HALMA: HUMANLIKE ABSTRACTION LEARNING MEETS AFFORDANCE IN RAPID PROBLEM SOLVING

Abstract

Humans learn compositional and causal abstraction, i.e., knowledge, in response to the structure of naturalistic tasks. When presented with a problem-solving task involving some objects, toddlers would first interact with these objects to reckon what they are and what can be done with them. Leveraging these concepts, they could understand the internal structure of this task, without seeing all of the problem instances. Remarkably, they further build cognitively executable strategies to rapidly solve novel problems. To empower a learning agent with similar capability, we argue there shall be three levels of generalization in how an agent represents its knowledge: perceptual, conceptual, and algorithmic. In this paper, we devise the very first systematic benchmark that offers joint evaluation covering all three levels. This benchmark is centered around a novel task domain, HALMA, for visual concept development and rapid problem solving. Uniquely, HALMA has a minimum yet complete concept space, upon which we introduce a novel paradigm to rigorously diagnose and dissect learning agents' capability in understanding and generalizing complex and structural concepts. We conduct extensive experiments on reinforcement learning agents with various inductive biases and carefully report their proficiency and weakness. 1

1. INTRODUCTION

Have you ever heard of Super Halma,foot_1 a fast-paced variant of Halma? In case you have not played Halma or its fast-paced variant before, we briefly introduce both of them here. Halma is a strategic board game, also known as Chinese checkers. The rules of Halma are minimal; it can be perspicuously explained using basic concepts of numbers and arithmetic. To win the game, one needs to transport pawns initially in one's own camp into the target camp. In each turn, a player could either move into an empty adjacent hole and end the play, or jump over an adjacent pawn, place on the opposite side of the jumped pawn, and recursively apply this jump rule till the end of the play. While the standard rules allow hopping over only a single adjacent occupied position at a time, Super Halma allows pieces to catapult over multiple adjacent occupied positions in a line when hopping; see an illustration in Fig. 1 . We will use the term Halma to specifically refer to Super Halma in the remainder of the paper. Now, imagine you are teaching your preschool cousin, Ada, to play Halma. Since she has not yet formed a complete notion of natural numbers or arithmetic, verbally explaining the rules to her will render in vain. Alternatively, you can play with her while providing scarce supervisions, e.g., if a move is allowed; you can even reward her when she successfully moves a pawn to the target camp.



We will make HALMA and tested agents publicly accessible upon publication. See https://en.wikipedia.org/wiki/Chinese_checkers#Variants for details.1



Figure 1: Illustration of the Super Halma playing task. By playing the game with scarce supervision, Ada should be able to learn basic concepts of numbers and arithmetic, such as concepts with both (a) valid and (b) invalid actions (jumps).

