A category theory perspective on compositionality and (the development of) cognitive capacity

A category theory perspective on compositionality and (the development of) cognitive capacity Steven Phillips (steve@ni.aist.go.jp) Mathematical Neuroinformatics Group, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8568 JAPAN Abstract A remarkable property of human cognition is the systematic co-occurrence of certain cognitive abilities. One challenge for cognitive science is to determine the (computational) princi- ples that derive this property as part of a broader goal of es- tablishing the foundations of cognitive architecture (i.e. the basic processes and modes of combination affording cogni- tive capacity) for a science of cognition. This paper continues a category theory approach to compositionality and cognitive capacity that posits universal construction (e.g., products) as a fundamental cognitive architectural component. As shown here, products can be modeled in other frameworks, thereby providing a link between an abstract computational principle and a concrete cognitive resource needed for particular capac- ities. For example, a network of weighted connections imple- menting a categorical product uses fewer resources when the number of task instances sharing a common product structure is greater than two; otherwise it is more economic to realize each instance independently. This cross-over may explain why human cognition is not always systematic: the cost of universal construction may not outweigh its expected gain. Keywords: category theory; cognitive capacity; composition- ality; systematicity, similarity; product; universal construction Introduction A remarkable property of human cognition is the systematic co-occurrence of certain cognitive abilities. For example, if one has the ability to infer square as the first shape in the pair (square, triangle), then one also has the ability to infer triangle as the first shape in the pair (triangle, square), assuming that squares and triangles are recognizable. This property is called systematicity (Fodor & Pylyshyn, 1988), and is characterized as having capacity c 1 if and only if c 2 (McLaughlin, 2009), i.e., as an equivalence class of cognitive capacities. For cognitive science, a major challenge has been to ex- plain why cognitive capacity is systematically distributed in a particular (non-arbitrary) way. The main theoretical frame- works propose some form of compositionality to explain sys- tematic cognitive capacity. The classical (symbol system) explanation is that cognitive processes are sensitive to gram- matical structure (Fodor & Pylyshyn, 1988). Thus, the two shape capacities (above) are inseparable because they involve a common process: say, (p, q) → p, where p and q are sym- bols for squares and triangles, on the assumption of having component processes , 4 → p, q for recognizing squares and triangles. 1 Similarly, a connectionist (neural network) account of capacity can make use of common processes in 1 Systematicity pertains to “molecular” not “atomic” capacities, hence that an architecture permits the recognition of squares, but not triangles, is not a counterexample to having the systematicity property (Fodor & Pylyshyn, 1988). the form of activation units and weighted connections mod- eling cognitive processes that are sensitive to spatial struc- ture (Phillips & Wilson, to appear). In this case, the two shape capacities factor through a common (sub)network of weighted connections that map (neuronal) vector representa- tions of shape pairs to first shapes. Although classical and connectionist models can be constructed such that capacity c 1 if and only if c 2 , systematicity doesn’t necessarily follow from classical, or connectionist principles: one can also de- vise models from those principles such that c 1 , but not c 2 . Additional, ad hoc (i.e. arbitrary) assumptions are needed to exclude models that lack systematicity. So, classical and connectionist theories (principles) fail to fully explain sys- tematicity (Aizawa, 2003). Another approach to the systematicity problem (Phillips & Wilson, 2010, 2011, 2012) used a mathematical theory of structure, called category theory (Mac Lane, 2000). In cat- egory theory, a universal construction relates a collection of arrows (interpreted as cognitive processes) via a common me- diating arrow (cognitive process) in a unique way. Hence, all capacities are indivisibly linked to this mediating arrow as the common component. The mediating arrow is associated with an equivalence class of systematically related cognitive capacities (Phillips & Wilson, 2011, Text S4). This paper further explores a categorical approach to com- positionality and cognitive capacity by looking at the relation- ship between universality and cognitive resources. First, our category theory approach to the systematicity problem, and related models, are recalled to motivate its continued use here. Then, the relationship between a specific universal construc- tion (product) and the resources needed for implementation are examined: fewer resources are needed when the number of task instances sharing a product is greater than two; other- wise, it is cheaper to realize each task instance independently. The implications of this relationship for development (learn- ing) are discussed in the final section. Compositionality and universal constructions Definitions of category and product are provided in this sec- tion. Other definitions are provided in Appendix A. Deeper introductions to category theory can be found in many books on the subject (see, e.g., Mac Lane, 2000; Simmons, 2011). Category theory and cognition Category theory starts with a definition of a category, which is a collection of objects and relations between objects, called arrows (or morphisms, or maps). The category theory ap- proach to cognition presented here regards a cognitive archi-