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Computability
About: Computability is a research topic. Over the lifetime, 2829 publications have been published within this topic receiving 85162 citations.
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TL;DR: Set-Constrained Delivery Broadcast (SCD-broadcast) as mentioned in this paper is a new communication abstraction, which allows each process to broadcast messages and deliver a sequence of sets of messages in such a way that, if a process delivers a set of messages including a message m and later delivers another message including m, no process can deliver first a set m including m and then another message m.
Abstract: This paper introduces a new communication abstraction, called Set-Constrained Delivery Broadcast (SCD-broadcast), whose aim is to provide its users with an appropriate abstraction level when they have to implement objects or distributed tasks in an asynchronous message-passing system prone to process crash failures. This abstraction allows each process to broadcast messages and deliver a sequence of sets of messages in such a way that, if a process delivers a set of messages including a message m and later delivers a set of messages including a message m ' , no process delivers first a set of messages including m ' and later a set of message including m. After having presented an algorithm implementing SCD-broadcast, the paper investigates its programming power and its computability limits. On the "power" side it presents SCD-broadcast-based algorithms, which are both simple and efficient, building objects (such as snapshot and conflict-free replicated data), and distributed tasks. On the "computability limits" side it shows that SCD-broadcast and read/write registers are computationally equivalent.
11 citations
Posted Content•
TL;DR: This note reviews the results of the application of domain theory to the quantum setting and presents some new thoughts in this field.
Abstract: Classically domain theory is a rigourous mathematical structure to describe denotational semantics for programming languages and to study the computability of partial functions. Recently, the application of domain theory has also been extended to the quantum setting. In this note we review these results and we present some new thoughts in this field.
11 citations
30 Jun 2006
TL;DR: A real variant of the classical word problem for groups is presented which is established reducible to and from the real Halting Problem and how additional real constants increase the power of a BSS machine is analyzed.
Abstract: Most of the existing work in real number computation theory concentrates on complexity issues rather than computability aspects. Though some natural problems like deciding membership in the Mandelbrot set or in the set of rational numbers are known to be undecidable in the Blum-Shub-Smale (BSS) model of computation over the reals, there has not been much work on different degrees of undecidability. A typical question into this direction is the real version of Post's classical problem: Are there some explicit undecidable problems below the real Halting Problem?
In this paper we study three different topics related to such questions: First an extension of a positive answer to Post's problem to the linear setting. We then analyze how additional real constants increase the power of a BSS machine. And finally a real variant of the classical word problem for groups is presented which we establish reducible to and from (that is, complete for) the BSS Halting problem.
11 citations
TL;DR: The problems of converting programs to decision tables are investigated and extensions to the theory of computation and computability are suggested.
Abstract: The problems of converting programs to decision tables are investigated. Objectives of these conversions are mainly program debugging and optimization in practice. Extensions to the theory of computation and computability are suggested.
11 citations
Posted Content•
TL;DR: In this paper, a rigorous model of human computation and associated measures of complexity is proposed to better understand what humans can compute in their heads, where the adversary is restricted to at most 10^24 (Avogadro number of) steps.
Abstract: What can humans compute in their heads? We are thinking of a variety of Crypto Protocols, games like Sudoku, Crossword Puzzles, Speed Chess, and so on. The intent of this paper is to apply the ideas and methods of theoretical computer science to better understand what humans can compute in their heads. For example, can a person compute a function in their head so that an eavesdropper with a powerful computer --- who sees the responses to random input --- still cannot infer responses to new inputs? To address such questions, we propose a rigorous model of human computation and associated measures of complexity. We apply the model and measures first and foremost to the problem of (1) humanly computable password generation, and then consider related problems of (2) humanly computable "one-way functions" and (3) humanly computable "pseudorandom generators".
The theory of Human Computability developed here plays by different rules than standard computability, and this takes some getting used to. For reasons to be made clear, the polynomial versus exponential time divide of modern computability theory is irrelevant to human computation. In human computability, the step-counts for both humans and computers must be more concrete. Specifically, we restrict the adversary to at most 10^24 (Avogadro number of) steps. An alternate view of this work is that it deals with the analysis of algorithms and counting steps for the case that inputs are small as opposed to the usual case of inputs large-in-the-limit.
11 citations