Kamil Khadiev

Very Narrow Quantum OBDDs and Width Hierarchies for Classical OBDDs

In the paper we investigate a model for computing of Boolean functions – Ordered Binary Decision Diagrams (OBDDs), which is a restricted version of Branching Programs. We present several results on the comparative complexity for several variants of OBDD models. We present some results on the comparative complexity of classical and quantum OBDDs. We consider a partial function depending on a parameter k such that for any k > 0 this function is computed by an exact quantum OBDD of width 2, but any classical OBDD (deterministic or stable bounded-error probabilistic) needs width 2 k + 1. We consider quantum and classical nondeterminism. We show that quantum nondeterminism can be more efficient than classical nondeterminism. In particular, an explicit function is presented which is computed by a quantum nondeterministic OBDD with constant width, but any classical nondeterministic OBDD for this function needs non-constant width. We also present new hierarchies on widths of deterministic and nondeterministic OBDDs. We focus both on small and large widths.

Very narrow quantum OBDDs and width hierarchies for classical OBDDs

In the paper we investigate Ordered Binary Decision Diagrams (OBDDs)–a model for computing Boolean functions. We present a series of results on the comparative complexity for several variants of OBDDmodels.• We present results on the comparative complexity of classical and quantum OBDDs. We consider a partial function depending on a parameter k such that for any k > 0 this function is computed by an exact quantum OBDD of width 2, but any classical OBDD (deterministic or stable bounded-error probabilistic) needs width 2k+1.• We consider quantum and classical nondeterminism. We show that quantum nondeterminismcan bemore efficient than classical nondeterminism. In particular, an explicit function is presented that is computed by a quantum nondeterministic OBDD of constant width but any classical nondeterministic OBDD for this function needs non-constant width.• We also present new hierarchies on widths of deterministic and nondeterministic OBDDs.

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

We consider Quantum OBDD model. It is restricted version of read-once Quantum Branching Programs, with respect to “width” complexity. It is known that maximal complexity gap between deterministic and quantum model is exponential. But there are few examples of such functions. We present method (called “reordering”), which allows to build Boolean function g from Boolean Function f, such that if for f we have gap between quantum and deterministic OBDD complexity for natural order of variables, then we have almost the same gap for function g, but for any order. Using it we construct the total function REQ which deterministic OBDD complexity is \(2^{\varOmega (n/log n)}\) and present quantum OBDD of width \(O(n^2)\). It is bigger gap for explicit function that was known before for OBDD of width more than linear. Using this result we prove the width hierarchy for complexity classes of Boolean functions for quantum OBDDs.

Width hierarchy for k -OBDD of small width

In this paper was explored well known model k-OBDD. There are proven width based hierarchy of classes of boolean functions which computed by k-OBDD. The proof of hierarchy is based on sufficient condition of Boolean function’s non representation as k-OBDD and complexity properties of Boolean function SAF. This function is modification of known Pointer Jumping (PJ) and Indirect Storage Access (ISA) functions.

Extension of the hierarchy for k-OBDDs of small width

In this paper we explore the well-known k-OBDD model of branching programs. We develop a method of representation of the k-OBDD computation process as an “automata-communication protocol” computation process. Our method allows us to extend the hierarchy proved by Bolling-Sauerhoff-Sieling-Wegener in 1996 for k-OBDDs. Moreover, using the PJM function (a modification of well-known PJ and ISA functions), we prove a new hierarchy.

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

The paper examines hierarchies for nondeterministic and deterministic ordered read-ktimes Branching programs. The currently known hierarchies for deterministic k-OBDD models of Branching programs for k = o(n1/2/log3/2n) 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 ktimes Branching programs for

Exact Affine Counter Automata

k = o\left( {\sqrt {\log n} /\log \log n} \right)

Classical and Quantum Computations with Restricted Memory.

k=o(logn/loglogn) 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.