On Relations Between Counting Communication Complexity Classes Matthias Krause Christoph Meinel Universitat Mannheim FB IV{Informatik Seminargebaude A5 Universitat Trier D{68131 Mannheim D{54286 Trier Stephan Waack Institut fur Numerische und Angewandte Mathematik Georg{August{Universitat Gottingen Lotzestr. 16{18, D{37083 Gottingen
Carsten Damm FB IV{Informatik Universitat Trier D{54286 Trier
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Proposed running head: Counting Communication Complexity Address of the author to whom proofs should be send: Stephan Waack Institut fur Numerische und Angewandte Mathematik Georg{August{Universitat Gottingen Lotzestr. 16{18 D{37083 Gottingen e-mail:
[email protected]
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Abstract
We develop upper and lower bound arguments for counting acceptance modes of communication protocols. A number of separation results for counting communication complexity classes is established. This extends the investigation of the complexity of communication between two processors in terms of complexity classes initiated by Babai, Frankl, and Simon [Proc. 27th IEEE FOCS 1986, pp. 337{347] and continued in several papers (e.g., Halstenberg and Reischuk [Journ. of Comput. and Syst. Sci. 41(1990), pp. 402{429], Karchmer et al. [Journ. of Comput. and Syst. Sci. 49(1994), pp. 247{257] More precisely, it will be shown that the communication complexity classes MODpP cc and MODq P cc are incomparable with regard to inclusion, for all pairs of distinct prime numbers p and q . The same is true for PP cc and MODm P cc , for any number m 2. Moreover, nondeterminism and modularity are incomparable to a large extend. On the other hand, if m = pl1 : : : plrr is the prime decomposition of m 2, then the complexity classes MODm P cc and MOD(m) P cc coincide, where (m) = p1 : : : pr . The results are obtained by characterizing the modular and probabilistic communication complexity in terms of the minimum rank of matrices ranging over certain equivalence classes. Methods from algebra and analytic geometry are used. Key words. communication complexity class, protocol, nondeterminism, probabilism, modularity, upper bound, lower bound AMS subject classi cation. 68Q22 1
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1 Introduction Communication complexity plays an important role in theoretical studies: many of the known lower-bound results are obtained by analyzing the communication between various parts of the computational device. This concerns area-time tradeo s for VLSI-computations ([1], [9]), time-space tradeo s for Turing machines, width-length tradeo s for oblivious and usual branching programs and -branching programs ([2], [12]). Moreover, lower bounds on the depth of monotone circuits ([16]), structural results in designing pseudorandom sequences ([4]), and lower bounds on the size of special threshold circuits of depth 3 ([8]) should be mentioned in this connection. Babai, Frankl, and Simon in [3] introduced the investigation of the complexity of communication between two processors in terms of complexity classes. They showed some analogies between Turing machine classes like P , NP , PP , etc. and the corresponding communication complexity classes P cc, NP cc , PP cc , etc. Halstenberg and Reischuk in [6] and [7] studied di erent measures of communication complexity of discrete functions. In this paper we study counting acceptance modes for nondeterministic communication protocols introduced in [7]. Certain types of counting communication complexity classes were also studied in [11]. The arising complexity classes are analogues of Turing machine based complexity classes. These were extensively studied in the last years and many oracle separations are known (cf., e.g., [5]). Counting acceptance modes for communication protocols have also proved to be useful for proving lower bounds for certain types of depth restricted circuits (see [10], [13], [17]). Further, in [20] it has been shown that all problems computable by constant depth, polynomial size circuits with modm -gates for arbitrary integers m, i.e., ACC -functions, are contained in certain counting communication complexity classes. This paper is organized as follows. In Section 2 the model is introduced. Section 4 is focused on nding complete problems for the various counting communication complexity classes. These are target functions for the separation results. Sections 5 and 6 are the heart of the paper as to the methods used. We derive rank arguments for proving upper and lower bounds there. We use the characterization results of Section 5 to describe the relations between the complexity classes under consideration: 1. For m 2 the classes PP cc and MODm P cc are incomparable with regard to inclusion (Section 7, Theorem 18). This extends the result MOD2P cc 6 PP cc proved in [7] with the -discriminator method. 2. If m = pl1 : : : plrr is the prime factorization of m then the mod(m)-mode is as powerful as the modm-mode, where (m) = p1 : : : pr (Section 8 , Theorem 5.) 3. The computational powers of the acceptance modes modp and modq are incomparable in the case of p 6= q being prime numbers. If, moreover, m is a proper multiple but not a power of the prime number p, then the mode modm is strictly more powerful than the mode modp, i.e., MODp P cc MODm P cc (Section 8, Theorem 24). 4. Nondeterminism and modularity are incomparable to a large extent (Section 9, Theorem 27). 1
