Yoram Moses

A guide to completeness and complexity for modal logics of knowledge and belief

Abstract We review and re-examine possible-worlds semantics for propositional logics of knowledge and belief with three particular points of emphasis: (1) we show how general techniques for finding decision procedures and complete axiomatizations apply to models for knowledge and belief, (2) we show how sensitive the difficulty of the decision procedure is to such issues as the choice of modal operators and the axiom system, and (3) we discuss how notions of common knowledge and distributed knowledge among a group of agents fit into the possible-worlds framework, As far as complexity is concerned, we show, among other results, that while the problem of deciding satisfiability of an S5 formula with one agent is NP-complete, the problem for many agents is PSPACE-complete. Adding a distributed knowledge operator does not change the complexity, but once a common knowledge operator is added to the language, the problem becomes complete for exponential time.

Knowledge and common knowledge in a byzantine environment: crash failures

By analyzing the states of knowledge that the processors attain in an unreliable system of a simple type, we capture some of the basic underlying structure of such systems. In particular, we study what facts become common knowledge at various points in the execution of protocols in an unreliable system. This characterizes the simultaneous actions that can be carried out in such systems. For example, we obtain a complete characterization of the number of rounds required to reach simultaneous Byzantine agreement, given the pattern in which failures occur. From this we derive a new protocol for this problem that is optimal in all runs, rather than just always matching the worst-case lower bound. In some cases this protocol attains simultaneous Byzantine agreement in as few as two rounds. We also present a nontrivial simultaneous agreement problem called bivalent agreement for which there is a protocol that always halts in two rounds. Our analysis applies to simultaneous actions in general, and not just to Byzantine agreement. The lower bound proofs presented here generalize and simplify the previously known proofs.

Knowledge and common knowledge in a distributed environment

We argue that the right way to understand distributed protocols is by considering how messages change the state of knowledge of a system. We present a hierarchy of knowledge states that a system may be in, and discuss how communication can move the systems state of knowledge of a fact up the hierarchy. Of special interest is the notion of common knowledge. Common knowledge is an essential state of knowledge for reaching agreements and coordinating action. We show that in practical distributed systems, common knowledge is not attainable. We introduce various relaxations of common knowledge that are attainable in many cases of interest. We describe in what sense these notions are appropriate, and discuss their relationship to each other. We conclude with a discussion of the role of knowledge in distributed systems.

Knowledge-based programs

Summary. Reasoning about activities in a distributed computer system at the level of the knowledge of individuals and groups allows us to abstract away from many concrete details of the system we are considering. In this paper, we make use of two notions introduced in our recent book to facilitate designing and reasoning about systems in terms of knowledge. The first notion is that of a knowledge-based program. A knowledge-based program is a syntactic object: a program with tests for knowledge. The second notion is that of a context, which captures the setting in which a program is to be executed. In a given context, a standard program (one without tests for knowledge) is represented by (i.e., corresponds in a precise sense to) a unique system. A knowledge-based program, on the other hand, may be represented by no system, one system, or many systems. In this paper, we provide a sufficient condition for a knowledge-based program to be represented in a unique way in a given context. This condition applies to many cases of interest, and covers many of the knowledge-based programs considered in the literature. We also completely characterize the complexity of determining whether a given knowledge-based program has a unique representation, or any representation at all, in a given finite-state context.

Fully Polynomial Byzantine Agreement for Processors in Rounds

This paper presents a polynomial-time protocol for reaching Byzantine agreement in t + 1 rounds whenever n > 3t, where n is the number of processors and t is an a priori upper bound on the number of failures. This resolves an open problem presented by Pease, Shostak, and Lamport in 1980. An early-stopping variant of this protocol is also presented, reaching agreement in a number of rounds that is proportional to the number of processors that actually fail.

A Layered Analysis of Consensus

This paper introduces a simple notion of layering as a tool for analyzing well-behaved runs of a given model of distributed computation. Using layering, a model-independent analysis of the consensus problem is performed and then applied to proving lower bounds and impossibility results for consensus in a number of familiar and less familiar models. The proofs are simpler and more direct than existing ones, and they expose a unified structure to the difficulty of reaching consensus. In particular, the proofs for the classical synchronous and asynchronous models now follow the same outline. A new notion of connectivity among states in runs of a consensus protocol, called potence connectivity, is introduced. This notion is more general than previous notions of connectivity used for this purpose and plays a key role in the uniform analysis of consensus.

Coordinated consensus in dynamic networks

We study several variants of coordinated consensus in dynamic networks. We assume a synchronous model, where the communication graph for each round is chosen by a worst-case adversary. The network topology is always connected, but can change completely from one round to the next. The model captures mobile and wireless networks, where communication can be unpredictable. In this setting we study the fundamental problems of eventual, simultaneous, and Δ-coordinated consensus, as well as their relationship to other distributed problems, such as determining the size of the network. We show that in the absence of a good initial upper bound on the size of the network, eventual consensus is as hard as computing deterministic functions of the input, e.g., the minimum or maximum of inputs to the nodes. We also give an algorithm for computing such functions that is optimal in every execution. Next, we show that simultaneous consensus can never be achieved in less than n - 1 rounds in any execution, where n is the size of the network; consequently, simultaneous consensus is as hard as computing an upper bound on the number of nodes in the network. For Δ-coordinated consensus, we show that if the ratio between nodes with input 0 and input 1 is bounded away from 1, it is possible to decide in time n-Θ(√ nΔ), where Δ bounds the time from the first decision until all nodes decide. If the dynamic graph has diameter D, the time to decide is min{O(nD/Δ),n-Ω(nΔ/D)}, even if D is not known in advance. Finally, we show that (a) there is a dynamic graph such that for every input, no node can decide before time n-O(Δ0.28n0.72); and (b) for any diameter D = O(Δ), there is an execution with diameter D where no node can decide before time Ω(nD / Δ). To our knowledge, our work constitutes the first study of Δ-coordinated consensus in general graphs.

Cheating husbands and other stories: A case study of knowledge, action, and communication

The relationship between knowledge and action is a fundamental one: a processor in a computer network (or a robot or a person, for that matter) should base its actions on the knowledge (or information) it has. One of the main uses of communication is passing around information that may eventually be required by the receiver in order to decide upon subsequent actions. Understanding the relationship between knowledge, action, and communication is fundamental to the design of computer network protocols, intelligent robots, etc. By looking at a number of variants of thecheating husbands puzzle, we illustrate the subtle relationship between knowledge, communication, and action in a distributed environment.

Towards a theory of knowledge and ignorance: preliminary report

Communication in a distributed system changes the state of knowledge of the processors. But what is a state of knowledge? Here we attempt to characterize this notion. In the case where an agent’s information is completely described by a formula, we give a number of equivalent ways to characterize the agent’s state of knowledge, as well as an algorithm for computing the formulas that are true in this state. The relationship between this work and related works by Stark, Konolige, and Moore is discussed.

A characterization of eventual Byzantine agreement

We investigate eventual Byzantine agreement (EBA) in the crash and omission failure models. The emphasis is on characterizing optimal EBA protocols in terms of the states of knowledge required by the processors in order to attain EBA. It is well known that common knowledge among the nonfaulty processors is a necessary and sufficient condition for attaining simultaneous Byzantine agreement (SBA). We define a new variant of common knowledge, which we call continual common knowledge, in terms of which we can characterize necessary and sufficient conditions for attaining EBA. Using our characterization, we provide a technique that allows us to start with any EBA protocol, apply a certain construction twice, and arrive at an optimal EBA protocol.