Decentralised collaborative action: cryptoeconomics in space
Murdoch J. Gabbay
TL;DR
This paper introduces semitopology as a mathematical framework to model decentralized collaborative action in cryptoeconomics. It treats participants as points and actionable coalitions as open sets, i.e., collections $O \subseteq \mathsf{P}$ closed under arbitrary unions (but not necessarily under finite intersections). The main insight is that antiseparation properties, captured by intertwined points, yield consensus guarantees that hold regardless of protocol details, enabling reasoning about fork-resistance in distributed systems. The authors illustrate with formal examples of binary consensus and majority coalitions, and discuss connections to real-world crypto networks, bridges, and heterogeneous participants. They outline future directions, including time-evolving semitopologies and the use of semitopological logics to specify and verify consensus protocols such as Paxos.
Abstract
Blockchains and peer-to-peer systems are part of a trend towards computer systems that are "radically decentralised", by which we mean that they 1) run across many participants, 2) without central control, and 3) are such that qualities 1 and 2 are essential to the system's intended use cases. We propose a notion of topological space, which we call a "semitopology", to help us mathematically model such systems. We treat participants as points in a space, which are organised into "actionable coalitions". An actionable coalition is any set of participants who collectively have the resources to collaborate (if they choose) to progress according to the system's rules, without involving any other participants in the system. It turns out that much useful information about the system can be obtained \emph{just} by viewing it as a semitopology and studying its actionable coalitions. For example: we will prove a mathematical sense in which if every actionable coalition of some point p has nonempty intersection with every actionable coalition of another point q -- note that this is the negation of the famous Hausdorff separation property from topology -- then p and q must remain in agreement. This is of practical interest, because remaining in agreement is a key correctness property in many distributed systems. For example in blockchain, participants disagreeing is called "forking", and blockchain designers try hard to avoid it. We provide an accessible introduction to: the technical context of decentralised systems; why we build them and find them useful; how they motivate the theory of semitopological spaces; and we sketch some basic theorems and applications of the resulting mathematics.