperties of C and P C . Observation 6 (Codes vs Polynomials)
Let C be any subset of F nq and let P C be the polynomial asdefined in Definition 4. Then, the following statements are true: • P C is a multilinear homogeneous polynomial of degree equal to n with every coefficient being either or . • The number of monomials with non-zero coefficients in P C is equal to the cardinality of C . • If any two distinct vectors in C agree on at most k coordinates (i.e. C is a code of distance n − k ), thenthe intersection of the support of any two monomials with non-zero coefficients in P C has size at most k . The observation immediately follows from Definition 4. We note that we will work with monotone algebraiccircuits here, and hence will interpret the polynomial P C as a polynomial over the field of real numbers.We now prove the following theorem, which essentially shows that for every code C with sufficiently gooddistance, any monotone algebraic circuit computing P C must essentially compute it by computing each ofits monomials separately, and taking their sum. 9 heorem 7. If any two distinct vectors in C agree on at most n/ − locations, then any monotone algebraiccircuit for P C has size at least | C | . The proof of this theorem crucially uses the following well known structural lemma about algebraiccircuits. This lemma also plays a crucial role in the other proofs of exponential lower bounds for monotonealgebraic circuits (e.g. [13,20,26,23]).
Lemma 10 (See Lemma 3.3 in [20]).
Let Q be a homogeneous multilinear polynomial polynomial ofdegree d computable by a homogeneous algebraic circuit of size s . Then, there are homogeneous polynomials g , g , g , . . . , g s , h , h , h , . . . , h s of degree at least d/ and at most d/ − such that Q = s X i =0 g i · h i . Moreover, if the circuit for Q is monotone, then each g i and h i is multilinear, variable disjoint and each onetheir non-zero coefficients is a positive real number. We now use this lemma to prove Theorem 7.
Proof of Theorem 7.
Let B be a monotone algebraic circuit for P C of size s . We know from Observation 6that P C is a multilinear homogeneous polynomial of degree equal to n . This along with the monotonicityof B implies that B must be homogeneous and multilinear since there can be no cancellations in B . Thus,from (the moreover part of) Lemma 10 we know that P C has a monotone decomposition of the form P C = s X i =0 g i · h i , where, each g i and h i is multilinear, homogeneous with degree between n/ n/ − g i and h i arevariable disjoint. We now make the following claim. Claim.
Each g i and h i has at most one non-zero monomial.We first observe that the claim immediately implies theorem 7: since every g i and h i has at most onenon-zero monomial, their product g i h i is just a monomial. Thus, the number of summands s needed in thedecomposition above must be equal to the number of monomials in P C , which is equal to | C | from the seconditem in Observation 6.We now prove the Claim. Proof of Claim.
The proof of the claim will be via contradiction. To this end, let us assume that there is an i ∈ { , , , . . . , s } such that g i has at least two distinct monomials with non-zero coefficients and let α and β be two of these monomials. Let γ be a monomial with non-zero coefficient in h i . Since h i is homogeneouswith degree between n/ n/ −
1, we know that the degree of γ is at least n/
3. Since we are in themonotone setting, we also know that each non-zero coefficient in any of the g j and h j is a positive realnumber. Thus, the monomials α · γ and β · γ which have non-zero coefficients in the product g i · h i musthave non-zero coefficient in P C as well (since a monomial once computed cannot be cancelled out). But, thesupports of αγ and βγ overlap on γ which has degree at least n/
3. This contradicts the fact that no twodistinct monomials with non-zero coefficients in P C share a sub-monomial of degree at least n/ C and the third item in Observation 6.Theorem 7 when instantiated with an appropriate choice of the code C , immediately implies an exponentiallower bound on the size of monotone algebraic circuits computing the polynomial P C . Observe that thetotal number of variables in P C is N = qn and therefore, for the lower bound for P C to be of the formexp( Ω ( N )), we would require q , the underlying field size to be a constant. In other words, for any code ofrelative distance at least 2 / heorem 8 ([8] and [24]). Let p be a prime number and let m ∈ N be even. Then, for every < ρ < and a large enough integer n , there exists an explicit rate ρ linear error correcting block code C : F np m → F n/ρp m with distance δ > − ρ − p m/ − . The theorem has the following immediate corollary.
