💻 computer science

Decentralised collaborative action: cryptoeconomics in space

This paper proposes using "semitopologies," a novel mathematical framework where participants are modeled as points and their collaborative groups as "actionable coalitions," to analyze decentralized systems and prove that specific intersection properties of these coalitions guarantee agreement among participants, thereby preventing critical issues like blockchain forking.

Original authors: Murdoch J. Gabbay

Published 2026-06-24

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Murdoch J. Gabbay

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: A New Way to Map Groups

Imagine you are trying to understand how a massive, chaotic crowd of people can agree on something without a boss telling them what to do. This is the world of blockchains and peer-to-peer networks. Usually, mathematicians use a tool called "topology" (the study of shapes and spaces) to map how things connect.

The author, Murdoch Gabbay, says the old math tools don't quite fit these new systems. So, he invented a new mathematical shape called a "semitopology."

Think of a semitopology as a map of who can work together, rather than a map of who is standing next to whom.

1. The Cast of Characters: Points and "Actionable Coalitions"

In this new math world:

Points are the people (or computers) in the system.
Open Sets are called "Actionable Coalitions."

The Analogy: The Tango Dance Floor
Imagine a dance floor full of people.

In a normal city map, you might group people by who is standing in the same room.
In this paper's map, you group people by who can dance together.

An "Actionable Coalition" is any group of people who could legally and technically dance a tango together if they wanted to.

It doesn't mean they are dancing right now.
It doesn't mean they have to dance perfectly.
It just means they have the capacity to collaborate to move forward.

For example, in a blockchain, a "coalition" might be a group of people holding enough digital tokens to make a decision. In a real-life example, it might be a group of people strong enough to lift a heavy rock together.

2. The Core Problem: Staying in Sync (No "Forks")

The biggest fear in these decentralized systems is forking.

The Analogy: Imagine a group of friends trying to decide where to eat. Half the group decides on Pizza, and the other half decides on Tacos. Now, the group is split. They can't function as one team anymore. In blockchain, this is a disaster.

The paper asks: How can we mathematically guarantee that two people will eventually agree, even if they never talk directly to each other?

3. The Magic Rule: "Intertwined"

The paper introduces a concept called "Intertwined."

The Analogy: The Overlapping Circles
Imagine two people, Alice and Bob.

If Alice can only dance with a group that overlaps with the group Bob can dance with, they are "intertwined."
Because their groups overlap, there is a person in the middle who knows what Alice is doing and what Bob is doing.
If Alice decides on Pizza, and Bob decides on Tacos, the person in the middle (the overlap) creates a conflict. To avoid this conflict and keep dancing, they are mathematically forced to agree on the same menu.

The paper proves a simple but powerful rule: If two people are "intertwined," they cannot legally disagree. If they both follow the rules, they must end up with the same answer.

4. Why This Math is Different (The "Semi" in Semitopology)

Traditional math (standard topology) has a strict rule: If Group A and Group B are both valid dance groups, then the people they have in common (the intersection) must also be a valid dance group.

The Paper's Twist:
In the real world of decentralized systems, this isn't always true.

Example: Imagine a bridge connecting two different blockchains (like Ethereum and Tezos).
- Group A (Ethereum users) can act together.
- Group B (Tezos users) can act together.
- But if you take the "intersection" (the people who are on both), they might not be able to act alone without the "bridge" node.
The author's Semitopology allows for these "broken" intersections. It says, "We don't need the overlap to be a perfect group; we just need to know that the groups touch."

This makes the math much more flexible and better suited for messy, real-world systems where rules can differ between groups.

5. What Did They Actually Prove?

The paper doesn't invent a new blockchain or a new coin. Instead, it provides a mathematical lens to look at existing systems.

The Result: If you can draw a map of a system where everyone is "intertwined" (their groups overlap enough), you can mathematically prove that the system cannot fork.
The Benefit: You don't need to know the complex code or the specific rules of the blockchain to know this. You just need to look at the structure of the groups. If the structure is "intertwined," agreement is guaranteed by the shape of the system itself.

Summary

The paper suggests that to understand how decentralized groups (like blockchains) stay together without a boss, we should stop looking at them as computers and start looking at them as dance partners.

By mapping out who can dance with whom (Actionable Coalitions), and checking if those dance circles overlap enough (Intertwinedness), we can use a new kind of math (Semitopology) to prove that the group will stay in sync and avoid splitting apart. It turns the chaotic problem of "how do strangers agree?" into a clean geometric puzzle.

Technical Summary: Decentralised Collaborative Action: Cryptoeconomics in Space

Problem Statement

The paper addresses the challenge of mathematically modeling decentralised collaborative action within modern computing systems, specifically in the context of cryptoeconomics. Unlike traditional distributed systems that often rely on centralised control or homogeneous rule sets, cryptoeconomic systems (e.g., blockchains, peer-to-peer networks) are characterised by being:

Decentralised: Composed of distinct parts without a central controller.
Permissionless: Participants can join or leave at will.
Heterogeneous: Participants may legitimately follow different rules or consensus mechanisms (e.g., in a blockchain bridge scenario).

