Stanislav Zák

An Exponential Lower Bound for One-Time-Only Branching Programs

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A Superpolynomial Lower Bound for (1, +k(n))-Branching Programs

By (1,+k(n))-branching programs (b. p.s) we mean those b. p.s which during each of their computations are allowed to test at most k(n) input bits repeatedly. For a Boolean function J computable within polynomial time a trade-off has been proven between the number of repeatedly tested bits and the size of each b. p. P which computes J. If at most ≫√n/48(log(c(n)))2⌋ — 1 repeated tests are allowed then the size of P is at least c(n). This yields superpolynomial lower bounds for e. g. (1, +√n/48(log(n)loglog(n))2) -b. p.s and for (1, +√n/48(log(n))4)-b. p.s.

A Hierarchy for (1, +k)-Branching Programs with Respect of k

Branching programs (b. p.s) or decision diagrams are a general graph-based model of sequential computation. The b. p.s of polynomial size are a nonuniform counterpart of LOG. Lower bounds for different kinds of restricted b. p.s are intensively investigated. An important restriction are so called k-b. p.s, where each computation reads each input bit at most k times. Although, for more restricted syntactic k-b.p.s, exponential lower bounds are proven and there is a series of exponential lower bounds for 1-b. p.s, this is not true for general (nonsyntactic) k-b.p.s, even for k = 2. Therefore, so called (1, +k)-b. p.s are investigated. For some explicit functions, exponential lower bounds for (1, +k)-b. p.s are known. Investigating the syntactic (1,+k)-b. p.s, Sieling has found functions fn,k which are polvnomially easy for syntactic (1,+k)-b. p.s, but exponentially hard for syntactic (1,+(k-1))-b. p.s. In the present paper, a similar hierarchy with respect to k is proven for general (nonsyntactic) (1, +k)-b. p.s.

On uncertainty versus size in branching programs

We propose an information-theoretic approach to proving lower bounds on the size of branching programs. The argument is based on Kraft type inequalities for the average amount of uncertainty about (or entropy of) a given input during the various stages of computation. The uncertainty is measured by the average depth of so-called `splitting trees? for sets of inputs reaching particular nodes of the program.We first demonstrate the approach for read-once branching programs. Then, we introduce a strictly larger class of so-called `balanced? branching programs and, using the suggested approach, prove that some explicit Boolean functions cannot be computed by balanced programs of polynomial size. These lower bounds are new since some explicit functions, which are known to be hard for most previously considered restricted classes of branching programs, can be easily computed by balanced branching programs of polynomial size.

Some Notes on the Information Flow in Read-Once Branching Programs

In this paper we describe a lower bounds argument for read-once branching programs which is not just a standard cut-and-paste. The argument is based on a more subtle analysis of the information flow during the individual computations. Although the same lower bound can be also obtained by standard arguments, our proof may be promising because (unlike the cut-and-paste argument) it can potentially be extended to more general models.