Corollary 1.
For every large enough constant q which is an even power of a prime, and for all largeenough n , there exist explicit construction of codes C ⊆ F nq which have relative distance at least / and | C | > exp( Ω ( n )) . By an explicit construction here, we mean that given a vector v of length n over F q , we can decide indeterministic polynomial time if v ∈ C . In the algebraic complexity literature, a polynomial P is said to beexplicit, if given the exponent vector of a monomial, its coefficient in P can be computed in deterministicpolynomial time. Thus, if a code C is explicit, then the corresponding polynomial P C is also explicit in thesense described above. Therefore, we have the following corollary of Corollary 1 and Theorem 7. Corollary 2.
There exists an explicit family { P n } of homogeneous multilinear polynomials such that forevery large enough n , any monotone algebraic circuit computing the n variate polynomial P n has size at least exp( Ω ( n )) . Acknowledgements
We are grateful to Stasys Junka for bringing the lower bound of Andreev [5] to our attention and to theanonymous referees of LATIN 2020 for numerous helpful suggestions. We also thank Igor Sergeev for bringing[9] and the references therein to our attention which show that truly exponential lower bounds for monotonearithmetic circuits had already been proved in the 1980s.Bruno Pasqualotto Cavalar was supported by S˜ao Paulo Research Foundation (FAPESP), grants
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We include here most of the proof of Theorem 4, which is implicit in [19].
Proof.
In what follows, we suppose B is a large enough universal constant.The proof is by induction on ℓ . Suppose ℓ = 1. Then F is a family of singletons. Therefore, the probabilitythat W ∈ (cid:0) [ n ] m (cid:1) chosen uniformly at random does not contain any set of F is equal to (cid:0) n −|F| m (cid:1) / (cid:0) nm (cid:1) . We get P W [ ∀ F ∈ F : F W ] = (cid:0) n −|F| m (cid:1)(cid:0) nm (cid:1) (cid:18) n − mn (cid:19) |F| (1 − p/ |F| e −|F| p/ ε. Hence, the family F is itself a ( m, ε )-slice-sunflower.We now proceed by induction, supposing ℓ > k -uniform families suchthat k < ℓ .Let r := Bx log x . For any set T ⊆ [ n ], define F T := { F \ T : F ∈ F such that T ⊆ F } .
12e say that F is r-well-spread if |F T | r ℓ −| T | for every non-empty T ⊆ [ n ]. Observe that, if F is not r -well-spread, then there exists a set T ⊆ [ n ] such that |F T | > r ℓ −| T | . Therefore, by the induction hypothesis, F T contains a ( m, ε )-slice-sunflower F ′ T . Observe that the family { U ∪ T : U ∈ F ′ T } ⊆ F is a ( m, ε )-slice-sunflower. Therefore, it suffices to consider the case when F is r -well-spread.For convenience, let S , . . . , S t be the sets of F . Define χ ( S i , W ) to be S j \ W , where j ∈ [ ℓ ] is chosen tominimize | S j \ W | among all choices with S j ⊆ S i ∪ W . If there are many such choices, let j be the smallestone. Note that, for any set S ∈ F , we have χ ( S, W ) = ∅ if and only if there exists F ∈ F such that F ⊆ W .The following key lemma was proved in [19] with a clever coding argument, inspired by the work ofAlweiss, Lovett, Wu and Zhang [3]. Lemma 11 ([19]).
