Semitopology: distributed collaborative action via topology, algebra, and logic

2024-01-16

arXiv.org

AI Summary

This paper addresses the problem of decentralized, asynchronous, and voluntary state updates in distributed systems—where no central authority governs coordination. To formalize such collaborative dynamics, it proposes a novel mathematical framework termed “semi-topological structure.” Methodologically, it fundamentally reinterprets topology by decoupling the conventional requirement that open sets be closed under finite intersections; instead, it replaces open sets with “actionable coalitions”—subsets of participants capable of jointly updating system state—thereby grounding the structure directly in distributed collaboration semantics. The framework integrates generalized point-set topology, semilattices and action algebras, and modal logic to yield a verifiable semantic model of cooperative behavior. Theoretically rigorous and self-contained, it provides a unified formal foundation for locality, autonomy, and asynchrony in peer-to-peer networks, blockchain systems, and other decentralized architectures, enabling principled modeling and analysis of consensus protocols, resource exchange mechanisms, and decentralized agreement algorithms.

Technology Category

• Multiagent Systems: Coordination and Collaboration
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Distributed Search

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Decentralized Web and Fediverse systems
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We introduce semitopologies, a generalisation of point-set topology that removes the restriction that intersections of open sets need necessarily be open. The intuition is that points are participants in some distributed system, and an open set is a collection of participants that can collaborate to update their local state by taking a distributed collaborative action; we call this an actionable coalition. What constitutes an actionable coalition depends on what actions we want to model. Intuitive examples include 'a group of people that is collectively strong enough to lift a rock', where the state update is very simply 'holding rock low' to 'holding rock high' and this update is common to all participants in the actionable coalition. Or, consider 'two people wishing to barter a can of juice for a bar of chocolate', in which case the coalition is any such pair and the state updates differ between participants to flip them between 'has/has no juice' and 'has/has no chocolate'. A characteristic of these systems is that state updates are local to the coalition, voluntary, may vary between participants, and are not assumed subject to permission or synchronisation by a central authority. Peer-to-peer computer networks, including filesharing and blockchain systems, provide motivating examples from computing. This monograph presents a comprehensive view of semitopologies which includes point-set semitopology, algebra, and logic inspired by these considerations. This is interesting in and of itself and it provides a conceptual framework within which to understand a useful class of distributed systems.

Problem

• Generalizes point-set topology for distributed collaborative actions
• Models actionable coalitions in decentralized systems without central authority
• Provides framework for state updates in peer-to-peer networks

Innovation

• Generalizes topology for distributed collaborative actions
• Models actionable coalitions for state updates
• Applies to peer-to-peer networks and blockchain