Power Of Nondeterminism Research Articles

The Power of Nondeterminism in Self-Assembly

We investigate the role of nondeterminism in Winfree's abstract Tile Assembly Model (aTAM), which was conceived to model artificial self-assembling systems constructed from DNA. Of particular practical importance is to find tile systems that minimize resources such as the number of distinct tile types, each of which corresponds to a set of DNA strands that must be custom-synthesized in actual implementations of the aTAM. We seek to identify to what extent the use of nondeterminism in tile systems affects the resources required by such shape-building algorithms. We first show a molecular computability theoretic result: there is an infinite shape S that is uniquely assembled by a tile system but not by any deterministic tile system. We then show an analogous phenomenon in the finitary molecular complexity theoretic case: there is a finite shape S that is uniquely assembled by a tile system with c tile types, but every deterministic tile system that uniquely assembles S has more than c tile types. In fact we extend the technique to derive a stronger (classical complexity theoretic) result, showing that the problem of finding the minimum number of tile types that uniquely assemble a given finite shape is Sigma-P-2-complete. In contrast, the problem of finding the minimum number of deterministic tile types that uniquely assemble a shape was shown to be NP-complete by Adleman, Cheng, Goel, Huang, Kempe, Moisset de Espanes, and Rothemund (Combinatorial Optimization Problems in Self-Assembly, STOC 2002). The conclusion is that nondeterminism confers extra power to assemble a shape from a small tile system, but unless the polynomial hierarchy collapses, it is computationally more difficult to exploit this power by finding the size of the smallest tile system, compared to finding the size of the smallest deterministic tile system.

Cluster computing and the power of edge recognition

Though complexity theory already extensively studies path-cardinality-based restrictions on the power of nondeterminism, this paper is motivated by a more recent goal: To gain insight into how much of a restriction, it is of nondeterminism to limit machines to have just one contiguous (with respect to some simple order) interval of accepting paths. In particular, we study the robustness—the invariance under definition changes—of the cluster class CL#P. This class contains each #P function that is computed by a balanced Turing machine whose accepting paths always form a cluster with respect to some length-respecting total order with efficient adjacency checks. The definition of CL#P is heavily influenced by the defining paper’s focus on (global) orders. In contrast, we define a cluster class, CLU#P, to capture what seems to us a more natural model of cluster computing. We prove that the naturalness is costless: CL#P = CLU#P. Then we exploit the more natural, flexible features of CLU#P to prove new robustness results for CL#P and to expand what is known about the closure properties of CL#P. The complexity of recognizing edges—of an ordered collection of computation paths or of a cluster of accepting computation paths—is central to this study. Most particularly, our proofs exploit the power of unique discovery of edges—the ability of nondeterministic functions to, in certain settings, discover on exactly one (in some cases, on at most one) computation path a critical piece of information regarding edges of orderings or clusters.

On the power of randomized multicounter machines

One-way two-counter machines represent a universal model of computation. Here we consider the polynomial-time classes of multicounter machines with a constant number of reversals and separate the computational power of nondeterminism, randomization and determinism. For instance, we show that polynomial-time one-way multicounter machines, with error probability tending to zero with growing input length, can recognize languages that cannot be accepted by polynomial-time nondeterministic two-way multicounter machines with a bounded number of reversals. A similar result holds for the comparison of determinism and one-sided-error randomization, and of determinism and Las Vegas randomization.

On the power of nondeterminism and Las Vegas randomization for two-dimensional finite automata

The goal of this work is to investigate the computational power of nondeterminism and Las Vegas randomization for two-dimensional finite automata. The following three results are the main contribution of this paper: (i) Las Vegas (three-way) two-dimensional finite automata are more powerful than (three-way) two-dimensional deterministic ones. (ii) Three-way two-dimensional nondeterministic finite automata are more powerful than three-way two-dimensional Las Vegas finite automata. (iii) There is a strong hierarchy based on the number of computations (as measure of the degree of nondeterminism) for three-way two-dimensional finite automata. These results contrast with the situation for one-way and two-way finite automata, where all these computation modes have the same acceptance power, and the differences may occur only in the sizes of automata. Results (i) and (ii) provide the first such simultaneous acceptance separation between nondeterminism, Las Vegas, and determinism for a computing model.

The Power of Nondeterminism and Randomness for Oblivious Branching Programs

Abstract. This paper abstracts and generalizes the known approaches for proving lower bounds on the size of various variants of oblivious branching programs (oblivious BPs for short), providing an easy-to-use technique which works for all nondeterministic and randomized modes of acceptance. The technique is applied to obtain the following results concerning the power of nondeterminism and randomness for oblivious BPs: