Showing papers on "Turing machine published in 1991"
TL;DR: The existence of a finite neural network, made up of sigmoidal neurons, which simulates a universal Turing machine, composed of less than 10 5 synchronously evolving processors, interconnected linearly is shown.
388 citations
TL;DR: A mechanism of interneuronal connections in which the stationary state of a chemical neuron is determined by the state of other neurons in a homogeneous chemical system and is thus a "hardware" chemical implementation of neural networks.
Abstract: We propose a reversible reaction mechanism with a single stationary state in which certain concentrations assume either high or low values dependent on the concentration of a catalyst. The properties of this mechanism are those of a McCulloch-Pitts neuron. We suggest a mechanism of interneuronal connections in which the stationary state of a chemical neuron is determined by the state of other neurons in a homogeneous chemical system and is thus a "hardware" chemical implementation of neural networks. Specific connections are determined for the construction of logic gates: AND, NOR, etc. Neural networks may be constructed in which the flow of time is continuous and computations are achieved by the attainment of a stationary state of the entire chemical reaction system, or in which the flow of time is discretized by an oscillatory reaction. In another article, we will give a chemical implementation of finite state machines and stack memories, with which in principle the construction of a universal Turing machine is possible.
278 citations
03 Jan 1991
TL;DR: It is shown that PP is closed under a variety of polynomial-time truth-table reductions and in complexity theory include the definite collapse and (assuming P ? PP) separation of certain query hierarchies over PP.
Abstract: In this seminal paper on probabilistic Turing machines, Gill asked whether the class PP is closed under intersection and union. We give a positive answer to this question. We also show that PP is closed under a variety of polynomial-time truth-table reductions. Consequences in complexity theory include the definite collapse and (assuming P ? PP) separation of certain query hierarchies over PP. Similar techniques allow us to combine several threshold gates into a single threshold gate. Consequences in the study of circuits include the simulation of circuits with a small number of threshold gates by circuits having only a single threshold gate at the root (perceptrons) and a lower bound on the number of threshold gates that are needed to compute the parity function.
215 citations
TL;DR: It is shown that it is undecidable to which class in the hierarchy a cellular automaton belongs and whether all spatially periodic configurations evolve to a fixed point, and there is no computable bound on the period lengths of these configurations.
197 citations
TL;DR: A class of shift-like dynamical systems is presented that displays a wide variety of behaviours, including periodic points, basins of attraction, and time series, and it is shown that they can be embedded in smooth maps in R2, or smooth flows in R3.
Abstract: A class of shift-like dynamical systems is presented that displays a wide variety of behaviours. Three examples are presented along with some general definitions and results. A correspondence with Turing machines allows one to discuss issues of predictability and complexity. These systems possess a type of unpredictability qualitatively stronger than that which has been previously discussed in the study of low-dimensional chaos, and many simple questions about their dynamics are undecidable. The author discusses the complexity of various sets they generate, including periodic points, basins of attraction, and time series. Finally, he shows that they can be embedded in smooth maps in R2, or smooth flows in R3.
177 citations
TL;DR: The semi-unboundedness property leads to the definition of semi- unbounded fan-in circuit families and suggests new characterizations of LOGCFL on the following models: alternating Turing machines, nondeterministic auxiliary pushdown automata, and bounded fan- in Boolean circuits.
151 citations
TL;DR: The complexity of decision problems that can be solved by a polynomial-time Turing machine that makes a bounded number of queries to an NP oracle is studied and the Boolean hierarchy and the bounded query hierarchies either stand or collapse together are studied.
142 citations
02 Jan 1991
TL;DR: In this article, the complexity of finite functions in deterministic and non-deterministic Turing machines has been discussed, where the exact action taken is governed by the current state, the symbols read, and the next-move function of the machine.
Abstract: Publisher Summary This chapter discusses the complexity of finite functions A deterministic Turing machine consists of a finite control and a finite collection of tapes each with a head for reading and writing The finite control is a finite collection of states A tape is an infinite list of cells each containing a symbol Initially, all tapes have blanks except for the first, which contains the input string Once started, the machine goes from state to state, reading the symbols under the heads, writing new ones, and moving the heads The exact action taken is governed by the current state, the symbols read, and the next-move function of the machine This continues until a designated halt state is entered The machine indicates its output by the halting condition of the tapes In a nondeterministic Turing machine, the next-move function is multivalued There can be several computations on a given input and several output values
119 citations
TL;DR: The main purpose of this paper is to survey several properties of alternating, nondeterministic, and deterministic two-dimensional Turing machines (including two- dimension finite automata and marker automata), and to briefly survey cellular types of two- dimensional automata.
