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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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Cited by 14 (8 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.
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.
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 4 (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 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
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.
The effects of input enhancement and interactive video viewing on the development of pragmatic awareness and use in the beginning Spanish L2 classroom
, 2002
"... ..."
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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Cited by 1 (1 self)
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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 ∩
Effectively open real functions
 J. Complexity
"... 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. ..."
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Cited by 1 (0 self)
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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 an according list for f −1 [V]. Analogously, effective openness requires the mapping U↦ → f[U] on open real subsets to be effective. The present work reveals important and rich classes of functions to be effectively open and thus applicable in the foundations of solid modeling to computations on regular sets. We also address the general relation between effective openness and computability of functions. 1
Undecidability Everywhere?
, 1996
"... We discuss the question of if and how undecidability might be translatable into physics, in particular with respect to prediction and description, as well as to complementarity games. 1 1 Physics after the incompleteness theorems There is incompleteness in mathematics [22, 63, 65, 13, 9, 12, 51 ..."
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We discuss the question of if and how undecidability might be translatable into physics, in particular with respect to prediction and description, as well as to complementarity games. 1 1 Physics after the incompleteness theorems There is incompleteness in mathematics [22, 63, 65, 13, 9, 12, 51]. That means that there does not exist any reasonable (consistent) finite formal system from which all mathematical truth is derivable. And there exists a "huge" number [11] of mathematical assertions (e.g., the continuum hypothesis, the axiom of choice) which are independent of any particular formal system. That is, they as well as their negations are compatible with the formal system. Can such formal incompleteness be translated into physics or the natural sciences in general? Is there some question about the nature of things which is provable unknowable for rational thought? Is it conceivable that the natural phenomena, even if they occur deterministically, do not allow their complete d...