On separating the read-k-times branching program hierarchy

Abstract

WC obtain an exponential separation between consecutive levels in the hierarchy of classes of functions computable by syntactic read-k-times branching programs of polynomial size, for all k > 0, as conjectured by various authors [24, 22, 161, For every k, we exhibit two explicit functions that can be computed by linear-sized read-(k+l)-times branching programs but require size to be computed by any read-k-times branching program. The result actually gives the strongest possible separation -the exponential lower bound applies to both nondeterministic read-k-times branching programs and randomized read-k-times branching programs with Zsided error E, for some E > 0. The only previously known results are the separation between k = 1 and k = 2 [6] and a aopnration of non-deterministic read-k from deterministic road-(kink/ In2 + C), where C is some appropriate constant, for each k [IS]. A simple corollary of our results is that randomization is not more powerful than non-determinism for read-k-times branching programs, A combinatorial result that we prove along the way is a "hash-mixing lemma" [12] for fnmilics of hash functions that are almost universal, which may be of independent interest.

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“…The result in this direction that we will use in our paper is the following. (This was proved in more general forms in [9], [13], and [6], and also follows from the results of [10].) Suppose that the rank of the matrix B is r and x resp.…”

“…The algebraic part of this proof (Lemma 3.11) is a theorem proved by Borodin, Razborov, and Smolensky [9] (and in more general forms by Jayram [13] and Beame, Saks, and Jayram [6]). We reduce the problem of giving a quadratic form with the required properties to a question about the ranks of the submatrices (or minors) of the matrix generating the quadratic form in a similar way as is done in [6].…”

“…The following lemma, in more general forms, is proved in [9], [13], [6], and also follows from the results of [10]. To make the paper more self contained we provide a proof.…”

“…Assume that, as before, x and y independently run over large sets and now we want to guarantee that the x T By is not constant. Motivated by similar considerations, quadratic forms were studied by Borodin, Razborov, and Smolensky [9], Jayram [13], and Beame, Saks, and Thatachar [6]. The result in this direction that we will use in our paper is the following.…”

“…The first is the time segmentation method which later was supplemented by the technique of considering changes of the values of variables on two disjoint sets: [8], [6], [4]. The second is the development of the algebraic techniques about quadratic forms based on matrices with rigidity properties, providing explicitly defined functions which were suitable for the lower bound proof techniques mentioned in the first direction of developments: [9], [13], [6]. The present paper uses the techniques of both of these directions.…”