Theory Of Turing Machines Research Articles

Molecular system for an exponentially fast growing programmable synthetic polymer

In this paper, we demonstrate a molecular system for the first active self-assembly linear DNA polymer that exhibits programmable molecular exponential growth in real time, also the first to implement “internal” parallel insertion that does not rely on adding successive layers to “external” edges for growth. Approaches like this can produce enhanced exponential growth behavior that is less limited by volume and external surface interference, for an early step toward efficiently building two and three dimensional shapes in logarithmic time. We experimentally demonstrate the division of these polymers via the addition of a single DNA complex that competes with the insertion mechanism and results in the exponential growth of a population of polymers per unit time. In the supplementary material, we note that an “extension” beyond conventional Turing machine theory is needed to theoretically analyze exponential growth itself in programmable physical systems. Sequential physical Turing Machines that run a roughly constant number of Turing steps per unit time cannot achieve an exponential growth of structure per time. In contrast, the “active” self-assembly model in this paper, computationally equivalent to a Push-Down Automaton, is exponentially fast when implemented in molecules, but is taxonomically less powerful than a Turing machine. In this sense, a physical Push-Down Automaton can be more powerful than a sequential physical Turing Machine, even though the Turing Machine can compute any computable function. A need for an “extended” computational/physical theory arises, described in the supplementary material section S1.

On the Semidecidability of the Remote State Estimation Problem

In this article, we consider the decision problem associated with the task of remotely estimating the state of a dynamic plant via a noisy communication channel. Given a machine-readable description of the plant’s and channel’s characteristics, does there exist an algorithm that decides whether remote state estimation is possible? From an analytic point of view, this problem has been shown to involve the zero-error capacity of the communication channel. By applying results from Turing machine theory and zero-error coding, we analyze several related variants of the decision problem mentioned above. Our analysis also incorporates a weakened form of the state estimation objective, which has been shown to depend on the classical Shannon Capacity instead. In the broadest sense, our results yield a fundamental limit to the capabilities of computer-aided design tools and adaptive autonomous systems, assuming they are based on digital hardware.

The turing machine theory for some spinal cord and brain condition, A toxicological - antidotic depurative approach

Aim of this work is to produce a general theory related an new depurative strategy to be devalued for reduce or delay some spinal cord and brain degenerative and inflammatory chronic disease or acute traumatic condition.

Verifiable Stream Computation and Arthur--Merlin Communication

In the setting of streaming interactive proofs (SIPs), a client (verifier) needs to compute a given function on a massive stream of data, arriving online, but is unable to store even a small fraction of the data. It outsources the processing to a third party service (prover) but is unwilling to blindly trust answers returned by this service. Thus, the service cannot simply supply the desired answer; it must convince the verifier of its correctness via a short interaction after the stream has been seen. In this work we study “barely interactive” SIPs. Specifically, we show that one or two rounds of interaction suffice to solve several query problems---including index, median, nearest neighbor search, pattern matching, and range counting---with polylogarithmic space and communication costs. Such efficiency with $O(1)$ rounds of interaction was thought to be impossible based on previous work. On the other hand, we initiate a formal study of the limitations of constant-round SIPs by introducing a new hierarchy of communication models called online interactive proofs (OIPs). The online nature of these models is analogous to the streaming restriction placed upon the verifier in a SIP. We give upper and lower bounds that (1) characterize, up to quadratic blowups, every finite level of the OIP hierarchy in terms of other well-known communication complexity classes, (2) separate the first four levels of the hierarchy, and (3) reveal that the hierarchy collapses to the fourth level. Our study of OIPs reveals marked contrasts and some parallels with the classic Turing machine theory of interactive proofs, establishes limits on the power of existing techniques for developing constant-round SIPs, and provides a new characterization of (nononline) Arthur--Merlin communication in terms of an online model.

Automata theory based on complete residuated lattice-valued logic: Turing machines

Automata theory based on complete residuated lattice-valued logic, called L-valued finite automata (L-VFAs), has been established by the second author in 2001. In view of the importance of Turing machines, in this paper, we establish a theory of Turing machines based on complete residuated lattice-valued logic, which is a continuation of L-VFAs. First, we give the definition of L-valued nondeterministic Turing machines (L-NTMs), and observe that the multitape L-NTMs have the same language-recognizing power as the single-tape L-NTMs. We give some related properties of L-valued Turing machines, and discuss computing with fuzzy letters via L-valued Turing machines. Second, we introduce the concepts of L-valued recursively enumerable languages and L-valued recursive languages, and obtain some equivalent relations. Some results concerning the characterization of n-recursively enumerable sets are given, and the super-computing power of L-valued Turing machines is investigated. We also prove that L-valued deterministic Turing machines and L-NTMs are not equivalent in the sense of recognizing or deciding languages. Finally, we show that there is no universal L-valued Turing machine. However, a universal L-valued Turing machine exists if the membership degrees of L-valued sets are restricted to a finite complete residuated lattice with universal bounds 0 and 1.

