Generalized Turing Machine Research Articles

Logical depth for reversible Turing machines with an application to the rate of decrease in logical depth for general Turing machines

The logical depth of a reversible Turing machine equals the shortest running time of a shortest program for it. This is applied to show that the result in [1] is valid notwithstanding the error noted in Corrigendum [7].

Systems with Partially Freezed Components

We study a particular class of switching systems called systems with partially freezed components. Fix an autonomous system of differential equations and consider a family of systems obtained from it by replacing some of components of the right-hand side by zeros (thus, “freezing” some components of a solution). We obtain the flow of a system with partially freezed components by fixing a sequence of switching times and of the corresponding sets of “active” components (i.e., the components that are not “freezed”). Such a sequence is called a freezing sequence. Systems with partially freezed components appear, for example, in the study of generalized Turing machines. In this paper, we are mostly interested in stability properties of such systems; our main goal is to relate such properties to properties of the generating system and freezing sequences.

Quantum turing machine and brain model represented by Fock space

The adaptive dynamics is known as a new mathematics to treat with a complex phenomena, for example, chaos, quantum algorithm and psychological phenomena. In this paper, we briefly review the notion of the adaptive dynamics, and explain the definition of the generalized Turing machine (GTM) and recognition process represented by the Fock space. Moreover, we show that there exists the quantum channel which is described by the GKSL master equation to achieve the Chaos Amplifier used in [M. Ohya and I. V. Volovich, J. Opt. B 5(6) (2003) 639., M. Ohya and I. V. Volovich, Rep. Math. Phys. 52(1) (2003) 25.]

Hierarchic Information Systems in a Search for Methods to Transcend Limitations of Complexity

The ability of an agent (natural or artificial) to overcome limitations caused by complexity can be identified with intelligence. Yet, the study of complexity is dominated by the issues not really associated with authentic intelligence. In search of the methods to overcome limitations of complexity it is necessary to find a sufficiently general conceptual framework for its study, to identify its characteristics and sources, and then to explore alternatives to the currently used methods. The present paper is using for this purpose the conceptual framework of information, its integration, and dynamics developed by the author in his earlier publications. Using this framework, complexity is characterized in both quantitative and qualitative (structural) ways, and in both static and dynamic perspectives. The main objective is to propose an approach to transcending limitations of complexity through reverse engineering of the effectiveness in overcoming complexity by natural, living organisms. Since the most striking characteristic of life is its hierarchic organization, the use of multi-level hierarchic information systems in information processing is explored. Theoretical design of such multi-level systems becomes possible with the use of generalized Turing machines (symmetric or s-machines) as components performing concurrent computation on all levels of the hierarchy.

Turing machines on represented sets, a model of computation for Analysis

We introduce a new type of generalized Turing machines (GTMs), which are intended as a tool for the mathematician who studies computability in Analysis. In a single tape cell a GTM can store a symbol, a real number, a continuous real function or a probability measure, for example. The model is based on TTE, the representation approach for computable analysis. As a main result we prove that the functions that are computable via given representations are closed under GTM programming. This generalizes the well known fact that these functions are closed under composition. The theorem allows to speak about objects themselves instead of names in algorithms and proofs. By using GTMs for specifying algorithms, many proofs become more rigorous and also simpler and more transparent since the GTM model is very simple and allows to apply well-known techniques from Turing machine theory. We also show how finite or infinite sequences as names can be replaced by sets (generalized representations) on which computability is already defined via representations. This allows further simplification of proofs. All of this is done for multi-functions, which are essential in Computable Analysis, and multi-representations, which often allow more elegant formulations. As a byproduct we show that the computable functions on finite and infinite sequences of symbols are closed under programming with GTMs. We conclude with examples of application.

Modelling of Complex Systems: Systems as Dataflow Machines

We develop a unified functional formalism for modelling complex systems, that is to say systems that are composed of a number of heterogeneous components, including typically software and physical devices. Our approach relies on non-standard analysis that allows us to model continuous time in a discrete way. S ystems are defined as generalized Turing machines with temporized input, internal and output mechanisms. Behaviors of systems are represented by transfer functions. A transfer function is said to be implementable if it is associated with a system. This notion leads us to define a new class - which is natural in our framework - of computable functions on (usual) real numbers. We show that our definitions are robust: on one hand, the class of implementable transfer functions is closed under composition; on the other hand, the class of computable functions in our meaning includes analytical functions whose coefficients are computable in the usual way, and is closed under addition, multiplication, differentiation and integration. Our class of computable functions also includes solutions of dynamical and Hamiltonian systems defined by computable functions. Hence, our notion of system appears to take suitably into account physical systems.

MINSKY'S SMALL UNIVERSAL TURING MACHINE

Marvin L. Minsky constructed a 4-symbol 7-state universal Turing machine in 1962. It was first announced in a postscript to [2] and is also described in [3, Sec. 14.8]. This paper contains everything that is needed for an understanding of his machine, including a complete description of its operation. Minsky's machine remains one of the minimal known universal Turing machines. That is, there is no known such machine which decreases one parameter without increasing the other. However, Rogozhin [6], [7] has constructed seven universal machines with the following parameters: [Formula: see text] His 4-symbol 7-state machine is somewhat different from Minsky's, but all of his machines use a construction similar to that used by Minsky. The following corrections should be noted: First machine, for q 6 00Lq 1 read q 6 00Lq 7; second machine, for q 4 11Rq 4 read q 4 11Rq 10; last machine, for q 2 b 2 bLq 2 read [Formula: see text]. A generalized Turing machine with 4 symbols and 7 states, closely related to Minsky's, was constructed and used in [5].

An O( T log T) reduction from RAM computations to satisfiability

A new method is given for obtaining a boolean expression whose satisfiability is equivalent to the existence of an accepting computation of some nondeterministic machine. Although starting from random access machines, this method gives an expression of the same O( T log T) length as the best reduction from general Turing machines.

Quantization of molecular motion at the synapse and its consequences as contrasted with the generalized turing machine

Using the most general models, we show that molecular motion concerned with transmission of the nervous impulse across the synapse is indeterminate (in the sense of Heisenberg). This occurs despite the assumption of a minimal uncertainty product and the taking into account of the nature of the random walk. Consequences of the interposition of an indeterminate process in nervous system function are detailed by investigation of the operation of a “quantized” Turing machine. Such a generalized algorithm has been taken by some investigators to be the starting point for building analogues to brain function. The brain, with a demonstrated indeterminate basis for at least one of its “levels” of operation, has capabilities or modes of operation not able to be mimed by the most general calculating engine.