A Conceptual Complexity Metric Based on Representational Rank

A conceptual complexity metric based on representational rank is proposed. Rank is the number of entities that are bound into a representation, and is related to the number of dimensions, which is a measure of complexity. Each rank corresponds to a class of neural nets. The ranks and typical concepts which belong to them, are: Rank 0, elemental association; Rank 1, content-specific representations and configural associations; Rank 2, unary relations, class membership, variable-constant bindings; Rank 3, binary relations, proportional analogies; Rank 4, ternary relations, transitivity and hierarchical classification; Rank 5, quaternary relations, proportion and the balance scale. Rank 6, quinary relations. Rank 0 can be performed by 2-layered nets, rank 1 by 3-layered nets, and ranks 2-6 by tensor products of the corresponding number of vectors. All animals with nervous systems perform rank 0, vertebrates perform rank 1, primates perform rank 2-3, but ranks 4-6 are uniquely human. Rank also increases with age. We want to propose a theory of conceptual complexity in cognition of humans and higher animals, and to show how properties of major classes of cognitive tasks can be derived from the theory. There are systematic differences between psychological processes depending on their complexity. For example, tasks that are performed by using basic processes such as association have different properties from tasks that require higher cognitive processes. Representational rank corresponds to the number of components of a representation, given that the components retain their identity when bound in a more complex representation. The ranks are summarised in Figure 1. Each rank comprises an equivalence class of cognitive processes of equal structural complexity, and higher ranks correspond to more complex tasks. Each representational rank can be identified with empirical indicators, and with a class of neural nets which can be used to predict properties of the levels. Ranks 0 and 1 are associative while Ranks 26 meet the criteria for symbolic thought (Halford, Wilson, & Phillips, submitted; Phillips & Halford, in press; Phillips, Halford, & Wilson, 1995). Rank 0 corresponds to Elemental associations , which comprise links between pairs of entities: E1 → E2 They are Rank 0 because they can be modelled by nets without representation other than input and output, and they can be implemented by 2-layered nets. Rank 0 can be assessed by any associative learning test, and can be performed by virtually all animals with nervous systems. Rank 1 corresponds to Configural associ ations , which entail links in which one cue is modified by another. They have the form: E1, E2 → E3 Rank 1 would be indicated by configural learning, without isomorphic transfer, by representation of objects, without symbolic representation, and by inferences that do not go beyond the immediate spatio-temporal frame. This level of performance has been demonstrated in a number of mammals (Pearce, 1994; Rescorla, Grau, & Durlach, 1985; Rudy & Sutherland, 1989) and may also be possible in other vertebrates. It can be modeled by 3-layered nets.