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Hypercomputation and the Physical ChurchTuring Thesis
, 2003
"... A version of the ChurchTuring Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logicomathematical, proposition. We review the main approaches to computation beyond Turing ..."
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A version of the ChurchTuring Thesis states that every e#ectively realizable physical system can be defined by Turing Machines (`Thesis P'); in this formulation the Thesis appears an empirical, more than a logicomathematical, proposition. We review the main approaches to computation beyond Turing definability (`hypercomputation'): supertask, nonwellfounded, analog, quantum, and retrocausal computation. These models depend on infinite computation, explicitly or implicitly, and appear physically implausible; moreover, even if infinite computation were realizable, the Halting Problem would not be a#ected. Therefore, Thesis P is not essentially di#erent from the standard ChurchTuring Thesis.
Halting probability amplitude of quantum computers
 Journal of Universal Computer Science
, 1995
"... The classical halting probability to quantum computations. introduced by Chaitin is generalized Chaitin's [1,2,3] is a magic number. It is a measure for arbitrary programs to take a nite number of execution steps and then halt. It contains the solution for all halting problems, and hence to question ..."
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Cited by 9 (7 self)
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The classical halting probability to quantum computations. introduced by Chaitin is generalized Chaitin's [1,2,3] is a magic number. It is a measure for arbitrary programs to take a nite number of execution steps and then halt. It contains the solution for all halting problems, and hence to questions codable into halting problems, such asFermat's theorem. It contains the solution for the question of whether or not a particular exponential Diophantine equation has in nitely many ora nite number of solutions. And, since is provable \algorithmically incompressible," it is MartinLof/Chaitin/Solovay random. Therefore, is both: a mathematicians \fair coin, " and a formalist's nightmare. Here, is generalized to quantum computations. Consider a (not necessarily universal) quantum computer C and its ith program pi, which, at time t 2 Z, can be described by a quantum state [4, 5,6,7,
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.
Complexity: A LanguageTheoretic Point of View
, 1995
"... this paper (see the discussion in [51, 58, 70, 126, 127, 120, 121, 130]); in what follows we shall superficially review this topic in connection with the related question: can computers think? ..."
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this paper (see the discussion in [51, 58, 70, 126, 127, 120, 121, 130]); in what follows we shall superficially review this topic in connection with the related question: can computers think?
A Quantum Mechanical Look At Time Travel and Free Will
, 2001
"... Consequences of the basic and most evident consistency requirementthat measured events cannot happen and not happen at the same timeare reviewed. Particular emphasis is given to event forecast and event control. As a consequence, particular, very general bounds on the forecast and control of e ..."
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Cited by 2 (2 self)
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Consequences of the basic and most evident consistency requirementthat measured events cannot happen and not happen at the same timeare reviewed. Particular emphasis is given to event forecast and event control. As a consequence, particular, very general bounds on the forecast and control of events within the known laws of physics result. These bounds are of a global, statistical nature and need not aect singular events or groups of events. We also present a quantum mechanical model of time travel and discuss chronology protection schemes. Such models impose restrictions upon certain capacities of event control.
On the Brightness of the Thomson Lamp. A Prolegomenon to Quantum Recursion Theory
, 2009
"... Some physical aspects related to the limit operations of the Thomson lamp are discussed. Regardless of the formally unbounded and even infinite number of “steps” involved, the physical limit has an operational meaning in agreement with the Abel sums of infinite series. The formal analogies to accele ..."
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Some physical aspects related to the limit operations of the Thomson lamp are discussed. Regardless of the formally unbounded and even infinite number of “steps” involved, the physical limit has an operational meaning in agreement with the Abel sums of infinite series. The formal analogies to accelerated (hyper) computers and the recursion theoretic diagonal methods are discussed. As quantum information is not bound by the mutually exclusive states of classical bits, it allows a consistent representation of fixed point states of the diagonal operator. In an effort to reconstruct the selfcontradictory feature of diagonalization, a generalized diagonal method allowing no quantum fixed points is proposed.