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48
TWIN PARADOX AND THE LOGICAL FOUNDATION OF RELATIVITY THEORY
, 2005
"... Abstract. We study the foundation of spacetime theory in the framework of firstorder logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for spacetime theory (or relativity). First we recall a ..."
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Abstract. We study the foundation of spacetime theory in the framework of firstorder logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for spacetime theory (or relativity). First we recall a simple and streamlined FOLaxiomatization Specrel of special relativity from the literature. Specrel is complete with respect to questions about inertial motion. Then we ask ourselves whether we can prove the usual relativistic properties of accelerated motion (e.g., clocks in acceleration) in Specrel. As it turns out, this is practically equivalent to asking whether Specrel is strong enough to “handle ” (or treat) accelerated observers. We show that there is a mathematical principle called induction (IND) coming from real analysis which needs to be added to Specrel in order to handle situations involving relativistic acceleration. We present an extended version AccRel of Specrel which is strong enough to handle accelerated motion, in particular, accelerated observers. Among others, we show that the Twin Paradox becomes provable in AccRel, but it is not provable without IND. Key words: twin paradox, relativity theory, accelerated observers, firstorder logic, axiomatization, foundation of relativity theory 1.
Logic of spacetime and relativity theory
, 2006
"... 2.1 Motivation for special relativistic kinematics in place of Newtonian kinematics......................... 4 ..."
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2.1 Motivation for special relativistic kinematics in place of Newtonian kinematics......................... 4
Zeno machines and hypercomputation
 Theoretical Computer Science
"... This paper reviews the ChurchTuring Thesis (or rather, theses) with reference to their origin and application and considers some models of “hypercomputation”, concentrating on perhaps the most straightforward option: Zeno machines (Turing machines with accelerating clock). The halting problem is br ..."
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This paper reviews the ChurchTuring Thesis (or rather, theses) with reference to their origin and application and considers some models of “hypercomputation”, concentrating on perhaps the most straightforward option: Zeno machines (Turing machines with accelerating clock). The halting problem is briefly discussed in a general context and the suggestion that it is an inevitable companion of any reasonable computational model is emphasised. It is suggested that claims to have “broken the Turing barrier ” could be toned down and that the important and wellfounded rôle of Turing computability in the mathematical sciences stands unchallenged.
Characteristics of discrete transfinite time Turing machine models: halting times, stabilization times, and . . .
, 2008
"... ..."
Abstract geometrical computation: Turingcomputing ability and undecidability
, 2004
"... In the Cellular Automata (CA) literature, discrete lines inside (discrete) spacetime diagrams are often idealized as Euclidean lines in order to analyze a dynamics or to design CA for special purposes. In this article, we present a parallel analog model of computation corresponding to this ideali ..."
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In the Cellular Automata (CA) literature, discrete lines inside (discrete) spacetime diagrams are often idealized as Euclidean lines in order to analyze a dynamics or to design CA for special purposes. In this article, we present a parallel analog model of computation corresponding to this idealization: dimensionless signals are moving on a continuous space in continuous time generating Euclidean lines on (continuous) spacetime diagrams. Like CA, this model is parallel, synchronous, uniform in space and time, and uses local updating. The main difference is that space and time are continuous and not discrete (i.e. R instead of Z). In this article, the model is restricted to Q in order to remain inside Turingcomputation theory. We prove that our model can carry out any Turingcomputation through twocounter automata simulation and provide some undecidability results.
Constraints on Hypercomputation, in
 Logical Approaches to Computational Barriers: Second Conference on Computability in Europe, CiE 2006
, 2006
"... “To infinity, and beyond!”, Buzz Lightyear, Toy Story, Pixar, 1995. Many attempts to transcend the fundamental limitations to computability implied by the Halting Problem for Turing Machines depend on the use of forms of hypercomputation that draw on notions of infinite or continuous, as opposed to ..."
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“To infinity, and beyond!”, Buzz Lightyear, Toy Story, Pixar, 1995. Many attempts to transcend the fundamental limitations to computability implied by the Halting Problem for Turing Machines depend on the use of forms of hypercomputation that draw on notions of infinite or continuous, as opposed to bounded or discrete, computation. Thus, such schemes may include the deployment of actualised rather than potential infinities of physical resources, or of physical representations of real numbers to arbitrary precision. Here, we argue that such bases for hypercomputation are not materially realisable and so cannot constitute new forms of effective calculability. 1
Visualizing some ideas about Gödeltype rotating universes
, 2008
"... Some kinds of physical theories describe what our universe looks like. Other kinds of physical theories describe instead what the universe could be like independently of the properties of the actual universe. This second kind aims for the “basic laws of physics” in some sense which we will not make ..."
