Hierarchies of turing machines with restricted tape alphabet size

It is shown that for any real constants b>a≥0, multitape Turing machines operating in space L1(n)=[bnr] can accept more sets than those operating in space L2(n)=[anr] provided the number of work tapes and tape alphabet size are held fixed. It is also shown that Turing machines with k+1 work tapes are more powerful than those with k work tapes if the tape alphabet size and the amount of work space are held constant

Finite automata with multiplication

AbstractA finite automaton with multiplication (FAM) is a finite automaton with a register which is capable of holding any positive rational number. The register can be multiplied by any of a fixed number of rationals and can be tested for value 1. Closure properties and decision problems for various types of FAM's (e.g. two-way, one-way, nondeterministic, deterministic) are investigated. In particular, it is shown that the languages recognized by two-way deterministic FAM's are of tape complexity log n and time complexity n3. Some decision problems related to vector addition systems are also studied

An Efficient and Exponentially Accurate Parallel h-p Spectral Element Method for Elliptic Problems on Polygonal Domains - The Dirichlet Case

For smooth problems spectral element methods (SEM) exhibit exponential convergence and have been very successfully used in practical problems. However, in many engineering and scientific applications we frequently encounter the numerical solutions of elliptic boundary value problems in non-smooth domains which give rise to singularities in the solution. In such cases the accuracy of the solution obtained by SEM deteriorates and they offer no advantages over low order methods. A new Parallel h-p Spectral Element Method is presented which resolves this form of singularity by employing a geometric mesh in the neighborhood of the corners and gives exponential convergence with asymptotically faster results than conventional methods. The normal equations are solved by the Preconditioned Conjugate Gradient (PCG) method. Except for the assemblage of the resulting solution vector, all computations are done on the element level and we don't need to compute and store mass and stiffness like matrices. The technique to compute the preconditioner is quite simple and very easy to implement. The method is based on a parallel computer with distributed memory and the library used for message passing is MPI. Load balancing issues are discussed and the communication involved among the processors is shown to be quite small