2 Hatad, Jukna & Pudlak gate, namely we hall tudy the ize of depth-three circuit. The technique we hall ue ha two ource. The rt one i a \nite" verion

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1 TOP-DOWN LOWER BOUNDS FOR DEPTH-THREE CIRCUITS J. Hatad, S. Jukna and P. Pudlak Abtract. We preent a top-down lower bound method for depth-three ^ _ :-circuit which i impler than the previou method and in ome cae give better lower bound. In particular, we prove that depth-three ^ _ :-circuit that compute parity (or majority) require ize at leat 2 0:618:::pn (or 2 0:849:::pn, repectively). Thi i the rt imple proof of a trong lower bound by a top-down argument for non-monotone circuit. Key word. Computational complexity mall-depth circuit. Subject claication. 68Q25. Warning: Eentially thi paper ha been publihed in Computational Complexity and i hence ubject to copyright retriction. It i for peronal ue only. 1. Introduction To prove lower bound in variou computational model i till one of the major challenge in complexity theory. In pite of ome recent progre, there are till no trong lower bound for general Boolean circuit. Even wore, there i not even any well dened line of attack of thi problem where the hope of progre i ubtantial and well founded. In view of thi ituation, it i crucial to get a better undertanding of exiting technique for proving lower bound and in particular, to undertand exactly in what ituation a particular technique might be ued and how the bound obtained by one method relate to bound obtained by another method. In thi paper, we will be concerned with a pecial cae of the general problem about the tradeo between the ize and depth of circuit with ^ _ :

2 2 Hatad, Jukna & Pudlak gate, namely we hall tudy the ize of depth-three circuit. The technique we hall ue ha two ource. The rt one i a \nite" verion of the topological approach propoed by Siper (1985). Given a hallow circuit for a boolean function f the idea i to combine rejecting computation on input in f ;1 (0) into an incorrect rejecting computation on an input in f ;1 (1): Thi incorrect rejecting computation i obtained a a \limit" of correct one. The econd one, introduced by Karchmer and Wigderon (1990), i baed on the fact that the circuit depth i equivalent to the number of bit needed to be exchanged to olve a particular combinatorial game. Both proof technique correpond in a natural way to a top-down argument for circuit. Such topdown argument have been uccefully applied to uual (fan-in 2) monotone circuit (Karchmer & Wigderon 1990, Raz & Wigderon 1990) and to bounded depth (unbounded fan-in) monotone circuit (Klawe et al. 1984). Here, we hall apply uch an argument to nonmonotone bounded depth circuit. There have been a number of reult for mall-depth circuit (Ajtai 1983, Furt et al. 1984, Yao 1985, Hatad 1989, Razborov 1987, Smolenky 1987), where uperpolynomial and exponential lower bound on circuit ize have been proved for imple Boolean function like parity and majority. Nontrivial reult have been obtained for circuit of depth up to (log n= log log n) (Hatad 1989, Razborov 1987, Smolenky 1987). All thoe paper have uedeentially bottom-up argument, i.e., tarting at the input and analyzing the circuit level by level. When witching to a top-down argument, we improve the reult for circuit of depth three. The bound we get are tronger (only the contant change in mot cae, but ome of the old reult were tight up to the value of the contant), and the argument i imple in that it only ue more or le tandard combinatoric. Our reult are only the rt tep in thi direction. It would be of great value if the top-down approach could be extended to greater depth, or if one could prove alower bound larger than 2 contp n for depth three. The paper i organized a follow: In Section 2, we etablih the baic connection between circuit and the combinatorial problem we analyze. Our preentation will be baed on limit, but the reader hould bear in mind that there i an equivalent game-theoretical interpretation. In Section 3, we prove lower bound for depth-three circuit computing parity and majority. Thee bound give better value for the contant in the exponent than bound obtained by previou method. In Section 4, we improve the bound on the cot of witching from a 3 -circuit to a 3 -circuit. Thee bound contitute an improvement alo in the aymptotic dependence. We end in Section 5 with a number of open problem.

3 Lower bound for depth-three circuit 3 A` k d 2. Circuit and limit -circuit (rep. a ` k d -circuit) i a depth-d unbounded fan-in circuit C over f^ _g of ize ` with the top gate ^ (rep. _) and with bottom fan-in bounded by k, i.e., each gate next to the bottom ha at mot k input. A uual, we aume that literal (input or negated input) are at the bottom and that each level conit of the ame type of gate. We alo ay that C i a ` k+ d -circuit if each gate next to the bottom ha at mot k negated input (the total number of input to the gate may ben). We will be particularly intereted in 3 -circuit. (Note that both parity and majority are elfdual if the number of input n i odd, thu our lower bound hold alo for 3 -circuit.) The behavior of uch circuit can be decribed in purely combinatorial term uing the following notion of \limit vector" introduced by Siper (1985, 1991). Our modication \lower limit" i a technical concept which enable u to get better contant in lower bound. We hall ue the following notation: [n] =f1 ::: ng and [n] k = fs [n] : jsj = kg: Alo, we let xj S denote the retriction of x to the et S for vector x 2 f0 1g n and S [n]. Definition 2.1. Let B f0 1g n be a et of vector. A vector y 2f0 1g n i a k-limit for a et B if, for any ubet of indice S 2 [n] k, there exit avector x 2 B uch that x 6= y and yj S = xj S : If x > y intead of x 6= y, we call y a lower k-limit for B. We ay that the pair (A B) of ubet of f0 1g n ha the property P (k `) if, for any coloring of B by ` color, there i a color cla B 0 B uch that the et A contain at leat one k-limit for B 0 : If the ame hold with \k-limit" replaced by \lower k-limit", then we ay that (A B) ha the property P + (k `): We alo ay that a circuit C eparate the pair (A B) ifc compute 1 (rep. 0) on input from A (rep. B). The following lemma i eentially due to Siper. Lemma 2.2. Let A B f0 1g n, A\B =. If the pair (A B) ha the property P (k `), then it cannot be eparated by a ` k 3 -circuit. If, moreover, the pair ha the property P + (k `), then it alo cannot be eparated by a ` k+ 3 -circuit. Proof. Let C be a ` k 3 -circuit eparating (A B): Then, the _-gate g i (i `) feeding into the top gate of C eparate pair (A B i ) o that [ì=1b i = B:

4 4 Hatad, Jukna & Pudlak Suppoe the pair (A B) ha the property P (k `): Then, for ome i 0 `, the et A contain a vector y which i a k-limit for the et B i0 : The gate g i0 mut reject all x 2 B i0 : We how that in thi cae g i0, and hence the whole circuit C i forced to incorrectly reject the limit y: Let F be the family coniting of the et of indice of input to the ^-gate feeding into g i0 : Since g i0 i an _-gate, we know that all thee ^-gate compute 0 on all x 2 B i0 : Since et in F are of cardinality at mot k and y i a k-limit for B i0 we alo have that on each S 2 F, the vector y coincide with at leat one vector x S 2 B i0 : Therefore, every ^-gate feeding into g i0 mut compute 0 on y alo, and thu, g i0 (y) =0 a contradiction. To prove the econd claim, it i enough to take F to be the family coniting of the et of indice of negated input, and ue the additional property x S y: Take an^-gate h feeding into g i0, and let S be the correponding et of negated input to h: We knowthaty coincide on thee input with ome vector x S for which h(x S )=0: If ome negated variable feeding in h compute 0 on x S, then it doe the ame on y and hence, h(y) = 0: Otherwie, the 0 i produced on x S by ome not negated variable. Since y x S thi variable mut produce 0 on y alo, and hence, h(y) =0: 2 Now we can explain our method. Suppoe a function f can be computed by a 3 -circuit C: 1. Firt, we apply a retriction to reduce the bottom fan-in of the circuit C: 2. Then, we prove the exitence of a limit in A = fx : f(x) =1g for every uciently large ubet of B = fx : f(x) =0g and apply Lemma 2.2. The reduction of the bottom fan-in i quite tandard. For now, let u point out that it i not neceary to ue random retriction (ee Lemma 3.2), and that both the retriction and the limit can be found by determinitic method. Thu, the whole method avoid randomne. It turn out that there i a claical reult in combinatoric which can be ued to prove the exitence of a (lower) limit. It i convenient nowtowitchto et-theoretical language, namely, from now on we look at a vector x 2f0 1g n a the correponding ubet X = fi : x i =1g of [n]: A cover of a family of et F i a et which interect every member of F: The minimum cardinality ofa cover i denoted by (F): For a et Y and a family of et F let F Y = fx n Y : X 2F X Y g and let F4Y denote the family of all ymmetric dierence (X n Y ) [ (Y n X) for X 2 F: In thee term, a et Y i a k-limit for F i (F 4 Y ) k +1, and alower k-limit i (F Y ) k +1.

5 Lower bound for depth-three circuit 5 Recall that a family of et X 1 ::: X k+1 i a unower with (k +1) petal and core Y if, for every i 6= j X i \ X j = Y: Clearly, iff ha uch a unower, then (F Y ) k i.e., the core Y i alower (k ; 1)-limit for F: Theorem 2.3. (Erdo & Rado 1960) Let F be a family with more than (k ; 1) et of cardinality. Then F contain a unower with k petal. Thi theorem can be directly applied to get a bound of 2 (n1=3 ) for the majority function. It proof can be eaily modied to get the following theorem which give the ame bound for the parity function. Theorem 2.4. Let 2 be an even (rep. odd) integer and let F be a family with more than n =2 k =2 13(;1) (rep. more than n (;1)=2 k (+1)=2 13 ) 24 24(;1) et of cardinality : Then F contain a unower with k +1petal and with an odd (rep. even) core. In order to get bound cloer to the optimal one, one need to conider limit. The proof of the exitence of limit given below i very imilar to the proof of the above Theorem 2.3 and 2.4. Therefore, we leave out the proof of Theorem The lower bound for parity and majority Let F (n k ) denote the function dened by 8 < n =2 k =2 24 F (n k ) = : n (;1)=2 k (+1)=2 24(;1) if i even if i odd: Lemma 3.1. Let F be a family of -element ubet of [n], 2, and uppoe that jfj > F (n k ): Then, there exit a lower k-limit Y [n] for F uch that jy j +1mod2: Proof. Uing the dicuion in the previou ection, we jut need to nd a et Y of the deired parity uch that (F Y ) k +1: The bai = 2 i trivial. In thi cae, F i the ordinary graph with more than F (n k 2) = kn=2 edge, and hence, it contain a vertex of degree at leat k +1:

6 6 Hatad, Jukna & Pudlak Suppoe now that the lemma i true for and prove it for +1: Take a family F of (+1)-element ubet of [n] with jfj >F(n k +1). For x 2 [n] let F x = fx : x 2 X 2Fg: Cae 1: +1 iodd ( i even). If (F) k +1,we are done ince we can take Y = (which i even). Otherwie, there exit an x 2 [n] for which the following relation hold: jf x j jfj k > F (n k +1) k = 1 k n=2 k (+2)=2 2 4 = F (n k ): The family F x conit of even et and by induction, ((F x ) Y 0) k +1for ome odd Y 0 : Thu, (F Y ) k +1where Y = Y 0 [fxg i even. Cae 2: +1 i even ( i odd). There exit an x 2 [n] which icontained in at leat ( +1)jFj=n et of F: For thi x, we have the following relation: jf x j > ( +1)F (n k +1) n = ( +1) n (+1)=2 k (+1)=2 = F (n k ): n 2 4 ( ; 1)( +1) The family F x conit of odd et and by induction, ((F x ) Y 0) k +1 for ome even Y 0 : Thu, (F Y ) k +1where Y = Y 0 [fxg i odd. 2 The following lemma will be ued to reduce the bottom fan-in. Lemma 3.2. Let k ` be poitive integer. Let F be a family of ` ubet of [n] each of cardinality more than k: Suppoe that the following inequality hold: `< n +1 k : (3.1) m +1 Then, there exit a ubet T [n] uch that jt j n ; m and T interect every et in F: Proof. We contruct the et T via the following \greedy" procedure. Let F 1 = F: For each i, 1 i n ; m, include in T the element x i 2 [n] which occur in the larget number of et of F i, then remove all the et containing x i from F i to obtain F i+1. Set deleted after i tep interect the et fx 1 ::: x i g: The ize of F i i bounded from above by jf 1 j 1 ; k n 1 ; k 1 ; n ; 1 k n ; i +2 :