A survey of recursive analysis and Moore’s notion of real computation

The theory of analog computation aims at modeling computational systems that evolve in a continuous space. Unlike the situation with the discrete setting there is no unified theory of analog computation. There are several proposed theories, some of them seem quite orthogonal. Some theories can be considered as generalizations of the Turing machine theory and classical recursion theory. Among such are recursive analysis and Moore’s class of recursive real functions. Recursive analysis was introduced by Turing (Proc Lond Math Soc 2(42):230–265, 1936), Grzegorczyk (Fundam Math 42:168–202, 1955), and Lacombe (Compt Rend l’Acad Sci Paris 241:151–153, 1955). Real computation in this context is viewed as effective (in the sense of Turing machine theory) convergence of sequences of rational numbers. In 1996 Moore introduced a function algebra that captures his notion of real computation; it consists of some basic functions and their closure under composition, integration and zero-finding. Though this class is inherently unphysical, much work have been directed at stratifying, restricting, and comparing it with other theories of real computation such as recursive analysis and the GPAC. In this article we give a detailed exposition of recursive analysis and Moore’s class and the relationships between them.

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.

A survey of state vectors

The theory of Turing machines, which is basic to proving the various limitations of computers, is quite well known. The theory of finite state automata, the basic hardware models such as Moore machines, Mealy machines, and their generalizations, are similarly well known. Our purpose here is to survey basic work done between 1958 and 1975 on still another mathematical model for computer science, the state vector model. In this model there are many variables, each with its own current value, rather than a single state; this makes the model closer to what happens in a real computer.

A numerical analysis of two point boundary value problem algorithms

The mathematical modeling of optimal control system problems is a method applied in industry to obtain correct electrical and mechanical design parameters once the system equations have been derived. The algorithms required to implement the control loop for these applications must provide stable, relatively accurate, efficient solutions. The purpose of this paper is to address the computational characteristics which would concern a system designer in the consideration of the selection of an effective algorithm to implement a two-point boundary value problem solution. Three Invariant Imbedding Algorithms are evaluated for a worst case and a best case problem by an adaptation of four methods of analysis. The areas of computer science, numerical analysis and Turing Machine Theory are drawn upon in these methods to implement and compare the computational form of the algorithms. The four analysis techniques indicated consistent results for the three two-point boundary value problem algorithms considered. Applications of two-point boundary value problem algorithms occur in problems of nuclear reactor heat transfer, pollution control, fluidics, vibration and magnetics.

On the complexity of ω-type Turing acceptors

Turing machines are considered as recognizers of sets of infinite (ω-type) sequences, so called ω-languages. The basic results on such ω-type Turing acceptors were presented in a preceding paper. This paper focuses on the theory of deterministic ω-type Turing acceptors (ω-DTA's) which turns out to be crucially different from the ‘classical’ theory of Turing machines. It is shown that there exists no ω-DTA which is universal for all ω-DTA's. Two infinite complexity hierarchies for ω-DTA's are established, the ‘states hierarchy’, corresponding to the number of states in the machine, and the ‘designated sets hierarchy’, corresponding to the number of designated sets of states used in the recognition. Concrete examples of ω-languages characterizing each of the complexity classes are exhibited. Two additional examples of interesting ω-languages are presented: 1. (i) An ω-language which is ‘inherently non-deterministic’, i.e. can be recognized by a non-deterministic Turing acceptor but by no deterministic acceptor. 2. (ii) An ω-language which cannot be recognized even by a non-deterministic Turing acceptor. The above examples are constructed without using diagonalization. Oscillating ω-DTA's, i.e. ω-DTA's which are allowed to oscillate on ω-inputs, are also considered and are shown to be strictly more powerful than non-oscillating ω-DTA's, yet strictly less powerful than non-deterministic ω-Turing acceptors.

ω-Computations on Turing machines

The paper develops the theory of Turing machines as recognizers of infinite (ω-type) input tapes. Various models of ω-type Turing acceptors are considered, varying mainly in their mechanism for recognizing ω-tapes. A comparative study of the models is made. It is shown that regardless of the ω-recognition model considered, non-deterministic ω-Turing acceptors are strictly more powerful than their deterministic counterparts. Canonical forms are obtained for each of the ω-Turing acceptor models. The corresponding families of ω-sets are studied; normal forms and algebraic characterizations are derived for each family.

Structured turing machines

A structured decomposition theorem for Turing machines is given. The nature of the building blocks and the form of the connections allowed suggest a parallel to the Bohm-Jacopini theorem on structured flowcharts. Thus in a broadest sense, there is obtained an independent machine-theoretic restatement of the fundamental precepts of structured programming. At the same time, the characteristics of the decomposition offer several obvious advantages over known results of this type. In particular, the building blocks (the “simple” machines) are seen to perform total and regular word functions. Furthermore, the connections themselves should prove to be useful as pedagogical tools in the Turing machine theory and as a theoretical framework for top-down machine (or algorithmic) design.