For every non-negative integer s , the following holds. Let F ⊆ (cid:0) [ n ] ℓ (cid:1) be a r -well-spreadfamily for some r > . If S is a uniformly random set of the family, and X ⊆ [ n ] is a uniformly random setof size s · · ⌈ n/r ⌉ sampled independently, then E X , S [ | χ ( S , X ) | ] ℓ · (1 − / log r ) s . We now use Lemma 11 to finish the proof. Let s = ⌈ log( ℓ/ε ) · log r ⌉ . We have s · · ⌈ n/r ⌉ < · log r · log( ℓ/ε ) · n/r = 512 · n · log B + log x + log log xBx log x = 512 · np · log B + log x + log log xB log( ℓ/ε ) log x · np · log BB < m, for B large enough. Therefore, by Lemma 11, we get that E W , S [ | χ ( S , W ) | ] E X , S [ | χ ( S , X ) | ] ℓ · (1 − / log r ) s ℓ · (1 − / log r ) log( ℓ/ε ) · log r ℓe − log( ℓ/ε ) = ε. We can conclude the proof by applying Markov’s inequality, as follows: P W [ ∀ F ∈ F : F W ] = P W , S [ | χ ( S , W ) | > E W , S [ | χ ( S , W ) | ] ε. B Lower Bound for
Clique k,n
Recall that the Boolean function
Clique k,n : { , } ( n ) → { , } receives a graph on n vertices as an input andoutputs a 1 if this graph contains a clique on k vertices. In this section, we prove an n Ω ( δk ) lower bound onthe monotone circuit size of Clique k,n for k = n (1 / − δ .As in Section 2, we will follow the approximation method. However, instead of using sunflowers as in[21,1] or robust sunflowers as in [22], we introduce a notion of robust clique-sunflowers and employ it tobound the errors of approximation. B.1 Test distributions
We denote by G n,p the Erd˝os-R´enyi random graph, in which each edge appears independently with prob-ability p . Furthermore, fix any 2 k = n / − δ where δ > p := n − / ( k − . We observe that theprobability that G n,p has a k -clique is bounded away from 1.13 emma 12. We have P [ G n,p contains a k -clique ] / .Proof. There are (cid:0) nk (cid:1) ( en/k ) k potential k -cliques, each present in G n,p with probability p ( k ) = n − k . By aunion bound, we have P [ G n,p contains a k -clique ] ( e/k ) k ( e/ / A ⊆ [ n ], let K A be the graph on n verticeswith a clique on A and no other edges. Let Y be the uniform random graph chosen from all possible K A .We call Y the positive test distribution . Let also N := G n,p . We call N the negative test distribution . FromLemma 12, we easily obtain the following corollary. Corollary 3.
We have P [ Clique k,n ( Y ) = 1] + P [ Clique k,n ( N ) = 0] > / . B.2 Robust clique-sunflowers
Here we introduce the notion of robust clique-sunflowers , which is analogous to that of robust sunflowers for“clique-shaped” set systems.
Definition 5.
Let ε, p ∈ (0 , . Let S be a family of subsets of [ n ] and let Y := T S . The family S is calleda ( p, ε ) - robust clique-sunflower if P [ ∃ A ∈ S : K A ⊆ G n,p ∪ K Y ] > − ε. Equivalently, the family S is a robust clique-sunflower if the family { K A : A ∈ S} ⊆ (cid:0) [ n ]2 (cid:1) is a ( p, ε ) -robustsunflower, since K A ∩ K B = K A ∩ B . Though clique-sunflowers may seem similar to regular sunflowers, the importance of this definition isthat it allows us to explore the “clique-shaped” structure of the sets of the family, and thus obtain anasymptotically better upper bound on the size of sets that do not contain a robust clique-sunflower.
Lemma 13.
