359 research outputs found
Computability of Operators on Continuous and Discrete Time Streams
A stream is a sequence of data indexed by time. The behaviour of natural and artificial systems can be modelled bystreams and stream transformations. There are two distinct types of data stream: streams based on continuous time and streamsbased on discrete time. Having investigated case studies of both kinds separately, we have begun to combine their study in aunified theory of stream transformers, specified by equations. Using only the standard mathematical techniques of topology, wehave proved continuity properties of stream transformers. Here, in this sequel, we analyse their computability. We use the theoryof computable functions on algebras to design two distinct methods for defining computability on continuous and discrete timestreams of data from a complete metric space. One is based on low-level concrete representations, specifically enumerations, andthe other is based on high-level programming, specifically âwhileâ programs, over abstract data types. We analyse when thesemethods are equivalent. We demonstrate the use of the methods by showing the computability of an analog computing system.We discuss the idea that continuity and computability are important for models of physical systems to be âwell-posedâ
Compact manifolds with computable boundaries
We investigate conditions under which a co-computably enumerable closed set
in a computable metric space is computable and prove that in each locally
computable computable metric space each co-computably enumerable compact
manifold with computable boundary is computable. In fact, we examine the notion
of a semi-computable compact set and we prove a more general result: in any
computable metric space each semi-computable compact manifold with computable
boundary is computable. In particular, each semi-computable compact
(boundaryless) manifold is computable
Verifying proofs in constant depth
In this paper we initiate the study of proof systems where verification of proofs proceeds by NC circuits. We investigate the question which languages admit proof systems in this very restricted model. Formulated alternatively, we ask which languages can be enumerated by NC functions. Our results show that the answer to this problem is not determined by the complexity of the language. On the one hand, we construct NC proof systems for a variety of languages ranging from regular to NP-complete. On the other hand, we show by combinatorial methods that even easy regular languages such as Exact-OR do not admit NC proof systems. We also present a general construction of proof systems for regular languages with strongly connected NFA's
A Computer Verified Theory of Compact Sets
Compact sets in constructive mathematics capture our intuition of what
computable subsets of the plane (or any other complete metric space) ought to
be. A good representation of compact sets provides an efficient means of
creating and displaying images with a computer. In this paper, I build upon
existing work about complete metric spaces to define compact sets as the
completion of the space of finite sets under the Hausdorff metric. This
definition allowed me to quickly develop a computer verified theory of compact
sets. I applied this theory to compute provably correct plots of uniformly
continuous functions.Comment: This paper is to be part of the proceedings of the Symbolic
Computation in Software Science Austrian-Japanese Workshop (SCSS 2008
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