7 Lower bound for depth-three circuit 7 We need to how that the bound i le than 1 for i = n ; m +1: Thi follow from the following etimate 1 ; k 1 ; k k 1 ; n n ; 1 n ; (n ; m +1)+2 = 1 ; k 1 ; k 1 ; k n n ; 1 m +1 e ; n k ; k n;1 ;:::; k m+1 e ;k(ln(n+1);ln(m+1)) n +1 ;k = : 2 m +1 Theorem 3.3. Any depth-three circuit computing the parity of n variable ha ize at leat 2 cp n;o( p n) where c =1=( p 2e ln 2) = 0:618 ::: : Proof. Let ` be the minimal ize of a depth-three circuit computing the parity of n variable. W.l.o.g., we can aume that it i a 3 -circuit. We rt ue Lemma 3.2 to reduce the bottom fan-in. Say that an ^-gate on the bottom i bad if it ha more than k negated input. Let F be the family of et of indice of negated input to bad ^-gate. Thi family ha no more than ` et. By Lemma 3.2, for every m atifying the inequality (3.1), there exit a ubet T of n ; m indice which interect every et in F: Thu, the aignment of the contant 1 to all the variable with indice in T evaluate all bad gate to 0. The remaining gate are good, i.e., each ha no more than k negated input. Therefore, for any k atifying the following inequality: k ln ` ln ((n +1)=(m +1)) (3.2) there i a ` k+ 3 -circuit C which compute the parity of m variable. Let 2 m=2 be an even integer (to be pecied later). The circuit C mut, in particular, eparate the pair (A B), where A f0 1g m i the et of all odd vector and B i the et of all vector with exactly one. By Lemma 2.2, the pair (A B) doenothave the property P + (k `): Thi, in particular, mean that A contain no lower k-limit for an `;1 fraction of B: By Lemma 3.1, thi fraction cannot be larger than F (m k ): Therefore, for any k atifying (3.2), the ize ` mut atify the inequality ` jbj F (m k ) = m 2 =2 : (3.3) mk 2

8 8 Hatad, Jukna & Pudlak So, the deired lower bound for ` can be obtained by an appropriate choice of the parameter m and. The optimal bound ln ` = ( p n) i obtained for k = ( p n) and m = (n) and the computation i quite imple. However, we want to compute the contant explicitly therefore, we need to compute more preciely. In particular, we mut chooe the contant for the parameter k m: We hall ue the following etimate (for m=2): m me where = 1 p e ;2 =m 4 which can be eaily derived uing Stirling' formula n =n n e ;np 2n e n where 1=(12n +1)< n < 1=12n: Uing the etimate for m, the inequality (3.3) give the following bound ` p p 4 e 2 =m em 2e =2 ;=2 mk 2 1 em =2 e : 2 =m+1 k Taking logarithm, we obtain ln ` em ln 2 k ; 2 m ; 1= 2 ln em k ; O(1) ince = ( p n) and m = (n). The function ln (a=) attain it maximum for = a=e hence we take = m=k which give ln ` m=(2k) ; O(1) and, by (3.2), we have the following inequalitie: ln ` (ln `) 2 ln ` m ln ((n +1)=(m + 1)) m ln (n=m) ; O(1) ; O(1) 2ln` 2ln` m ln (n=m) m ln (n=m) ; O(ln `) = ; o(n) 2 2 m=n ln (n=m) p n ; o( p n): 2 The right hand ide ha a maximum for m=n = e ;1 : Hence, ` 2 cp n;o( p n) where c =1=( p 2e ln 2). 2 In cae of the majority function, we ue the following bound for the exitence of limit.

