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Linear lower bounds on unbounded fan-in boolean circuits
16 August 1985
Information Processing Letters 21 (1985) 71-74 North-Holland
LINEAR LOWER B O U N D S ON U N B O U N D E D FAN-IN BOOLEAN CIRCUITS * Juraj HROMKOVI(~ Department of Theoretical Cybernetics, Comenius University, 842 15 Bratislava, Czechoslovakia Communicated by W.M. Turski Received 5 October 1984 Revised 7 November 1984
Keywords: Combinational complexity, communication complexity, unbounded fan-in Boolean circuits
Introduction Proving lower bounds on the circuit complexity of a specific Boolean function seems to be one of the most challenging problems in complexity theory. Although most Boolean functions f: {0, 1} n --->{0, 1} have exponential complexity on fan-in two-bounded circuits containing AND, OR, and NEG gates, only linear lower bounds, for specific Boolean functions, are known [13,15,16]. The linear, lower bounds on fan-in two-bounded circuits with gates in an arbitrary basis (clearly, in the bases AND, OR, NEG tOO) were obtained in several papers [1,4,5,6,9,12]. We shall consider unbounded fan-in circuits in this paper. Chandra et al. [2] have shown the relation between unbounded fan-in circuits containing AND'S, OR'S, and NOT'S and parallel RAM (WRAM). A large number of nonpolynomial and exponential lower bounds on unbounded fan-in circuits with constant depth was obtained in [2,4,10,17]. We shall study a more general model of unbounded fan-in circuits as opposed to [2,4,10,17]. Besideg AND, Or,, and NEO gates we allow, for each commutative and associative Boolean function g: {0, 1} 2--* {0, 1), g-gates. So, for example, we have PLUS (mod 2) -gates. Such circuits will be
* This work was partly supported by an SPZV 1-5-7/7 Grant.
called CA-circuits in what follows. Clearly, NEG gates can have only one input. If a g-gate, for g different from NEG, has only one input, we consider that it computes the identical function. Now, considering the parity function we show that CA-circuits are more powerful than unbounded fan-in circuits studied till now. Lupanov [10] proved an exponential lower bound on unbounded fan-in circuits with depth 2 computing parity function. Furst et al. [4] have shown that there do not exist unbounded fan-in circuits with constant depth and polynomial size computing parity function. Clearly, CA-circuit computing parity functions need only one PLus-gate. So, its complexity is only 1. We still note that the linear lower bounds on fan-in two-bounded circuits in [13,15] were obtained for parity functions. We define the communication complexity of CA-circuits, and prove that it gives a lower bound on CA-circuit complexity (the number of gates). We note that the communication complexity introduced here differs essentially from the communication complexity studied in [3,7,8,11]. Let f : ( 0 , 1} 2m---~(0, 1} be a Boolean function such that f(x~, x2,...,X2m } = 1 iff x i = Xm+i for all i {1, 2 .... ,m}. We show that this function f, and all functions which can be obtained from f by the permutation of variables, have the communication complexity of CA-circuits equal to m. Obviously, in this way we obtain linear lower bounds on CA-circuits.
0020-0190/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)
71
Volume 21, N u m b e r 2
INFORMATION PROCESSING LETTERS
This paper is organized as follows. Section 1 involves the definition of the communication complexity of CA-circuits, and the Shannon characterisation of this complexity measure. In Section 2 the relation between the communication complexity and the size complexity of CA-circuits is given, and the n / 2 lower bounds for specific Boolean functions are proved.
