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An Internal Model Principle For Robots

Vadim K. Weinstein, Tamara Alshammari, Kalle G. Timperi, Mehdi Bennis, Steven M. LaValle

TL;DR

This paper mathematically describes the emergence of the robot's internal structure isomorphic or bisimulation equivalent to that of the environment, and shows that surprise minimization leads to having an internal model isomorphic to the environment.

Abstract

When designing a robot's internal system, one often makes assumptions about the structure of the intended environment of the robot. One may even assign meaning to various internal components of the robot in terms of expected environmental correlates. In this paper we want to make the distinction between robot's internal and external worlds clear-cut. Can the robot learn about its environment, relying only on internally available information, including the sensor data? Are there mathematical conditions on the internal robot system which can be internally verified and make the robot's internal system mirror the structure of the environment? We prove that sufficiency is such a mathematical principle, and mathematically describe the emergence of the robot's internal structure isomorphic or bisimulation equivalent to that of the environment. A connection to the free-energy principle is established, when sufficiency is interpreted as a limit case of surprise minimization. As such, we show that surprise minimization leads to having an internal model isomorphic to the environment. This also parallels the Good Regulator Principle which states that controlling a system sufficiently well means having a model of it. Unlike the mentioned theories, ours is discrete, and non-probabilistic.

An Internal Model Principle For Robots

TL;DR

This paper mathematically describes the emergence of the robot's internal structure isomorphic or bisimulation equivalent to that of the environment, and shows that surprise minimization leads to having an internal model isomorphic to the environment.

Abstract

When designing a robot's internal system, one often makes assumptions about the structure of the intended environment of the robot. One may even assign meaning to various internal components of the robot in terms of expected environmental correlates. In this paper we want to make the distinction between robot's internal and external worlds clear-cut. Can the robot learn about its environment, relying only on internally available information, including the sensor data? Are there mathematical conditions on the internal robot system which can be internally verified and make the robot's internal system mirror the structure of the environment? We prove that sufficiency is such a mathematical principle, and mathematically describe the emergence of the robot's internal structure isomorphic or bisimulation equivalent to that of the environment. A connection to the free-energy principle is established, when sufficiency is interpreted as a limit case of surprise minimization. As such, we show that surprise minimization leads to having an internal model isomorphic to the environment. This also parallels the Good Regulator Principle which states that controlling a system sufficiently well means having a model of it. Unlike the mentioned theories, ours is discrete, and non-probabilistic.
Paper Structure (7 sections, 20 theorems, 22 equations, 2 figures)

This paper contains 7 sections, 20 theorems, 22 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Connection to other disciplines and related work
  3. Contribution
  4. Surpriseless couplings and bisimulation
  5. Theory for General Deterministic Transition Systems
  6. Application to Internal and External Systems
  7. Examples

Key Result

Theorem 10

The coupling of $\mathcal{F}(U\times Y)$ to any environment $\mathcal{X}=(X,U,f,h)$ is surpriseless.

Figures (2)

  • Figure 1: The environment (a) has four states. Agent can move clockwise and counter-clockwise, but in states 1 and 4 nothing happens, if it tries to go left or right respectively. In (b) the binary tree of action observation sequences is depicted. In most nodes the sensor data is "white" but when the agent is in the left-most state, the data is "green". In (c) we show the minimal sufficient refinement of (b), and in (d) we have taken the quotient of (c) with respect to the refinement. It turns out to be isomorphic to (a), as predicted by Corollary \ref{['cor:Main1']}.
  • Figure 2: Same idea, as in Figure \ref{['fig:Indistinguishable']}, but with a circular environment. In this case Corollary \ref{['cor:GroupAction']} could be applied, with the group $\mathbb{Z}/4\mathbb{Z}$ acting on $\mathcal{X}$.

Theorems & Definitions (37)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • remark thmcounterremark
  • Definition 8
  • Definition 9
  • ...and 27 more