9 Lower bound for depth-three circuit 9 Lemma 3.4. Let F be a family of -element ubet of [n], 1: If jfj >k, then there exit a lower k-limit Y for F uch that jy j ; 1: Proof. The ame a that of Lemma 3.1 in Cae 1 uing k intead of F (m k ): The bai =1i trivial. Suppoe now that the lemma i true for and prove it for +1: Take a family F of ( + 1)-element ubet of [n] with jfj >k +1. If (F) k +1,we are done ince we can take Y = : Otherwie, there exit an x 2 [n] for which jf x j jfj=k > k and we can apply the induction hypothei. 2 Theorem 3.5. Any depth-three circuit computing the majority function ha ize at leat 2 dp n;o( p n) where d =1=p ln 4 = 0:849 ::: : Proof. Let ` be the minimal ize of a depth-three circuit computing :MAJ n the negation of majority (and hence, the minimal ize of a depth-three circuit computing MAJ n itelf). Since :MAJ n i elfdual (i.e., complementing the output and all input doe not change the function), we can w.l.o.g. aume that we have a 3 -circuit. The argument i imilar to that of Theorem 3.3 hence, let u only decribe how to modify that proof. Set m = n=2+ with n=2 and reduce the bottom fan-in uing Lemma 3.2, i.e., we now have m remaining variable and the bottom fan-in i bounded by k, where k i the mallet integer that atie inequality (3.1). Now our circuit C mut eparate the pair (A B), where A f0 1g m i the et of all vector with at mot ; 1 one and B i the et of all vector with exactly one. By Lemma 2.2 and 3.4, ` m k ; = 1 em p : (3.4) k Now, if we chooe = m=k, then (3.4) give ` = (e = p ) : However, we need to fulll (3.1) and hence we want to make ure that e n+1 m+1 k : If we take = k ln(2 ; ) with = O(n ;1=2 ) then ince = m=k we obtain k = q m= ln(2 ; ) which give the bound ` 2 dp n;o( p n), where d = q 1= ln 4 i approximately 0:849 ::: : 2 For the threhold function :T n (1;e ;1 )n, one can get the lower bound 2dp n;o( p n) with a lightly better contant d =1=( p e ln 2) = 0:875::: :

10 10 Hatad, Jukna & Pudlak 4. A lower bound for a function computable by a mall 3 -circuit In thi ection, we prove an optimal lower bound for a function computable by a mall 3 -circuit. Let S m be the boolean function with n = 2m variable dened a follow: _ m^ S m (x y) = (x i j _ y i j ): i=1 j=1 We hall how that thi function require 3 -circuit of ize 2 (p n) while it ha a 3 -circuit of ize O(n): By a reult of Klawe et al. (1984), thi function ha 3 -circuit of ize 2 O(p n) thu the bound i optimal (up to the contant in the exponent). Lemma 4.1. If f = W V m i=1 j=1 x i j following inequality hold: i computed by a ` k+ 3 -circuit, then the m ` : k Proof. Aume to the contrary that ` < (m=k) : Any ` k+ 3 -circuit for f mut eparate the pair (A B), where A f0 1g m i the et of all vector with at mot ; 1 one and B i the et of m vector with exactly one killing all ^0 inf. By Lemma 2.2, thi pair (A B) doenot have the property P + (k `). Thi, in particular, mean that there exit a ubet B 0 B uch that jb 0 jjbj=` = m =` > k and no vector in A i a lower k-limt for B 0 a contradiction with Lemma Theorem 4.2. Any 3 -circuit computing S p n=2 pn=2 with c =0:453 ::: : p ha ize at leat n 2c Proof. In order to apply Lemma 4.1, we need only decreae the fan-in on the rt level. In particular, we need to make ure that no gate on the rt level ha more than k negated input. The mot natural way to do thi i to randomly x one variable from each pair x i j y i j to 1. If an ^-gate contain both x i j and y i j negatively for ome i j, then it i alway reduced to 0. Otherwie, an ^-gate with more than k negated input i reduced to 0 with probability at leat 1 ; 2 ;k;1. Thi mean that if k p c n, then with probability atleat a