114 citations
TL;DR: This paper solves an open problem raised by Stern (1985) — “Is finite-automaton aperiodicity PSPACE-complete?” by providing an affirmative answer and characterize the exact complexity of two other problems considered by Stern: (1) dot-depth-one language recognition and (2) piecewise testable language recognition.
105 citations
TL;DR: An optimal $\Omega (n^2 )$ lower bound is shown for the time-space product of any R branching program that determines those values which occur exactly once in a list of n integers in the range $1,R$ where $R \geqq n$.
Abstract: An optimal $\Omega (n^2 )$ lower bound is shown for the time-space product of any R branching program that determines those values which occur exactly once in a list of n integers in the range $[1,R]$ where $R \geqq n$. This $\Omega (n^2 )$ tradeoff also applies to the sorting problem and thus improves the previous time-space tradeoffs for sorting. Because the R-way branching program is such a powerful model, these time-space product tradeoffs also apply to all models of sequential computation that have a fair measure of space such as off-line multitape Turing machines and off-line log-cost random access machines (RAMs).
01 Jun 1991
TL;DR: In this article, the authors provide a mathematical framework for the analysis of discrete dynamical systems with a discrete state spaces. And they find that Hilbert spaces are the appropriate spaces in which to embed the dynamics.
Abstract: We provide a mathematical framework for the analysis of discrete dynamical systems. These are systems with a discrete state spaces. We find that Hilbert spaces are the appropriate spaces in which to embed the dynamics. In this formulation, it is possible to define and write equations of motion for the system. Although Hilbert spaces are generally associated with quantum systems in physics, they are employed here in a purely classical manner; both dynamical evolution and results of measurements are just as in classical mechanics. Two examples analysed are Euclid''s algorithm and the Turing machine.
19 Nov 1991
TL;DR: The models appear as a most natural transposition in the realm of process algebras of the well known Time Petri Nets of Merlin and Farber and are proposed as a simple, sound and effective basis for timed extensions of the LOTOS standard.
Abstract: A LOTOS-like timed process algebra is first introduced, which offers operators for specifying the urgency of a specified action, but also of an interaction involving two or more processes, and other fundamental time-related behaviours. The formal semantics of the language consists of two independent sets of inference rules which handle, respectively, the occurrence of actions and the passing of time. The language can specify in a natural way the “wait-until-timeout” scenario, and, due to its time related operators, it can simulate Turing machines. A refinement is then presented where one can specify time intervals for the occurrence of actions and interactions. The models appear as a most natural transposition in the realm of process algebras of the well known Time Petri Nets of Merlin and Farber and, as such, are proposed as a simple, sound and effective basis for timed extensions of the LOTOS standard.
TL;DR: By means of exponential lower and polynomial upper bounds for read-once-only Ω-branching programs, the logarithmic space-bounded complexity classes L e, NL e, co-NL e and P e for eraser Turing machines are separated.
01 Jan 1991
TL;DR: This survey contains both old and very recent results in non-quantitative aspects of inductive inference of total recursive functions in selected directions of research performed at the University of Latvia, concentrated on the more explored areas such as the inference of indices innon-Goedel computable numberings.
Abstract: This survey contains both old and very recent results in non-quantitative aspects of inductive inference of total recursive functions. The survey is not complete. The paper was written to stress some of the main results in selected directions of research performed at the University of Latvia rather than to exhaust all of the obtained results. We concentrated on the more explored areas such as the inference of indices in non-Goedel computable numberings, the inference of minimal Goedel numbers, and the specifics of inference of minimal indices in Kolmogorov numberings.
TL;DR: It is shown that there are no unbounded monotone increasing nondeterministically space constructible functions with supn→∞s (n)/log(n)=0, so the space constructibility cannot be used to separate nondeterministic space from deterministic one.
Abstract: The open problem of nondeterministic space constructibility for sublogarithmic functions is resolved. We show that there are no unbounded monotone increasing nondeterministically space constructible functions with supn→∞s(n)/log(n)=0. Consequently, the space constructibility cannot be used to separate nondeterministic space from deterministic one, since functions like loglog(n), and \(\sqrt {\log (n)}\)are not space constructible by nondeterministic Turing machines.
