Thinking about Gödel and Turing - Essays on Complexity, 1970 - 2007
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The Halting Probability Omega: Irreducible Complexity in Pure Mathematics∗
Some Gödel centenary reflections on whether incompleteness is really serious, and whether mathematics should be done somewhat differently, based on using algorithmic complexity measured in bits of information. Introduction: What is mathematics? It is a pleasure for me to be here today giving this talk in a lecture series in honor of Frederigo Enriques. Enriques was a great believer in mathemati...
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question P ? = NP for Infinite Time Turing Machines, and several variants on it, are treated in e.g. [Sc], [DeHaSc], and [HaWe]. Besides time complexity, we may also try to look at issues of space complexity in ITTMs. However, because an ITTM contains tapes of length ω, and all nontrivial ITTM computations will use the entire, ω-length tape, simply measuring the space complexity by counting the...
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The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requirement regarding basic operations implies Church’s Thesis, namely, that the only numeric function...
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In this paper we develop an approach to the notion of computable functionals in a very abstract setting not restricted to Turing or, say, polynomial computability. We assume to start from some basic class of domains and a basic class of functions deened on these domains. (An example may be natural numbers with polytime computable functions). Then we deene what are \all" corresponding functional...
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We discuss the influence of Gödel and his results on the surfacing of the rigorous notion of computability afforded by Turing. We also address the debate about the limits of Artificial Intelligence spurned by Roger Penrose, on the basis of Gödel's theorems, and the views of Gödel himself. We conclude by touching upon the use of logic as a tool with which to approach the description of mind.
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