11 Lower bound for depth-three circuit 11 half, all gate that originally had at leat k + 1 negated input will be reduced to 0, and in particular, uch a retriction exit. Let u note that we can avoid randomne alo here, uing a tandard trick. Namely, to each gate g with a>knegated input which do not contain both x i j and y i j negatively for ome i j, aign a weight w(g) =2 ;a. Note that we can diregard the gate which originally have no more than k negated input and thoe gate which contain both x i j and y i j negatively for ome i j. The former are allowed to remain and the latter are alway reduced to 0. We determine the retriction piece by piece and at each point we let the weight of a gate be 0 if it ha already been reduced to 0 and 2 ;b otherwie, where b i the number of negated variable that come from pair where we o far have not determined which variable to x. In other word, the weight i the probability that the gate will not be reduced to 0 if we make random choice in the future. We now determine the value of the retriction on the pair x i j y i j by calculating the total weight in the cae when we et x i j to 1 and y i j to 1, repectively. By denition, the average of thee two number i the current total weight, and hence, one of the alternative will give at mot the ame weight. We x thi choice and then continue with the next pair. The aignment contructed in thi way give a nal weight which i at mot one half and ince the retriction i completely determined, it mut be 0 hence, we have reduced all the gate in quetion to 0. Uing Lemma 4.1, we thu get a lower bound for the ize of a 3 -circuit computing S m : Chooing = m = of the theorem. 2 ` min 2 k m : (4.1) k q m=2 and k = c p n with c =0:453 ::: give the bound Note p that the previou reult (Hatad 1989) gave only a lower bound log n) for the ize of 3 -circuit computing a function which ha a 4-2 (n1=6 = circuit of ize O(n): 5. Concluion and open problem The combinatorial technique that we have ued are very imple. Therefore, we hope that by applying more complicated argument, it will be poible to get

12 12 Hatad, Jukna & Pudlak ubtantially more. We hall lit ome problem which we conider important and give motivation for them. Problem 1. Prove a uperpolynomial lower bound for depth larger than three uing a top-down argument. A top-down approach ha been uccefully applied in the cae of monotone circuit (Klawe et al. 1984). What we do for depth three i quite imilar. Therefore, it i poible that our argument can be extended to larger depth. Problem 2. Prove alower bound 2 (n ) with >1=2 for depth-three circuit. More generally, prove uch a bound with > 1=(d ; 1) for depth-d circuit. Such bound would give nonlinear lower bound for formula ize uing the reduction of Klawe et al. (1984). In fact, if a function ha a lower bound 2 (n ) with axed>0 for all depth d then it i not in NC 1 : Problem 3. Prove a bound for ` k 3 -circuit with k log ` = (n): Uing the above technique, we can prove a bound larger than 2 (pn) but the product k log ` i alway O(n): Improving it i intereting, becaue if one (n= log log n) could eventually prove a lower bound ` = 2 for k n where i an arbitrary mall poitive contant, then we would have a nonlinear lower bound for depth O(log n) circuit (with fan-in 2) by a reult of Valiant (1977). Another reaon why uch an improvementwould be intereting i the poibility to prove non-trivial pace-time trade-o. Take a non-determinitic Turing machine computing f in time T and pace S: Uing the idea of Theorem 1in (Borodin et al. 1993), one can prove the following: if f ha the property P (k `), then S T =(klog `): Detail can be found in (Jukna 1994). Problem 4. Determine the aymptotical complexity of depth-three circuit for majority. p The bet upper bound for the majority function i 2 O n log n uing monotone circuit (ee Klawe et al. 1984). For k n, the bound in Lemma 3.4 i optimal: take mutually dijoint ubet of [n] A 1 ::: A, each of cardinality k, and dene F = fx [n] : jx \ A i j =1 8ig : Then jfj = k but (F Y ) k

