Restricted Branching Programs Research Articles

A Polynomial-Time Construction of a Hitting Set for Read-Once Branching Programs of Width 3

Recently, an interest in constructing pseudorandom or hitting set generators for restricted branching programs has increased, which is motivated by the fundamental issue of derandomizing space-bounded computations. Such constructions have been known only in the case of width 2 and in very restricted cases of bounded width. In this paper, we characterize the hitting sets for read-once branching programs of width 3 by a so-called richness condition. Namely, we show that such sets hit the class of read-once conjunctions of DNF and CNF (i.e. the weak richness). Moreover, we prove that any rich set extended with all strings within Hamming distance of 3 is a hitting set for read-once branching programs of width 3. Then, we show that any almost O(log n)-wise independent set satisfies the richness condition. By using such a set due to Alon et al. (1992) our result provides an explicit polynomial-time construction of a hitting set for read-once branching programs of width 3 with acceptance probability ɛ > 5/6. We announced this result at conferences more than ten years ago, including only proof sketches, which motivated a number of subsequent results on pseudorandom generators for restricted read-once branching programs. This paper contains our original detailed proof that has not been published yet.

Incremental Branching Programs

We propose a new model of restricted branching programs specific to solving GEN problems, which we call incremental branching programs. We show that syntactic incremental branching programs capture previously studied models of computation for the problem GEN, namely marking machines (Cook, S.A. in J. Comput. Syst. Sci. 9(3):308–316, 1974) and Poon’s extension (Proc. of the 34th IEEE Symp. on the Foundations of Computer Science, pp. 218–227, 1993) of jumping automata on graphs (Cook, S.A., Rackoff, C.W. in SIAM J. Comput. 9:636–652, 1980). We then prove exponential size lower bounds for our syntactic incremental model, and for some other variants of branching program computation for GEN. We further show that nondeterministic syntactic incremental branching programs are provably stronger than their deterministic counterpart when solving a natural NL-complete GEN sub-problem. It remains open if syntactic incremental branching programs are as powerful as unrestricted branching programs for GEN problems.

A large lower bound on the query complexity of a simple boolean function

Combinatorial property testing, initiated formally by Goldreich, Goldwasser, and Ron (1998) and inspired by Rubinfeld and Sudan (1996), deals with the relaxation of decision problems. Given a property P the aim is to decide whether a given input satisfies the property P or is far from having the property. For a family of boolean functions f = ( f n ) the associated property is the set of 1-inputs of f. Here, the known lower bounds on the query complexity of properties identified by boolean functions representable by (very) restricted branching programs of small size is improved up to Ω ( n 1 / 2 ) , where n is the input length.

A hierarchy result for read-once branching programs with restricted parity nondeterminism

Restricted branching programs are considered in complexity theory in order to study the space complexity of sequential computations and in applications as a data structure for Boolean functions. In this paper ( ⊕ , k ) -branching programs and ( ∨ , k ) -branching programs are considered, i.e., branching programs starting with a ⊕ - (or ∨ -)node with a fan-out of k whose successors are k read-once branching programs. This model is motivated by the investigation of the power of nondeterminism in branching programs and of similar variants that have been considered as a data structure. Lower bound methods and hierarchy results for polynomial size ( ⊕ , k ) - and ( ∨ , k ) -branching programs with respect to k are presented.

A Lower Bound Technique for Nondeterministic Graph-Driven Read-Once-Branching Programs and Its Applications

We present a new lower bound technique for a restricted branching program model, namely for nondeterministic graph-driven read-once branching programs (g.d.-BP1s). The technique is derived by drawing a connection between ω-nondeterministic g.d.-BP1s and ω-nondeterministic communication complexity (for the nondeterministic acceptance modes ω∈{⋁,⋀,⊕}). We apply the technique in order to prove an exponential lower bound for integer multiplication for ω-nondeterministic well-structured g.d.-BP1s. (For ω=⊕ an exponential lower bound was already obtained in [5] by using a different technique.) Further, we use the lower bound technique to prove for an explicitly defined function which can be represented by polynomial size ω-nondeterministic BP1s that it has exponential complexity in the ω-nondeterministic well-structured g.d.-BP1 model for ω∈{⋁,⊕}. This answers an open question from Brosenne et al., whether the nondeterministic BP1 model is in fact more powerful than the well-structured graph-driven variant.

Hierarchy theorems for kOBDDs and kIBDDs

Almost the same types of restricted branching programs (or binary decision diagrams BDDs) are considered in complexity theory and in applications like hardware verification. These models are read-once branching programs (free BDDs) and certain types of oblivious branching programs (ordered and indexed BDDs with k layers). The complexity of the satisfiability problem for these restricted branching programs is investigated and tight hierarchy results are proved for the classes of functions representable by k layers of ordered or indexed BDDs of polynomial size.

Completeness and Non-Completeness Results with Respect to Read-Once Projections

Several models of restricted branching programs or binary decision diagrams have a lot of applications in hardware verification. These models are investigated here from a complexity theoretical viewpoint. Because of depth restrictions projections are not suitable as reduction type and have to be restricted to read-once projections. Several types of polynomial-size binary decision diagrams have complete problems with respect to polynomial read-once projections. On the contrary it can be proved that the classes of polynomial-size decision trees and polynomial-size read-once branching programs or free binary decision diagrams do not have such complete problems.

New Lower Bounds and Hierarchy Results for Restricted Branching Programs

In unrestricted branching programs all variables may be tested arbitrarily often on each path. But exponential lower bounds are only known if on each path the number of tests of each variable is bounded. We examine branching programs in which for each path the number of variables that are tested more than once is bounded bykbut we do not bound the number of tests of those variables. Using a new lower bound method we can prove that such branching programs become more powerful by increasingkonly by 1: Fork⩽(1−ε)(n/3)(1/3)/log2/3n, whereε>0, we exhibit Boolean functions that can be represented in polynomial size ifkvariables may be tested more than once on each path, but only in exponential size ifk−1 variables may be tested more than once on each path. Therefore, we obtain a tight hierarchy.