02 Jan 1991
TL;DR: This chapter discusses algebraic complexity theory, which unites two quite different traditions, that of straight-line program or arithmetic circuit or computation sequence and that of computation tree.
Abstract: Publisher Summary This chapter discusses algebraic complexity theory. Complexity theory, as a project of lower bounds and optimality, unites two quite different traditions. The first comes from mathematical logic and the theory of recursive functions. In this, the basic computational model is the Turing machine. The second tradition has developed from questions of numerical algebra. The problems in this typically have a fixed finite size. Consequently, the computational model is based on something like an ordinary computer that however is supplied with the ability to perform any arithmetic operation with infinite precision and that in turn is required to deliver exact results. The formal model is that of straight-line program or arithmetic circuit or computation sequence, more generally that of computation tree.
01 Jun 1991
TL;DR: It is found that under certain conditions the strategies can tolerate errors in the sensory data, recover from pathological states, and converge.
Abstract: A paradigm of iterative, interactive scene segmentation and simplification of random heaps of unknown objects via vision and manipulation is introduced. The scene simplification is based on the graph operations of vertex and edge removal. These operations are defined isomorphic to the pick and push manipulation actions. Sensors are used as graph generators and the manipulator is used as the decomposing mechanism of the graphs. The model is a nondeterministic finite-state Turing machine. A vision system, a manipulator, and force/torque and other sensory input are integrated into a robot work cell. Experiments conducted to test convergence and error recovery of four different strategies are discussed. It is found that under certain conditions the strategies can tolerate errors in the sensory data, recover from pathological states, and converge. >
29 Mar 1991
TL;DR: This paper shows that, for a slightly modified Turing machine model, the low level deterministic and nondeterministic space bounded complexity classes are different and shows that Savitch and Immerman-Szelepcsenyi theorems do not hold in the range lg lg n to lG n.
Abstract: In this paper we review the key results about space bounded complexity classes, discuss the central open problems and outline the relevant proof techniques. We show that, for a slightly modified Turing machine model, the low level deterministic and nondeterministic space bounded complexity classes are different. Furthermore, for this computation model, we show that Savitch and Immerman-Szelepcsenyi theorems do not hold in the range lg lg n to lg n. We also discuss some other computation models to bring out and clarify the importance of space constructibility and establish some results about these models. We conclude by enumerating a few open problems which arise out of the discussion.
TL;DR: In this paper, the authors expose and discuss Penrose's thesis that nature produces harnessable noncomputable processes, but none at the classical level, and suggest a partial counterexample to it, based on aGedanken experiment about an undecidable family of integrable Hamiltonian systems that could lead to a sort of idealized solution to the Halting problem for Turing machines.
Abstract: We expose and discussPenrose's thesis: “Nature produces harnessable noncomputable processes, but none at the classical level.” We then suggest a partial counterexample to it, based on aGedanken experiment about an undecidable family of integrable Hamiltonian systems that could lead to a sort of idealized solution to the Halting problem for Turing machines.
03 Jun 1991
TL;DR: An enhanced timed process algebra called ρ2, which includes only five operators, and preserves the expressivity of ρ1, is proposed: it combines mutual disabling, choice, parallel composition with synchronization, and pure interleaving, into a unique, general-purpose, parametric binary operator.
Abstract: A timed process algebra called ρ1 is introduced, which offers operators for specifying time-dependent behaviours and, in particular, the urgency of a given (inter-)action involving one or more processes. The formal semantics of the language is given in a style similar to the one adopted by Tofts and Moller for TCCS: two independent sets of inference rules are provided, which handle, respectively, the occurrence of actions and the passing of time. The language, partly inspired to LOTOS, can specify in a natural way the “wait-until-timeout” scenario, and we prove that, due to its two time-related operators, it can simulate Turing machines. The formalism appears as a most natural transposition in the realm of process algebras of an expressivity-preserving subset of the well known Time Petri Nets of Merlin and Farber. An enhanced timed process algebra called ρ2, which includes only five operators, and preserves the expressivity of ρ1, is then proposed: it combines mutual disabling, choice, parallel composition with synchronization, and pure interleaving, into a unique, general-purpose, parametric binary operator.
