Quantifying Morphological Computation

  • [PDF] K. Zahedi and N. Ay, “Quantifying morphological computation,” Entropy, vol. 15, iss. 5, p. 1887–1915, 2013.
    [Bibtex]
    @article{Zahedi2013aQuantifying,
    Author = {Zahedi, Keyan and Ay, Nihat},
    Issn = {1099-4300},
    Journal = {Entropy},
    Number = {5},
    Pages = {1887--1915},
    Pdf = {http://www.mdpi.com/1099-4300/15/5/1887},
    Title = {Quantifying Morphological Computation},
    Volume = {15},
    Year = {2013}}

Abstract

The field of embodied intelligence emphasises the importance of the morphology and environment with respect to the behaviour of a cognitive system. The contribution of the morphology to the behaviour, commonly known as morphological computation, is well-recognised in this community. We believe that the field would benefit from a formalisation of this concept as we would like to ask how much the morphology and the environment contribute to an embodied agent’s behaviour, or how an embodied agent can maximise the exploitation of its morphology within its environment. In this work we derive two concepts of measuring morphological computation, and we discuss their relation to the Information Bottleneck Method. The first concepts asks how much the world contributes to the overall behaviour and the second concept asks how much the agent’s action contributes to a behaviour. Various measures are derived from the concepts and validated in two experiments that highlight their strengths and weaknesses.

http://www.mdpi.com/1099-4300/15/5/1887

In a Nutshell

The following paragraphs will present a very compressed version of the paper. For the full content, please use one of the links presented above.

This is the first paper to discuss information-theoretic quantifications of morphological computation. The focus of this early work was on quantifications that can be evaluated by an embodied agent, i.e., which can be computed from information that is intrinsically available. To derive these measure, we first require a causal model of the sensorimotor loop (citation), which is presented next:


Sensorimotor Loop

The sensorimotor loop describes the interaction of the brain/controller with the sensors, actuators and environment. Everything that is external to be agent, i.e., body and environment, is captured in a single random variable, that we call world. Based on the causal diagram, we can now describe the two concepts that were first introduced in this paper.

Concept 1:
Morphological Computation as the Negative Effect of the Action on the Behaviour

To visualise this concept, we compare the following two one-step versions of the causal model presented in the image above:


Visualisation of Concept 1

The left-hand side is the one-step version of the causal diagram, where W refers to the world (body and environment), S refers to the sensor signal, A refers to the actuator signal and W’ is the next world state. The first concept starts with the assumption that the systems shows maximal morphological computation, i.e., that the behaviour of the systems is independent of the action. An example for such a system is the Passive Dynamic Walker. For the first concept, we now assume that the action had no influence on the next world state (shown on the right hand side). If the observation (left-hand side) diverges from the assumption (right-hand side), then the action had an influence on the behaviour, and hence, less morphological computation should be measured. This lead to the following definition:

     \begin{align*} \mathsf{MC}_\mathsf{A} & = \langle D(p(w'|w,a)||p(w'|w))\\ & = I(W';A|W) \end{align*}

Concept 2:
Morphological Computation as the Positive Effect of the World on Itself

To visualise this concept, we compare the following two one-step versions of the causal model presented in the image above:


Visualisation of Concept 2

The left-hand side is the one-step version of the causal diagram, where W refers to the world (body and environment), S refers to the sensor signal, A refers to the actuator signal and W’ is the next world state. The second concept starts with the assumption that the systems shows minimal morphological computation, i.e., that the behaviour of the systems is fully determined by the action. An example for such a system is a grid world in which the agent can jump from cell to cell without any limitations. For the second concept, we now assume that the previous world state W had no influence on the next world state (shown on the right hand side). If the observation (left-hand side) diverges from the assumption (right-hand side), then the world had an influence on itself , and hence, morphological computation should be measured. This lead to the following definition:

     \begin{align*} \mathsf{MC}_\mathsf{W} & = \langle D(p(w'|w,a)||p(w'|a))\\ & = I(W';W|A) \end{align*}

Adaptations to the Intrinsic Perspective

The focus of this paper was to develop quantification that can be used on intrinsically available information only, i.e., the random variables S,A, and C. There quantifications are given here without further discussion. For details, please read the paper. The link is provided at the top of this page.

Intrinsic Variations of the first concept:

     \begin{align*} \mathrm{ASOC}_\mathrm{A} & = \sum_{s',s,a} p(s',s,a) D(p(s'|s,a)||p(s'|s))\\ \mathrm{C}_\mathrm{A} & =  1 -         \frac{1}{\log_2|\mathcal{S}|}D(p(s'|\mathrm{do}(a))||p(s'|\mathrm{do}(s))) \end{align*}

Intrinsic Variations of the second concept:

     \begin{align*} \mathrm{ASOC}_\mathrm{W} & = \sum_{s',s,a} p(s',s,a) D(p(s'|s,a)||p(s'|a)) \\ \mathrm{C}_\mathrm{W} & = \sum_{s',s} p(s'|s)p(s) \log_2 \frac{p(s'|s)}{\tilde{p}(s'|s)} \end{align*}

Numerical simulations:

The properties of each quantification are evaluated based on a parametrised binary model of the sensorimotor loop.

Conclusions

This paper is the first introduction of quantifications for Morphological Computation.

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