1. The communication complexity of CA-circuits First, let us formally introduce CA-circuits. A CA-circuit is a directed, acyclic graph with no b o u n d on the indegree and the outdegree of nodes. It has exactly one node with outdegree O, called the output node, and k >/1 nodes with indegree O. The nodes with indegree 0 are entries, and are labelled with the input variables x a , . . . , x k. Each non-entry node is labelled by g, where g: (0, 1} 2 -~ (0, 1) is a commutative, associative Boolean function. Each node with indegree I can be labelled by NEG tOO. The number of non-entry nodes (gates) of a CA-circuit S is called the size of S, and denoted c(S). The CA-circuit complexity of a Boolean function h, c(h), is min(c(S)IS computes
h). Now, let us define the communication complexity of Boolean functions according to CA-circuits. Let S be a CA-circuit computing a function h of n = 2m variables. Let S have k gates. Obviously, S has 2m entries. Let ~ be the partition of the set of entries (x 1, x 2. . . . ,XEm} into two equal-sided sets A~ and A 2. Let a be the partition of k non-entry nodes gl, g2,--.,gk into two sets B 1 and B2 with no bound on their power (for example, B 1 can be equal to (gl,---, gk } and B2 can be the empty set). Let %(S) denote the number of edges between the nodes from A 1 k) B~ and the nodes from A 2 k.) B 2 in the CA-circuit S. Then, the communication complexity of S according to 'rr is ~r(S)= mind(cry(S)}, where the minimum is taken over all partitions a of the nodes ga,---, gk- The communication complexity of a Boolean function h according to ,rr is 'rr(h)= rain{ ~r(S) IS computes h}. The communication complexity of h, com(h), is defined as max(~r(h) I ~r is the partition of (x 1. . . . . x 2m } into two equal-sided sets}. 72
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We give some Shannon characteristics of the communication complexity of CA-circuits in the following two assertions. Lemma 1.1. For each Boolean function of 2m variables, com(h) ~< m holds. Proof. Let S be a CA-circuit computing h with entries x a, x 2 , . . . ,X2m and gates gl, g2,---, gk- For every partition ~r of (x 1. . . . ,X2m } we construct a CA-circuit S' computing h such that ~r(S')~< m. Let ~r be a partition of ( x l , . . . , x z m } into two sets A a = ( z l , . . . , Z m } and A 2 = (Yl,''',Ym}" We construct a CA-circuit S' with gates g l , - - - , g k , Z1,... ,Z m from the CA-circuit S in the following way. We replace all entries z x , . . . , z ~ of S by the gates Z ~ , . . . , Z m respectively which we label by OR. Then we give one edge from z i (as entry of S') to Z i for all i --- 1 , . . . , m. Clearly, Z1,..., Z m compute identical functions which implies that S' computes the same function h as S. Let us consider the partition et of G = {gl,---,gk, Z1 . . . . . Z m } to the sets B 1 = 0 and B 2 = G. So, ¢r~(S')= m, and it follows that v(h)~< m. [] Theorem 1.2. Almost all Boolean functions of 2m variables require communication complexity m.
] Proof. It can easdy be seen that the communication complexity of CA-circuits is a special case of the general communication complexity measure studied in [3,7,11], i.e., for each CA-circuit S with the partitions 'rr, ~x, a protocol (see [3]) simulating the computation of S with the communication complexity %(S) can be constructed. As Papadimitriou et al., considering the general model of communication complexity, proved [11] that almost all Boolean functions of 2m variables have communication complexity m, Theorem 1.2 holds. []
2. The lower bounds on CA-circuits complexity In the following we shall show that the communication complexity of CA-circuits gives the lower bound on the complexity of CA-circuits.
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T h e o r e m 2.1. Let h be a Boolean function. Then
