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Can Newtonian systems, bounded in space, time, mass and energy compute all functions?
"... In the theoretical analysis of the physical basis of computation there is a great deal of confusion and controversy (e.g., on the existence of hypercomputers). First, we present a methodology for making a theoretical analysis of computation by physical systems. We focus on the construction and anal ..."
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Cited by 13 (4 self)
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In the theoretical analysis of the physical basis of computation there is a great deal of confusion and controversy (e.g., on the existence of hypercomputers). First, we present a methodology for making a theoretical analysis of computation by physical systems. We focus on the construction and analysis of simple examples that are models of simple subtheories of physical theories. Then we illustrate the methodology, by presenting a simple example for Newtonian Kinematics, and a critique that leads to a substantial extension of the methodology. The example proves that for any set A of natural numbers there exists a 3dimensional Newtonian kinematic system MA, with an infinite family of particles Pn whose total mass is bounded, and whose observable behaviour can decide whether or not n ∈ A for all n ∈ N in constant time. In particular, the example implies that simple Newtonian kinematic systems that are bounded in space, time, mass and energy can compute all possible sets and functions on discrete data. The system is a form of marble run and is a model of a small fragment of Newtonian Kinematics. Next, we use the example to extend the methodology. The marble run shows that a formal theory for computation by physical systems needs strong conditions on the notion of experimental procedure and, specifically, on methods for the construction of equipment. We propose to extend the methodology by defining languages to express experimental procedures and the construction of equipment. We conjecture that the functions computed by experimental computation in Newtonian Kinematics are “equivalent” to those computed by algorithms, i.e. the partial computable functions.
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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Cited by 4 (2 self)
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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.
Controller synthesis and ordinal automata
 In ATVA’06, Lecture Notes in Computer Science
, 2006
"... Abstract. Ordinal automata are used to model physical systems with Zeno behavior. Using automata and games techniques we solve a control problem formulated and left open by Demri and Nowak in 2005. It involves partial observability and a new synchronization between the controller and the environment ..."
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Cited by 3 (0 self)
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Abstract. Ordinal automata are used to model physical systems with Zeno behavior. Using automata and games techniques we solve a control problem formulated and left open by Demri and Nowak in 2005. It involves partial observability and a new synchronization between the controller and the environment. 1
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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Cited by 1 (1 self)
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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.
Small Turing universal signal machines
, 906
"... This article aims at providing signal machines as small as possible able to perform any computation (in the classical understanding). After presenting signal machines, it is shown how to get universal ones from Turing machines, cellularautomata and cyclic tag systems. Finally a halting universal si ..."
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This article aims at providing signal machines as small as possible able to perform any computation (in the classical understanding). After presenting signal machines, it is shown how to get universal ones from Turing machines, cellularautomata and cyclic tag systems. Finally a halting universal signal machine with 13 metasignals and 21 collision rules is presented. 1
Author manuscript, published in "4th Conf. Computability in Europe (CiE~'08) (abstracts and extended abstracts of unpublished papers), Greece (2008)" Abstract geometrical computation
, 2010
"... beyond the ..."