Stylised Tensor Product Network Picture

Bill Wilson's Projects in Cognitive Modelling

Pullback diagram

These projects are joint in various combinations with:

Graeme/Brett/Bill/Steve at work
Left to right: Graeme, Brett, Bill, and Steve on research retreat.

Applying Category Theory and Relational Algebra in Aspects of Cognitive Modelling

Some aspects of cognitive modelling transcend representational implementations, whether symbolic or subsymbolic. This work looks for unexpected commonalities arising from the deep mathematical structure of cognitive tasks. This may help to explain why certain tasks are, for no other obvious reason, of similar difficulty.

Some past grants

Publications on cognitive modelling using tensor product networks

Modelling Human Problem Solving Using Evolutionary Computation Techniques

This project, with John Sweller and Nadine Marcus, aims to model John's theory about the development of human problem solving solving skills in a particular area (e.g. solving routine algebra problems or routine physics problems) from a searched-based method at the start to a schemata-based method as expertise grows. The model will use evolutionary computation for the search-based version, and we will investigate the development of the schemata in this context.

Models of relational processing

With Nadine Marcus, I am investigating the problem of using feedforward nets and other neural network models to simulate human relational processing, and in particular the multi-dimensional access property of human relational (semantic) memory.

By multi-dimensional access, we refer to the ability of humans who know a fact like "Jane likes pizza" to answer a range of questions, including "Does Jane like pizza?", "What does Jane like?", "Who likes pizza?" and even "What is the relationship between Jane and pizza?" (though I wouldn't try that one on a five-year-old!) and beyond - "Who likes what?", etc.

A PDP model of human analogical reasoning

The aim of this study is to further develop and test a Parallel Distributed Processing (PDP) model of analogical reasoning, called the Structured Tensor Analogical Reasoning (STAR) model (Halford, Wilson, Guo, Gayler, Wiles & Stewart, 1994). The 1994 STAR model can simulate solution of analogical reasoning problems, but this project will seek to extend the model in the following ways:

The acquisition and processing of relational knowledge

Relational knowledge lies at the core of most higher cognitive processes. It is the basis of mathematical thinking. Number operations, addition and multiplication, may be thought of as ternary relations, since each consists a set of ordered 3-tuples such as +(2,3,5), +(3,4,7), *(2,3,6), *(3,4,12) etc. Relational knowledge is also important in science generally, and the conceptual sophistication which is essential to expertise depends on relational knowledge (e.g. the relationship between force, mass, and acceleration). Relations are the basis of all structured knowledge, since a structure consists of a set of elements on which one or more relations is defined (i.e. a structure is an ordered pair (S,R), where S is a set of elements and R is a set of relations on subsets of S).

Within the psychological literature, the concepts of transitivity, class inclusion, and configurational concepts (oddity, conditional discrimination, transverse patterning, and negative patterning) are all relational concepts. Analogical reasoning, which has been shown to be of fundamental importance to higher cognition in humans is a map from base to target, both of which are coded in terms of relations.

The concept of relational knowledge is also relevant to certain distinctions of historical and contemporary importance. For example, Gestalt Psychology emphasised relations and structure, in opposition to associationism, which did not use relations as a fundamental explanatory construct. As we will see, symbolic processing generally can be defined as processing of relations, in contrast to imaginal codes, implicit processing, or subsymbolic processing. Norman's (1986) observation that "People do seem to have at least two modes of operation, one rapid, efficient, subconscious, the other slow, serial, and conscious." can be accounted for largely by the distinction between associative and relational processing.

However, as Smith (1989) has noted, "... despite all the empirical work, and the importance of relational concepts, there is no unified framework for thinking about their structure and about how they develop". Though these distinctions have been of considerable importance, the precise properties of each mode of processing have remained somewhat obscure.

Our work has indicated that these modes can be defined mathematically, and their essential properties can be captured in neural net models. The first step to doing this is to provide a precise definition of relational processing, then to derive its properties, and contrast them with alternative modes of processing.

The aims of this project are:

Maintained by Bill Wilson
Contact details here.

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