218 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Topological Characterization of Task Solvability in General Models of Computation
The famous asynchronous computability theorem (ACT) relates the existence of
an asynchronous wait-free shared memory protocol for solving a task with the
existence of a simplicial map from a subdivision of the simplicial complex
representing the inputs to the simplicial complex representing the allowable
outputs. The original theorem relies on a correspondence between protocols and
simplicial maps in round-structured models of computation that induce a compact
topology. This correspondence, however, is far from obvious for computation
models that induce a non-compact topology, and indeed previous attempts to
extend the ACT have failed.
This paper shows that in every non-compact model, protocols solving tasks
correspond to simplicial maps that need to be continuous. It first proves a
generalized ACT for sub-IIS models, some of which are non-compact, and applies
it to the set agreement task. Then it proves that in general models too,
protocols are simplicial maps that need to be continuous, hence showing that
the topological approach is universal. Finally, it shows that the approach used
in ACT that equates protocols and simplicial complexes actually works for every
compact model.
Our study combines, for the first time, combinatorial and point-set
topological aspects of the executions admitted by the computation model
Semitopology: a new topological model of heterogeneous consensus
A distributed system is permissionless when participants can join and leave
the network without permission from a central authority. Many modern
distributed systems are naturally permissionless, in the sense that a central
permissioning authority would defeat their design purpose: this includes
blockchains, filesharing protocols, some voting systems, and more. By their
permissionless nature, such systems are heterogeneous: participants may only
have a partial view of the system, and they may also have different goals and
beliefs. Thus, the traditional notion of consensus -- i.e. system-wide
agreement -- may not be adequate, and we may need to generalise it.
This is a challenge: how should we understand what heterogeneous consensus
is; what mathematical framework might this require; and how can we use this to
build understanding and mathematical models of robust, effective, and secure
permissionless systems in practice?
We analyse heterogeneous consensus using semitopology as a framework. This is
like topology, but without the restriction that intersections of opens be open.
Semitopologies have a rich theory which is related to topology, but with its
own distinct character and mathematics. We introduce novel well-behavedness
conditions, including an anti-Hausdorff property and a new notion of `topen
set', and we show how these structures relate to consensus. We give a
restriction of semitopologies to witness semitopologies, which are an
algorithmically tractable subclass corresponding to Horn clause theories,
having particularly good mathematical properties. We introduce and study
several other basic notions that are specific and novel to semitopologies, and
study how known quantities in topology, such as dense subsets and closures,
display interesting and useful new behaviour in this new semitopological
context
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
The Computational Power of Distributed Shared-Memory Models with Bounded-Size Registers
The celebrated Asynchronous Computability Theorem of Herlihy and Shavit (STOC
1993 and STOC 1994) provided a topological characterization of the tasks that
are solvable in a distributed system where processes are communicating by
writing and reading shared registers, and where any number of processes can
fail by crashing. However, this characterization assumes the use of
full-information protocols, that is, protocols in which each time any of the
processes writes in the shared memory, it communicates everything it learned
since the beginning of the execution. Thus, the characterization implicitly
assumes that each register in the shared memory is of unbounded size. Whether
unbounded size registers are unavoidable for the model of computation to be
universal is the central question studied in this paper. Specifically, is any
task that is solvable using unbounded registers solvable using registers of
bounded size? More generally, when at most processes can crash, is the
model with bounded size registers universal? These are the questions answered
in this paper
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