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2 Preliminaries Let X and Y be disjoint sets. We consider the following two-party communication game on X Y : Two players, which we denote by X and Y, have to compute the value of a function f : X Y ! f0; 1g on an input (x; y), where X has only access to x 2 X and Y has only access to y 2 Y . The players share a blackboard, where they can exchange bits subject to a given communication protocol. In the deterministic version of a protocol at each step the next bit to be communicated is completely determined by the input and the communication history (the string on the blackboard). Although, when describing protocols we will do this mostly in an informal way, we need a formal de nition. Recall the following standard notation. Given a directed full binary tree with a distinguished root, we label the outgoing edges of each inner node by 0 and 1, respectively. Then each node in the tree will be denoted by the string that describes the path from the root to this node. In particular, the root of the tree is denoted by the empty string . Formally de ned a deterministic (two-party ) communication protocol P on X Y is given by a nite directed rooted tree whose inner nodes are partitioned into X -nodes and Y -nodes. Let Z be any of X and Y. Each Z -node v is labeled by a mapping bv : Z ! f0; 1g. We write bv (x; y) to uniquely denote bv (x) or bv (y) without specifying whether v is type X or Y. Each node is a state in the communication process. The process starts at the root. The type of the actual node determines the player to communicate next. The value bv (x; y) is the bit to be communicated by the player in turn upon seeing his part of the input (x; y) and the communication history v. A computation (of length t) of protocol P on input (x; y) 2 X Y is given by a leaf w = w1w2 wt of the tree such that for 1 i t holds wi = bw w wi? (x; y) (using the convention w0 = ). The last bit of a computation is called the output of the computation on (x; y). We call a computation accepting if the output is 1 and rejecting otherwise. Observe that for each input (x; y) there is exactly one computation of a given deterministic communication protocol on (x; y). The function fP : X Y ! f0; 1g that maps (x; y) to the output of this single computation of P on (x; y) is called the function computed by P . We will base our constructions on nondeterministic protocols. In a nondeterministic protocol the bit to be announced by the player in turn need not be uniquely de ned by the proper part of the input and the communication history. This is accomplished by setting the range of the mappings bv to ff0g; f1g; f0; 1gg for inner nodes v of the tree. Computations of the protocol on input (x; y) are leaves w = w1w2 wt of the tree that ful ll wi 2 bw w wi? (x; y) for 1 i t. In the sequel the term \communication protocol" refers to both, deterministic and nondeterministic protocols. A nondeterministic protocol may have several accepting or rejecting computations on the same input. Given a protocol P let accP (x; y) and rejP (x; y), respectively, denote the number of di erent accepting and rejecting, respectively, computations of P on input (x; y). Using (counting ) acceptance modes we will interpret these numbers in various ways to de ne the function computed by the protocol. Any function : IN IN ! f0; 1g de nes an acceptance mode in the following way. We 0
0
1
1
5
1
1
say: P -computes function f : X Y ! f0; 1g if for all (x; y) 2 X Y holds
f (x; y) = 1 () (accP (x; y); rejP (x; y)) = 1: In particular we will consider the following acceptance modes for nondeterministic protocols: the nondeterministic mode: n( ; ) = 1 () > 0, the co-nondeterministic mode: co-n( ; ) = 1 () = 0, modular modes: mod m( ; ) = 1 () 6= 0 mod m for some xed positive integer m, and the probabilistic mode: prob( ; ) = 1 () > , where and stand for nonnegative integers. Observe that for deterministic protocols all mentioned acceptance modes (except for the co-nondeterministic one) lead to the same computed function. To enable uni ed terminology we may speak of the deterministic mode d, although it is understood that determinism is a property of the protocol not of a special acceptance mode. Further, we will often speak of -protocols, instead of nondeterministic protocols with acceptance mode . The complexity of a communication is its length, i.e., the depth of the leaf in the protocol tree. The complexity (or length ) of a protocol is the maximum complexity of its computations. The -communication complexity c(f ) of a function f : X Y ! f0; 1g is the minimum complexity of a communication protocol on X Y that -computes f . Especially we speak of the deterministic, nondeterministic, co-nondeterministic, mod m-, and probabilistic communication complexity of a function. We will use the notations cd(f ), cn(f ), cco-n(f ), cmod m(f ), and cprob(f ) to denote these.
3 Counting communication complexity classes Throughout we will consider the case where X = Y = n for some nite alphabet . Further, without loss of generality we will assume all computations of a given protocol to be of the same length. Let L be a language containing only strings of even length. A -communication protocol P for L is a sequence (Pn ) of communication protocols, such that for each n protocol Pn -computes Ln, where Ln is the characteristic function of L \ 2n , i.e., Ln equals 1 on L \ 2n and vanishes outside this set. As argued in [3] the natural unit with respect to which to measure communication complexity is log n (log here and elsewhere denotes logarithm to base 2). Hence a polynomial time communication protocol is a sequence P = (Pn ) such that for all n Pn is a communication protocol on n n with complexity at most (log n)c, where c is a constant independent of n. Given a speci c acceptance mode : IN IN ! f0; 1g we de ne P cc as being the class of all languages for which exist polynomial time nondeterministic -communication protocols. Classes of this type will be called counting communication complexity classes. Especially we consider: 6
P cc = fL j 9k 8n : cd(Ln ) (log n)k g NP cc = fL j 9k 8n : cn(Ln ) (log n)k g co-NP cc = fL j 9k 8n : cco-n(Ln) (log n)k g MODm P cc = fL j 9k 8n : cmod m(Ln ) (log n)k g PP cc = fL j 9k 8n : cprob(Ln ) (log n)k g. The above de ned classes are classes of languages, however, they can be seen also as classes of sequences of 0-1-functions (which we call problems ) by identifying a set with its characteristic function. We will in the sequel freely switch between both concepts. It will be convenient to consider also the following class introduced in [3]: #P cc = f(accPn )n2IN j P = (Pn ) is a polynomial time communication protocol g.