Referenced automata and metaregular families

It is well-known that the set of strings (over some alphabet) definable by a read-only Turing machine in time intervals equal to the lengths of the strings is just a regular set over the given alphabet. The situation changes however if the machine is permitted to interrogate an oracle in the manner prescribed by Turing. We recall that in Turing machine theory [2] a computed function f is said to be recursive in a predicate ~ , if the change of state of the machine is permitted to depend upon whether or not the current number n (on the tape) is in ~ . As formalized in [2], there is no loss of computing time involved in consulting and responding to the oracle, but this desirable feature of the oracle may be lost if a particular model of realization (of the oracular operation) is adopted. In what follows, we shall be concerned with the notion of definability (of classes of sets) which results when a one-way read-only Turing machine is permitted to interrogate--in a style different from that of Turing--an oracle in the form of an additional tape R on which the oracular information is stored. R may be viewed as a (one-way infinite) sequence 7a7~73 over an alphabet F. ( I f / ' i s considered as a subset of{0, 1} n then/ may be taken to represent a finite set of predicates 9~i, i = 1,..., n.) We permit the read-only Turing machine to ask, at the ith step of its operation, not whether the number on the input tape is in the oracle's predicate, but whether i is in the oracle's predicate(s). Thus we seek to model our construction on the type of computation which relies upon a table look-up, but which, at each next step of computation, may know only the next table entry. Thus, in contradistinction to Turing's oracle, the device here (which we call a referenced automaton) may be regarded as a real-time oracle. With respect to some initial positioning of the machine (with respect to the reference tape) a set of input strings is defined (accepted); if the initial position is changed, the corresponding machine-reference tape combination defines another set of input strings. Our interest is in the family of sets thus defined. (Such families we call metaregular families.) The referenced automata have recently been considered by Agasandyan [1] and Salomaa [5]. They studied the generalization of regular sets obtained from a time-

Marvin L. Minsky. Recursive unsolvability of Post's problem of “Tag” and other topics in the theory of Turing machines. Annals of mathematics, second series, vol. 74 (1961), pp. 437–455.

Marvin L. Minsky. Recursive unsolvability of Post's problem of “Tag” and other topics in the theory of Turing machines. Annals of mathematics, second series, vol. 74 (1961), pp. 437–455. - Volume 31 Issue 4

Flow diagrams, turing machines and languages with only two formation rules

In the first part of the paper, flow diagrams are introduced to represent inter ah mappings of a set into itself. Although not every diagram is decomposable into a finite numbm of given base diagrams, this becomes hue at a semantical level due to a suitable extension of the given set and of the basic mappings defined in it. Two normalization methods of flow diagrams are given. The first has |hree base diagrams; the second, only two. In the second part of the paper, the second method is applied to 'lhe theory of Turing machines. With every Turing maching provided with a two-way half-tape, ihere is associated a similar machine, doing essentially 'lhe same job, but working on a tape obtained from the first one by interspersing alternate blank squares. The new machine belongs to the family, elsewhere introduced, generated by composition and iteration from the two machines X and R. That family is a proper subfamily of the whole family of Turing machines.

A Perspective View of Discrete Automata and Their Design

Some relationships between the theory of Turing machines and the theory of finite automata are discussed. An attempt is made to indicate some relationships between these mathematical theories and the physical devices called digital computers.

Systems Approach to Theory of Computing Systems

A formal system is introduced which is capable of representing a universal Turing machine. This enables the input, state-output representation of the computing systems that allows the study on a unified basis of the larger systems involving computer-like subsystems. It has been shown how some problems and concepts associated with the theory of Turing machines can be applied to engineering system and vice-versa. This has been made possible by using the proposed formal system.

Recursive Unsolvability of Post's Problem of "Tag" and other Topics in Theory of Turing Machines

The equivalence of the notions of effective computability as based (1) on formal systems (e.g., those of Post), and (2) on computing machines (e.g., those of Turing) has been shown in a number of ways. The main results of this paper show that the same notions of computability can be realized within (1) the highly restricted monogenic formal systems called by Post the Tag systems, and (2) within a peculiarly restricted variant of Turing machine which has two tapes, but can neither write on nor erase these tapes. From these, or rather from the arithmetization device used in their construction, we obtain also an interesting basis for recursive function theory involving programs of only the simplest arithmetic operations. We show first how Turing machines can be regarded as programmed computers. Then by defining a hierarchy of programs which perform certain arithmetic transformations, we obtain the representation in terms of the restricted two-tape machines. These machines, in turn, can be represented in terms of Post normal canonical systems in such a way that each instruction for the machine corresponds to a set of productions in a system which has the monogenic property (for each string in the Post system just one production can operate). This settles the questions raised