Real-time computing without stable states: a new framework for neural computation based on perturbations

A key challenge for neural modeling is to explain how a continuous stream of multimodal input from a rapidly changing environment can be processed by stereotypical recurrent circuits of integrate-and-fire neurons in real time. We propose a new computational model for real-time computing on time-varying input that provides an alternative to paradigms based on Turing machines or attractor neural networks. It does not require a task-dependent construction of neural circuits. Instead, it is based on principles of high-dimensional dynamical systems in combination with statistical learning theory and can be implemented on generic evolved or found recurrent circuitry. It is shown that the inherent transient dynamics of the high-dimensional dynamical system formed by a sufficiently large and heterogeneous neural circuit may serve as universal analog fading memory. Readout neurons can learn to extract in real time from the current state of such recurrent neural circuit information about current and past inputs that may be needed for diverse tasks. Stable internal states are not required for giving a stable output, since transient internal states can be transformed by readout neurons into stable target outputs due to the high dimensionality of the dynamical system. Our approach is based on a rigorous computational model, the liquid state machine, that, unlike Turing machines, does not require sequential transitions between well-defined discrete internal states. It is supported, as the Turing machine is, by rigorous mathematical results that predict universal computational power under idealized conditions, but for the biologically more realistic scenario of real-time processing of time-varying inputs. Our approach provides new perspectives for the interpretation of neural coding, the design of experiments and data analysis in neurophysiology, and the solution of problems in robotics and neurotechnology.     

Undecidability in the Imitation Game

This paper considers undecidability in the imitation game, the so-called Turing Test. In the Turing Test, a human, a machine, and an interrogator are the players of the game. In our model of the Turing Test, the machine and the interrogator are formalized as Turing machines, allowing us to derive several impossibility results concerning the capabilities of the interrogator. The key issue is that the validity of the Turing test is not attributed to the capability of human or machine, but rather to the capability of the interrogator. In particular, it is shown that no Turing machine can be a perfect interrogator. We also discuss meta-imitation game and imitation game with analog interfaces where both the imitator and the interrogator are mimicked by continuous dynamical systems.     

Natural halting probabilities,partial randomness,and zeta functions

We introduce the zeta number, natural halting probability, and natural complexity of a Turing machine and we relate them to Chaitin’s Omega number, halting probability, and program-size complexity. A classification of Turing machines according to their zeta numbers is proposed: divergent, convergent, and tuatara. We prove the existence of universal convergent and tuatara machines. Various results on (algorithmic) randomness and partial randomness are proved. For example, we show that the zeta number of a universal tuatara machine is c.e. and random. A new type of partial randomness, asymptotic randomness, is introduced. Finally we show that in contrast to classical (algorithmic) randomness—which cannot be naturally characterised in terms of plain complexity—asymptotic randomness admits such a characterisation.     

One Head Machines from a symbolic approach

We consider the Turing Machine as a dynamical system and we study a particular partition projection of it. In this way, we define a language (a subshift) associated to each machine. The classical definition of Turing Machines over a one-dimensional tape is generalized to allow for a tape in the form of a Cayley Graph. We study the complexity of the language of a machine in terms of realtime recognition by putting it in relation with the structure of its tape. In this way, we find a large set of realtime subshifts some of which are proved not to be deterministic in realtime. Sofic subshifts of this class correspond to machines that cannot make arbitrarily large tours. We prove that these machines always have an ultimately periodic behavior when starting with a periodic initial configuration, and this result is proved for any Cayley Graph.     

Continuous-time symmetric Hopfield nets are computationally universal

We establish a fundamental result in the theory of computation by continuous-time dynamical systems by showing that systems corresponding to so-called continuous-time symmetric Hopfield nets are capable of general computation. As is well known, such networks have very constrained Lyapunov-function controlled dynamics. Nevertheless, we show that they are universal and efficient computational devices, in the sense that any convergent synchronous fully parallel computation by a recurrent network of n discrete-time binary neurons, with in general asymmetric coupling weights, can be simulated by a symmetric continuous-time Hopfield net containing only 18n + 7 units employing the saturated-linear activation function. Moreover, if the asymmetric network has maximum integer weight size w(max) and converges in discrete time t*, then the corresponding Hopfield net can be designed to operate in continuous time Theta(t*/epsilon) for any epsilon > 0 such that w(max)2(12n) 相似文献   