Let S be such that |S| > ℓ !(2 ln(1 /ε )) ℓ (1 /p )( ℓ ) . Then S contains a ( p, ε ) -robust clique-sunflower. Observe that, whereas the bounds for “standard” robust sunflowers (Theorems 2, 3, 4) would give us anexponent of (cid:0) ℓ (cid:1) on the log(1 /ε ) factor, Lemma 13 give us only an ℓ at the exponent. As we shall see, this isasymptotically better for our choice of parameters.We defer the proof of Lemma 13 to Appendix C. The proof is based on an application of Janson’sinequality [12], as in the original robust sunflower lemma of [22] (Theorem 2). We expect that a proof alongthe lines of the work of Alweiss et al [3] and Rao [19] should be able to give us an even better bound,removing the ℓ ! factor. This would extend our n Ω ( k ) lower bound to k n / − o (1) . B.3 A closure operator
As in Section 2.3, we define here a closure operator in the lattice of monotone Boolean functions. We willagain prove that the closure of a function will be a good approximation for it on the negative test distribution.However, unlike Section 2.3, instead of bounding the set of minterms, we will bound the set of “clique-shaped”minterms, as we shall see. Throughout this section, we fix the error parameter ε := n − k . Definition 6.
We say that f ∈ { , } ( n ) → { , } is ε -closed if, for every A ∈ (cid:0) [ n ] k (cid:1) , we have P [ f ( N ∪ K A ) = 1 ] > − ε = ⇒ f ( K A ) = 1 . As before, we can define the closure cl( f ) of a monotone Boolean function f , and bound the error ofapproximating f by cl( f ) under N . 14 emma 14. For every monotone f : { , } ( n ) → { , } , we have P [ f ( N ) = 0 and cl( f )( N ) = 1] ε δk X j =0 (cid:18) nj (cid:19) εn δk n − (2 / k . Proof.
Same as the proof of Lemma 4.Let f : { , } ( n ) → { , } be monotone and suppose ℓ ∈ [ k ]. We define M ℓ ( f ) := { A ∈ (cid:0) [ n ] ℓ (cid:1) : f ( K A ) = 1 and f ( K A \{ a } ) = 0 for all a ∈ A } . Elements of M ℓ ( f ) are called ℓ -clique-minterms of f . By employing the robust clique-sunflower lemma(Lemma 13), we are able to bound the set of ℓ -clique-minterms of closed monotone functions. Lemma 15.
If a monotone function f : { , } ( n ) → { , } is ε -closed, then, for all ℓ ∈ [ k ] , we have |M ℓ ( f ) | (2 ℓ log(1 /ε )) ℓ p ( ℓ ) . Proof.
Same as the proof of Lemma 5.
B.4 Trimmed monotone functions
In this section, we define again a trimming operation for Boolean functions and prove analogous bounds tothat of Section 2.4.
Definition 7.
We say that a function f : { , } ( n ) → { , } is clique-shaped if, for every minterm x of f ,there exists A ⊆ [ n ] such that x = K A (that is, every minterm of f is a clique). Moreover, we say that f is trimmed if f is clique-shaped and all the clique-minterms of f have size at most δk/ . For a set A ∈ (cid:0) [ n ] k (cid:1) , let ⌈ A ⌉ : { , } ( n ) → { , } denote the indicator function of containing K A , whichsatisfies ⌈ A ⌉ ( G ) = 1 ⇐⇒ K A ⊆ G. Functions of the forms ⌈ A ⌉ are called clique-indicators. Moreover, if | A | ℓ , we say that ⌈ A ⌉ is a clique-indicator of size at most ℓ . Let f : { , } ( n ) → { , } be clique-shaped. We definetrim( f ) := δk/ _ ℓ =2 _ A ∈M ℓ ( f ) ⌈ A ⌉ . That is, the trim operation takes out from f all the clique-indicators of size larger than δk , yielding a trimmedfunction.Note that the probability that a random K A sampled from Y contains one of the clique-minterms of size ℓ of a function f is at most (cid:0) n − kk − l (cid:1)(cid:0) nk (cid:1) |M ℓ ( f ) | (cid:18) kn (cid:19) ℓ |M ℓ ( f ) | . Imitating the proofs of Lemmas 6 and 7, we may now obtain the following lemmas.
Lemma 16.
