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Computability theory
, 2004
"... Nature was computing long before humans started. It is the algorithmic content of the universe makes it an environment we can survive in. On the other hand, computation has been basic to civilisation from the earliest times. But computability? Computability theory is computation with consciousness, ..."
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Cited by 83 (6 self)
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Nature was computing long before humans started. It is the algorithmic content of the universe makes it an environment we can survive in. On the other hand, computation has been basic to civilisation from the earliest times. But computability? Computability theory is computation with consciousness, and entails the huge step from doing computation to observing and analysing the activity, and understanding something about what we can and cannot compute. And then — using the knowledge acquired as a stepping stone to a better understanding of the world we live in, and to new and previously unexpected computational strategies. It is relatively recently that computability graduated from being an essential element of our daily lives to being a concept one could talk about with precision. Computability as a theory originated with the work of Gödel, Turing, Church and others in the 1930s. The idea that reasoning might be essentially algorithmic goes back to Gottfried Leibniz — as he says in The Art of Discovery (1685), [24, p.51]:
Composition and Submachine Concepts for Sequential ASMs
 Computer Science Logic (Proceedings of CSL 2000), volume 1862 of LNCS
, 2000
"... We define three composition and structuring concepts which... ..."
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Cited by 29 (13 self)
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We define three composition and structuring concepts which...
Algorithmic randomness, quantum physics, and incompleteness
 Proceedings of the Conference “Machines, Computations and Universality” (MCU’2004), number 3354 in Lecture Notes in Computer Science
, 2006
"... When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is almost certainly wrong. Arthur C. Clarke ..."
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Cited by 15 (2 self)
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When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is almost certainly wrong. Arthur C. Clarke
Abstract State Machines: A unifying view of models of computation and of system design frameworks
 ANNALS OF PURE AND APPLIED LOGIC
, 2005
"... We capture the principal models of computation and specification in the literature by a uniform set of transparent mathematical descriptions which—starting from scratch—provide the conceptual basis for a comparative study. ..."
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Cited by 13 (7 self)
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We capture the principal models of computation and specification in the literature by a uniform set of transparent mathematical descriptions which—starting from scratch—provide the conceptual basis for a comparative study.
Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
, 2009
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A Review of Mathematical Knowledge Management ⋆
"... Abstract. Mathematical Knowledge Management (MKM), as a field, has seen tremendous growth in the last few years. This period was one where many research threads were started, and the field was defining itself. We believe that we are now in a position to use the MKM body of knowledge as a means to de ..."
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Cited by 6 (0 self)
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Abstract. Mathematical Knowledge Management (MKM), as a field, has seen tremendous growth in the last few years. This period was one where many research threads were started, and the field was defining itself. We believe that we are now in a position to use the MKM body of knowledge as a means to define what MKM is, what it worries about, etc. In this paper, we review the literature of MKM and gather various metadata from these papers. After offering some definitions surrounding MKM, we analyse the metadata we have gathered from these papers, in an effort to cast more light on the field of MKM and its evolution.
The Church–Turing thesis: consensus and opposition
 Logical Approaches to Computational Barriers: Second Conference on Computability in Europe, CiE 2006
, 2006
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FROM DESCARTES TO TURING: THE COMPUTATIONAL CONTENT OF SUPERVENIENCE
"... Mathematics can provide precise formulations of relatively vague concepts and problems from the real world, and bring out underlying structure common to diverse scientific areas. Sometimes very natural mathematical concepts lie neglected and not widely understood for many years, before their fundame ..."
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Cited by 4 (4 self)
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Mathematics can provide precise formulations of relatively vague concepts and problems from the real world, and bring out underlying structure common to diverse scientific areas. Sometimes very natural mathematical concepts lie neglected and not widely understood for many years, before their fundamental relevance is recognised and their explanatory power is fully exploited. The notion of definability in a structure is such a concept, and Turing’s [77] 1939 model of interactive computation provides a fruitful context in which to exercise the usefulness of definability as a powerful and widely applicable source of understanding. In this article we set out to relate this simple idea to one of the oldest and apparently least scientifically approachable of problems — that of realistically modelling how mental properties supervene on physical ones.
Romańczuk U.: On Dynamical Systems of Large Girth or Cycle Indicator and their applications to Multivariate Cryptography
 Artificial Intelligence, Evolutionary Computing and Metaheuristics, In the footsteps of Alan Turing Series: Studies in Computational Intelligence
, 2013
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Incomputability, Emergence and the Turing Universe
"... Amongst the huge literature concerning emergence, reductionism and mechanism, there is a role for analysis of the underlying mathematical constraints. Much of the speculation, confusion, controversy and descriptive verbiage might be clarified via suitable modelling and theory. The key ingredients we ..."
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Amongst the huge literature concerning emergence, reductionism and mechanism, there is a role for analysis of the underlying mathematical constraints. Much of the speculation, confusion, controversy and descriptive verbiage might be clarified via suitable modelling and theory. The key ingredients we bring to this project are the mathematical notions of definability and invariance, a computability theoretic framework in a realworld context, and within that, the modelling of basic causal environments via Turing’s 1939 notion of interactive computation over a structure described in terms of reals. Useful outcomes are: a refinement of what one understands to be a causal relationship, including nonmechanistic, irreversible causal relationships; an appreciation of how the mathematically simple origins of incomputability in definable hierarchies are materialised in the real world; and an understanding of the powerful explanatory role of current computability theoretic developments. The theme of this article concerns the way in which mathematics can structure everyday discussions around a range of important issues — and can also reinforce intuitions about theoretical links between different aspects of the real world. This fits with the widespread sense of excitement and expectation felt in many fields — and of a corresponding confusion — and of a tension characteristic of a Kuhnian paradigm shift. What we have below can be seen as tentative steps towards the sort of mathematical modelling needed for such a shift to be completed. In section 1, we outline the decisive role mathematics played in the birth of modern science; and how, more recently, it has helped us towards a better understanding of the nature and limitations of the scientific enterprise. In section 2, we review how the mathematics brings out inherent contradictions in the Laplacian model of scientific activity. And we look at some of the approaches to dealing