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45
Incomputability in Nature
"... To what extent is incomputability relevant to the material Universe? We look at ways in which this question might be answered, and the extent to which the theory of computability, which grew out of the work of Godel, Church, Kleene and Turing, can contribute to a clear resolution of the current conf ..."
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Cited by 19 (11 self)
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To what extent is incomputability relevant to the material Universe? We look at ways in which this question might be answered, and the extent to which the theory of computability, which grew out of the work of Godel, Church, Kleene and Turing, can contribute to a clear resolution of the current confusion. It is hoped that the presentation will be accessible to the nonspecialist reader.
Real Hypercomputation and Continuity
, 2005
"... By the sometimes socalled Main Theorem of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of hypercomputation allow for the effective evaluation of also discontinuous f: R → R. More precisely the present work considers the following t ..."
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Cited by 12 (1 self)
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By the sometimes socalled Main Theorem of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of hypercomputation allow for the effective evaluation of also discontinuous f: R → R. More precisely the present work considers the following three superTuring notions of real function computability: – relativized computation; specifically given oracle access to the Halting Problem H ≡T ∅ ′ or its jump H ′ ≡T ∅ ′ ′; – encoding input x ∈ R and/or output y = f(x) in weaker ways also related to the Arithmetic Hierarchy; – nondeterministic computation. It turns out that any f: R → R computable in the first or second sense is still necessarily continuous whereas the third type of hypercomputation does provide the required power to evaluate for instance the discontinuous Heaviside function.
The construction and validation of rating scales for oral tests in English as a Foreign Language. Unpublished
, 1993
"... The present research investigates the principles upon which rating scales in oral testing are constructed and used, and the subsequent claims of reliability and validity made for them. The research addresses two main questions: (i) can rating scales be constructed on the basis of an analysis of a da ..."
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Cited by 9 (3 self)
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The present research investigates the principles upon which rating scales in oral testing are constructed and used, and the subsequent claims of reliability and validity made for them. The research addresses two main questions: (i) can rating scales be constructed on the basis of an analysis of a database of actual student speech; and (ii) are scales produced on the basis of student speech superior to those produced using apriori methods? A corpus of spoken data was collected and analyzed. Discriminant analysis was used in order to isolate factors which could discriminate between students of different ability levels, and a Fluency and an Accuracy rating scale constructed. The Fluency rating scale was seen to be the most stable in the construction phase of the study. Forty seven students took three tasks. Video recordings were rated by five raters on the two rating scales and the English Language Testing Service rating scale.
Set Theory and Physics
 FOUNDATIONS OF PHYSICS, VOL. 25, NO. 11
, 1995
"... Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of soli ..."
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Cited by 9 (7 self)
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Inasmuch as physical theories are formalizable, set theory provides a framework for theoretical physics. Four speculations about the relevance of set theoretical modeling for physics are presented: the role of transcendental set theory (i) hr chaos theory, (ii) for paradoxical decompositions of solid threedimensional objects, (iii) in the theory of effective computability (ChurchTurhrg thesis) related to the possible "solution of supertasks," and (iv) for weak solutions. Several approaches to set theory and their advantages and disadvatages for" physical applications are discussed: Cantorian "naive" (i.e., nonaxiomatic) set theory, contructivism, and operationalism, hr the arrthor's ophrion, an attitude of "suspended attention" (a term borrowed from psychoanalysis) seems most promising for progress. Physical and set theoretical entities must be operationalized wherever possible. At the same thne, physicists shouM be open to "bizarre" or "mindboggling" new formalisms, which treed not be operationalizable or testable at the thne of their " creation, but which may successfully lead to novel fields of phenomenology and technology.
Computable Invariance
 THEORETICAL COMPUTER SCIENCE
, 1996
"... In Computable Analysis each computable function is continuous and computably invariant, i.e. it maps computable points to computable points. On the other hand, discontinuity is a sufficient condition for noncomputability, but a discontinuous function might still be computably invariant. We investig ..."
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Cited by 8 (3 self)
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In Computable Analysis each computable function is continuous and computably invariant, i.e. it maps computable points to computable points. On the other hand, discontinuity is a sufficient condition for noncomputability, but a discontinuous function might still be computably invariant. We investigate algebraic conditions which guarantee that a discontinuous function is sufficiently discontinuous and sufficiently effective such that it is not computably invariant. Our main theorem generalizes the First Main Theorem ouf PourEl & Richards (cf. [20]). We apply our theorem to prove that several setvalued operators are not computably invariant.
