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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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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]:
How can Nature help us compute
 SOFSEM 2006: Theory and Practice of Computer Science – 32nd Conference on Current Trends in Theory and Practice of Computer Science, Merin, Czech Republic, January 21–27
, 2006
"... Abstract. Ever since Alan Turing gave us a machine model of algorithmic computation, there have been questions about how widely it is applicable (some asked by Turing himself). Although the computer on our desk can be viewed in isolation as a Universal Turing Machine, there are many examples in natu ..."
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Abstract. Ever since Alan Turing gave us a machine model of algorithmic computation, there have been questions about how widely it is applicable (some asked by Turing himself). Although the computer on our desk can be viewed in isolation as a Universal Turing Machine, there are many examples in nature of what looks like computation, but for which there is no wellunderstood model. In many areas, we have to come to terms with emergence not being clearly algorithmic. The positive side of this is the growth of new computational paradigms based on metaphors for natural phenomena, and the devising of very informative computer simulations got from copying nature. This talk is concerned with general questions such as: • Can natural computation, in its various forms, provide us with genuinely new ways of computing? • To what extent can natural processes be captured computationally? • Is there a universal model underlying these new paradigms?
Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory
, 2009
"... ..."
Oracles and Advice as Measurements
"... Abstract. In this paper we will try to understand how oracles and advice functions, which are mathematical abstractions in the theory of computability and complexity, can be seen as physical measurements in Classical Physics. First, we consider how physical measurements are a natural external source ..."
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Abstract. In this paper we will try to understand how oracles and advice functions, which are mathematical abstractions in the theory of computability and complexity, can be seen as physical measurements in Classical Physics. First, we consider how physical measurements are a natural external source of information to an algorithmic computation. We argue that oracles and advice functions can help us to understand how the structure of space and time has information content that can be processed by Turing machines (after Cooper and Odifreddi [10] and Copeland and Proudfoot [11, 12]). We show that nonuniform complexity is an adequate framework for classifying feasible computations by Turing machines interacting with an oracle in Nature. By classifying the information content of such an oracle using Kolmogorov complexity, we obtain a hierarchical structure for advice classes. 1
Definability as hypercomputational effect
 Applied Mathematics and Computation
"... The classical simulation of physical processes using standard models of computation is fraught with problems. On the other hand, attempts at modelling realworld computation with the aim of isolating its hypercomputational content have struggled to convince. We argue that a better basic understandin ..."
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The classical simulation of physical processes using standard models of computation is fraught with problems. On the other hand, attempts at modelling realworld computation with the aim of isolating its hypercomputational content have struggled to convince. We argue that a better basic understanding can be achieved through computability theoretic deconstruction of those physical phenomena most resistant to classical simulation. From this we may be able to better assess whether the hypercomputational enterprise is proleptic computer science, or of mainly philosophical interest.
The Mathematician's Bias  and the Return to Embodied Computation. In H. Zenil (Ed.) A Computable Universe. Understanding Computation & Exploring Nature As Computation. World Scientific: New York/London/Singapore
, 2012
"... There are growing uncertainties surrounding the classical model of computation established by Gödel, Church, Kleene, Turing and others in the 1930s onwards. The mismatch between the Turing machine conception, and the experiences of those more practically engaged in computing, has parallels with th ..."
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There are growing uncertainties surrounding the classical model of computation established by Gödel, Church, Kleene, Turing and others in the 1930s onwards. The mismatch between the Turing machine conception, and the experiences of those more practically engaged in computing, has parallels with the wider one between science and those working creatively or intuitively out in the ‘real ’ world. The scientific outlook is more flexible and basic than some understand or want to admit. The science is subject to limitations which threaten careers. We look at embodiment and disembodiment of computation as the key to the mismatch, and find Turing had the right idea all along – amongst a productive confusion of ideas about computation in the real and the abstract worlds. When we get out of bed in the morning, we approach a complicated world of information with a determination not just to survive the day – though that may be hard enough: we mean to “compute ” our way towards various vaguely defined
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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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.
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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Cited by 2 (1 self)
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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
Five views of hypercomputation
"... We overview different approaches to the study of hypercomputation and other investigations on the plausibility of the physical Church–Turing thesis. We propose five thesis to classify investigation in this area. Sly does it. Tiptoe catspaws. Slide and creep. ..."
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We overview different approaches to the study of hypercomputation and other investigations on the plausibility of the physical Church–Turing thesis. We propose five thesis to classify investigation in this area. Sly does it. Tiptoe catspaws. Slide and creep.
The Incomputable Alan Turing
 In the Proceedings of
"... The last century saw dramatic challenges to the Laplacian predictability which had underpinned scientific research for around 300 years. Basic to this was Alan Turing’s 1936 discovery (along with Alonzo Church) of the existence of unsolvable problems. This paper focuses on incomputability as a power ..."
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The last century saw dramatic challenges to the Laplacian predictability which had underpinned scientific research for around 300 years. Basic to this was Alan Turing’s 1936 discovery (along with Alonzo Church) of the existence of unsolvable problems. This paper focuses on incomputability as a powerful theme in Turing’s work and personal life, and examines its role in his evolving concept of machine intelligence. It also traces some of the ways in which important new developments are anticipated by Turing’s ideas in logic. This paper is based on the talk given on 5th June 2004 at the conference at Manchester University marking the 50th anniversary of Alan Turing’s death. It is published by the British Computer Society on