An anatomy of interaction: co-occurrences and entanglements

Polynomial Functors: A Mathematical Theory of Interaction

This monograph is a study of the category of polynomial endofunctors on the category of sets and its applications to modeling interaction protocols and dynamical systems. We assume basic categorical background and build the categorical theory from the ground up, highlighting pictorical techniques and concrete examples to build intuition and provide applications.

Interactions that reshape the interfaces of the interacting parties

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.

Coeffects: Context-aware programming languages

Interactive essay that explains theory of coeffects and lets you type-check and run sample programs.

Operators

When building a Bonsai program, you chain together reactive operators to create new observable sequences. There are many different operators, which can create all kinds of observable sequences. These operators can be roughly grouped into different categories, depending on their shared characteristics.

Rabrg/artificial-life

A simple (300 lines of code) reproduction of Computational Life: How Well-formed, Self-replicating Programs Emerge from Simple Interaction

Alexander Lerchner, The Abstraction Fallacy: Why AI Can Simulate But Not Instantiate Consciousness - PhilPapers

Computational functionalism dominates current debates on AI consciousness. This is the hypothesis that subjective experience emerges entirely from abstract causal topology, regardless of the underlying physical substrate. We argue this view ...

Substrates conference series - Substrates 2026

An increasing number of researchers see their work as interactive authoring tools or software substrates for interactive computational media. By talking about “authoring tools”, we remove the divide between programmers and users; “software substrates” let us look beyond conventional programming languages and systems; and “interactive computational media” promises a more malleable and adaptable notion of tools for thought we are striving for. This workshop aims to bring together a wide range of perspectives on these matters.

J Notation as a Tool of Thought

Kenneth Iverson’s 1964 language, APL, won him the Turing Award. His award lecture, Notation as a Tool of Thought, argued that better notations would lead people to deeper insights about mathematics. He provided a number of examples ranging across linear algebra, arithmetic, probability, and logic. Unfortunately, most of the mathematics he covers isn’t relevant to programming. However, his core idea still applies, and changing how we describe programs changes how we think about them.

A theory of type polymorphism in programming

The aim of this work is largely a practical one. A widely employed style of programming, particularly in structure-processing languages which impose n…

Alexander Lerchner, The Abstraction Fallacy: Why AI Can Simulate But Not Instantiate Consciousness - PhilArchive

Computational functionalism dominates current debates on AI consciousness. This is the hypothesis that subjective experience emerges entirely from abstract causal topology, regardless of the underlying physical substrate. We argue this view ...

Software in the natural world: A computational approach to hierarchical emergence

Understanding the functional architecture of complex systems is crucial to illuminate their inner workings and enable effective methods for their prediction and control. Recent advances have introduced tools to characterise emergent macroscopic levels; however, while these approaches are successful in identifying when emergence takes place, they are limited in the extent they can determine how it does. Here we address this limitation by developing a computational approach to emergence, which characterises macroscopic processes in terms of their computational capabilities. Concretely, we articulate a view on emergence based on how software works, which is rooted on a mathematical formalism that articulates how macroscopic processes can express self-contained informational, interventional, and computational properties. This framework establishes a hierarchy of nested self-contained processes that determines what computations take place at what level, which in turn delineates the functional architecture of a complex system. This approach is illustrated on paradigmatic models from the statistical physics and computational neuroscience literature, which are shown to exhibit macroscopic processes that are akin to software in human-engineered systems. Overall, this framework enables a deeper understanding of the multi-level structure of complex systems, revealing specific ways in which they can be efficiently simulated, predicted, and controlled.

The Abstraction Fallacy: Why AI Can Simulate But Not Instantiate Consciousness

Computational functionalism dominates current debates on AI consciousness. This is the hypothesis that subjective experience emerges entirely from abstract causal topology, regardless of the underlying physical substrate. We argue this view fundamentally mischaracterizes how physics relates to information. We call this mistake the Abstraction Fallacy. Tracing the causal origins of abstraction reveals that symbolic computation is not an intrinsic physical process. Instead, it is a mapmaker-dependent description. It requires an active, experiencing cognitive agent to alphabetize continuous physics into a finite set of meaningful states. Consequently, we do not need a complete, finalized theory of consciousness to assess AI sentience—a demand that simply pushes the question beyond near-term resolution and deepens the AI welfare trap. What we actually need is a rigorous ontology of computation. The framework proposed here explicitly separates simulation (behavioral mimicry driven by vehicle causality) from instantiation (intrinsic physical constitution driven by content causality). Establishing this ontological boundary shows why algorithmic symbol manipulation is structurally incapable of instantiating experience. Crucially, this argument does not rely on biological exclusivity. If an artificial system were ever conscious, it would be because of its specific physical constitution, never its syntactic architecture. Ultimately, this framework offers a physically grounded refutation of computational functionalism to resolve the current uncertainty surrounding AI consciousness.

A Pattern Language for Interfaces | Notion

System → Programs → Apps

Theorizing Protocolization II: Atomic Protocol Questions

Solving real coordination problems to discover the formal laws of protocols.