Lemma 1 1. NP cc =
fL j 9 : 9f = (fn)n2IN 2 #P cc : L = f(x; y) j 9n : (x; y) 2 n n ^ fn(x; y) > 0gg; 2. co-NP cc =
fL j 9 : 9f = (fn)n2IN 2 #P cc : L = f(x; y) j 9n : (x; y) 2 n n ^ fn(x; y) = 0gg; 3. MODm P cc =
fL j 9 : 9f = (fn)n2IN 2 #P cc : L = f(x; y) j 9n : (x; y) 2 n n^fn (x; y) 6= 0 mod mgg:
Proof: Obvious from the de nitions. Let f = (fn : n n ! IN)n2IN; g = (gn : n n ! IN)n2IN be sequences of functions. Let k be a nonnegative integer. The sum and product, respectively, of f and g are the families f + g = (fn + gn )n2IN, f g = (fn gn )n2IN, where these operations are de ned on individual members of thesequences in the obvious way. Similarly we de ne the kth binomial f f n coecient of f as k = k n2IN.
Lemma 2 Let k be a nonnegative integer. #P cc is closed under sum, f product, andcckth cc binomial coecient, i.e., if f; g belong to #P , then f + g; f g , and k belong to #P as well.
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Proof: Given protocols P and Q on n n with accP = fn ; accQ = gn and lengths s
and t, respectively. We construct protocols of lengths at most polynomial in s and t whose number fn (x;y) of accepting computations on (x; y) are fn(x; y) + gn (x; y); fn(x; y) gn (x; y), and k , respectively. Similar constructions are known for Turing machines (cf. [5]). For the sum generate a root v with bv (x; y) = f0; 1g and identify v0 and v1 with the roots of P and of Q. For the product connect a the root of a copy of Q to each accepting computation of P . For the second binomial coecient consider the \ordered twofold iteration" of P , i.e., connect the root of a copy of P to each accepting computation of P and rede ne the mappings bv as follows: For inner nodes v of this tree bv is identical to the corresponding mapping of the original node in P . De ne bw (x; y) = 1 for leaves w of the tree, for which w = w0w00, where w0 and w00 are accepting computations of P and w0 is lexicographically smaller than w00. De ne bw (x; y) = 0 for all other leaves w of the tree. For the kth binomial coecient consider the ordered k-fold iteration of P . We call the protocols constructed in the proof sum and product of P and Q and kth P binomial coecient of P , respectively, and denote them by P +Q, P Q, and k , respectively. Sum and product easily generalize to any constant number of operands. So we can consider protocols or sequences of protocols like P k (kth power of P ) or kP (sum of k copies of P ). Further we will denote by 1 the deterministic protocol that on each input lets player X announce 1 and stop. Hence protocols p(P1; : : : ; Pr ) can be considered, where P1; : : :; Pr are arbitrary protocols or sequences of protocols and p is a xed polynomial with nonnegative integer coecients. For a given protocol P we will denote by 1 ? P the protocol which is obtained from P by switching accepting and rejecting computations.
Lemma 3 Let P be a nondeterministic communication protocol on n n and let p be a prime. Then for each input (x; y) 2 n n holds: 1. if P 0 = 1 ? P then accP 0 (x; y) = rejP (x; y) and rejP 0 (x; y) = accP (x; y), 2. if P 0 = 1 + 2P then accP 0 (x; y) = 6 rejP 0 (x; y) and accP 0 (x; y) > rejP 0 (x; y) () accP (x; y) > rejP (x; y); (
mod p if accP (x; y) 0 mod p 3. if P 0 = P p?1 then acc (x; y) 10 mod p otherwise. P0
Proof: The only nontrivial observation is Claim 3, which relies on Fermat's Little Theorem. In a standard way we will make use of the closure properties mentioned in Lemma 2 and Lemma 3. 8
Lemma 4
1. If m1; m2 are positive integers and m1 divides m2, then MODm P cc MODm P cc : 2. Let m1 ; : : :; mr be pairwise coprime positive integers and m = m1 : : : mr . Then for any language L holds L 2 MODm P cc if and only if there are languages L1 2 MODm P cc ; : : :; Lr 2 MODmr P cc such that L = L1 [ : : : [ Lr . 3. If p is prime, then MODp P cc = MODpk P cc for any k. 1
2
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Sketch of proof: The proof relies on the above lemmas and the following observations: 1. If f 2 #P cc and m2=m1 = d then d f 2 #P cc . 2. Let f1; : : : ; fr 2 #P cc and let e1; : : : ; er be orthogonal idempotents in ZZ=m1ZZ ZZ=mr ZZ. Then f = e1f1 + + er fr 2 #P cc and for (x; y) 2 n n holds f (x; y) 0 mod m () 9ifi(x; y) 0 mod mi. Hence for languages L1; : : :; Lr as in the claim holds L = L1 [ : : : [ Lr 2 MODm P cc . If, on the other hand, some protocol P computes L modulo m, de ne Li as the language that is computed modulo mi by P . 3. For k = 2: let f 2 #P cc . Based on the identity kl = (k?l+1)(lk! ?l+2)k for nonnegative integers k l and on divisibility considerations one proves that for (x; y) 2 n n and a = f (x; y) holds ! a 2 a 0 mod p () a 0 mod p and p 0 mod p: Routine checking (based on Fermat's Little Theorem) shows that the latter is true if and only if 0 !!p?11 a A 0 mod p: 1 + (p ? 1)(1 + (p ? 1) ap?1) @1 + (p ? 1) p For general k iterate the construction using
a
!
a 0 mod pk () a 0 mod p and pk?1 0 mod p which is proved from the above by induction. A consequence of Lemma 4 is: Theorem 5 Let m = pl1 : : : plrr be the prime factorization of the natural number m. Then MODm P cc = MOD(m)P cc ; where (m) = p1 : : : pr . 1
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4 Complete problems We consider the following type of reductions introduced in [3]: Let L0 0, L00 00 be languages containing only binary strings of even length. We say that L0 is rectangular reducible to L00, i there are some constant c and sequences c 0 n 00 (log n ) (rn)n2IN and (ln)n2IN of transformations ln; rn : ! such that for all n and for all 0 n 0 n 0 00 (x; y) 2 we have (x; y) 2 L () (ln(x); rn(y)) 2 L . We write L0 rec L00. It is not hard to see that for every : IN2 ! f0; 1g the complexity class P cc is closed under rectangular reductions. A language L00 is said to be complete for a communication complexity class C cc, i L00 2 C cc and for any L0 2 C cc holds L0 rec L00. It turns out that all counting communication complexity classes have complete languages. We describe them as sequences of 0-1-functions. Let : IN2 ! f0; 1g be an arbitrary counting acceptance mode. The inner product with respect to is the problem -IP = (-IP2n : f0; 1g2n f0; 1g2n ! f0; 1g)n2IN , where
-IP2n(x11; : : :; x1n; x01; : : :; x0n; y11; : : :; y1n; y01; : : : ; y0n)= Pnj=1 x1j y1j ; Pnj=1 x0j y0j :
Theorem 6 -IP is complete for P cc with respect to rectangular reductions. Proof: We describe a -protocol of length dlog ne +2 for -IP2n. It works within 2 rounds. Round 1: Player X nondeterministically chooses an index l 2 f1; 0g. If xlk = 0, for all k 2 f1; : : :; ng, then the computation stops rejecting. Otherwise X chooses nondeterministically
some index k such that xlk = 1 and announces (l; k) encoded in binary. Round 2: If ylk = 0, then the computation stops rejecting. Otherwise player Y announces the one-bit-message 1P? l. Clearly, there are nk=1 x1k y1k accepting and Pnk=1 x0k y0k rejecting computations on the input (x11; : : : ; x1n; x01; : : : ; x0n; y11; : : : ; y1n; y01; : : :; y0n). By de nition the protocol -computes the function -IP2n. It remains to show that if f = (fn : 0n 0n ! f0; 1g)n2IN 2 P cc , then f is rectangular reducible to -IP. Let P be a -protocol of length t = (log n)O(1) for fn (without loss of generality we assume all computations to be of the same length). We x enumerations cb1; : : :cb2t? of all 0; 1-sequences of length t whose last bit is b and de ne the two transformations ln; rn : 0n ! t f0; 1g2 ; for all (x; y) 2 0n 0n by 1
ln(x) = ( PX (x; c11); : : :; PX (x; c12t? ); PX (x; c01); : : :; PX (x; c02t? )) rn (y) = ( PY (y; c11); : : :; PY (y; c12t? ); PY (y; c01); : : : ; PY (y; c02t? )); 1
1
1
1
where PX and PY are the characteristic functions of valid computations on the input from X's and Y's viewpoint. More explicit: for w 2 f0; 1gt de ne PX (x; w) = 1 if there is some y 2 0n, such that w is a computation of P on input (x; y). Similarly PY is de ned. Since PX (x; w) PY (y; w) = 1 if and only if w is a computation of P on (x; y), the claim follows. 10
We will consider also the following problems: inner product mod m: IPm = (IPmn : (ZZ=mZZ)n (Z=mZ )n ! f0; 1g)n2IN : 1 if Pn xiyi 6 0 mod m; m IPn (x1; : : : xn; y1; : : : ; yn) := 0 if Pin=1 x y 0 mod m. i=1 i i (The function 1 ? IPmn is the orthogonality test mod m ORTmn studied e.g. in [13].) Boolean inner product: BIP = (BIPn : f0; 1gn f0; 1gn ! f0; 1g)n2IN: _n BIPn (x1; : : :xn ; y1; : : :; yn ) := (xi ^ yi): i=1
(Note that 1 ? BIPn is the set disjointness test SDTn (see e.g. [14]).)
Corollary 7 1. IPm is complete for MODm P cc , 2. ORTm is complete for MODm P cc , for prime m, 3. BIP is complete for NP cc , 4. SDT is complete for co-NP cc .
