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Semitopology: a new topological model of heterogeneous consensus

Murdoch GabbayGiuliano Losa
Mar 2023
A distributed system is permissionless when participants can join and leavethe network without permission from a central authority. Many moderndistributed systems are naturally permissionless, in the sense that a centralpermissioning authority would defeat their design purpose: this includesblockchains, filesharing protocols, some voting systems, and more. By theirpermissionless nature, such systems are heterogeneous: participants may onlyhave a partial view of the system, and they may also have different goals andbeliefs. Thus, the traditional notion of consensus -- i.e. system-wideagreement -- may not be adequate, and we may need to generalise it. This is a challenge: how should we understand what heterogeneous consensusis; what mathematical framework might this require; and how can we use this tobuild understanding and mathematical models of robust, effective, and securepermissionless systems in practice? We analyse heterogeneous consensus using semitopology as a framework. This islike topology, but without the restriction that intersections of opens be open. Semitopologies have a rich theory which is related to topology, but with itsown distinct character and mathematics. We introduce novel well-behavednessconditions, including an anti-Hausdorff property and a new notion of `topenset', and we show how these structures relate to consensus. We give arestriction of semitopologies to witness semitopologies, which are analgorithmically tractable subclass corresponding to Horn clause theories,having particularly good mathematical properties. We introduce and studyseveral other basic notions that are specific and novel to semitopologies, andstudy how known quantities in topology, such as dense subsets and closures,display interesting and useful new behaviour in this new semitopologicalcontext.