Speed-Up Theorems in Type-2 Computation

Speed-up Theorems in Type-2 Computation Using Oracle Turing Machines

A classic result known as the speed-up theorem in machineindependent complexity theory shows that there exist some computable functions that do not have best programs for them [2, 3]. In this paper we lift this result into type-2 computation. Although the speed-up phenomenon is essentially inherited from type-1 computation, we observe that a direct application of the original proof to our type-...

A Matrix Splitting Perspective on Planning with Options

We show that the Bellman operator underlying the options framework leads to a matrix splitting, an approach traditionally used to speed up convergence of iterative solvers for large linear systems of equations. Based on standard comparison theorems for matrix splittings, we then show how the asymptotic rate of convergence varies as a function of the inherent timescales of the options. This new ...

Priority Arguments in Complexity Theory

machine-independent) complexity theory is basically a theory which establishes limits on what may be said about the complexity of specific recursive functions. (For example, the speed-up theorem [1] shows that it is not always possible to discover close upper and lower bounds for the complexity of a function.) As such, its basic proof method is diagonalization. Sacks, Spector, etc. [2], [3] hav...

Limit theorems and absorption problems for quantum random walks in one dimension

In this paper we consider limit theorems, symmetry of distribution, and absorption problems for two types of one-dimensional quantum random walks determined by 2 × 2 unitary matrices using our PQRS method. The one type was introduced by Gudder in 1988, and the other type was studied intensively by Ambainis et al. in 2001. The difference between both types of quantum random walks is also clarified.

Common Fixed Point Theorems of Integral Type in Modular Spaces