Interactions that reshape the interfaces of the interacting parties

2026-02-20

AI Summary

Traditional dynamic interaction models struggle to capture scenarios where system interfaces evolve during interaction—such as cell differentiation, sensor degradation, or the dynamic increase in neural network output resolution. This work proposes modeling such adaptively reshaping interfaces using coinductive polynomial trees, constructing their monoidal closed category PolyTr and extending it to the bicategory OrgTr. The framework formally incorporates interface dynamism into interaction semantics for the first time, employing (u ◁ u)-coalgebras and a state-driven action-update mechanism to realize interface evolution, while remaining compatible with the original Org model via constant-tree embeddings. As a demonstration, the approach successfully models the adaptive resolution growth in progressive generative adversarial networks.

Technology Category

• Cognitive Modeling & Cognitive Systems: Adaptive Behavior
• Knowledge Representation and Reasoning: Action, Change, and Causality
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

Polynomial functors model systems with interfaces: each polynomial specifies the outputs a system can produce and, for each output, the inputs it accepts. The bicategory $\mathbb{O}\mathbf{rg}$ of dynamic organizations \cite{spivak2021learners} gives a notion of state-driven interaction patterns that evolves over time, but each system's interface remains fixed throughout the interaction. Yet in many systems, the outputs sent and inputs received can reshape the interface itself: a cell differentiating in response to chemical signals gains or loses receptors; a sensor damaged by its input loses a channel; a neural network may grow its output resolution during training. Here we introduce *polynomial trees*, elements of the terminal $(u\triangleleft u)$-coalgebra where $u$ is the polynomial associated to a universe of sets, to model such systems: a polynomial tree is a coinductive tree whose nodes carry polynomials, and in which each round of interaction -- an output chosen and an input received -- determines a child tree, hence the next interface. We construct a monoidal closed category $\mathbf{PolyTr}$ of polynomial trees, with coinductively-defined morphisms, tensor product, and internal hom. We then build a bicategory $\mathbb{O}\mathbf{rgTr}$ generalizing $\mathbb{O}\mathbf{rg}$, whose hom-categories parametrize morphisms by state sets with coinductive action-and-update data. We provide a locally fully faithful functor $\mathbb{O}\mathbf{rg}\to\mathbb{O}\mathbf{rgTr}$ via constant trees, those for which the interfaces do not change through time. We illustrate the generalization by suggesting a notion of progressive generative adversarial networks, where gradient feedback determines when the image-generation interface grows to a higher resolution.

Problem

• polynomial functors
• dynamic interfaces
• state-driven interaction
• interface evolution
• coinductive systems

Innovation

• polynomial trees
• coinductive interaction
• dynamic interfaces
• monoidal closed category
• progressive GANs