Reality of Affordances: �A Category-Theoretic Approach

Ryuzo Hirota¹, Hayato Saigo² and Shigeru Taguchi³

¹ The University of Tokyo

² Nagahama Institute of Bio-Science and Technology

³ Hokkaido University

ALIFE Conference 2024 @ Copenhagen, Denmark

Introduction

• Creating an autonomous system must be accompanied by designing its environment with affordances.
• Affordance: an opportunity for action offered by the environment to an organism (Gibson, 1979)
• Understanding affordance is crucial for ALife study.
• However, formalizing affordance has been challenging, partly because of an apparent paradox inherent in the concept: affordances are both relational and objective??
• We resolve this paradox using category theory.

Affordance: Resource or Relation?

• “an affordance is neither an objective property nor a subjective property; or it is both if you like.” (Gibson, 1979)
• Resource Theory: Affordances are resources that exist already there independently of the organisms (e.g., Reed, 1995).
• But.. how is the emergence of new affordance possible? (Kauffman)
• Relation Theory: Affordances are “relations between the abilities of animals and features of the environment” (Chemero, 2003; 2009)
• But.. this runs the risk of making affordances too subjective and arbitrary (i.e., non-real)
• Category theory enables us to understand affordances as both resources (objective) and relations.

Category: a “network of relationships”¹

• Collection of “objects” and “arrows” (or “morphisms”).
• You can “compose” arrows through objects.
• Every object is characterized only in terms of the arrows it has to/from other objects and itself.
• aligned with Gibson‘s “ecological” approach

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¹ from David Spivak’s talk: https://youtu.be/cJ46AOEOc14?si=JAVZ8OGvbCN6QlM6

Functor: Mapping between categories

• Mapping a category to category (objects to objects, arrows to arrows) that preserves the structure.
• Functors represent “ways of viewing” a category in terms of another category (Fong and Spivak, 2019)
• Measuring with a scale, modeling, counting, etc.

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Natural Transformation

• Mapping form functor to functor in a way that preserves the structural consistency.
• A natural transformation expresses how the “view” changes if you change the “way of viewing” (i.e., perspective) from one to another.
• unit conversion, scale transformation, coordination transformation, etc.

(Fong and Spivak, 2019)

t: F → G; t₁ = c, t₂ = g

Category-theoretic formalization of the reality of affordances

• Based on Warren (1984)
• We consider two categories:
• The “category of height” (H) of a step
• “A → B” means “A is lower than B”
• The “category of climbability” (C) of a step
• “X → Y” means “X is easier than Y”
• IC : Impossible to Climb
• PC : Possible to Climb (but not easy)
• EC : Easy to Climb

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H

C

Category-theoretic formalization of the reality of affordances

• We define two functors from H to C: F₁ and F₂
• F₁ and F₂ represent the perspective of taller people and shorter people resp.
• Functoriality = monotonicity (“the higher, the harder”)
• They capture the relational aspect of affordances.
• “A man can bite into an apple but not a rock; he can get a grip on a handle but not on a wall; ... He measures these features of the environment by the standard of his body.” (Gibson, 1977; p. 79)

Category-theoretic formalization of the reality of affordances

• There is a natural transformation t from F₁ to F₂
• This represents the structure of how the climbability of a step with a given height would change if the perspective changed from F₁ to F₂ .
• This captures the objective aspect of affordances
• What is objective is the lawful structure of transformations between different perspectives (cf. Rietveld and Kiverstein, 2014)

Discussion and further works

• Parallelism to the shift in Modern Physics:
• From entity (“ism”) to transformation (“ifm”)
• Dependence on the perspective of the observer does not make things less objective and “real”.
• Relation to other frameworks
• Category theory was created for connecting different frameworks.
• Autonomy (operational closure) as a “monoid” (Hirota et al., 2023)
• Mathematical (dynamical, statistical) formalizations of sensorimotor contingency (e.g., Buhrmann et al., 2013; Ay and Löhr, 2015)
• Theory of Indeterminate Natural Transformations (TINT) (Fuyama, Saigo and Takahashi, 2020)

Conclusion

• Using category theory, we can formalize the reality of affordances as both objective and relational in a very simple but rigorous and consistent way.
• Even the most basic concepts of category theory (category, functor and natural transformation) can be very useful for us.
• Category theory can be a common language for interdisciplinary research fields, including Alife.
• Come join us!