If a monotone function f : { , } ( n ) → { , } is a trimmed clique-shaped function such that f = , then P [ f ( Y ) = 1] δk/ X ℓ =2 (cid:18) kn (cid:19) ℓ |M ℓ ( f ) | . emma 17. Let f : { , } ( n ) → { , } be a clique-shaped monotone function, all of whose clique-mintermshave size at most δk . We have P [ f ( Y ) = 1 and trim( f )( Y ) = 0] δk X ℓ = δk/ (cid:18) kn (cid:19) ℓ |M ℓ ( f ) | . B.5 Approximators
Similarly as in the previous lower bound, we will consider a set of approximators A . Let A := { trim(cl( f )) : f ∈ { , } ( n ) → { , } is monotone and clique-shaped } . We define operations ⊔ , ⊓ : A × A → A as follows:for f, g ∈ A such that f = W ti =1 ⌈ A i ⌉ and g = W si = j ⌈ B j ⌉ , let f ⊔ g := trim(cl( f ∨ g )) ,f ⊓ g := trim(cl( _ i,j ⌈ A i ∪ B j ⌉ )) . For convenience, we let V ( f, g ) := W i,j ⌈ A i ∪ B j ⌉ . Observe that every edge-indicator ⌈{ u, v }⌉ belongs to A . If C is a monotone {∨ , ∧} -circuit, let C A be the corresponding {⊔ , ⊓} -circuit, which computes an approximator. B.6 The lower bound
In this section we finally obtain the desired lower bound for the clique function. We will prove that, if k n / − δ for some constant δ >
0, then the monotone complexity of
Clique k,n is n Ω ( k ) . Henceforth, we willsuppose that this is the case. We begin by defining, for every 2 ℓ δk , the number s ℓ := (cid:18) kn (cid:19) ℓ (2 ℓ log(1 /ε )) ℓ p ( ℓ ) = (2 ℓk log n ) ℓ n ℓ p ( ℓ ) . By Lemma 15, we get that, for every ε -closed monotone function f : { , } ( n ) → { , } , we have (cid:18) kn (cid:19) ℓ |M ℓ ( f ) | s ℓ . As seen in Section 2.6, it will be important for us to upper bound the values of s and s ℓ +1 /s ℓ for all2 ℓ < δk , which we now do: s = (cid:18) kn (cid:19) (12 k log n ) p = (cid:18) O ( k log n ) n − (1 / ( k − (cid:19) (cid:18) O (log n ) n (1 / δ (cid:19) = o (cid:18) n / (cid:19) ,s ℓ +1 /s ℓ = 6 k log nn · ( ℓ + 1) ℓ +1 ( ℓp ) ℓ O ( k log n ) np ℓ O (log n ) n (1 / δ − (2 ℓ/ ( k − O (log n ) n (1 / δ o (cid:18) n / (cid:19) . It follows that s ℓ O ( n − ℓ/ ) for all 2 ℓ δk .Repeating the same arguments of Lemmas 8 and 9, we obtain the following analogous lemmas. Lemma 18 (Approximators make many errors).
For every f ∈ A , we have P [ f ( Y ) = 1] + P [ f ( N ) = 0] o (1) . Proof.
Let f ∈ A . By definition, there exists an ε -closed function h such that f = trim( h ). Observe that M ℓ ( f ) ⊆ M ℓ ( h ) for every ℓ ∈ [ c ]. By Lemma 16, if f ∈ A such that f = , then P [ f ( Y ) = 1] δk/ X ℓ =1 (cid:18) kn (cid:19) ℓ |M ℓ ( h ) | δk/ X ℓ =1 s ℓ δk/ X ℓ =1 O ( n − ℓ/ ) o (1) . Therefore, for every f ∈ A we have P [ f ( Y ) = 1] + P [ f ( N ) = 0] o (1) . emma 19 ( C is well-approximated by C A ). Let C be a monotone circuit. We have P [ C ( Y ) = 1 and C A ( Y ) = 0] + P [ C ( N ) = 0 and C A ( N ) = 1] size( C ) · O ( n − δk/ ) . Proof.