com(h) 4 c(h). Proof. To prove T h e o r e m 2.1 we need to show that for all partitions "rr it follows that ~r(h) 4 c(h). The p r o o f is d o n e by contradiction. Let us assume that there exists a CA-circuit S c o m p u t i n g h such that c(S) < rr(h) for some ~r. T h e n we shall construct a CA-circuit S' computing h such that % ( S ' ) < 'rr(h) for a partition a. Let S be a CA-circuit c o m p u t i n g h such that c(S) < "rr(h). So, there exists a partition ct of the gates of S such that c ( S ) < %(S). Let ~ divide entries into sets A 1 and A 2 , and let tx divide gates into sets B~ and B2. Since the n u m b e r of edges between A I U B 1 and A 2 t_) B 2 is greater than c(S), there should exist at least two edges f r o m A~ U B~ to 132 (or f r o m A 2 U B2 to B~), which leads to the same gate G with a label g in B2 (or B1). Let us construct a CA-circuit S' in the following way. We add a new node H labelled by g and a new edge f r o m H to G. The above considered group of edges going to G from A1 t2 B 1 (respectively A 2 U B2) is replaced b y the group of edges with the same begin going to H. If we put tx' such that it divides the gates of S' into B 1 W ( H ) (respectively B1) and B 2 ( r e s p e c t i v e l y B 2 I..) (H}), we have ~r,~,(S') < %(S). Since g is a c o m m u t a t i v e a n d associative function, S' c o m p u t e s h. Clearly, doing the construction introduced until a g r o u p f r o m B 1 t2 A 1 (or f r o m A 2 u 132) to one gate from B2 (or B~) exists, we can construct f r o m S a CA-circuit S 1 whose c o m m u n i c a t i o n complexity according to ,rr is at most c(S). This is a contradiction with the assumption c(S) < ~-(h). [] Using T h e o r e m 2.1 we give the lower b o u n d n / 2 , for some functions of n variables, on the CA-circuit complexity. T h e o r e m 2.2. Let f b e a Boolean function of n = 2m
variables such that f(xl, x 2 , . . . , X m , Yl, Y2,--.,Ym) = 1 iff Xi = Yi for all i = 1, 2 , . . . , m. Then c(f) >>.m. Proof. According to T h e o r e m 2.1 it is sufficient to prove c o m ( f ) = m. Let us prove this fact by contradiction. Let ~r be the partition of {x 1, x 2 , . . . , x m, Yl,
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Y2. . . . ,Ym} into sets A 1 = (xl, x 2 . . . . ,Xm} a n d A 2 = (Yl, Y2. . . . , Ym} and let ~r(f)~< m - 1. T h e n there exists a CA-circuit computing f with the p r o p e r t y %(S) ~< m - 1, for some partition ct of the gates of S into two sets B 1 and B2. We shall show that no such S computes f. Without loss of generality we can assume that B2 involves the output node. For each input z = (al, a 2 , . . . , a m , bl, b2 . . . . . b m ) ~ {0, 1} 2m we can consider the c o m m u n i c a t i o n ClC2...Ck, for cj (0, 1}, which contains all bits going on the k ~ m - 1 edges from A1 U B 1 to B2 and from A 2 U B2 to B 1 during the c o m p u t a t i o n of S on the input z. Since the n u m b e r of different c o m m u n i c a t i o n s is b o u n d e d by 2 m- 1, there exist two distinct inputs zl = (bl, b 2 , - - - , bin, bl, b 2 , . - . , bin) a n d z 2 = (d~, d 2. . . . , d m, d 1, d 2 . . . . , d i n ) with the same c o m m u n i c a t i o n c. It can easily be seen that S has the same c o m m u n i c a t i o n c for the inputs ul = (b 1, b 2 , . . . , b m, d 1, d 2 , . . . , d m ) a n d u 2 = (d 1, d 2 . . . . ,drn, b 1, b 2 , . . . , b m ) . It implies that if S computes 1 for the inputs zl, z2, then it should c o m p u t e 1 for the inputs u~, u 2 (the a r g u m e n t s of the output node in B2 are y~, Y2,-",Ym a n d the part of c o m m u n i c a t i o n bits); this is a contradiction with the fact that f(ul) = f(u2) = 0. [] Corollary 2.3. Let h be a Boolean function o f 2m variables which can be obtained from the function f as considered in Theorem 2.2 by the permutation of variables. Then c(h) >~ m. Proof. Obviously, choosing a suitable ,rr, ~r(h) > / m can be proved in the same way as in the p r o o f of T h e o r e m 2.2. [] W e conclude this paper by making two remarks. The function h of 2m variables from Corollary 2.3 can be c o m p u t e d with the CA-circuits containing m N E G - P L u s gates and o n e AND gate. All lower b o u n d s on the general m o d e l of c o m m u n i c a t i o n complexity and on the m o d e l with a fixed partition too [9,11] are lower b o u n d s on CA-circuits complexity as well.
References [1] N. Blare, Boolean functions requiring 3n network size, Theoret. Comput. Sci. 28 (1984) 337-345.
