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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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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
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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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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.
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
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.
THE MINIMAL EDEGREE PROBLEM IN FRAGMENTS OF PEANO ARITHMETIC
"... Abstract. We study the minimal enumeration degree (edegree) problem in models of fragments of Peano arithmetic (PA) and prove the following results: In any model M of Σ2 induction, there is a minimal enumeration degree if and only if M is a nonstandard model. Furthermore, any cut in such a model ha ..."
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Abstract. We study the minimal enumeration degree (edegree) problem in models of fragments of Peano arithmetic (PA) and prove the following results: In any model M of Σ2 induction, there is a minimal enumeration degree if and only if M is a nonstandard model. Furthermore, any cut in such a model has minimal edegree. By contrast, this phenomenon fails in the absence of Σ2 induction. In fact, whether every Σ2 cut has minimal edegree is independent of the Σ2 bounding principle. 1.
Total Degrees and Nonsplitting Properties of 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. The main results below include a proof that any high total edegree below 0 ′ e is splittable over any low edegree ..."
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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. The main results below include a proof that any high total edegree below 0 ′ e is splittable over any low edegree below it, and a construction of a Π 0 1 edegree unsplittable over a ∆2 edegree below it. In [ACK] it was shown that using semirecursive sets one can construct minimal pairs of edegrees by both effective and uniform ways, following which new results concerning the local distribution of total edegrees and of the degrees of semirecursive sets enabled one to proceed, via the natural embedding of the Turing degrees in the enumeration degrees, to results concerning embeddings of the diamond lattice in the edegrees. A particularly striking application of these techniques was a relatively simple derivation of a strong generalisation of the Ahmad Diamond Theorem. This paper extends the known constraints on further progress in this direction, such as the result of Ahmad and Lachlan [AL98] showing the existence of a nonsplitting ∆ 0 2 edegree> 0e, and the recent result of Soskova [Sos07] showing that 0 ′ e is unsplittable in the Σ 0 2 edegrees above some Σ 0 2 edegree < 0 ′ e. This work also relates to results (e.g. Cooper and Copestake [CC88]) limiting the local distribution of total edegrees. For further background concerning enumeration reducibility and its degree structure, the reader is referred to Cooper [Co90], Sorbi [Sor97] or Cooper [Co04], chapter 11. Theorem 1 If a < h ≤ 0 ′ , a is low and h is total and high then there is a low total edegree b such that a ≤ b < h.
, # (. " /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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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.
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
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.