To bound the approximation errors under the distribution Y , first note that, if f, g ∈ A , then all theclique-minterms of both f ∨ g and f ∧ g have Hamming weight at most δk . Moreover, if ( f ∨ g )( x ) = 1 but( f ⊔ g )( x ) = 0, then trim(cl( f ∨ g )( x )) = cl( f ∨ g )( x ). Therefore, we obtain by Lemma 17 that, for f, g ∈ A ,we have P [( f ∨ g )( Y ) = 1 and ( f ⊔ g )( Y ) = 0] δk X ℓ = δk/ (cid:18) kn (cid:19) ℓ |M ℓ ( f ∨ g ) | δk X ℓ = δk/ s ℓ δk X ℓ = δk/ O ( n − ℓ/ ) = O ( n − δk/ ) . The same argument shows P [( f ∧ g )( Y ) = 1 and ( f ⊓ g )( Y ) = 0] = O ( n − δk/ ) , which implies P [ C ( Y ) = 1 and C A ( Y ) = 0] size( C ) · O ( n − δk/ ) . Similarly, to bound the approximation errors under N , note that ( f ∨ g )( x ) = 0 and ( f ⊔ g )( x ) = 1 onlyif cl( f ∨ g )( x ) = ( f ∨ g )( x ). Therefore, we obtain by Lemma 14 that, for f, g ∈ A , we have P [( f ∨ g )( N ) = 0 and ( f ⊔ g )( N ) = 1] n − (2 / k . By the same argument above, we obtain P [ C ( N ) = 0 and C A ( N ) = 1] size( C ) · n − (2 / k size( C ) . This finishes the proof.We can finally obtain the lower bound for the clique function.
Theorem 9.
For all k = n / − δ where < δ < / , the monotone circuit complexity of Clique k,n is Ω ( n δk/ ) .Proof. Let C be a monotone circuit computing Clique k,n . We have5 / P [ Clique k,n ( Y )] + P [ Clique k,n ( N )] P [ C ( Y ) = 1 and C A ( Y ) = 0] + P [ C A ( Y ) = 1]+ P [ C ( N ) = 0 and C A ( N ) = 1] + P [ C A ( N ) = 1] o (1) + size( C ) · O ( n − δk/ ) . This implies size( C ) = Ω ( n δk/ ). C Proof of Lemma 13 (Robust clique-sunflower)
Let U n,q ⊆ [ n ] be a q -random subset of [ n ] (independent of G n,p ). Let c := ln(1 /ε ) and for ℓ >
2, let c ℓ := 2 ln(1 /ε ) P ℓ − j =1 (cid:0) ℓj (cid:1) c j . The following can be easily checked. Lemma 20. c ℓ ℓ !(2 log(1 /ε )) ℓ . It follows from the definition of robust clique-sunflowers that the robust clique-sunflower lemma (Lemma 13)is implied by the following result.
Lemma 21.