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[2] A.K. Chandra, L.J. Stockmeyer and U. Vishkin, A complexity theory for unbounded fan-in parallelism, Proc. 23rd Ann. Symp. on Foundations of Computer Science (1982) 1-14. [3] P. [)uriC, Z. Galil and G. Schnitger, Lower bounds on communication complexity, Proc. 16th Ann. ACM Symp. on Theory of Computing (1984) 81-91. [4] M. Furst, J.B. Saxe and M. Sipser, Parity, circuit and the polynomial time hierarchy, Proc. 22nd Ann. IEEE Symp. on Foundations of Computer Science (1981) 260-270. [5] L.H. Harper and J.E. Savage, Complexity made simple, Proc. Internat. Symp. on Combinatorial Theory, Rome (1973) 2 - 1 5 . [6] L.H. Harper, W.N. Hsieh and J.E. Savage, A class of Boolean functions with linear combinational complexity, Theoret. Comput. Sci. 1 (2) (1975) 161-183. [7] J. Hromko;¢i~, Communication complexity, Proc. llth Internat. Coll. on Automata, Languages and Programming, Lecture Notes in Computer Science 172 (Springer, Berlin, 1984) 235-246. [8] J. Hromkovi~, Relation between Chomsky hierarchy and communication complexity hierarchy, Acta Mathematica Universitatis Comenianae XLIV (1985) to appear. [9] B.M. Kloss and B.A. Maly~ev, Ocenki sloLnosti nekatorych klassov funkcij (in Russian), Seria Matem. Mech. 4 (1965) 45-55.
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[10] O. Lupanov, Implementing the algebra of logic functions in terms of bounded depth formulas in the basis +, *, - , English translation in: Soy. Phys.-Dokl. 6 (2) (1961) 107-108. [11] C.H. Papadimitriou and M. Sipser, Communication complexity, Proc. 14th Ann. ACM Symp. on Theory of Computing (1982) 330-337. [12] W.J. Paul, A 2.5n-lower bound on the combinational complexity of Boolean functions, SIAM J. Comput. 6 (1977) 427-443. [13] N.P. Rex]kin, Dokazaterstvo minimalnosti nekatorych schem iz funkcionalnych elementov (in Russian), Probl. Kibernetiki 23 (1970) 83-101. [14] J.E. Savage, The Complexity of Computing (Wiley-Interscience, New York, 1976). [15] C.P. Schnorr, Zwei lineare untere Schranken fur die Komplexit~t Boolescher Funktionen (in German), Computing 13 (1974) 155-171. [16] C.P. Schnorr, A 3n-lower bound on the network complexity of Boolean functions, Theoret. Comput. Sci. 10 (1980) 83-92. [17] A.C. Yao, Lower bounds by probabilistic arguments, Proc. 24th Ann. IEEE Symp. on Foundations of Computer Science (1983) 420-428.
Information Processing Letters 21 (1985) 71-74 North-Holland
LINEAR LOWER B O U N D S ON U N B O U N D E D FAN-IN BOOLEAN CIRCUITS * Juraj HROMKOVI(~ Department of Theoretical Cybernetics, Comenius University, 842 15 Bratislava, Czechoslovakia Communicated by W.M. Turski Received 5 October 1984 Revised 7 November 1984
Keywords: Combinational complexity, communication complexity, unbounded fan-in Boolean circuits
Introduction Proving lower bounds on the circuit complexity of a specific Boolean function seems to be one of the most challenging problems in complexity theory. Although most Boolean functions f: {0, 1} n --->{0, 1} have exponential complexity on fan-in two-bounded circuits containing AND, OR, and NEG gates, only linear lower bounds, for specific Boolean functions, are known [13,15,16]. The linear, lower bounds on fan-in two-bounded circuits with gates in an arbitrary basis (clearly, in the bases AND, OR, NEG tOO) were obtained in several papers [1,4,5,6,9,12]. We shall consider unbounded fan-in circuits in this paper. Chandra et al. [2] have shown the relation between unbounded fan-in circuits containing AND'S, OR'S, and NOT'S and parallel RAM (WRAM). A large number of nonpolynomial and exponential lower bounds on unbounded fan-in circuits with constant depth was obtained in [2,4,10,17]. We shall study a more general model of unbounded fan-in circuits as opposed to [2,4,10,17]. Besideg AND, Or,, and NEO gates we allow, for each commutative and associative Boolean function g: {0, 1} 2--* {0, 1), g-gates. So, for example, we have PLUS (mod 2) -gates. Such circuits will be
* This work was partly supported by an SPZV 1-5-7/7 Grant.
