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32
The many forms of hypercomputation
 Applied Mathematics and Computation
, 2006
"... This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess. ..."
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This paper surveys a wide range of proposed hypermachines, examining the resources that they require and the capabilities that they possess.
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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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.
Hypercomputability of quantum adiabatic processes: facts versus prejudices
 http://arxiv.org/quantph/0504101
, 2005
"... Abstract. We give an overview of a quantum adiabatic algorithm for Hilbert’s tenth problem, including some discussions on its fundamental aspects and the emphasis on the probabilistic correctness of its findings. For the purpose of illustration, the numerical simulation results of some simple Diopha ..."
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Abstract. We give an overview of a quantum adiabatic algorithm for Hilbert’s tenth problem, including some discussions on its fundamental aspects and the emphasis on the probabilistic correctness of its findings. For the purpose of illustration, the numerical simulation results of some simple Diophantine equations are presented. We also discuss some prejudicial misunderstandings as well as some plausible difficulties faced by the algorithm in its physical implementations. “To believe otherwise is merely to cling to a prejudice which only gives rise to further prejudices... ” 1
BioSteps Beyond Turing
 BIOSYSTEMS
, 2004
"... Are there `biologically computing agents' capable to compute Turing uncomputable functions? It is perhaps tempting to dismiss this question with a negative answer. Quite the opposite, for the first time in the literature on molecular computing we contend that the answer is not theoretically ..."
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Are there `biologically computing agents' capable to compute Turing uncomputable functions? It is perhaps tempting to dismiss this question with a negative answer. Quite the opposite, for the first time in the literature on molecular computing we contend that the answer is not theoretically negative. Our results will be formulated in the language of membrane computing (P systems). Some mathematical results presented here are interesting in themselves. In contrast with most speedup methods which are based on nondeterminism, our results rest upon some universality results proved for deterministic P systems. These results will be used for building "accelerated P systems". In contrast with the case of Turing machines, acceleration is a part of the hardware (not a quality of the environment) and it is realised either by decreasing the size of "reactors" or by speedingup the communication channels.
On the existence of a new family of diophantine equations for Ω
 Fundamenta Informaticae
"... Abstract. We show how to determine the kth bit of Chaitin’s algorithmically random real number, Ω, by solving k instances of the halting problem. From this we then reduce the problem of determining the kth bit of Ω to determining whether a certain Diophantine equation with two parameters, k and N, ..."
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Abstract. We show how to determine the kth bit of Chaitin’s algorithmically random real number, Ω, by solving k instances of the halting problem. From this we then reduce the problem of determining the kth bit of Ω to determining whether a certain Diophantine equation with two parameters, k and N, has solutions for an odd or an even number of values of N. We also demonstrate two further examples of Ω in number theory: an exponential Diophantine equation with a parameter, k, which has an odd number of solutions iff the kth bit of Ω is 1, and a polynomial of positive integer variables and a parameter, k, that takes on an odd number of positive values iff the kth bit of Ω is 1.
Using biased coins as oracles
, 2004
"... Abstract. While it is well known that a Turing machine equipped with the ability to flip a fair coin cannot compute more that a standard Turing machine, we show that this is not true for a biased coin. Indeed, any oracle set X may be coded as a probability pX such that if a Turing machine is given a ..."
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Abstract. While it is well known that a Turing machine equipped with the ability to flip a fair coin cannot compute more that a standard Turing machine, we show that this is not true for a biased coin. Indeed, any oracle set X may be coded as a probability pX such that if a Turing machine is given a coin which lands heads with probability pX it can compute any function recursive in X with arbitrarily high probability. We also show how the assumption of a nonrecursive bias can be weakened by using a sequence of increasingly accurate recursive biases or by choosing the bias at random from a distribution with a nonrecursive mean. We conclude by briefly mentioning some implications regarding the physical realisability of such methods.
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
A Formalization of the ChurchTuring Thesis for StateTransition Models
"... Abstract. Our goal is to formalize the ChurchTuring Thesis for a very large class of computational models. Specifically, the notion of an “effective model of computation ” over an arbitrary countable domain is axiomatized. This is accomplished by modifying Gurevich’s “Abstract State Machine ” postu ..."
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Abstract. Our goal is to formalize the ChurchTuring Thesis for a very large class of computational models. Specifically, the notion of an “effective model of computation ” over an arbitrary countable domain is axiomatized. This is accomplished by modifying Gurevich’s “Abstract State Machine ” postulates for statetransition systems. A proof is provided that all models satisfying our axioms, regardless of underlying data structure—and including all standard statetransition models—are equivalent to (up to isomorphism), or weaker than, Turing machines. To allow the comparison of arbitrary models operating over arbitrary domains, we employ a quasiordering on computational models, based on their extensionality. LCMs can do anything that could be described as “rule of thumb ” or “purely mechanical”.... This is sufficiently well established that it is now agreed amongst logicians that “calculable by means of an LCM” is the correct accurate rendering of such phrases. 1
Scaleinvariant cellular automata and selfsimilar Petri nets
 THE EUROPEAN PHYSICAL JOURNAL B
, 2009
"... Two novel computing models based on an infinite tessellation of spacetime are introduced. They consist of recursively coupled primitive building blocks. The first model is a scaleinvariant generalization of cellular automata, whereas the second one utilizes selfsimilar Petri nets. Both models are ..."
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Two novel computing models based on an infinite tessellation of spacetime are introduced. They consist of recursively coupled primitive building blocks. The first model is a scaleinvariant generalization of cellular automata, whereas the second one utilizes selfsimilar Petri nets. Both models are capable of hypercomputations and can, for instance, “solve” the halting problem for Turing machines. These two models are closely related, as they exhibit a stepbystep equivalence for finite computations. On the other hand, they differ greatly for computations that involve an infinite number of building blocks: the first one shows indeterministic behavior, whereas the second one halts. Both models are capable of challenging our understanding of computability, causality, and spacetime.
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.