Some Bounds on the Computational Power of Piecewise Constant Derivative Systems

We study the computational power of Piecewise Constant Derivative (PCD) systems. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous time. We show that the computation time of these machines can be measured either as a discrete value, called discrete time, or as a continuous value, called continuous time. We relate the two notions of time for general PCD systems. We prove that general PCD systems are equivalent to Turing machines and linear machines in finite discrete time. We prove that the languages recognized by purely rational PCD systems in dimension d in finite continuous time are precisely the languages of the (d-2) th level of the arithmetical hierarchy. Hence the reachability problem of purely rational PCD systems of dimension d in finite continuous time is Σ _d-2 -complete. Received May 1997, and in final form May 1998.     

P systems with active membranes: trading time for space

We consider recognizer P systems having three polarizations associated to the membranes, and we show that they are able to solve the PSPACE-complete problem Quantified 3SAT when working in polynomial space and exponential time. The solution is uniform (all the instances of a fixed size are solved by the same P system) and uses only communication rules: evolution rules, as well as membrane division and dissolution rules, are not used. Our result shows that, as it happens with Turing machines, this model of P systems can solve in exponential time and polynomial space problems that cannot be solved in polynomial time, unless P = SPACE.     

基于量子逻辑的图灵机及其通用性

基于量子逻辑的自动机理论是量子计算模型的一个重要研究方向.该文研究了基于量子逻辑的图灵机(简称量子图灵机)及其一些变形,给出了包括非确定型量子图灵机l-VTM,确定型量子图灵机l-VDTM以及相应类型的多带量子图灵机,并引入量子图灵机基于深度优先与宽度优先识别语言的两种不同定义方式,证明了这两种定义方式在量子逻辑意义下是不等价的.进一步证明了l-VTM、l-VDTM与相应类型的多带量子图灵机之间的等价性.其次,给出了量子递归可枚举语言及量子递归语言的定义,并给出了二者的层次刻画,证明了l-VTM与l-VDTM不等价,但两者作为量子递归语言的识别器是等价的.最后,文中讨论了基于量子逻辑的通用图灵机的存在性问题,给出了一套合理编码系统,证明了基于量子逻辑的通用图灵机在其所取值的正交模格无限时不存在,而在其所取值的正交模格有限时是存在的.     

Unwinding performance and power on Colossus, an unconventional computer

In 1944 the computing machine known as Colossus became operational in support of British cryptanalysis and decryption of German High Command wireless traffic. This first electronic digital and very unconventional computer was not a stored-program general purpose computer in today’s terms, despite printed claims to the contrary. At least one of these asserts Colossus was a Turing machine. While an appropriate Turing machine can simulate the operation of Colossus, that is not an argument for generality of computation. Nor does the behavior of Colossus resemble that of a Turing machine, much less a universal Turing machine (UTM). Nonetheless, we shall see that a UTM could have been implemented on a clustering of the ten Colossus machines installed at Bletchley Park, England, by the end of WWII in 1945. Improvements require even fewer machines. Several advances in input, output, speed, processing, and applications—within the hardware capability of the time and respectful of the specification of Colossus—are also offered.     

Universal dynamical systems

This paper is concerned with the problem of finding a universal dynamical system for all dynamical systems on separable metric spaces. Special care is given to exhibit a universal dynamical system which was used to motivate the definition of a dynamical system. We establish that this class of dynamical systems is topologically as narrow as a system describable by a first-order partial differential equation. We find that a classical solution space of this partial differential equation will serve as the phase space of a universal system for dynamical systems on locally compact separable metric spaces. In fact, the functions in this solution space areC and vanish at infinity. For the remaining dynamical systems on separable metric spaces we find a universal system similar to the shift system exhibited by Bebutov. The marked difference is that there is no restriction on the set of rest points. Further comments concerning the history of this problem follow some basic definitions given in the introduction.     

A Note on Maxwell's Demon and Universal Computation

Abstract

An analogy between a Maxwellian Demon capable of regulating the passage of particles between two chambers and a Turing machine capable of manipulating a tape and its symbols, is made explicit. It is shown that a slightly modified Maxwell's Demon can simulate a Universal Turing Machine.     

Abstract geometrical computation 4: Small Turing universal signal machines

This paper provides several very small signal machines able to perform any computation—in the classical understanding—generated from Turing machines, cellular automata and cyclic tag systems. A halting universal signal machine with 13 meta-signals and 21 collision rules is presented (resp. 15 and 24 for a robust version). If infinitely many signals are allowed to be present in the initial configuration, five meta-signals and seven collision rules are enough to achieve non-halting weak universality (resp. six and nine for a robust version).     

Approximation and universality of fuzzy Turing machines

Fuzzy Turing machines are the formal models of fuzzy algorithms or fuzzy computations. In this paper we give several different formulations of fuzzy Turing machine, which correspond to nondeterministic fuzzy Turing machine using max-＊ composition for some t-norm＊ （or NFTM＊, for short）, nondeterministic fuzzy Turing machine （or NFTM）, deterministic fuzzy Turing machine （or DFTM）, and multi-tape versions of fuzzy Turing machines. Some distinct results compared to those of ordinary Turing machines are obtained. First, it is shown that NFTM＊, NFTM, and DFTM are not necessarily equivalent in the power of recognizing fuzzy languages if the t-norm＊ does not satisfy finite generated condition, but are equivalent in the approximation sense. That is to say, we can approximate an NFTM＊ by some NFTM with any given accuracy; the related constructions are also presented. The level characterization of fuzzy recursively enumerable languages and fuzzy recursive languages are exploited by ordinary r.e. languages and recursive languages. Second, we show that universal fuzzy Turing machine exists in the approximated sense. There is a universal fuzzy Turing machine that can simulate any NFTM＊ on it with a given accuracy.     

Universality and programmability of quantum computers

Manin, Feynman, and Deutsch have viewed quantum computing as a kind of universal physical simulation procedure. Much of the writing about quantum logic circuits and quantum Turing machines has shown how these machines can simulate an arbitrary unitary transformation on a finite number of qubits. The problem of universality has been addressed most famously in a paper by Deutsch, and later by Bernstein and Vazirani as well as Kitaev and Solovay. The quantum logic circuit model, developed by Feynman and Deutsch, has been more prominent in the research literature than Deutsch’s quantum Turing machines. Quantum Turing machines form a class closely related to deterministic and probabilistic Turing machines and one might hope to find a universal machine in this class. A universal machine is the basis of a notion of programmability. The extent to which universality has in fact been established by the pioneers in the field is examined and this key notion in theoretical computer science is scrutinised in quantum computing by distinguishing various connotations and concomitant results and problems.     

On efficient deterministic simulation of turing machine computations below logaspace

It is shown that a bounded language accepted by a nondeterministic Turing machine in spaceS(n o(logn) can be accepted by a deterministic Turing machine in space MAX {(s(nn))², logn}. This can be interpreted as extending Savitch's algorithm below logspace. Results of Alt can be used to show that in the general case the indicated space bound cannot be improved. The result can also be interpreted as a sharpening in the special case of bounded languages of the results of Monien and Sudborough.     

On alternation

Summary Every alternating t(n)-time bounded multitape Turing machine can be simulated by an alternating t(n)-time bounded 1-tape Turing machine. Every nondeterministic t(n)-time bounded 1-tape Turing machine can be simulated by an alternating (n + (t(n)) ^1/2)-timebounded 1-tape Turing machine. For wellbehaved functions t(n) every nondeterministic t(n)-time bounded 1-tape Turing machine can be simulated by a deterministic ((nlogn)^1/2 + (t(n))^1/2)-tape bounded off-line Turing machine. These results improve or extend results by Chandra-Stockmeyer, Lipton-Tarjan and Paterson. A preliminary version of this paper was presented at the 19th IEEE-FOCS     

Minimization strategies for maximally parallel multiset rewriting systems

Maximally parallel multiset rewriting systems (MPMRS) give a convenient way to express relations between unstructured objects. The functioning of various computational devices may be expressed in terms of MPMRS (e.g., register machines and many variants of P systems). In particular, this means that MPMRS are Turing universal; however, a direct translation leads to quite a large number of rules. Like for other classes of computationally complete devices, there is a challenge to find a universal system having the smallest number of rules. In this article we present different rule minimization strategies for MPMRS based on encodings and structural transformations. We apply these strategies to the translation of a small universal register machine (Korec (1996) [9]) and we show that there exists a universal MPMRS with 23 rules. Since MPMRS are identical to a restricted variant of P systems with antiport rules, the results we obtained improve previously known results on the number of rules for those systems.