LEARNING ALGEBRAIC REPRESENTATION FOR ABSTRACT SPATIAL-TEMPORAL REASONING Anonymous authors Paper under double-blind review

Abstract

Is intelligence realized by connectionist or classicist? While connectionist approaches have achieved superhuman performance, there has been growing evidence that such task-specific superiority is particularly fragile in systematic generalization. This observation lies in the central debate (Fodor et al., 1988; Fodor & McLaughlin, 1990) between connectionist and classicist, wherein the latter continually advocates an algebraic treatment in cognitive architectures. In this work, we follow the classicist's call and propose a hybrid approach to improve systematic generalization in reasoning. Specifically, we showcase a prototype with algebraic representations for the abstract spatial-temporal reasoning task of Raven's Progressive Matrices (RPM) and present the ALgebra-Aware Neuro-Semi-Symbolic (ALANS 2 ) learner. The ALANS 2 learner is motivated by abstract algebra and the representation theory. It consists of a neural visual perception frontend and an algebraic abstract reasoning backend: the frontend summarizes the visual information from object-based representations, while the backend transforms it into an algebraic structure and induces the hidden operator on-the-fly. The induced operator is later executed to predict the answer's representation, and the choice most similar to the prediction is selected as the solution. Extensive experiments show that by incorporating an algebraic treatment, the ALANS 2 learner outperforms various pure connectionist models in domains requiring systematic generalization. We further show that the algebraic representation learned can be decoded by isomorphism and used to generate an answer.

1. INTRODUCTION

"Thought is in fact a kind of Algebra." -William James (James, 1891) Imagine you are given two alphabetical sequences of "c, b, a" and "d, c, b", and asked to fill in the missing element in "e, d, ?". In nearly no time will one realize the answer to be c. However, more surprising for human learning is that, effortlessly and instantaneously, we can "freely generalize" (Marcus, 2001) the solution to any partial consecutive ordered sequences. While believed to be innate in early development for human infants (Marcus et al., 1999) , such systematic generalizability has constantly been missing and proven to be particularly challenging in existing connectionist models (Lake & Baroni, 2018; Bahdanau et al., 2019) . In fact, such an ability to entertain a given thought and semantically related contents strongly implies an abstract algebra-like treatment (Fodor et al., 1988) ; in literature, it is referred to as the "language of thought" (Fodor, 1975) , "physical symbol system" (Newell, 1980), and "algebraic mind" (Marcus, 2001). However, in stark contrast, existing connectionist models tend only to capture statistical correlation (Lake & Baroni, 2018; Kansky et al., 2017; Chollet, 2019) , rather than providing any account for a structural inductive bias where systematic algebra can be carried out to facilitate generalization. This contrast instinctively raises a question-what constitutes such an algebraic inductive bias? We argue that the foundation of the modeling counterpart to the algebraic treatment in early human development (Marcus, 2001; Marcus et al., 1999) lies in algebraic computations set up on mathematical axioms, a form of formalized human intuition and the starting point of modern mathematical reasoning (Heath et al., 1956; Maddy, 1988) . Of particular importance to the basic building blocks of algebra is the Peano Axiom (Peano, 1889). In the Peano Axiom, the essential components of algebra, the algebraic set and corresponding operators over it, are governed by three statements: (1) the existence of at least one element in the field to study ("zero" element), (2) a successor function that is recursively applied to all elements and can, therefore, span the entire field, and (3) the principle of