called CA-circuits in what follows. Clearly, NEG gates can have only one input. If a g-gate, for g different from NEG, has only one input, we consider that it computes the identical function. Now, considering the parity function we show that CA-circuits are more powerful than unbounded fan-in circuits studied till now. Lupanov [10] proved an exponential lower bound on unbounded fan-in circuits with depth 2 computing parity function. Furst et al. [4] have shown that there do not exist unbounded fan-in circuits with constant depth and polynomial size computing parity function. Clearly, CA-circuit computing parity functions need only one PLus-gate. So, its complexity is only 1. We still note that the linear lower bounds on fan-in two-bounded circuits in [13,15] were obtained for parity functions. We define the communication complexity of CA-circuits, and prove that it gives a lower bound on CA-circuit complexity (the number of gates). We note that the communication complexity introduced here differs essentially from the communication complexity studied in [3,7,8,11]. Let f : ( 0 , 1} 2m---~(0, 1} be a Boolean function such that f(x~, x2,...,X2m } = 1 iff x i = Xm+i for all i {1, 2 .... ,m}. We show that this function f, and all functions which can be obtained from f by the permutation of variables, have the communication complexity of CA-circuits equal to m. Obviously, in this way we obtain linear lower bounds on CA-circuits.
0020-0190/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)
71
Volume 21, N u m b e r 2
INFORMATION PROCESSING LETTERS
This paper is organized as follows. Section 1 involves the definition of the communication complexity of CA-circuits, and the Shannon characterisation of this complexity measure. In Section 2 the relation between the communication complexity and the size complexity of CA-circuits is given, and the n / 2 lower bounds for specific Boolean functions are proved.
1. The communication complexity of CA-circuits First, let us formally introduce CA-circuits. A CA-circuit is a directed, acyclic graph with no b o u n d on the indegree and the outdegree of nodes. It has exactly one node with outdegree O, called the output node, and k >/1 nodes with indegree O. The nodes with indegree 0 are entries, and are labelled with the input variables x a , . . . , x k. Each non-entry node is labelled by g, where g: (0, 1} 2 -~ (0, 1) is a commutative, associative Boolean function. Each node with indegree I can be labelled by NEG tOO. The number of non-entry nodes (gates) of a CA-circuit S is called the size of S, and denoted c(S). The CA-circuit complexity of a Boolean function h, c(h), is min(c(S)IS computes
h). Now, let us define the communication complexity of Boolean functions according to CA-circuits. Let S be a CA-circuit computing a function h of n = 2m variables. Let S have k gates. Obviously, S has 2m entries. Let ~ be the partition of the set of entries (x 1, x 2. . . . ,XEm} into two equal-sided sets A~ and A 2. Let a be the partition of k non-entry nodes gl, g2,--.,gk into two sets B 1 and B2 with no bound on their power (for example, B 1 can be equal to (gl,---, gk } and B2 can be the empty set). Let %(S) denote the number of edges between the nodes from A 1 k) B~ and the nodes from A 2 k.) B 2 in the CA-circuit S. Then, the communication complexity of S according to 'rr is ~r(S)= mind(cry(S)}, where the minimum is taken over all partitions a of the nodes ga,---, gk- The communication complexity of a Boolean function h according to ,rr is 'rr(h)= rain{ ~r(S) IS computes h}. The communication complexity of h, com(h), is defined as max(~r(h) I ~r is the partition of (x 1. . . . . x 2m } into two equal-sided sets}. 72
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We give some Shannon characteristics of the communication complexity of CA-circuits in the following two assertions. Lemma 1.1. For each Boolean function of 2m variables, com(h) ~< m holds. Proof. Let S be a CA-circuit computing h with entries x a, x 2 , . . . ,X2m and gates gl, g2,---, gk- For every partition ~r of (x 1. . . . ,X2m } we construct a CA-circuit S' computing h such that ~r(S')~< m. Let ~r be a partition of ( x l , . . . , x z m } into two sets A a = ( z l , . . . , Z m } and A 2 = (Yl,''',Ym}" We construct a CA-circuit S' with gates g l , - - - , g k , Z1,... ,Z m from the CA-circuit S in the following way. We replace all entries z x , . . . , z ~ of S by the gates Z ~ , . . . , Z m respectively which we label by OR. Then we give one edge from z i (as entry of S') to Z i for all i --- 1 , . . . , m. Clearly, Z1,..., Z m compute identical functions which implies that S' computes the same function h as S. Let us consider the partition et of G = {gl,---,gk, Z1 . . . . . Z m } to the sets B 1 = 0 and B 2 = G. So, ¢r~(S')= m, and it follows that v(h)~< m. [] Theorem 1.2. Almost all Boolean functions of 2m variables require communication complexity m.
] Proof. It can easdy be seen that the communication complexity of CA-circuits is a special case of the general communication complexity measure studied in [3,7,11], i.e., for each CA-circuit S with the partitions 'rr, ~x, a protocol (see [3]) simulating the computation of S with the communication complexity %(S) can be constructed. As Papadimitriou et al., considering the general model of communication complexity, proved [11] that almost all Boolean functions of 2m variables have communication complexity m, Theorem 1.2 holds. []
2. The lower bounds on CA-circuits complexity In the following we shall show that the communication complexity of CA-circuits gives the lower bound on the complexity of CA-circuits.
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INFORMATION PROCESSING LETTERS
T h e o r e m 2.1. Let h be a Boolean function. Then
com(h) 4 c(h). Proof. To prove T h e o r e m 2.1 we need to show that for all partitions "rr it follows that ~r(h) 4 c(h). The p r o o f is d o n e by contradiction. Let us assume that there exists a CA-circuit S c o m p u t i n g h such that c(S) < rr(h) for some ~r. T h e n we shall construct a CA-circuit S' computing h such that % ( S ' ) < 'rr(h) for a partition a. Let S be a CA-circuit c o m p u t i n g h such that c(S) < "rr(h). So, there exists a partition ct of the gates of S such that c ( S ) < %(S). Let ~ divide entries into sets A 1 and A 2 , and let tx divide gates into sets B~ and B2. Since the n u m b e r of edges between A I U B 1 and A 2 t_) B 2 is greater than c(S), there should exist at least two edges f r o m A~ U B~ to 132 (or f r o m A 2 U B2 to B~), which leads to the same gate G with a label g in B2 (or B1). Let us construct a CA-circuit S' in the following way. We add a new node H labelled by g and a new edge f r o m H to G. The above considered group of edges going to G from A1 t2 B 1 (respectively A 2 U B2) is replaced b y the group of edges with the same begin going to H. If we put tx' such that it divides the gates of S' into B 1 W ( H ) (respectively B1) and B 2 ( r e s p e c t i v e l y B 2 I..) (H}), we have ~r,~,(S') < %(S). Since g is a c o m m u t a t i v e a n d associative function, S' c o m p u t e s h. Clearly, doing the construction introduced until a g r o u p f r o m B 1 t2 A 1 (or f r o m A 2 u 132) to one gate from B2 (or B~) exists, we can construct f r o m S a CA-circuit S 1 whose c o m m u n i c a t i o n complexity according to ,rr is at most c(S). This is a contradiction with the assumption c(S) < ~-(h). [] Using T h e o r e m 2.1 we give the lower b o u n d n / 2 , for some functions of n variables, on the CA-circuit complexity. T h e o r e m 2.2. Let f b e a Boolean function of n = 2m
variables such that f(xl, x 2 , . . . , X m , Yl, Y2,--.,Ym) = 1 iff Xi = Yi for all i = 1, 2 , . . . , m. Then c(f) >>.m. Proof. According to T h e o r e m 2.1 it is sufficient to prove c o m ( f ) = m. Let us prove this fact by contradiction. Let ~r be the partition of {x 1, x 2 , . . . , x m, Yl,
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Y2. . . . ,Ym} into sets A 1 = (xl, x 2 . . . . ,Xm} a n d A 2 = (Yl, Y2. . . . , Ym} and let ~r(f)~< m - 1. T h e n there exists a CA-circuit computing f with the p r o p e r t y %(S) ~< m - 1, for some partition ct of the gates of S into two sets B 1 and B2. We shall show that no such S computes f. Without loss of generality we can assume that B2 involves the output node. For each input z = (al, a 2 , . . . , a m , bl, b2 . . . . . b m ) ~ {0, 1} 2m we can consider the c o m m u n i c a t i o n ClC2...Ck, for cj (0, 1}, which contains all bits going on the k ~ m - 1 edges from A1 U B 1 to B2 and from A 2 U B2 to B 1 during the c o m p u t a t i o n of S on the input z. Since the n u m b e r of different c o m m u n i c a t i o n s is b o u n d e d by 2 m- 1, there exist two distinct inputs zl = (bl, b 2 , - - - , bin, bl, b 2 , . - . , bin) a n d z 2 = (d~, d 2. . . . , d m, d 1, d 2 . . . . , d i n ) with the same c o m m u n i c a t i o n c. It can easily be seen that S has the same c o m m u n i c a t i o n c for the inputs ul = (b 1, b 2 , . . . , b m, d 1, d 2 , . . . , d m ) a n d u 2 = (d 1, d 2 . . . . ,drn, b 1, b 2 , . . . , b m ) . It implies that if S computes 1 for the inputs zl, z2, then it should c o m p u t e 1 for the inputs u~, u 2 (the a r g u m e n t s of the output node in B2 are y~, Y2,-",Ym a n d the part of c o m m u n i c a t i o n bits); this is a contradiction with the fact that f(ul) = f(u2) = 0. [] Corollary 2.3. Let h be a Boolean function o f 2m variables which can be obtained from the function f as considered in Theorem 2.2 by the permutation of variables. Then c(h) >~ m. Proof. Obviously, choosing a suitable ,rr, ~r(h) > / m can be proved in the same way as in the p r o o f of T h e o r e m 2.2. [] W e conclude this paper by making two remarks. The function h of 2m variables from Corollary 2.3 can be c o m p u t e d with the CA-circuits containing m N E G - P L u s gates and o n e AND gate. All lower b o u n d s on the general m o d e l of c o m m u n i c a t i o n complexity and on the m o d e l with a fixed partition too [9,11] are lower b o u n d s on CA-circuits complexity as well.