The core problem is that these systems lack a unified mathematical framework to reason about consensus (agreement) and forking (disagreement) based solely on their structural properties. Existing topological models (point-set topology) are insufficient because they require the intersection of open sets to be open, a property that does not hold for "actionable coalitions" in heterogeneous systems. The paper seeks to determine if macro-level correctness properties (like the impossibility of forking) can be derived from the abstract structure of participant coalitions without analyzing specific algorithms or protocols.

Methodology

The authors propose a new mathematical abstraction called a semitopology, which generalises point-set topology to model decentralised systems.

1. Semitopological Spaces

A semitopology is defined as a pair $(P, \text{Open}(P))$ , where:

$P$ is a set of points representing participants.
$\text{Open}(P)$ is a collection of subsets of $P$ representing actionable coalitions.

An actionable coalition is defined as a set of participants who collectively possess the resources and legal entitlement to collaborate to update their local states, independently of the rest of the system.

Axioms of Semitopology:

The empty set $\emptyset$ and the total set $P$ are in $\text{Open}(P)$ .
The union of any collection of open sets is an open set.
Crucial Distinction: The intersection of two open sets is not required to be open. This reflects the reality that two valid coalitions may not form a valid coalition when combined.

2. Modeling Consensus

The paper maps the concept of binary consensus (agreeing on a value $t$ or $f$ ) to continuity in the semitopological space.

A value assignment is a function $f: P \to \{t, f\}$ .
The assignment is continuous at a point $p$ if there exists an open neighbourhood $O$ containing $p$ where $f$ is constant.
Consensus = Continuity: If a participant updates their state, they must do so in collaboration with an actionable coalition (an open set) where the value is agreed upon.

3. Antiseparation Properties

The paper introduces antiseparation properties, which are the negation of the Hausdorff separation property in standard topology.

Intertwined ( $p \between p'$ ): Two points are intertwined if every open neighbourhood of $p$ intersects every open neighbourhood of $p'$ .
Transitively Intertwined: Two points are related by the transitive closure of the intertwined relation.

Key Contributions and Results

1. Theoretical Framework: Semitopology

The paper establishes semitopology as a rigorous framework for decentralised systems. By treating actionable coalitions as open sets, the authors demonstrate that the structural properties of these coalitions dictate system behavior.

2. Theorem on Agreement (Theorem 3.4)

The central mathematical result states:

If two points $p$ and $p'$ are intertwined, and a value assignment $f$ is continuous at both points, then $f(p) = f(p')$ .
If $p$ and $p'$ are transitively intertwined and $f$ is continuous everywhere, then $f(p) = f(p')$ .

Implication: If the structure of actionable coalitions ensures that participants are intertwined (or transitively intertwined), then any legal progress made by one participant must eventually be agreed upon by all others. This proves that forking is impossible under these structural conditions, regardless of the specific consensus algorithm used.

3. Application to Real-World Systems

The paper applies this framework to concrete examples:

Simple Majority Systems: In a system where actionable coalitions are majorities, all points are intertwined. Thus, forking is impossible.
Chain-like Systems (Example 2.4): Even in systems where coalitions do not directly intersect (e.g., overlapping triplets $\{2i, 2i+1, 2i+2\}$ ), points can be transitively intertwined, ensuring global agreement.
Blockchain Bridges (Example 3.6): The paper models a bridge between Ethereum and Tezos. It shows that if the individual chains are internally intertwined, the bridging node (which requires consensus from both sides to act) makes the combined system transitively intertwined, preventing forks across the bridge.

4. Connection to Arrow's Theorem

The paper notes a connection to social choice theory, proving that every semitopology has an "actionable kernel." If this kernel makes a decision, all other participants must follow. This is presented as a distributed systems analogue to Arrow's Impossibility Theorem, suggesting the existence of "dictator sets" in decentralised systems.

Significance and Claims

The paper claims that semitopology provides a "general mathematics of (anti)separation" that allows for the prediction of macro-level system properties (specifically consensus and the impossibility of forking) based solely on the topology of actionable coalitions.

Algorithm Independence: The results hold for any concrete algorithm. If the structural condition of intertwining is met, agreement is guaranteed without needing to inspect the protocol's internal logic.
Handling Heterogeneity: Unlike standard topology, semitopology accommodates heterogeneous systems where participants follow different rules, making it uniquely suited for modelling blockchain bridges and complex peer-to-peer networks.
Practical Utility: The authors state that these logics are already being used to formally specify, reason about, and debug consensus protocols (e.g., axiomatising Paxos and identifying errors in Heterogeneous Paxos).

The paper concludes that while decentralised systems appear chaotic due to the lack of central control, semitopology reveals an underlying order defined by the intersection properties of actionable coalitions, offering a new lens for designing and verifying cryptoeconomic systems.