13 Lower bound for depth-three circuit 13 for any ety [n], jy j ; 1: We do not know the optimal value for k > n: It i poible that for uch parameter, one can obtain a bound larger than 2 (pn) : It eem that what i needed i the following. Recall the property P introduced in Section 2. In the above lower bound we alway took the color cla B 0 with the larget cardinality and looked for a limit there. In many cae, thi may be not the bet choice. We need an argument which ue the whole partition, not jut one cla. Acknowledgement We would like to thank Ingo Wegener for being an initial link of communication between u which reulted in the preent work. We would alo like to thank an anonymou referee for many helpful comment. The econd and third author acknowledge upport from the Alexander von Humboldt Foundation. The econd author alo acknowledge upport from DFG grant Me 1077/5-2. Reference M. Ajtai, 1 1-formulae on nite tructure. Ann. Pure and Appl. Logic 24 (1983), 1{48. A. Borodin, A. Razborov and R. Smolenky,On lower bound for read-k-time branching program. Computational Complexity 3 (1993), 1{18. P. Erdo and R. Rado, Interection theorem for ytem of et. J. London Math. Soc. 35 (1960), 85{90. M. Furt, J. Saxe and M. Siper, Parity, circuit and the polynomial time hierarchy. Math. Sytem Theory 17 (1984), 13{27. J. Hatad, Almot Optimal Lower Bound for Small Depth Circuit. Advance in Computing Reearch, ed. S. Micali, Vol 5 (1989), 143{170. S. Jukna, Finite limit and lower bound for circuit ize. Tech. Rep. 94{06, Informatik, Univerity oftrier, M. Karchmer and A. Wigderon, Monotone circuit for connectivity require uper-logarithmic depth. SIAM J. Dic. Math. 3 (1990), 255{265.

14 14 Hatad, Jukna & Pudlak M. Klawe, W.J. Paul, N. Pippenger, M. Yannakaki, On monotone formulae with retricted depth. In Proc. Sixteenth Ann. ACM Symp. Theor. Comput., 1984, 480{487. R. Raz and A. Wigderon, Monotone circuit for matching require linear depth. In Proc. Twenty-econd Ann. ACM Symp. Theor. Comput., 1990, 287{292. A. A. Razborov, Lower bound for the ize of circuit of bounded depth with bai f^ g: Math. Note of the Academy of Science of the USSR 41:4 (1987), 333{338. M. Siper, Private communication, M. Siper, A topological view of ome problem in complexity theory. In Colloq. Math. Soc. Jano Bolyai 44 (1985), 387{391. R. Smolenky, Algebraic method in the theory of lower bound for Boolean circuit complexity. In Proc. Nineteenth Ann. ACM Symp. Theor. Comput., 1987, 77{82. L.G. Valiant, Graph-theoretic argument in low level complexity. In Proc. Sixth Conf. Math. Foundation of Computer Science, Lecture Note in Computer Science, 1977, Springer-Verlag, 162{176. A.C. Yao, Separating the polynomial time hierarchy by oracle. In Proc. Twentyixth Ann. IEEE Symp. Found. Comput. Sci., 1985, 1{10. Manucript received October 24, 1993 J. Hatad Royal Intitute of Technology Stockholm, SWEDEN johanh@nada.kth.e P. Pudlak Mathematical Intitute Prague, CZECH REPUBLIC pudlak@cearn.bitnet pudlak@earn.cvut.cz S. Jukna Intitute of Mathematic Vilniu, LITHUANIA Current addre of S. Jukna: Univerity oftrier Trier, GERMANY jukna@ti.uni-trier.de