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$G$-systems and 4E Cognitive Science

Vadim K. Weinstein

TL;DR

The paper develops $G$-systems, a discrete-time dynamical-systems framework with a coupling operation $*$, to ground 4E cognitive science in rigorous mathematics. It defines reducibility and emergence, and introduces a dependence atom $=\!(A;B)$ to capture when one set of variables determines another over a context $ extbf{C}$, along with a decomposition theorem linking these notions. Intervention and causality are formulated via a substitution operation $\mathbf{g}[\mathbf{s}]$ and a Pearl-like causal influence relation $c(A,B)$, enabling analysis of agent–environment loops beyond acyclic graphs. Collectively, the work provides a principled foundation for analyzing attunement, affordances, and emergent cognition in embodied, environment-coupled systems, with potential applications to robotics, AI, and cognitive modeling.

Abstract

We introduce a class dynamical systems called $G$-systems equipped with a coupling operation. We use $G$-systems to define the notions of dependence (borrowed from dependence logic) and causality (borrowed from Pearl) for dynamical systems. As a converse to coupling we define decomposition or ``reducibility''. We give a characterization of reducibility in terms of the dependence "atom". We do all this with the motivation of developing mathematical foundations for 4E cognitive science, see introductory sections.

$G$-systems and 4E Cognitive Science

TL;DR

The paper develops -systems, a discrete-time dynamical-systems framework with a coupling operation , to ground 4E cognitive science in rigorous mathematics. It defines reducibility and emergence, and introduces a dependence atom to capture when one set of variables determines another over a context , along with a decomposition theorem linking these notions. Intervention and causality are formulated via a substitution operation and a Pearl-like causal influence relation , enabling analysis of agent–environment loops beyond acyclic graphs. Collectively, the work provides a principled foundation for analyzing attunement, affordances, and emergent cognition in embodied, environment-coupled systems, with potential applications to robotics, AI, and cognitive modeling.

Abstract

We introduce a class dynamical systems called -systems equipped with a coupling operation. We use -systems to define the notions of dependence (borrowed from dependence logic) and causality (borrowed from Pearl) for dynamical systems. As a converse to coupling we define decomposition or ``reducibility''. We give a characterization of reducibility in terms of the dependence "atom". We do all this with the motivation of developing mathematical foundations for 4E cognitive science, see introductory sections.
Paper Structure (10 sections, 4 theorems, 7 equations, 2 figures)

This paper contains 10 sections, 4 theorems, 7 equations, 2 figures.

Table of Contents

  1. Philosophical motivation
  2. Example: Tetrapus
  3. A brief introduction to 4E
  4. Technical motivation
  5. $G$-systems and coupling
  6. History $I$-spaces as special cases
  7. Reducibility and emergence
  8. Interpretation in 4E cognition
  9. Dependence
  10. Intervention and causality

Key Result

Lemma 1

Suppose that $(G,\bullet)$ is associative and commutative. Let $X,Y$ be some sets, $\alpha,\alpha'\colon G^X\to G^X$, and $\beta,\beta'\colon G^Y\to G^Y$. Then

Figures (2)

  • Figure 1: (a) Tetrapus swimming in a container. (b) Palpating the wall. Occationally entering holes with a tentacle. (c) If the tentacle can bend on the other side, it triggers a pulling behaviour. This bending can only happen if the tentacle can go deep enough into the hole. (d) Tetrapus ends up pulling itself through the hole. Since the thickness of the tentacle matches the size of the beak, the hole is of the right diameter and the pull is successful.
  • Figure 2: Agent-environment dynamics: The classical model

Theorems & Definitions (15)

  • Lemma 1
  • proof
  • Definition 1
  • Definition 2
  • Definition 3
  • Lemma 2
  • proof
  • Definition 4
  • Definition 5
  • Theorem 1: Emergent, not reducible
  • ...and 5 more