Second-Order Systematicity of Associative Learning: A Paradox for Classical Compositionality and a Coalgebraic Resolution

Systematicity is a property of cognitive architecture whereby having certain cognitive capacities implies having certain other “structurally related” cognitive capacities. The predominant classical explanation for systematicity appeals to a notion of common syntactic/symbolic structure among the systematically related capacities. Although learning is a (second-order) cognitive capacity of central interest to cognitive science, a systematic ability to learn certain cognitive capacities, i.e., second-order systematicity, has been given almost no attention in the literature. In this paper, we introduce learned associations as an instance of second-order systematicity that poses a paradox for classical theory, because this form of systematicity involves the kinds of associative constructions that were explicitly rejected by the classical explanation. Our category theoretic explanation of systematicity resolves this problem, because both first and second-order forms of systematicity are derived from the same categorical construction: universal morphisms, which generalize the notion of compositionality of constituent representations to (categorical) compositionality of constituent processes. We derive a model of systematic associative learning based on (co)recursion, which is an instance of a universal construction. These results provide further support for a category theory foundation for cognitive architecture.

entity and each (directed) edge is labeled with the strength of association from the source associate to the target associate. Note that the returned graph also represents a (first-order) function, which justifies regarding the learning of associations as a second-order function (second-order cognitive capacity). Lists can be built up inductively-a list is either the empty list, or an item prepended to a list. This inductive construction provides the basis for a recursive model of systematic learning of paired associates, i.e., if there are no more paired associates, then return the constructed network, else add the current paired associate to the network obtained from the learning of the remaining list of associates. The following two examples provide a conceptual guide to this (dual) algebraic approach, and also motivate our coalgebraic model. Example 1 (Summing). The summing of a list of numbers can be performed by a type of function called a fold over lists, denoted fold (0, +), which is an instance of a catamorphism [1]. The first argument, 0, assigns the result of zero to the empty list, and the second argument, +, adds the head (first element) of the list to the fold of the tail (remaining elements) of the list. Thus, for example, A categorical treatment of recursion starts with a suitable endofunctor F : C → C from which we construct the category Alg(F ), whose objects are F -algebras, i.e., the pairs (X, ϕ) consisting of objects X and morphisms ϕ : F (X) → X in C, and morphisms are F -algebra homomorphisms, i.e., the , called a catamorphism and denoted (|ϕ| ), since it is uniquely determined by ϕ [1].
Thus an initial algebra is an initial object (dual to terminal object) in Alg(F ).
In greater detail, a catamorphism k : Catamorphism k is denoted (|β| ), by banana brackets [2], since k is determined by β. Catamorphism is also called fold. For lists built from elements of A, catamorphisms are given by We also write (|I v , f | ) as fold (I v , f ).
For our purposes we employ an endofunctor on the category Set for list-related constructions, i.e., • X is the set of (labeled) directed graphs (association networks) G, where each graph g ∈ G is a pair (E, V ) consisting of a set of edges E and a set of vertices V , and each edge is a triple (s, σ, t),

Second-order systematicity of associative learning
4 where s and t are the source and target vertices and σ is the strength of association; • I v is the function I e that assigns the empty list to the empty graph e; and • f is the function assoc that takes a paired associate and a network graph to returns a network is also (recursively) obtained by the composition of three functions whenever the list is not empty. The third function constructs the associative network from the current paired associates and the network (recursively) constructed from the previous list. Thus, for our example, the pair (bread, butter) is added to the network constructed from the list [(fork, knife), (knife, butter)], which is constructed by adding (fork, knife) to the network constructed from the list [(knife, butter)], which is constructed by adding (knife, butter) to the network constructed from the empty list, which is just the empty graph. So, the catamorphism constructs the following network, with associative strengths σ i : Second-order systematicity says that if we can learn one collection of word associations (e.g., in English) then we can learn another collection of associations (e.g., in Japanese), assuming of course capacity for English and Japanese words. Such instances of associative learning are catamorphisms from Second-order systematicity of associative learning 5 the same initial algebra, as described by Diagram 3, i.e., replace X, which is instantiated as the set of graphs on English words in the discussion above, with the set of graphs on Japanese words. Moreover, initial algebras are instances of universal morphisms, specifically initial morphisms.