One-Way Communication Complexity and the Nečiporuk Lower Bound on Formula Size

Hartmut Klauck
2007 SIAM journal on computing (Print)  
The Nečiporuk method for proving lower bounds on the size of Boolean formulae is reformulated in terms of one-way communication complexity. We investigate the scenarios of probabilistic formulae, nondeterministic formulae, and quantum formulae. In all cases we can use results about one-way communication complexity to prove lower bounds on formula size. In the latter two cases we newly develop the employed lower bounds for communication complexity. The main results are as follows: A polynomial
more » ... ze gap between probabilistic/quantum and deterministic formulae. A near-quadratic size gap between nondeterministic formulae with access to less resp. (a log factor) more than a (polynomial) threshold on the number of nondeterministic bits. A near quadratic lower bound on quantum formula size, as well as a polynomial separation between the sizes of quantum formulae with and without multiple read random inputs. In the case of quantum formulae we construct a programmable quantum gate with large success probability to validate the lower bound method. The methods for quantum and probabilistic formulae employ a variant of the Nečiporuk bound in terms of the VC-dimension. 1 derive a variation of the Nečiporuk bound in terms of randomized communication complexity and, using results from that area, a combinatorial variant involving the VC-dimension. Applying this lower bound we show a near-quadratic lower bound for probabilistic formula size (corollary 3.7). We also show that there is a function, for which probabilistic formulae are smaller by a factor of √ n than deterministic formulae and even Las Vegas (zero error) formulae (corollary 3.13). This is shown to be the maximal gap provable if the lower bound for deterministic formulae is proven with the Nečiporuk method. Furthermore we observe that the standard Nečiporuk bound actually asymptotically also holds for Las Vegas formulae. We then further generalize the Nečiporuk method for nondeterministic formulae and for quantum formulae. To apply these generalizations we have to provide lower bounds for one-way communication complexity with limited nondeterminism, and for quantum one-way communication complexity. For both measures lower bounds explicitly depending on the one-way restriction were unknown prior to this work. Since the communication problems we investigate are asymmetric (i.e., Bob receives much fewer inputs than Alice) our results show optimal separations between one-and two-round communication complexity for limited nondeterministic and for quantum communication complexity. Such separations have been known previously for deterministic and probabilistic protocols only, see [27, 37] . In the nondeterministic case we give a specific combinatorial argument for the communication lower bound (theorem 5.5). In the quantum case we give a general lower bound method based on the VC-dimension (theorem 5.9), that actually can be extended to the case where the players share prior entanglement. Furthermore we show that exact and Las Vegas quantum one-way communication complexity are never much smaller than deterministic one-way communication complexity for total functions (theorems 5.11/5.12). Then we are ready to give Nečiporuk style lower bound methods for nondeterministic formulae and quantum formulae. In the case of quantum formulae one more ingredient we need is the construction of a high confidence programmable quantum gate (theorem 4.21). Our results for formulae are then as follows. In the nondeterministic case we show that for an explicit function there is a threshold on the amount of nondeterminism, so that a near-quadratic size gap occurs between formulae allowed to make a certain amount of nondeterministic guesses, and formulae allowed a logarithmic factor of non-Communication Complexity and the Nečiporuk Method 3 deterministic guesses more. The threshold is polynomial in the input length (theorem 6.4). For quantum formulae we show a lower bound of Ω(n 2 / log n) beating the best previously known bound given in [38] (theorem 6.11). Also our bound applies to a more general model of quantum formulae, which are allowed to access multiple read random variables. This feature makes these generalized quantum formulae a proper generalization of both quantum formulae and probabilistic formulae. It turns out that we can give a Ω( √ n/ log n) separation between formulae with multiple read random variables and without this option, even if the former are classical and the latter are quantum (corollary 6.6). Thus quantum formulae as defined by Yao are not capable of efficiently simulating classical probabilistic formulae. We show that the VC-dimension variant of the Nečiporuk bound holds for generalized quantum formulae and the standard Nečiporuk bound holds for generalized quantum Las Vegas formulae (theorem 6.10). Preliminaries. 2.
doi:10.1137/s009753970140004x fatcat:nnlcg6twr5e4tgtftlfip6eiqu