5 Rank arguments for upper and lower bounds We prove in this section complexity bounds for counting acceptance modes based on ranks of matrices over semirings in a unifying treatment. Some of the arguments where known before for speci c acceptance modes (see [14]). Let R be any commutative semiring, i.e., a set R together with two binary operations + and (addition and multiplication) such that (R; +) and (R; ) are commutative monoids and multiplication distributes over addition. In a semiring R let 1R denote the multiplicative unit. Special semirings we consider in this paper are the non-negative integers IN, the Boolean semiring IB, the integers ZZ, the reals IR, and ZZ=mZZ, the integers modulo m. For any R in this collection the mapping 1 7! 1R can be extended to a unique semiring homomorphism from IN to R | the canonic homomorphism. Given a non-zero m n matrix M over R let the R-rank of M be the minimum k such that there exist an m k matrix A over R and a k n matrix B over R with M = A B . We denote the R-rank of M by rankR(M ). According to this de nition a rank 1 matrix is 11
a matrix M of shape vT w, where vT is a column vector and w is a row vector. Hence, we can give a second characterization of matrix ranks: k X rankR(M ) = minfk j 9 matrices A1; : : :; Ak of rank 1 such that M = Aig: i=1
The following properties of ranks over semirings are easy to prove. Lemma 8 Let M1; M2 be non-zero m n matrices over R and let h : R ! R0 be a semiring homomorphism. Then 1. rankR0 (h(M1)) rankR(M1 ), 2. rankR(M1 + M2) rankR(M2 ) + rankR(M2 ). Throughout we use the following notation: f denotes a function with domain n n , and M f denotes the corresponding communication matrix, i.e., the jjn jjn matrix with entry f (x; y) in row x and column y. This notation is not restricted to 0-1-valued functions. Rank based lower bound arguments rely on the following lemma. Lemma 9 Let P be a protocol on n n with complexity L. Then rankIN(M accP ) 2t?1; rankIN(M rejP ) 2t?1:
Proof: Obviously for each (x; y) 2 n n holds accP (x; y) = and
rejP (x; y) =
X
w2f0;1gL;wL =1
X w2f0;1gL;wL =0
PX (x; w) PY (y; w); PX (x; w) PY (y; w);
where PX and PY are de ned as in the proof of Theorem 6. Hence, M accP and M rejP both are products of matrices of sizes jjn 2t?1 and 2t?1 jjn. Additionally we will need the notion of variation ranks introduced in [13]: Let m be a positive integer. Two integer matrices A and B are said to be modm-equivalent if for all indices i, j , Aij 0 mod m () Bij 0 mod m: The modular variation rank var-rankZZ=mZZ(A) is the minimum of all numbers rankZZ=mZZB , where B is an integer matrix which is modm-equivalent to A. 12
Two real matrices A and B with non-zero entries are said to be order-equivalent if for all indices i and j holds Aij > 0 () Bij > 0: Let be a positive natural number and let A be a real matrix with non-zero entries. We de ne var-rank; (A) to be the minimum over all numbers rankIR(B ), where B is a matrix with entries in f1; 2; : : : ; g such that B is order-equivalent to A. We call var-rank; (A) the real variation rank with respect to . Now we are prepared to prove bounds for concrete acceptance modes. Except for the probabilistic acceptance mode the lower bounds are immediate consequences of Lemma 9 together with Lemma 8, Claim 1 applied to the appropriate canonic semiring homomorphism. Therefore we skip a detailed explanation for these lower bounds.
Proposition 10 (Mehlhorn, Schmidt [15]) Let R be an arbitrary semiring. Then cd(f ) log rankR(M f ) : We can fully characterize the modular communication complexity in terms of variation ranks. Proposition 11 Let m 2 be a natural number. Then cmod m(f ) = log var-rankZZ=mZZ(M f ) :
Proof: To prove the upper bound, we choose an integer matrix B which is modm-equivalent to M f , such that r = rankZZ=mZZB = var-rankZZ=mZZ(M f ). Then B = B (1) + : : : + B (r), where the B (k) have ZZ=mZZ-rank 1. This is equivalent to Bxy(k) Ux(k) Vy(k) mod m, for Ux(k); Vy(k) 2 f1; : : : ; mg, and for x; y = 1; : : : ; B . The players execute the following protocol P of length dlog re + 2dlog me + 1 on input (x; y) 2 n n : Player X chooses nondeterministically indices k, 1 k r, and l1, 1 l1 Ux(k), and announces (k; l1). Afterwards Y chooses nondeterministically some index l2, 1 l2 Vy(k), announces l2 and accepts. Clearly, there are Prk=1 Ux(k) Vy(k) Bxy mod m many accepting computations assigned to the input. Hence, f is modm-computed by P .
On the basis of Lemma 3 we can do better in the case of m = p being a prime number. In this case we denote as usual ZZ=pZZ by IFp.
Proposition 12 Let p be a prime number. Then
cmod p(f ) = log rankIFp (M f ) : 13
The upper bounds of the next proposition can be derived as in Proposition 11.
Proposition 13
1. cn(f ) = log rankIB (M f ) ,
2. cco-n(f ) = log rankIB (M 1?f ) .
Finally we consider the probabilistic mode. Lemma 14 Let J be the all-one matrix of size jjn jjn and let t be the length of a probabilistic communication protocol computing f . Then t log var-rank;2t (J ? 2M f ) ? 1: +1
Proof: Given a probabilistic protocol P of complexity t for f , we replace it according
to Lemma 3 by a protocol P 0 of complexity at most t +0 2 that for no input has the same P ?rejP 0 acc number of accepting and rejecting computations. M is obviously order-equivalent to J ? 2M f and its entries are bounded in absolute value by 2t+1. Further the real rank of the matrix is at most 2t+1 by Lemma 8 and Lemma 9 . Lemma 14 di ers considerably in structure from the other rank arguments in this section. Therefore we develop methods to estimate real variation ranks in the next section. The results might be of independent interest.
6 Estimates for the real variation rank
Throughout this section N is a positive integer. We will consider the real vector space IRN and, to stress the geometric nature of the arguments, attach a vector symbol to vectors from this space: ~x. PN We recall some notions q from linear algebra. Let h~x; ~yi = i=1 xiyi be the standard scalar product and k~xk = h~x;~xi the induced norm. If A = (aij )qis a real N N -matrix, then P ja j2 is the l -norm and kAk := sup fk A~ x k j k ~ x k = 1 g is the spectral norm , k A k 2 := 2 i;j ij tr(A) := Pi aii is the trace of the matrix A. A regular matrix A is called orthogonal if and only if A?1 = AT . If ~x and ~y are nonzero vectors, then let 6 (~x; ~y) denote the angle between them. The cosine of this angle is de ned by cos(6 (~x; ~y)) := k~xhk~x; ~yki~yk : 14
If ~x is a vector, and if U IRN is a nonzero linear subspace, then cos(6 (~x; U )) := maxfcos(6 (~x; ~y)) j ~y 2 U g: Let U : IRN ! U be the orthogonal projection of IRN onto the embedded subspace. Obviously, cos(6 (~x; U )) is always non-negative and cos(6 (~x; U )) = cos(6 (~x; U (~x)) if ~x 6? U ; 0 otherwise. We will use the following T well-known facts from linear algebra (see, e.g., [18]): 2 Fact 1: kAk2 = tr A A . Fact 2: p1N kAk2 = kAk if and only if A = d U , where 0 d 2 IR and U is an orthogonal matrix.
