The dynamical hypothesis is the claim that cognitive agents are dynamical systems. It stands opposed to the dominant computational hypothesis, the claim that cognitive agents are digital computers. This target article articulates the dynamical hypothesis and defends it as an open empirical alternative to the computational hypothesis. Carrying out these objectives requires extensive clarification of the conceptual terrain, with particular focus on the relation of dynamical systems to computers. Key words cognition, systems, dynamical systems, computers, computational systems, computability, modeling, time. Long Abstract The heart of the dominant computational approach in cognitive science is the hypothesis that cognitive agents are digital computers; the heart of the alternative dynamical approach is the hypothesis that cognitive agents are dynamical systems. This target article attempts to articulate the dynamical hypothesis and to defend it as an empirical alternative to the compu...

We investigate the Church-Kalmar-Kreisel-Turing Theses concerning theoretical (necessary) limitations of future computers and of deductive sciences, in view of recent results of classical general relativity theory. We argue that (i) there are several distinguished Church-Turing-type Theses (not only one) and (ii) validity of some of these theses depend on the background physical theory we choose to use. In particular, if we choose classical general relativity theory as our background theory, then the above mentioned limitations (predicted by these Theses) become no more necessary, hence certain forms of the Church-Turing Thesis cease to be valid (in general relativity). (For other choices of the background theory the answer might be dierent.) We also look at various "obstacles" to computing a non-recursive function (by relying on relativistic phenomena) published in the literature and show that they can be avoided (by improving the "design" of our future computer). We also ask ourselves, how all this reects on the arithmetical hierarchy and the analytical hierarchy of uncomputable functions.

Motivated partly by the resurgence of neural computation research, and partly by advances in device technology, there has been a recent increase of interest in analog, continuous-time computation. However, while special-case algorithms and devices are being developed, relatively little work exists on the general theory of continuous-time models of computation. In this paper, we survey the existing models and results in this area, and point to some of the open research questions. 1 Introduction After a long period of oblivion, interest in analog computation is again on the rise. The immediate cause for this new wave of activity is surely the success of the neural networks "revolution", which has provided hardware designers with several new numerically based, computationally interesting models that are structurally sufficiently simple to be implemented directly in silicon. (For designs and actual implementations of neural models in VLSI, see e.g. [30, 45]). However, the more fundamental...

In the theory of computation on topological algebras there is a considerable gap between so-called abstract and concrete models of computation. In concrete models, unlike abstract models, the computations depend on the representation of the algebra. First, we show that with abstract models, one needs algebras with partial operations, and computable functions that are both continuous and many-valued. This many-valuedness is needed even to compute single-valued functions, and so abstract models must be nondeterministic even to compute deterministic problems. Asanabstract model, we choose the “while”-array programming language, extended with a nondeterministic “countable choice ” assignment, called the WhileCC ∗ model. Using this, we introduce the concept of approximable many-valued computation on metric algebras. For our concrete model, we choose metric algebras with effective representations. Weprove: (1) for any metric algebra A with an effective representation α, WhileCC ∗ approximability implies computability in α, and (2) also the converse, under certain reasonable conditions on A.From (1) and (2) we derive an equivalence theorem between abstract and concrete computation on metric partial algebras. We give examples of algebras where this equivalence holds.

) Steven Awodey 1 Lars Birkedal 2y Dana S. Scott 2z 1 Department of Philosophy, Carnegie Mellon University 2 School of Computer Science, Carnegie Mellon University April 15, 1999 Abstract This work is a step toward developing a logic for types and computation that includes both the usual spaces of mathematics and constructions and spaces from logic and domain theory. Using realizability, we investigate a configuration of three toposes, which we regard as describing a notion of relative computability. Attention is focussed on a certain local map of toposes, which we study first axiomatically, and then by deriving a modal calculus as its internal logic. The resulting framework is intended as a setting for the logical and categorical study of relative computability. 1 Introduction We report here on the current status of research on the Logic of Types and Computation at Carnegie Mellon University [SAB + ]. The general goal of this research program is to develop a logical fra...

We investigate the computational power of continuous-time neural networks with Hopfield-type units. We prove that polynomial-size networks with saturated-linear response functions are at least as powerful as polynomially space-bounded Turing machines. 1 Introduction In a paper published in 1984 [11], John Hopfield introduced a continuoustime version of the neural network model whose discrete-time variant he had discussed in his seminal 1982 paper [10]. The 1984 paper also contains an electronic implementation scheme for the continuous-time networks, and an argument showing that for sufficiently large-gain nonlinearities, these behave similarly to the discrete-time ones, at least when used as associative memories. The power of Hopfield's discrete-time networks as general-purpose computational devices was analyzed in [17, 18]. In this paper we conduct a similar analysis for networks consisting of Hopfield's continuous-time units; however we are at this stage able to analyze only the gen...

We establish a new connection between the two most common traditions in the theory of real computation, the Blum-Shub-Smale model and the Computable Analysis approach. We then use the connection to develop a notion of computability and complexity of functions over the reals that can be viewed as an extension of both models. We argue that this notion is very natural when one tries to determine just how “difficult ” a certain function is for a very rich class of functions. 1

We investigate computability of convex sets restricted to rational inputs. Several quite different algorithmic characterizations are presented and proved to be equivalent, like the existence of effective approximations by polygons or effective line intersection tests. Also obtained are characterizations of recursively enumerable convex sets. 1 Introduction Convex sets play a prominent role in mathematical programming, computational geometry, convex analysis, and many other areas. There is a large number of papers dealing with polynomial time computable convex sets, but the basic question "Which convex sets are computable?" was scarcely studied. In this paper we characterize computable convex sets and show that several quite different approaches lead to the same notion. We consider computations on rational inputs, as e.g. in [4]. This has the advantage that the standard notions of computability and complexity on the natural numbers can be applied. We call a convex set A computable if t...

Abstract computable functions are defined by abstract finite deterministic algorithms on manysorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable functions on any many-sorted algebra; (ii) all functions effectively approximable by abstract computable functions on any metric algebra. We show that there exist universal algebraic specifications for all the classically computable functions on the set R of real numbers. The algebraic specifications used are mainly bounded universal equations and conditional equations. We investigate the initial algebra semantics of these specifications, and derive situations where algebraic specifications precisely define the computable functions.

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