A theory of one-tape (one-head) linear-time Turing machines is essentially different from its polynomial-time counterpart since these machines are closely related to finite state automata. This paper discusses structural-complexity issues of one-tape Turing machines of various types (deterministic, nondeterministic, reversible, alternating, probabilistic, counting, and quantum Turing machines) that halt in linear time, where the running time of a machine is defined as the length of any longest computation path. We explore structural properties of one-tape linear-time Turing machines and clarify how the machines' resources affect their computational patterns and power.

A theory of one-tape (one-head) linear-time Turing machines is essentially different from its polynomial-time counterpart since these machines are closely related to finite state automata. This paper discusses structural-complexity issues of one-tape Turing machines of various types (deterministic, nondeterministic, reversible, alternating, probabilistic, counting, and quantum Turing machines) that halt in linear time, where the running time of a machine is defined as the length of any longest computation path. We explore structural properties of one-tape linear-time Turing machines and clarify how the machines' resources affect their computational patterns and power.is short, then this machine recognizes only a regular language. Using the non-regularity measure of Dwork and Stockmeyer [13], the second claim asserts that any language accepted by a machine with short crossing sequences has constantly-bounded non-regularity. Extending Hennie's argument, Kobayashi [25] later showed that any language recognized by one-tape o(n log n)-time deterministic Turing machines should be regular as well. This time bound o(n log n) is actually optimal since certain one-tape O(n log n)-time deterministic Turing machines can recognize non-regular languages.Unlike polynomial-time computation, one-tape linear-time nondeterministic computation is sensitive to the definition of the machine's running time. Such sensitivity is also observed in average-case complexity theory [44]. By taking his weak definition that defines the running time of a nondeterministic Turing machine to be the length of a "shortest" accepting path, Michel [30] demonstrated that one-tape nondeterministic Turing machines running in linear time (in the sense of his weak definition) solve even NP-complete problems. Clearly, his weak definition gives an enormous power to one-tape nondeterministic machines and therefore it does not seem to offer any interesting features of time-bounded nondeterminism. On the contrary, the strong definition (in Michel's term) requires the running time to be the length of any "longest" (both accepting and rejecting) computation path. This strong definition provides us with a reasonable basis to study the effect of linear-time bounded computations. We therefore adopt his strong definition of running time and, throughout this paper, all one-tape time-bounded Turing machines are assumed to accommodate this strong definition. By expanding Kobayashi's result, we prove that one-tape o(n log n)-time nondeterministic Turing machines recognize only regular languages.The model of alternating Turing machines of Chandra, Kozen, and Stockmeyer [6] naturally expand the model of nondeterministic machines. The number of alternations of such an alternating Turing machine seems to enhance the computational power of the machine; however, our strong definition of running time makes it possible for us to prove that a constant number of alternations do not give any additional computational power to one-tape linear-time alternating Turing machines; namely...

One-particle eigenstates and eigenvalues of two-dimensional electrons in the strong magnetic field with short range impurity and impurities, cosine potential, boundary potential, and periodic array of short range potentials are obtained by magnetic von-Neumann lattice in which Landau level wave functions have minimum spatial extensions. We find that there is a dual correspondence between cosine potential and lattice kinetic term and that the representation based on the von-Neumann lattice is quite useful for solving the system's dynamics.

To help the study of constructing a formal neuron by computer, we propose a logic-based dynamical theory for a genesis of the biological threshold which specific proteins, such as ion channel proteins, or their networks can produce. By viewing such a protein or a protein network as a computational machine, the statements concerning the states of reaction chains which eventually activate or inactivate the protein are treated. By introducing dynamical systems, associated with an inference process based on statements with continuous truth values, we investigated invariant characters of such dynamics and we obtained a sigmoidal function for an invariant distribution function of the truth values. The domain of solutions of the functional equations regarded as representing the self-description of proteins or protein networks as a machine indicates the emergence of a threshold, namely the realization of dyadic value, 0 or 1, based on the continuous truth values. The results obtained may highlight the mechanism of neuronal threshold in a framework which differs from population dynamics. The derived dynamical systems may also provide a simple model of 'demon' rectifying the thermal fluctuations to drive unidirectional movements.

A theory of the integer quantum Hall effect ͑QHE͒ in realistic systems based on a von Neumann lattice is presented. We show that the momentum representation is quite useful and that the quantum Hall regime ͑QHR͒, which is defined by the propagator in the momentum representation, is realized. In the QHR, the Hall conductance is given by a topological invariant of the momentum space and is quantized exactly. The edge states do not modify the value and topological property of xy in the QHR. We next compute the distribution of current based on the effective action and generally find a finite amount of current in the bulk and the edge. Due to the Hall electric field in the bulk, breakdown of the QHE occurs. The critical electric field of the breakdown is proportional to B 3/2 and the proportional constant has no dependence on Landau levels in our theory, in agreement with recent experiments.