Lemma 15 If H is an N N -matrix with entries in f?1; 1g, and A = (xij ) is an N N matrix over IR such that A is order-equivalent to H , and 1 jaij j , for all 1 i; j N , then for any 1 holds var-rank; (H ) kkAkAk : 2 2
2
Proof: Let ~a; ~b 2 IRN be order-equivalent and 1 jaij and 1 jbij for all i 2 f1; : : : ; N g. Then N N X X h~a; ~bi jaij = 1 jaij 1 kak2: Analogously h~a; ~bi 1 kbk2, hence
i=1
i=1
cos2(6 (~a; ~b)) 1 :
(1)
Now consider some N N matrix B = (bij ) with 1 jbij j for all i 2 f1; : : :; N g and such that B is order-equivalent to H (and hence also to A). Let m = rankIRB: It suces to prove m kAkkAk . Denote the transposes of the rows of A and B by ~a1; : : :;~aN and ~b1; : : : ; ~bN . Let U be the subspace spanned by ~b1; : : : ; ~bN and let ~u1; : : : ; ~um be an orthonormal basis of U . We shall estimate the sum m N X X ha~i; u~j i2 S= 2 2
2 2
2
i=1 j =1
from below and above. By the properties of the orthogonal projection for all i holds: Pm ha~ ; u~ i2 2 cos (6 (~ai; U )) = j=1k~a ki 2 j : i By (1) we obtain P 1 cos2(6 (~a ; ~b )) mj=1ha~i; u~j i2 : i i k~a k2 i
15
Consequently, S can be estimated from below as follows: PN k~a k2 i=1 i S: 2 On the other hand, for 1 j m, PNi=1h~a; u~j i2 = PNi=1(~aTi ~uj )2 = kA~ kAk2; by PuNj kk~ai de nition of the spectral norm. Thus S m kAk2; and we conclude that i k mkAk2 or, equivalently, PN k~a k2 m i=1 kAik2 : The claim follows, since PNi=1 k~aik2 = tr(AT A) = kAk22 by Fact 1. =1
2
As an application we prove:
Proposition 16 Let f : n n ! f0; 1g and let A be any real matrix which is order equivalent to J ? 2M f , such that 1 jaij j , for all 1 i; j 2n . Further let A0 be any square submatrix of A. Then
0 cprob(f ) log kkAA0kk2 ? 21 log ? 1:
Proof: For the case A0 = A the assertion follows immediately from Lemma 14 and Lemma 15 by setting H = J ? 2M f and = 2t+1, where t = cprob(f ).
To complete the proof, observe that all arguments so far apply also to a di erent setting of communication games, where the players have to compute the function only on a squareshape subset R C n n. Let f 0 = f jRC , where R and C are the sets of rows 0k k A 0 0 and columns of A . Then we have cprob(f ) log kA0k ) ? 21 log ? 1, since A0 is order equivalent to J 0 ? 2M f 0 , where J 0 is the all-one matrix of appropriate size. Since obviously cprob(f ) cprob(f 0), we are done. 2
By Fact 2 the largest bound is achieved if A0 is of special shape: Corollary 17 Let f and A be as in Proposition 16. If A0 is an N 0 N 0-submatrix of A such that there is some d > 0 and an orthogonal matrix U such that A0 = d U 0 , then cprob(f ) log N 0 ? 21 log ? 1
16
7 Probabilistic versus modular versus nondeterministic mode Theorem 18
1. Let m > 1 be an arbitrary integer. Then PP cc and MODm P cc are incomparable with regard to inclusion. 2. NP cc and co-NP cc are incomparable with regard to inclusion and properly contained in PP cc . In order to prove the theorem we compute the modular as well as the probabilistic communication complexity of the orthogonality test function introduced in VSection 4 and the sequence equality function SEQ de ned by SEQn (x1; : : : ; xn; y1; : : : ; yn) = ni=1(1 ? ((xi + yi) mod 2)): To characterize the modular communication complexity of SEQ; the following lemma supplies a necessary and sucient condition (see Proposition 11). Lemma 19 Let IN denote the identity N N -matrix. Let m be a positive integer and let l m = p1 : : : plrr be its prime factorization. Then var-rankZZ=mZZ (IN ) = dN=re: 1
Proof: First we prove that dN=re is a lower bound. Let A0 be an N N matrix with entries
in ZZ=mZZ that is modm-equivalent to IN such that var-rankZZ=mZZ(IN ) = rankZZ=mZZ(A0) = m. By de nition of the rank there are integer matrices C and D of sizes N m and m N such that A0 = C D mod m. Then the integer matrix A = C D is modm-equivalent to IN and rankZZ(A) = m. By de nition we have, for all indices i; j , aii 6 0 mod m; and aij 0 mod m, if j 6= i: For all i 2 f1; : : : ; N g there is a k 2 f1; : : : ; rg such that aii 6 0 mod plkk . We conclude that there is a factor plkk of m such that aii 6 0 mod plkk for at least dN=re many i. Let for simplicity p := pk and l := lk . We consider a submatrix B obtained from A by deleting all rows and columns i for which aii 0 mod plkk . The remaining set of rows and columns is denoted by I . For each i 2 I let pl?i be the maximal power of p that divides aii. Hence aii 0 mod pl?i ; aii 6 0 mod pl?i +1; aij 0 mod pl ; for all j 2 I , j 6= i. It is sucient to show that det B 6= 0 where the determinant is computed over the reals. Let N 0 = #I . It is easy to see that PN 0 b1;1 : : : bN 0;N 0 6 0 mod pN 0 l+1? i i , but PN 0 b1;(1) : : : bN 0;(N 0) 0 mod pN 0 l+1? i i ; for all permutations of the set f1; : : :; N 0g di erent from the identity permutation. Consequently, PN 0 det B b1;1 : : : bN 0;N 0 6 0 mod pN 0l+1? i i : =1
=1
=1
17
To prove that dN=re is an upper bound let fi = m=(plii ), Fj = (f1; : : :; fj ), and Aj = FjT Fj for i; j 2 f1; : : : ; rg. Let A0 be the unique 0 0-matrix, which, of course, has rank 0. Clearly, Aj mod m is a j j -diagonal matrix of ZZ=mZZ-rank 1, for j 2 f1; : : : ; rg. Let A be an N N -matrix consisting of a chain of bN=rc copies of Ar (and perhaps an additional copy of Ar0 , r0 = N ? r bN=rc) arranged along the main diagonal and lled with zeroes outside this chain. It follows that A mod m is a diagonal N N -matrix, and that rankZZ=mZZ(A mod m) dN=re. Lemma 19 immediately yields: Proposition 20 For arbitrary m, we have
cmod m (SEQn) = (n): Let us consider the orthogonality test function ORTp, for p a prime number. First we need a technical lemma. Let Gnp IFnp be a maximal set of pairwise linear independent vectors. Since there are pn ? 1 non-zero vectors in IFnp and each one has p ? 1 non-zero scalar multiples, we have n jGnpj = pp??11 . Lemma 21 Let ~x; y~ 2 Gnp , ~x 6= ~y. Then 1. #f~z 2 Gnp j ORTpn(~z;~x) = ORTpn (~z; ~y) = 1g = pnp??1?1 ; 2
2. #f~z 2 Gnp j ORTpn(~z;~x) = ORTpn (~z; ~y) = 0g = pn?2 (p ? 1); 3. #f~z 2 Gnp j ORTpn(~z;~x) 6= ORTpn (~z; ~y)g = 2pn?2 :
Proof: The set fz 2 IFnp j ORTpn (~z;~x) = ORTpn(~z; ~y)g is a n ? 2 dimensional linear subspace n
of IFp . The rst claim counts a maximal set of pairwise independent non-zero vectors in this subspace. The second claim follows similarly with the help of the inclusion-exclusion principle. The third equality follows from the rst two.
Proposition 22 Let p be a prime number. Then cprob(ORTpn ) = (n):
18
Proof:
Let N 0 = ppn??11 . We de ne the following N 0 N 0-matrix A0 = (a~x;~y ) indexed by Gnp Gnp : ( a := p ? 1 if ORTpn (~x; ~y) = 1; n ? = a~x;~y := b := ? p +1 otherwise. p n? = (
2) 2
(
2) 2
p
Clearly, A0 is order-equivalent to an N 0 N 0-submatrix of J ? 2M ORTn . We prove that 0 T A A0 = d I for some d > 0. For each x 2 Gnp we have (A0T A0)~x;~x = a2 #f~y 2 Gnp j ORTpn (~x; ~y) = 1g + b2 #f~y 2 Gnp j ORTpn(~x; ~y) = 0g: It is easy to see, that this expression is independent from x. It remains to show that for ~x 6= ~y holds (A0T A0)~x;~y = 0. Obviously (A0T A0)~x;~y = u a2 + v b2 + v ab; where u; v; w are the numbers of vectors ~z 2 Gnp that are, respectively, orthogonal to both, to none, and to exactly one of ~x and ~y. Lemma 21 and some lines of routine checking show, that indeed (A0T A0)~x;~y = 0. The claim follows from Corollary 17 now.
Proof of Theorem 18 1. It is known that cprob(SEQn ) = O(log n) (see, e.g., [14]). Hence by Proposition 20 follows SEQ 2 PP cc n MODm P cc . On the other hand, for p a prime factor of m, it follows from Lemma 4, Corollary 7, and Proposition 22 that ORTp 2 MODm P cc n PP cc . 2. It is known that cn(SEQn ) = (n) ([14]) whereas it is easy to see that cco-n(SEQn ) = O(log n). Hence NP cc 6 co-NP cc and, by considering the negated functions, co-NP cc 6 NP cc. But PP cc is closed under complement by Lemma 3. To prove that NP cc is contained in PP cc , consider a protocol P of complexity t that computes some language in the nondeterministic acceptance mode. Protocol P + 2t obviously computes the same language probabilistically.
8 Relations between modular classes Proposition 23 Let p, q primes with p 6= q. Then
cmod p(ORTqn ) = (n) for in nitely many n: 19
Proof: By Proposition 12 it is sucient to show that
p rankIFq M ORTn = exp( (n)): As in Section 7 let Gnp be a maximal set of pairwise linear independent vectors of IFnp, p and consider the submatrix M 0 of M ORTn indexed by vectors in Gnp . Remember N 0 = (pn ? 1)=(p ? 1): We prove that, for in nitely many n, M 02 is invertible (over IFq = ZZ=qZZ) and, hence, has rank N = 2 (n) : For the moment consider M 0 as integer matrix. Since (M 02)~y;~z = #fx 2 Gnp j ORTpn(~y;~x) = 1 and ORTpn(~x;~z) = 1g we obtain 8 < a := pnp??1?1 if ~y = ~z 0 2 (M )~y;~z = : b := pnp??1?1 else Consequently, M 0 is a matrix of the form 1 0a B . . . b CC C: M 02 = B B@ b ... A a Let B be the N 0 N 0 matrix whose entries are all equal to b and let IN 0 be the N 0 0 N 0 identity0 matrix. The characteristic polynomial of B is p() = det(B ? IN 0 ) = (?1)N ( ? bN 0)N ?1 (see [18]). Therefore M 02 is invertible if and only if p(b ? a) 6 0 mod q. But since a ? b = pn?2 6 0 mod q the latter is the case if and only if pn?2 + N 0 b 6 0 (mod q): For n with n 0 (mod q) and n 2 (mod q ? 1) (2) holds N 0 b 0 (mod q) and, hence, pn?2 + N 0 b 6 0 (mod q): Indeed, due to Fermat's Little Theorem we have pn?2 p(q?1) 1 (mod q), 2 IN. Now, p 1 (mod q) yields N 0 = 1+ p + : : : + pn?1 n 0 (mod q), and p 6 1 (mod q) implies b = (pn?2 ? 1)=(p ? 1) 0 (mod q). Since gcd(q; q ? 1) = 1, we have p N 0 = exp( (n)) = rankIFq (M 0) rankIFq M ORTn ; for the in nitely many n 2 IN satisfying (2). 1
2
Proposition 23 together with Lemma 4 yields Theorem 24 Let p and q be two di erent prime numbers, and let m be a multiple of pq. 1. MODpP cc and MODq P cc are incomparable with regard to inclusion. 2. MODpP cc is properly contained in MODm P cc. 20
9 Nondeterminism versus modularity Proposition 25 Let p be prime. Then
cn(ORTpn) = cco-n(IPpn ) = (n):
Proof: By Proposition 13 it is sucient to show that
p rankIB M ORTn = exp( (n)):
Again, we represent each 1-dimensional subspace of IFnp by a non-zero vector and consider p the N 0 N 0 submatrix M 0 of M ORTn , N = (pn ? 1)=(p ? 1), indexed by elements from Gnp . Recall that a 0-1-matrix has Boolean rank 1 if there are sets A and B of rows and column, such that the entries inside A B are equal to 1 and all other entries are equal to 0. Let M 0jAB be a 1-homogeneous submatrix of M 0, i.e., a submatrix whose entries are equal to 1. Since ORTpn(~a; ~b) = 1 for any ~a 2 A, ~b 2 B we have ORTpn(~a;~x) = 1 for all vectors ~x from the linear subspace L(B ) generated by B . Consequently, the vectors ~a 2 A are elements of the orthogonal complement L(B )? of L(B ). Hence, #A #B (pn?k ? 1) (pk ? 1)=(p ? 1)2 with k = dim L(B ). Since (pn?k ? 1) (pk ? 1)=(p ? 1)2 [(pdn=2e ? 1)=(p ? 1)]2; we can estimate, for even n, the size of each 1-homogeneous submatrix of M 0 by #A #B [(pn=2 ? 1)=(p ? 1)]2: Now we are done. The number of 1-entries in M 0 equals (pn ? 1)(pn?1 ? 1)=(p ? 1)2: We have
p rankIB M ORTn rankIB(M 0) (pn ? 1)(pn?1 ? 1)=(pn=2 ? 1)2 = exp( (n));
since each 1-entry of M 0 has to be covered by at least one 1-homogeneous submatrix of M 0. The next proposition is based on the well-known fact that we can interchange the rows and the columns of the matrix M SDTn in such a way that we get a triangular matrix with 1's in its main diagonal (see [14]). 21
Proposition 26 Let p be a prime number. Then cmod p(SDTn ) = cmod p(BIPn) = (n):
Theorem 27 Let m be a positive natural number, and p be a prime number. Then 1. NP cc and MODp P cc as well as co-NP cc and MODp P cc are incomparable with respect to inclusion. 2. co-NP cc and MODm P cc are incomparable with respect to inclusion.
Proof: 1. By Corollary 7 and Proposition 25 and 26 we know that BIP 2 NP cc n MODpP cc ; ORTp 2 MODp P cc n NP cc ; cc cc SDT 2 co-NP n MODp P ; IPp 2 MODp P cc n co-NP cc : 2. It is easy to see that cco-n(SEQn) = O(log n). Hence, by Proposition 20 SEQ 2 co-NP cc n MODm P cc : On the other hand, let q be a prime number such that qjm. Then by Lemma 4 and by Proposition 25 IPp 2 MODm P cc n co-NP cc :
10 Conclusion
Let p and q be distinct primes and m > 0 a multiple of p q. The known relations between the counting communication complexity classes considered in this paper are shown in the following picture. Solid lines between classes indicate containment and dotted lines indicate non-comparability.
MODm P cc MODp
P cc
PP cc MODq P cc P cc 22
NP cc
co-NP cc
To complete the picture the following problems remain open: 1. For prime m the class MODm P cc is closed under complement. Is this true also for composite m? 2. Let m1 6= m2 be composite numbers and m1 j m2. Is MODm P cc properly contained in MODm P cc ? 1
2
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