Towards a Framework for Comparing the Complexity of Robotic Tasks

TL;DR

This paper introduces a formal framework for comparing the complexity of robotic tasks through a notion of reduction and a quantitative measure, supported by algorithms and examples.

Contribution

It formalizes task complexity comparison via reductions and proposes practical algorithms to estimate relative complexity in robotics.

Findings

The reduction notion satisfies key properties like reflexivity and transitivity.

The complexity measure is nonnegative and monotonic.

Algorithms successfully estimate task complexity in analytical and reinforcement learning scenarios.

Abstract

We are motivated by the problem of comparing the complexity of one robotic task relative to another. To this end, we define a notion of reduction that formalizes the following intuition: Task 1 reduces to Task 2 if we can efficiently transform any policy that solves Task 2 into a policy that solves Task 1. We further define a quantitative measure of the relative complexity between any two tasks for a given robot. We prove useful properties of our notion of reduction (e.g., reflexivity, transitivity, and antisymmetry) and relative complexity measure (e.g., nonnegativity and monotonicity). In addition, we propose practical algorithms for estimating the relative complexity measure. We illustrate our framework for comparing robotic tasks using (i) examples where one can analytically establish reductions, and (ii) reinforcement learning examples where the proposed algorithm can estimate the relative complexity between tasks.