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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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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.
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 8 (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.
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 3 (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
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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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
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 3 (2 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.
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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Cited by 2 (1 self)
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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.
Effectively Open Real Functions
, 2005
"... Abstract. A function f is continuous iff the preimage f −1 [V] of any open set V is open again. Dual to this topological property, f is called open iff the image f[U] of any open set U is open again. Several classical Open Mapping Theorems in Analysis provide a variety of sufficient conditions for ..."
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Abstract. A function f is continuous iff the preimage f −1 [V] of any open set V is open again. Dual to this topological property, f is called open iff the image f[U] of any open set U is open again. Several classical Open Mapping Theorems in Analysis provide a variety of sufficient conditions for openness. By the Main Theorem of Recursive Analysis, computable real functions are necessarily continuous. In fact they admit a wellknown characterization in terms of the mapping V ↦ → f −1 [V] being effective: Given a list of open rational balls exhausting V, a Turing Machine can generate a corresponding list for f −1 [V]. Analogously, effective openness requires the mapping U ↦ → f[U] on open real subsets to be effective. By effectivizing classical Open Mapping Theorems as well as from application of Tarski’s Quantifier Elimination, the present work reveals several rich classes of functions to be effectively open. 1
Under PhysicsMotivated Constraints, GenerallyNonAlgorithmic Computational Problems Become Algorithmically Solvable
"... Abstract. It is well known that many computational problems are, in general, not algorithmically solvable: e.g., it is not possible to algorithmically decide whether two computable real numbers are equal, and it is not possible to compute the roots of a computable function. We propose to constraint ..."
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Abstract. It is well known that many computational problems are, in general, not algorithmically solvable: e.g., it is not possible to algorithmically decide whether two computable real numbers are equal, and it is not possible to compute the roots of a computable function. We propose to constraint such operations to certain “sets of typical elements” or “sets of random elements”. In our previous papers, we proposed (and analyzed) physicsmotivated definitions for these notions. In short, a set T is a set of typical elements if for every definable sequences of sets An with An ⊇ An+1 and ∩
The effects of input enhancement and interactive video viewing on the development of pragmatic awareness and use in the beginning Spanish L2 classroom
, 2002
"... ..."
Series Preproceedings of the Workshop “Physics and Computation ” 2008
, 2008
"... In the 1940s, two different views of the brain and the computer were equally important. One was the analog technology and theory that had emerged before the war. The other was the digital technology and theory that was to become the main paradigm of computation. 1 The outcome of the contest between ..."
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In the 1940s, two different views of the brain and the computer were equally important. One was the analog technology and theory that had emerged before the war. The other was the digital technology and theory that was to become the main paradigm of computation. 1 The outcome of the contest between these two competing views derived from technological and epistemological arguments. While digital technology was improving dramatically, the technology of analog machines had already reached a significant level of development. In particular, digital technology offered a more effective way to control the precision of calculations. But the epistemological discussion was, at the time, equally relevant. For the supporters of the analog computer, the digital model — which can only process information transformed and coded in binary — wouldn’t be suitable to represent certain kinds of continuous variation that help determine brain functions. With analog machines, on the contrary, there would be few or no layers between natural objects and the work and structure of computation (cf. [4, 1]). The 1942–52 Macy Conferences in cybernetics helped to validate digital theory and logic as legitimate ways to think about the brain and the machine [4]. In particular, those conferences helped made McCullochPitts ’ digital model