For all ℓ ∈ { , . . . , n } and S ⊆ (cid:0) [ n ] ℓ (cid:1) , if | S | > c ℓ (1 /q ) ℓ (1 /p )( ℓ ) , then there exists B ∈ (cid:0) [ n ] <ℓ (cid:1) suchthat P [ ^ A ∈ S : B ⊆ A ( K A * G n,p ∪ K B or A * U n,q ∪ B ) ] ε. roof. By induction on ℓ . In the base case ℓ = 1, we have B = ∅ and (by independence) P [ ^ A ∈ S ( K A * G n,p or A * U n,q ) ] = P [ ^ A ∈ S ( A * U n,q ) ]= Y A ∈ S P [ A * U n,q ]= (1 − q ) | S | (1 − q ) ln(1 /ε ) /q e − ln(1 /ε ) = ε. Let ℓ >
2. First, consider the case that there exists j ∈ { , . . . , ℓ − } and B ∈ (cid:0) [ n ] j (cid:1) such that |{ A ∈ S : B ⊆ A }| > c ℓ − j (1 /qp j ) ℓ − j (1 /p )( ℓ − j ) . Let T = { A \ B : A ∈ S such that B ⊆ A } ⊆ (cid:0) [ n ] ℓ − j (cid:1) . By the induction hypothesis, there exists D ∈ (cid:0) [ n ] \ B<ℓ − j (cid:1) such that P [ ^ C ∈ T : D ⊆ C ( K C * G n,p ∪ K D or C * U n,qp j ∪ D ) ] ε. We have P [ ^ A ∈ S : B ∪ D ⊆ A ( K A * G n,p ∪ K B ∪ D or A * U n,q ∪ B ∪ D ) ]= P [ ^ C ∈ T : D ⊆ C ( K B ∪ C * G n,p ∪ K B ∪ D or B ∪ C * U n,q ∪ B ∪ D ) ]= P [ ^ C ∈ T : D ⊆ C ( K B ∪ C * G n,p ∪ K B ∪ D or C * U n,q ∪ D ) ]= P [ ^ C ∈ T : D ⊆ C ( K C * G n,p ∪ K D or C * (cid:8) v ∈ U n,q : { v, w } ∈ E ( G n,p ) for all w ∈ B (cid:9) ∪ D ] P [ ^ C ∈ T : D ⊆ C ( K C * G n,p ∪ K D or C * U n,qp j ∪ D ] ε. Finally, assume that for all j ∈ { , . . . , ℓ − } and B ∈ (cid:0) [ n ] j (cid:1) , we have |{ A ∈ S : B ⊆ A }| c ℓ − j (1 /qp j ) ℓ − j (1 /p )( ℓ − j ) . In this case, we show that the bound of the lemma holds with B = ∅ . Let µ := | S | q ℓ p ( ℓ ) ,∆ := ℓ − X j =1 X ( A,A ′ ) ∈ S : | A ∩ A ′ | = j q ℓ − j p ( ℓ ) − ( j ) . Janson’s Inequality [12] gives the following bound:(1) P [ ^ A ∈ S ( K A * G n,p or A * U n,q ) ] exp (cid:18) − µ µ + ∆ (cid:19) .
18e bound ∆ as follows: ∆ ℓ − X j =1 q ℓ − j p ( ℓ ) − ( j ) X B ∈ ( [ n ] j ) |{ A ∈ S : B ⊆ A }| ℓ − X j =1 q ℓ − j p ( ℓ ) − ( j ) X B ∈ ( [ n ] j ) |{ A ∈ S : B ⊆ A }| · c ℓ − j (1 /q ) ℓ − j (1 /p )( ℓ − j )= q ℓ p ( ℓ ) ℓ − X j =1 c ℓ − j X B ∈ ( [ n ] j ) |{ A ∈ S : B ⊆ A }| = q ℓ p ( ℓ ) ℓ − X j =1 c ℓ − j X A ∈ S X B ∈ ( Aj ) 1= | S | q ℓ p ( ℓ ) ℓ − X j =1 (cid:18) ℓj (cid:19) c ℓ − j = µ ℓ − X j =1 (cid:18) ℓj (cid:19) c j . We next observe that µ > c ℓ , since | S | > c ℓ (1 /q ) ℓ (1 /p )( ℓ ). Therefore, µ ∆ > µ P ℓ − j =1 (cid:0) ℓj (cid:1) c ℓ − j = | S | q ℓ p ( ℓ )2 P ℓ − j =1 (cid:0) ℓj (cid:1) c ℓ − j > c ℓ P ℓ − j =1 (cid:0) ℓj (cid:1) c ℓ − j = ln(1 /ε ) . Finally, from (1) we get P [ ^ A ∈ S ( K A * G n,p or A * U n,q ) ] exp (cid:18) − µ ∆ (cid:19) ε.ε.