On the hierarchies for deterministic, nondeterministic and probabilistic ordered read-k-times branching programs

Abstract

Abstract. The paper examines hierarchies for nondeterministic and deterministic ordered read-k-times Branching programs. The currently known hierarchies for deterministic k-OBDD models of Branching programs for k = o(n 1/2 / log 3/2 n) are proved by B. Bollig, M. Sauerhoff, D. Sieling, and I. Wegener in 1998. Their lower bound technique was based on communication complexity approach. For nondeterministic k-OBDD it is known that, if k is constant then polynomial size k-OBDD computes same functions as polynomial size OBDD (The result of Brosenne, Homeister and Waack, 2006). In the same time currently known hierarchies for nondeterministic read k-times Branching programs for k = o( √ log n/ log log n) are proved by Okolnishnikova in 1997, and for probabilistic read k-times Branching programs for k ≤ log n/3 are proved by Hromkovic and Saurhoff in 2003.We show that increasing k for polynomial size nodeterministic k-OBDD makes model more powerful if k is not constant. Moreover, we extend the hierarchy for probabilistic and nondeterministic k-OBDDs for k = o(n/ log n). These results extends hierarchies for read k-times Branching programs, but k-OBDD has more regular structure. The lower bound techniques we propose are a "functional description" of Boolean function presented by nondeterministic k-OBDD and communication complexity technique. We present similar hierarchies for superpolynomial and subexponential width nondeterministic k-OBDDs.Additionally we expand the hierarchies for deterministic k-OBDDs using our lower bounds for k = o(n/ log n). We also analyze similar hierarchies for superpolynomial and subexponential width k-OBDDs.

Lower Bounds and Hierarchies for Quantum Memoryless Communication Protocols and Quantum Ordered Binary Decision Diagrams with Repeated Test

“…Note that due to the definition, k-OBDD is polynomial width iff it is polynomial size. Similar hierarchies are known for classical cases [11,6,17,19]. But for k-QOBDD it is a new result.The paper has the following structure.…”

Reordering Method and Hierarchies for Quantum and Classical Ordered Binary Decision Diagrams

“…We proved similar almost tight hierarchy for polynomial size bounded error probabilistic k-OBDD with error at most 1/3 for k = o(n 1/3 / log n). And almost tight hierarchies for superpolynomial and subexponential size, these results improve results from [Kha16]. Note that, for example for nondeterministic k-OBDD we cannot get result better than [Kha16], because for constant k 1-OBDD of polynomial size and k-OBDD compute the same Boolean functions [BHW06].…”

“…Additionally, we prove almost tight hierarchy for k − OBDD of superpolynomial and subexponential size. These hierarchies improve known not tight hierarchy from [Kha16]. Our hierarchy is almost tight (with small gap), but for little bit smaller k. The proof of hierarchis is based on complexity properties of Boolean function Reordered Pointer Jumping, it is modification of Pointer Jumping function from [NW91], [BSSW98], is based on ideas of reordering method.…”

“…They proved that P-(k−1)OBDD P-kOBDD for k = o( √ n log 3/2 n). It is known extension of this hierarchy for k = o(n/ log 2 n) in papers [Kha16], [AK13]. But this hierarchy is not tight with the gap between classes in heararchy at least not constant.…”

“…Our hierarchy is almost tight (with small gap), but for little bit smaller k. The proof of hierarchis is based on complexity properties of Boolean function Reordered Pointer Jumping, it is modification of Pointer Jumping function from [NW91], [BSSW98], is based on ideas of reordering method. For probabilistic case it is not known tight hierarchy for polynomial size only for sublinear width [Kha16]. Additionally, for more general model Probabilistic k-BP Hromkovich and Sauerhoff in 2003 [HS03] proved the tight hierarchy for k ≤ log n/3.…”