This work presents a classification of weak models of distributed computing. We focus on deterministic distributed algorithms, and study models of computing that are weaker versions of the widely-studied port-numbering model. In the port-numbering model, a node of degree @math d receives messages through @math d input ports and sends messages through @math d output ports, both numbered with @math 1 , 2 , ? , d . In this work, @math VV c is the class of all graph problems that can be solved in the standard port-numbering model. We study the following subclasses of @math VV c : Now we have many trivial containment relations, such as @math SB ⊆ MB ⊆ VB ⊆ VV ⊆ VV c , but it is not obvious if, for example, either of @math VB ⊆ SV or @math SV ⊆ VB should hold. Nevertheless, it turns out that we can identify a linear order on these classes. We prove that @math SB ? MB = VB ? SV = MV = VV ? VV c . The same holds for the constant-time versions of these classes. We also show that the constant-time variants of these classes can be characterised by a corresponding modal logic. Hence the linear order identified in this work has direct implications in the study of the expressibility of modal logic. Conversely, one can use tools from modal logic to study these classes.
Modal logic (see ) has, of course, been applied previously in the context of distributed systems. For example, in their seminal paper, Halpern and Moses @cite_5 use modal logic to model epistemic phenomena in distributed systems. A distributed system @math gives rise to a (see ), whose set @math of domain points corresponds to the set of partial runs of @math , that is, finite sequences of global states of @math . For each processor @math of @math , there is an @math such that @math if and only if @math and @math are from the point of view of processor @math . This framework suits well for epistemic considerations.
@cite_5
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This work presents a classification of weak models of distributed computing. We focus on deterministic distributed algorithms, and study models of computing that are weaker versions of the widely-studied port-numbering model. In the port-numbering model, a node of degree @math d receives messages through @math d input ports and sends messages through @math d output ports, both numbered with @math 1 , 2 , ? , d . In this work, @math VV c is the class of all graph problems that can be solved in the standard port-numbering model. We study the following subclasses of @math VV c : Now we have many trivial containment relations, such as @math SB ⊆ MB ⊆ VB ⊆ VV ⊆ VV c , but it is not obvious if, for example, either of @math VB ⊆ SV or @math SV ⊆ VB should hold. Nevertheless, it turns out that we can identify a linear order on these classes. We prove that @math SB ? MB = VB ? SV = MV = VV ? VV c . The same holds for the constant-time versions of these classes. We also show that the constant-time variants of these classes can be characterised by a corresponding modal logic. Hence the linear order identified in this work has direct implications in the study of the expressibility of modal logic. Conversely, one can use tools from modal logic to study these classes. A real-world distributed system is rarely implemented as a standalone monolithic system. Instead, it is composed of multiple independent interacting components that together ensure the desired system-level specification. One can scale systematic testing to large, industrial-scale implementations by decomposing the system-level testing problem into a collection of simpler component-level testing problems. This paper proposes techniques for compositional programming and testing of distributed systems with two central contributions: (1) We propose a module system based on the theory of compositional trace refinement for dynamic systems consisting of asynchronously-communicating state machines, where state machines can be dynamically created, and communication topology of the existing state machines can change at runtime; (2) We present ModP, a programming system that implements our module system to enable compositional reasoning (assume-guarantee) of distributed systems. We demonstrate the efficacy of our framework by building two practical fault-tolerant distributed systems, a transaction-commit service and a replicated hash-table. ModP helps implement these systems modularly and validate them via compositional testing. We empirically demonstrate that the abstraction-based compositional reasoning approach helps amplify the coverage during testing and scale it to real-world distributed systems. The distributed services built using ModP achieve performance comparable to open-source equivalents. This paper presents a logical system in which various group-level epistemic actions are incorporated into the object language. That is, we consider the standard modeling of knowledge among a set of agents by multi-modal Kripke structures. One might want to consider actions that take place, such as announcements to groups privately, announcements with suspicious outsiders, etc. In our system, such actions correspond to additional modalities in the object language. That is, we do not add machinery on top of models (as in [1], but we reify aspects of the machinery in the logical language. The first case of the kind of logics we consider appears in Gerbrandy and Groeneveld [3]. They introduced a language in which one can faithfully represent all of the reasoning in examples such as the Muddy Children scenario. In that paper we find operators for updating worlds via announcements to groups of agents who are isolated from all others. We advance this by considering many more actions; and by using a more general semantics. Our logic contains the infinitary operators used in the standard modeling of common knowledge. We present a sound and complete logical system for the logic, and we study its expressive power. A @math -modal probability distribution over the discrete domain @math is one whose histogram has at most @math "peaks" and "valleys." Such distributions are natural generalizations of monotone ( @math ) and unimodal ( @math ) probability distributions, which have been intensively studied in probability theory and statistics. In this paper we consider the problem of (i.e., performing density estimation of) an unknown @math -modal distribution with respect to the @math distance. The learning algorithm is given access to independent samples drawn from an unknown @math -modal distribution @math , and it must output a hypothesis distribution @math such that with high probability the total variation distance between @math and @math is at most @math Our main goal is to obtain algorithms for this problem that use (close to) an information-theoretically optimal number of samples. We give an efficient algorithm for this problem that runs in time @math . For @math , the number of samples used by our algorithm is very close (within an @math factor) to being information-theoretically optimal. Prior to this work computationally efficient algorithms were known only for the cases @math Birge:87b,Birge:97 . A novel feature of our approach is that our learning algorithm crucially uses a new algorithm for as a key subroutine. The learning algorithm uses the property tester to efficiently decompose the @math -modal distribution into @math (near-)monotone distributions, which are easier to learn.