Radically Compositional Cognitive Concepts

by   Toby B. St Clere Smithe, et al.
University of Oxford

Despite ample evidence that our concepts, our cognitive architecture, and mathematics itself are all deeply compositional, few models take advantage of this structure. We therefore propose a radically compositional approach to computational neuroscience, drawing on the methods of applied category theory. We describe how these tools grant us a means to overcome complexity and improve interpretability, and supply a rigorous common language for scientific modelling, analogous to the type theories of computer science. As a case study, we sketch how to translate from compositional narrative concepts to neural circuits and back again.



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1 Introduction

How should we conceive of concepts, of cognition, or of circuit computation? We argue: compositionally. Composition is the tool by which we construct complex concepts: constantly, informally, and automatically. But it is not just our concepts that are compositional: it has long been noted that our cognitive architecture is modular (fodor1983modularity; marcus2018deep), and much of cognitive neuroscience relies on the well-tested assumption that this modularity maps onto the structure of the brain. Mathematics itself is increasingly recognized as compositional (kock2006synthetic; bauer2017hott), and strong compositionality is increasingly used in software engineering to improve correctness and code reusability (fischer2017haskell; haftmann2010higher)

. Despite this simplifying power, few models in cognitive science and machine learning are actually compositional, even when modelling compositionality itself (

eg., (chang2016compositional; piantadosi2016logical)).

We propose instead taking compositionality seriously, using the mathematics of composition—category theory—and show how doing so allows us to translate concepts between contexts and levels: from abstract concepts themselves (phillips2010categorial; bolt2019interacting) to their possible realization in a circuit model. This paper is an abridged version of a work in progress, provisionally to appear in Compositionality, and we defer many formal details and proofs to that manuscript. In §2 we introduce the background mathematics. In §LABEL:sec:concepts, we introduce the conceptual setting. §LABEL:sec:circ shows how to translate concepts to circuits, and §LABEL:sec:back suggests how to translate back again.

2 Category theory and the ‘structure of structure’

A category is a very simple structure, capturing only what is necessary to enforce compositionality:

Definition 1.

A category is a set of objects such that for any two objects there is a set of arrows obeying a composition rule: for any arrow and any arrow , there is a composite arrow . Every object has an identity arrow, .

Being so general, almost all concepts can be formalized categorically. For example, there is a category whose objects are parts of speech and whose arrows are grammatical relations; and a category of vector spaces and linear maps. Category theory can also be applied to itself: there is a category

of categories, whose arrows are called functors: