By Marco Giunti
This paintings elaborates an image of the appliance of dynamical the right way to the research of cognition. It argues that either computational platforms and connectionist networks are specified varieties of dynamical structures, and indicates how this dynamical method could be utilized to difficulties of cognition.
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Additional info for Computation, dynamics, and cognition
Let me now distinguish two more types of mathematical dynamical systems. MDS is strongly irreversible if and only if there are state transitions g' and gz and two different states x and y such that g'(x) = gz(y) and, for any time w, gw(x) y and gw(y) x; MDS is weakly irre- versible if and only if MDS is logically irreversible and MDS is not strongly irreversible. Many computational systems are strongly irreversible. By contrast, all dynamical models of mechanical systems are reversible, for the time set T of these mathematical dynamical systems is the set of the 30 Computation, Dynamics, and Cognition real numbers.
We then construct a second machine C* by relabeling the symbols and the internal states of C. and by switching left and right. It is intuitively clear that C* and C are essentially the same machine. We can give a formal proof of this fact by considering the two cascades that correspond to C and C*, for these two cascades turn out to be isomorphic in the sense of definition 2. EXAMPLE 4 (changing the alphabet of a Turing machine and switching left and right produces an isomorphic system) If C is an arbitrary Turing machine, let C* be a second Turing machine obtained by relabeling the symbols and the internal states of C, and by switching L and R.
These virtual architectures are supposedly defined in terms of the lower level workings of the implementing machine. The relations of emulation and realization allow us to make this idea precise. 1 will call this system the virtual MDS2 implemented by MDS 1 . The basic idea of this definition is that, first, we can identify each state y of MDS2 with u(y), where u(y) is the state (or set of states) of MDS1 that represents the state y of MDS2. Second, we can then use v(y, t) and gt1 to define the state transition g3t of the virtual system that corresponds to g t 2 (y), where gt2 is an arbitrary state transition of MDS2.
Computation, dynamics, and cognition by Marco Giunti