TL;DR: 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.
TL;DR: A variant of program-length complexity is used to formally define the structure of a binary string, where theructure of an object is taken to mean the aggregate of its projectible properties.
TL;DR: The first 4-symbol 7-state universal Turing machine was constructed by Minsky in 1962 as mentioned in this paper, which is the only known machine that decreases one parameter without increasing the other.
Abstract: 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: 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 q600Lq1 read q600Lq7; second machine, for q411Rq4 read q411Rq10; last machine, for q2b2bLq2 read . A generalized Turing machine with 4 symbols and 7 states, closely related to Minsky's, was constructed and used in [5].
08 Jul 1991
TL;DR: It is proven that for any given Turing machine there exists a recurrent neural network with local, second-order, and uniformly connected weights which can simulate it.
Abstract: In principle, a potentially infinitely large neural network (either in number of neurons or in the precision of a single neural activity) could possess an equivalent computational power to a Turing machine. The authors show such an equivalence of Turing machines to several explicitly constructed neural networks. It is proven that for any given Turing machine there exists a recurrent neural network with local, second-order, and uniformly connected weights (i.e., the weights connecting the second-order product of local 'input neurons' with their corresponding 'output neurons') which can simulate it. The numerical implementation and learning of such a neural Turing machine are also discussed. >
TL;DR: It is shown that for certain variations of the problems, they could be either polynomial-time computable or not polynomially computable, depending on different oracles, and the approach of relativization is attacked.
Abstract: The following problems about time-bounded program-size complexity are studied: (1) for a given string x, a size bound s, and a time bound t, whether there exists a Turing machine of size less than or equal to s that prints x in t moves; (2) for two given finite sets Y and Z of strings, a size bound s, and a time bound t, whether there exists a Turing machine of size less than or equal to s that operates in time t and accepts all $y \in Y$ and rejects all $z \in Z$. These problems are fundamental in complexity theory and feasible learning theory. The complexity of these problems is apparently between P and $NP$, but appears very difficult to classify precisely. These problems are attacked by the approach of relativization. It is shown that for certain variations of the problems, they could be either polynomial-time computable or not polynomial-time computable, depending on different oracles. Furthermore, there are oracles relative to which they are not complete for $NP$ under the polynomial-time Turing red...
08 Jul 1991
TL;DR: This work attempts to motivate, justify and survey the average case reduction theory, and shows how the theory can be motivated, justified and surveyed.
Abstract: We attempt to motivate, justify and survey the average case reduction theory.
09 Sep 1991
TL;DR: In this paper, it was shown that sets with no infinite subset in PPP are immune to ACO, and that a significant breakthrough will be necessary in order to prove a bound much better than PPP.
Abstract: We investigate sets that are immune to ACo; that is, sets with no infinite subset in ACo. First we show that such sets exist in PPP. Although this seems like a rather weak result (since ACo is an extremely weak complexity class and PPP contains the entire polynomial hierarchy) we also prove a somewhat surprising theorem, showing that a significant breakthrough will be necessary in order to prove a bound much better than PPP. Namely, we show that any answer to the question:
TL;DR: This work considers one-tape nondeterministic Turing machines, i.e. with a unique worktape on which the input is written, and exhibits nonregular languages accepted by such machines in time n + O(√nlogn) or n +O(log2 n), and an NP-complete language accepted in timen + O (√NLogn) by using a tight padding method.
01 Feb 1991
TL;DR: It is proved that OROW-PRAMs are a fairly robust model and recognize the same class of languages when the model is modified in several ways and that all kinds of PRAMs intertwine with the NC-hierarchy without timeloss.
Abstract: We analyze the owner concept for PRAMs. In OROW-PRAMs each memory cell has one distinct processor that is the only one allowed to write into this memory cell and one distinct processor that is the only one allowed to read from it. By symmetric pointer doubling, a new proof technique for OROW-PRAMs, it is shown that list ranking can be done in O(log n) time by an OROW-PRAM and that LOGSPACE\(\subseteq\) OROW-TIME(log n). Then we prove that OROW-PRAMs are a fairly robust model and recognize the same class of languages when the model is modified in several ways and that all kinds of PRAMs intertwine with the NC-hierarchy without timeloss. Finally it is shown that EREW-PRAMs can be simulated by OREW-PRAMs and ERCW-PRAMs by ORCW-PRAMs.