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...

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Computational mechanics, an approach to structural complexity, defines a process’s causal states and gives a procedure for finding them. We show that the causal-state representation—an E-machine—is the minimal one consistent with

To study quantum computation, it might be helpful to generalize structures from language and automata theory to the quantum case. To that end, we propose quantum versions of finite-state and push-down automata, and regular and context-free grammars. We nd analogs of several classical theorems, including pumping lemmas, closure properties, rational and algebraic generating functions, and Greibach normal form. We also show that there are quantum context-free languages that are not context-free, so QCFL != CFL.

Shannon's General Purpose Analog Computer (GPAC) is an elegant model of analog computation in continuous time. In this paper, we consider whether the set G of GPAC-computable functions is closed under iteration, that is, whether for any function f(x) 2 G there is a function F (x; t) 2 G such that F (x; t) = f t (x) for non-negative integers t. We show that G is not closed under iteration, but a simple extension of it is. In particular, if we relax the definition of the GPAC slightly to include unique solutions to boundary value problems, or equivalently if we allow functions x k (x) that sense inequalities in a dierentiable way, the resulting class, which we call G + k , is closed under iteration. Furthermore, G + k includes all primitive recursive functions, and has the additional closure property that if T (x) is in G+k , then any function of x computable by a Turing machine in T (x) time is also.

Continuous-valued recurrent neural networks can learn mechanisms for processing context-free languages. The dynamics of such networks is usually based on damped oscillation around fixed points in state space and requires that the dynamical components are arranged in certain ways. It is shown that qualitatively similar dynamics with similar constraints hold for a n b n c n , a context-sensitive language. The additional difficulty with a n b n c n , compared with the context-free language a n b n , consists of "counting up" and "counting down" letters simultaneously. The network solution is to oscillate in two principal dimensions, one for counting up and one for counting down. This study focuses on the dynamics employed by the Sequential Cascaded Network, in contrast with the Simple Recurrent Network, and the use of Backpropagation Through Time. Found solutions generalize well beyond training data, however, learning is not reliable. The contribution of this ...

This paper presents a theory that enables to interpret natural processes as special purpose analog computers. Since physical systems are naturally described in continuous time, a definition of computational complexity for continuous time systems is required. In analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems, discrete or continuous in time, on the basis of an intrinsic time scale of the system. Dissipative dynamical systems are classified into the computational complexity classes P d , Co-RP d , NP d

In order for neural networks to learn complex languages or grammars, they must have sufficient computational power or resources to recognize or generate such languages. Though many approaches to effectively utilizing the computational power of neural networks have been discussed, an obvious one is to couple a recurrent neural network with an external stack memory- in effect creating a neural network pushdown automata (NNPDA). This NNPDA generalizes the concept of a recurrent network so that the network becomes a more complex computing structure. This paper discusses in detail a NNPDA- its construction, how it can be trained and how useful symbolic information can be extracted from the trained network. To effectively couple the external stack to the neural network, an optimization method is developed which uses an error function that connects the learning of the state automaton of the neural network to the learning of the operation of the external stack: push, pop, and no-operation. To minimize the error function using gradient descent learning, an analog stack is designed such that the action and storage of information in the stack are continuous. One interpretation of a continuous stack is the probabilistic storage of and action on data. After training on sample strings of an unknown source grammar, a quantization procedure extracts from the analog stack and neural network a discrete pushdown automata (PDA). Simulations show that in learning deterministic context-free grammars- the balanced parenthesis language, 1 n 0 n, and the deterministic Palindrome- the extracted PDA is correct in the sense that it can correctly recognize unseen strings of arbitrary length. In addition, the extracted PDAs can be shown to be identical or equivalent to the PDAs of the source grammars which were used to generate the training strings.

The folding architecture together with adequate supervised training algorithms is a special recurrent neural network model designed to solve inductive inference tasks on structured domains. Recently, the generic architecture has been proven as a universal approximator of mappings from rooted labeled ordered trees to real vector spaces. In this article we explore formal correspondences to the automata (language) theory in order to characterize the computational power (representational capabilities) of different instances of the generic folding architecture. As the main result we prove that simple instances of the folding architecture have the computational power of at least the class of deterministic bottom-up tree automata. It is shown how architectural constraints like the number of layers, the type of the activation functions (first-order vs. higher-order) and the transfer functions (threshold vs. sigmoid) influence the representational capabilities. All proofs are carried out in a c...

The recent work on automata whose variables and parameters are real numbers (e.g., Blum, Shub, and Smale, 1989; Koiran, 1993; Bournez and Cosnard, 1996; Siegelmann, 1996; Moore, 1996) has focused largely on questions about computational complexity and tractability. It is also revealing to examine the metric relations that such systems induce on automata via the natural metrics on their parameter spaces. This brings the theory of computational classification closer to theories of learning and statistical modeling which depend on measuring distances between models. With this in mind, I develop a generalized method of identifying pushdown automata in one class of real-valued automata. I show how the real-valued automata can be implemented in neural networks. I then explore the metric organization of these automata in a basic example, showing how it fleshes out the skeletal structure of the Chomsky Hierarchy and indicates new approaches to problems in language learning and language typolog...