References [1] N. Blare, Boolean functions requiring 3n network size, Theoret. Comput. Sci. 28 (1984) 337-345.
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Volume 21, Number 2
INFORMATION PROCESSING LETTERS
[2] A.K. Chandra, L.J. Stockmeyer and U. Vishkin, A complexity theory for unbounded fan-in parallelism, Proc. 23rd Ann. Symp. on Foundations of Computer Science (1982) 1-14. [3] P. [)uriC, Z. Galil and G. Schnitger, Lower bounds on communication complexity, Proc. 16th Ann. ACM Symp. on Theory of Computing (1984) 81-91. [4] M. Furst, J.B. Saxe and M. Sipser, Parity, circuit and the polynomial time hierarchy, Proc. 22nd Ann. IEEE Symp. on Foundations of Computer Science (1981) 260-270. [5] L.H. Harper and J.E. Savage, Complexity made simple, Proc. Internat. Symp. on Combinatorial Theory, Rome (1973) 2 - 1 5 . [6] L.H. Harper, W.N. Hsieh and J.E. Savage, A class of Boolean functions with linear combinational complexity, Theoret. Comput. Sci. 1 (2) (1975) 161-183. [7] J. Hromko;¢i~, Communication complexity, Proc. llth Internat. Coll. on Automata, Languages and Programming, Lecture Notes in Computer Science 172 (Springer, Berlin, 1984) 235-246. [8] J. Hromkovi~, Relation between Chomsky hierarchy and communication complexity hierarchy, Acta Mathematica Universitatis Comenianae XLIV (1985) to appear. [9] B.M. Kloss and B.A. Maly~ev, Ocenki sloLnosti nekatorych klassov funkcij (in Russian), Seria Matem. Mech. 4 (1965) 45-55.
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[10] O. Lupanov, Implementing the algebra of logic functions in terms of bounded depth formulas in the basis +, *, - , English translation in: Soy. Phys.-Dokl. 6 (2) (1961) 107-108. [11] C.H. Papadimitriou and M. Sipser, Communication complexity, Proc. 14th Ann. ACM Symp. on Theory of Computing (1982) 330-337. [12] W.J. Paul, A 2.5n-lower bound on the combinational complexity of Boolean functions, SIAM J. Comput. 6 (1977) 427-443. [13] N.P. Rex]kin, Dokazaterstvo minimalnosti nekatorych schem iz funkcionalnych elementov (in Russian), Probl. Kibernetiki 23 (1970) 83-101. [14] J.E. Savage, The Complexity of Computing (Wiley-Interscience, New York, 1976). [15] C.P. Schnorr, Zwei lineare untere Schranken fur die Komplexit~t Boolescher Funktionen (in German), Computing 13 (1974) 155-171. [16] C.P. Schnorr, A 3n-lower bound on the network complexity of Boolean functions, Theoret. Comput. Sci. 10 (1980) 83-92. [17] A.C. Yao, Lower bounds by probabilistic arguments, Proc. 24th Ann. IEEE Symp. on Foundations of Computer Science (1983) 420-428.