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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]:
T.: “Elementary computable topology
 Journal of Universal Computer Science
, 2009
"... Abstract: We revise and extend the foundation of computable topology in the framework of Type2 theory of effectivity, TTE, where continuity and computability on finite sets by means of notations and representations. We start from a computable topological space, which is a T0space with a notation ..."
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Cited by 18 (6 self)
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Abstract: We revise and extend the foundation of computable topology in the framework of Type2 theory of effectivity, TTE, where continuity and computability on finite sets by means of notations and representations. We start from a computable topological space, which is a T0space with a notation of a base such that intersection is computable, and define a number of multirepresentations of the points and of the open, the closed and the compact sets and study their properties and relations. We study computability of boolean operations. By merely requiring “provability ” of suitable relations (element, nonempty intersection, subset) we characterize in turn computability on the points, the open sets (!), computability on the open sets, computability on the closed sets, the compact sets(!), and computability on the compact sets. We study modifications of the definition of a computable topological space that do not change the derived computability concepts. We study subspaces and products and compare a number of representations of the space of partial continuous functions. Since we are operating mainly with the base elements, which can be considered as regions for points (“pointless topology”), we study to which extent these regions can be filled with points (completions). We conclude with some simple applications including Dini’s Theorem as an example.
Emergence as a ComputabilityTheoretic Phenomenon
, 2008
"... In dealing with emergent phenomena, a common task is to identify useful descriptions of them in terms of the underlying atomic processes, and to extract enough computational content from these descriptions to enable predictions to be made. Generally, the underlying atomic processes are quite well un ..."
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Cited by 5 (2 self)
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In dealing with emergent phenomena, a common task is to identify useful descriptions of them in terms of the underlying atomic processes, and to extract enough computational content from these descriptions to enable predictions to be made. Generally, the underlying atomic processes are quite well understood, and (with important exceptions) captured by mathematics from which it is relatively easy to extract algorithmic content. A widespread view is that the difficulty in describing transitions from algorithmic activity to the emergence associated with chaotic situations is a simple case of complexity outstripping computational resources and human ingenuity. Or, on the other hand, that phenomena transcending the standard Turing model of computation, if they exist, must necessarily lie outside the domain of classical computability theory. In this talk we suggest that much of the current confusion arises from conceptual gaps and the lack of a suitably fundamental model within which to situate emergence. We examine the potential for placing emergent relations in a familiar context based on Turing’s 1939 model for interactive computation over structures described in terms of reals. The explanatory power of this model is explored, formalising informal descriptions in terms of mathematical definability and invariance, and relating a range of basic scientific puzzles to results and intractable problems in computability theory. In this talk
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.
, # (. " /01The Impossibility of an E¤ective Theory of Policy in a Complex Economy
, 2005
"... It is shown that for a ‘complex economy’, characterised in terms of a formal dynamical system capable of computation universality, it is impossible to devise an e¤ective theory of policy. ..."
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Cited by 4 (3 self)
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It is shown that for a ‘complex economy’, characterised in terms of a formal dynamical system capable of computation universality, it is impossible to devise an e¤ective theory of policy.
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
Extending and Interpreting Post’s Programme
, 2008
"... Computability theory concerns information with a causal – typically algorithmic – structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic in ..."
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Cited by 2 (2 self)
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Computability theory concerns information with a causal – typically algorithmic – structure. As such, it provides a schematic analysis of many naturally occurring situations. Emil Post was the first to focus on the close relationship between information, coded as real numbers, and its algorithmic infrastructure. Having characterised the close connection between the quantifier type of a real and the Turing jump operation, he looked for more subtle ways in which information entails a particular causal context. Specifically, he wanted to find simple relations on reals which produced richness of local computabilitytheoretic structure. To this extent, he was not just interested in causal structure as an abstraction, but in the way in which this structure emerges in natural contexts. Posts programme was the genesis of a more far reaching research project. In this article we will firstly review the history of Posts programme, and look at two interesting developments of Posts approach. The first of these developments concerns the extension of the core programme, initially restricted to the Turing structure of the computably enumerable sets of natural numbers, to the Ershov hierarchy of sets. The second looks at how new types of information coming from the recent growth of research into randomness, and the revealing of unexpected new computabilitytheoretic infrastructure. We will conclude by viewing Posts programme from a more general perspective. We will look at how algorithmic structure does not just emerge mathematically from information, but how that emergent structure can model the emergence of very basic aspects of the real world.
2004], There is no low maximal d.c.e. degree  corrigendum
 Math. Log. Quart
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SPLITTING AND JUMP INVERSION IN THE TURING DEGREES
"... Abstract. It is shown that for any computably enumerable degree a � = 0, any degree c � = 0, and any Turing degree s, if s ≥ 0 ′ , and c.e. in a, then there exists a c.e. degree x with the following properties, (1) x < a, c � ≤ x, (2) a is splittable over x, and (3) x ′ = s. This implies that th ..."
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Abstract. It is shown that for any computably enumerable degree a � = 0, any degree c � = 0, and any Turing degree s, if s ≥ 0 ′ , and c.e. in a, then there exists a c.e. degree x with the following properties, (1) x < a, c � ≤ x, (2) a is splittable over x, and (3) x ′ = s. This implies that the Sacks ’ splitting theorem and the Sacks ’ jump theorem can be uniformly combined. A corollary is that there is no atomic jump class consisting entirely of Harrington nonsplitting bases. 1.
Splitting and Nonsplitting in the Σ 0 2 Enumeration Degrees ∗
"... This paper continues the project, initiated in [ACK], of describing general conditions under which relative splittings are derivable in the local structure of the enumeration degrees, for which the Ershov hierarchy provides an informative setting. The main results below include a proof that any hig ..."
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This paper continues the project, initiated in [ACK], of describing general conditions under which relative splittings are derivable in the local structure of the enumeration degrees, for which the Ershov hierarchy provides an informative setting. The main results below include a proof that any high total edegree below 0 ′ e is splittable over any low edegree below it, a noncupping result in the high enumeration degrees which occurs at a low level of the Ershov hierarchy, and a ∅ ′′ ′priority construction of a Π 0 1 edegree unsplittable over a 3c.e. edegree below it. 1