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Some kinds of physical theories describe what our universe looks like. Other kinds of physical theories describe instead what the universe could be like independently of the properties of the actual universe. This second kind aims for the “basic laws of physics” in some sense which we will not make precise here (but cf. e.g. Malament [25, pp.9899]). The present paper belongs to the second kind. Moreover, it is even more abstract than this, namely it aims for visualizing or grasping some mathematical or logical aspects of what the universe could be like. The first six pages of this material are of a “sciencepopularizing ” character in the sense that first we recall a spacetime diagram from HawkingEllis [18] as “Godgiven truth”, i.e. we do not explain why the reader should believe that diagram. Then we derive carefully in an easily understandable visual manner an exciting, exotic consequence of that diagram: timetravel. This applies to the first six pages. The rest of this work is of a more ambitious character. The reader does not have to believe anything 1. We do our best to make the paper selfcontained and explain and visualize most of what we say. In more detail, this work consists of Sections 18. Section 1 (p.2) is the just mentioned “popular ” part. Section 2 (p.8) lays the foundation for discussing rotating universes. E.g. it shows how to visualize such spacetimes. The spacetime built up in this section is called the “Naive Spiral world”. Section 3 (p.19) is about nonexistence of a natural “now ” in Gödel’s universe GU. Section 4 (p.22) introduces corotating coordinates “transforming the rotation away”. Section 5 (p.29) refines the Gödeltype universe (obtained in Section 2). Section 6 (p.46) illustrates a fuller view of the refined version of GU. Section 7 (p.52) recoordinatizes the refined GU in order that the socalled gyroscopes do not rotate in this coordinatization. Section 8 (p.67) gives connections with the literature. E.g. it presents detailed computational comparison with the spacetime metric in Gödel’s papers. Section 9 (p.70) contains technical data about how we constructed the figures illustrating Gödel’s universe. 1 Not even the diagram recalled from HawkingEllis [18] in Figure 1 or any of the statements made in the first six pages.
Computational Power of Infinite Quantum Parallelism
 pp.2057–2071 in International Journal of Theoretical Physics vol.44:11
, 2005
"... Recent works have independently suggested that quantum mechanics might permit procedures that fundamentally transcend the power of Turing Machines as well as of ‘standard ’ Quantum Computers. These approaches rely on and indicate that quantum mechanics seems to support some infinite variant of class ..."
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Recent works have independently suggested that quantum mechanics might permit procedures that fundamentally transcend the power of Turing Machines as well as of ‘standard ’ Quantum Computers. These approaches rely on and indicate that quantum mechanics seems to support some infinite variant of classical parallel computing. We compare this new one with other attempts towards hypercomputation by separating (1) its computing capabilities from (2) realizability issues. The first are shown to coincide with recursive enumerability; the second are considered in analogy to ‘existence’ in mathematical logic. KEY WORDS: Hypercomputation; quantum mechanics; recursion theory; infinite parallelism.
What lies beyond the mountains? Computational systems beyond the Turing limit
 BULLETIN OF THE EUROPEAN ASSOCIATION FOR THEORETICAL COMPUTER SCIENCE
, 2005
"... Up to Turing power, all computations are describable by suitable programs, which correspond to the prescription by finite means of some rational parameters of the system or some computable reals. ¿From Turing power up we have computations that are not describable by finite means: computation without ..."
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Up to Turing power, all computations are describable by suitable programs, which correspond to the prescription by finite means of some rational parameters of the system or some computable reals. ¿From Turing power up we have computations that are not describable by finite means: computation without a program. When we observe natural phenomena and endow them with computational significance, it is not the algorithm we are observing but the process. Some objects near us may be performing hypercomputation: we observe them, but we will never be able to simulate their behaviour on a computer. What is then the profit of such a theory of computation to Science?
A Hardware Relaxation Paradigm for Solving NPHard Problems
"... Digital circuits with feedback loops can solve some instances of NPhard problems by relaxation: the circuit will either oscillate or settle down to a stable state that represents a solution to the problem instance. This approach differs from using hardware accelerators to speed up the execution of ..."
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Digital circuits with feedback loops can solve some instances of NPhard problems by relaxation: the circuit will either oscillate or settle down to a stable state that represents a solution to the problem instance. This approach differs from using hardware accelerators to speed up the execution of deterministic algorithms, as it exploits stabilisation properties of circuits with feedback, and it allows a variety of hardware techniques that do not have counterparts in software. A feedback circuit that solves many instances of Boolean satisfiability problems is described, with experimental results from a preliminary simulation using a hardware accelerator. Keywords: NPhard problem, Boolean satisfiability, digital circuit with feedback, relaxation, simulated annealing