Functionally compostional representations
Smolensky and others have developed "functionally compositional" representational schemes, which can account for systematicity without implementing a classical architecture; thereby, avoiding the dilemma.
Such representations can be built up from parts and can be broken back down, but they are non-concatenative—that is, they don’t explicitly contain or “token” their parts.

Supported by “Tensor Products Representations avoid the dilemma” Box 38.

Tim Van Gelder, 1990.

Notes: Van Gelder point out the Gdel numbers also exhibit nonclassical constituent structure. See sidebar, "The steps of Gdel's Proof" on Map 7.

Functional Compositionality: a representational scheme that can produce complex representations from parts and that can decompose a complex representation back down into those parts. The process can be repeated to create increasingly complex representations.

Concatenative Compositionality: functional compositionality, with the added feature that complex representations explicitly contain their parts. Parts are “literally present” or tokens in complex representations. Classical symbolic representations are concatentative in this sense.

Postulates of the Dynamical Approach to Cognition

1) Natural cognitive systems are dynamical systems.

2) The mathematics of dynamical systems provide a general framework for constructing and testing theories of cognition.

3) A dynamical system is a set of changing aspects of the world, represented by variables. “It is, in short, the essence of dynamical models of this kind to describe how processes unfold, moment by moment in real time” (van Gelder and Port, 1995, p.19).

4) A state of the system is the way the variables happen to be at a given point in time.

5) The state space of the system is a set of all states that the system might be in.

6) The behaviour of a dynamical system is governed by differential equations. The differential equations describe how the state of the system changes continuously through time.
7) Dynamical systems exhibit such features as attractors, limit cycles, complexity, bifurcation, and chaos. Many of these features can be visualised—to a point—in graphic presentations.

Note: the dynamical system approach has been applied to many aspects of mind, including development, language, perception, action, and the brain.

Proponents include: Walter Freeman, Timothy van Gelder, Christine Skarda, Robert Port, Christopher Zeeman, Jean Petitot, and Rene Thom. In some sense alll cognitive scientists are dynamicists, to the extent that they accept contemporary mathematics with its dynamical formalisms.

The postulates are adapted from Timothy van Gelder and Robert Port (1995). For a visual discussion of dynamics, see Ralph Abraham and Christopher Shaw (1982).

Artificial Intelligence »Artificial Intelligence
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The Connectionist Dilemma »The Connectionist Dilemma
Connectionist representations avoid the dilemma »Connectionist representations avoid the dilemma
Functionally compostional representations
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