The decimationHadamard transform of twolevel autocorrelation sequences
 IEEE Trans. Inform. Theory
, 2002
"... Abstract—A new method to study and search for twolevel autocorrelation sequences for both binary and nonbinary cases is developed. This method iteratively applies two operations: decimation and the Hadamard transform based on general orthogonal functions, referred to as the decimationHadamard tra ..."
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Cited by 7 (7 self)
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Abstract—A new method to study and search for twolevel autocorrelation sequences for both binary and nonbinary cases is developed. This method iteratively applies two operations: decimation and the Hadamard transform based on general orthogonal functions, referred to as the decimationHadamard transform (DHT). The second iterative DHT can transform one class of such sequences into another inequivalent class of such sequences, a process called realization. The existence and counting problems of the second iterative DHT are discussed. Using the second iterative DHT, and starting with a single binarysequence (when is odd), we believe one can obtain all the known twolevel autocorrelation sequences of period 2 1which have no subfield factorization. We have verified this for odd 17. Interestingly, no previously unknown examples were found by this process for any odd 17. This is supporting evidence (albeit weak) for the conjecture that all families of cyclic Hadamard difference sets of period 2 1 having no subfield factorization are now known, at least for odd. Experimental results are provided. Index Terms—Group characters, iterative decimationHadamard transform (DHT), orthogonal functions, trace representation, twolevel autocorrelation sequences. I.
Physicallyrelativized ChurchTuring Hypotheses. Applied Mathematics and Computation 215, 4
 in the School of Mathematics at the University of Leeds, U.K. © 2012 ACM 00010782/12/03 $10.00 march 2012  vol. 55  no. 3  communications of the acm 83
"... Abstract. We turn ‘the ’ ChurchTuring Hypothesis from an ambiguous source of sensational speculations into a (collection of) sound and welldefined scientific problem(s): Examining recent controversies, and causes for misunderstanding, concerning the state of the ChurchTuring Hypothesis (CTH), sug ..."
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Cited by 6 (0 self)
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Abstract. We turn ‘the ’ ChurchTuring Hypothesis from an ambiguous source of sensational speculations into a (collection of) sound and welldefined scientific problem(s): Examining recent controversies, and causes for misunderstanding, concerning the state of the ChurchTuring Hypothesis (CTH), suggests to study the CTH relative to an arbitrary but specific physical theory—rather than vaguely referring to “nature ” in general. To this end we combine (and compare) physical structuralism with (models of computation in) complexity theory. The benefit of this formal framework is illustrated by reporting on some previous, and giving one new, example result(s) of computability
Strong Determinism vs. Computability
 The Foundational Debate, Complexity and Constructivity in Mathematics and
, 1995
"... Are minds subject to laws of physics? Are the laws of physics computable? Are conscious thought processes computable? Currently there is little agreement as to what are the right answers to these questions. Penrose ([41], p. 644) goes one step further and asserts that: a radical new theory is indeed ..."
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Cited by 5 (1 self)
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Are minds subject to laws of physics? Are the laws of physics computable? Are conscious thought processes computable? Currently there is little agreement as to what are the right answers to these questions. Penrose ([41], p. 644) goes one step further and asserts that: a radical new theory is indeed needed, and I am suggesting, moreover, that this theory, when it is found, will be of an essentially noncomputational character. The aim of this paper is three fold: 1) to examine the incompatibility between the hypothesis of strong determinism and computability, 2) to give new examples of uncomputable physical laws, and 3) to discuss the relevance of Gödel’s Incompleteness Theorem in refuting the claim that an algorithmic theory—like strong AI—can provide an adequate theory of mind. Finally, we question the adequacy of the theory of computation to discuss physical laws and thought processes. 1
On the complexity of finding paths in a twodimensional domain I: Shortest paths
 in Proc. International Conference on Computability and Complexity in Analysis, 2003, Informatik Berichte 3028/2003, FernUniversität in
, 2004
"... The problem of finding a piecewise straightline path, with a constant number of line segments, in a twodimensional domain is studied in the Turing machinebased computational model and in the discrete complexity theory. It is proved that, for polynomialtime recognizable domains associated with po ..."
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The problem of finding a piecewise straightline path, with a constant number of line segments, in a twodimensional domain is studied in the Turing machinebased computational model and in the discrete complexity theory. It is proved that, for polynomialtime recognizable domains associated with polynomialtime computable distance functions, the complexity of this problem is equivalent to a discrete problem which is complete for Σ P 2, the second level of the polynomialtime hierarchy.
Analytical and Experimental Verification of a Flight Article for a Mach8 BoundaryLayer Experiment, NASA TM4733
 Computational Mechanics Publications, Southampton, United Kingdom
, 1996
"... for a Mach8 BoundaryLayer ..."