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A possible hypercomputational quantum algorithm
 SPIE) PROC. SPIE 5815 219–26
, 2005
"... The term ‘hypermachine’ denotes any data processing device (theoretical or that can be implemented) capable of carrying out tasks that cannot be performed by a Turing machine. We present a possible quantum algorithm for a classically noncomputable decision problem, Hilbert’s tenth problem; more spe ..."
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The term ‘hypermachine’ denotes any data processing device (theoretical or that can be implemented) capable of carrying out tasks that cannot be performed by a Turing machine. We present a possible quantum algorithm for a classically noncomputable decision problem, Hilbert’s tenth problem; more specifically, we present a possible hypercomputation model based on quantum computation. Our algorithm is inspired by the one proposed by Tien D. Kieu, but we have selected the infinite square well instead of the (onedimensional) simple harmonic oscillator as the underlying physical system. Our model exploits the quantum adiabatic process and the characteristics of the representation of the dynamical Lie algebra su(1, 1) associated to the infinite square well.
Computing a Turingincomputable problem from quantum computing
 quantph/0309198 (2003). 12, 2008 9:5 WSPC/INSTRUCTION FILE Transcending2 Transcending Turing computability 7
"... Abstract. A hypercomputation model named Infinite Square Well Hypercomputation Model (ISWHM) is built from quantum computation. This model is inspired by the model proposed by Tien D. Kieu [1] and solves an Turingincomputable problem. For the proposed model and problem, a simulation of its behavior ..."
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Abstract. A hypercomputation model named Infinite Square Well Hypercomputation Model (ISWHM) is built from quantum computation. This model is inspired by the model proposed by Tien D. Kieu [1] and solves an Turingincomputable problem. For the proposed model and problem, a simulation of its behavior is made. Furthermore, it is demonstrated that ISWHM is a universal quantum computation model. 1
Comparative Analysis of Hypercomputational Systems Submitted in partial fulfilment
"... In the 1930s, Turing suggested his abstract model for a practical computer, hypothetically visualizing the digital programmable computer long before it was actually invented. His model formed the foundation for every computer made today. The past few years have seen a change in ideas where philosoph ..."
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In the 1930s, Turing suggested his abstract model for a practical computer, hypothetically visualizing the digital programmable computer long before it was actually invented. His model formed the foundation for every computer made today. The past few years have seen a change in ideas where philosophers and scientists are suggesting models of hypothetical computing devices which can outperform the Turing machine, performing some calculations the latter is unable to. The ChurchTuring Thesis, which the Turing machine model embodies, has raised discussions on whether it could be possible to solve undecidable problems which Turing’s model is unable to. Models which could solve these problems, have gone further to claim abilities relating to quantum computing, relativity theory, even the modeling of natural biological laws themselves. These so called ‘hypermachines ’ use hypercomputational abilities to make the impossible possible. Various models belonging to different disciplines of physics, mathematics and philosophy, have been suggested for these theories. My (primarily researchoriented) project is based on the study and review of these different hypercomputational models and attempts to compare the different models in terms of computational power. The project focuses on the ability to compare these models of different disciplines on similar grounds and
From Hypocomputation to Hypercomputation
, 2008
"... Hypercomputational formal theories will, clearly, be both structurally and foundationally different from the formal theories underpinning computational theories. However, many of the maps that might guide us into this strange realm have been lost. So little work has been done recently in the area of ..."
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Hypercomputational formal theories will, clearly, be both structurally and foundationally different from the formal theories underpinning computational theories. However, many of the maps that might guide us into this strange realm have been lost. So little work has been done recently in the area of metamathematics, and so many of the previous results have been folded into other theories, that we are in danger of loosing an appreciation of the broader structure of formal theories. As an aid to those looking to develop hypercomputational theories, we will briefly survey the known landmarks both inside and outside the borders of computational theory. We will not focus in this paper on why the structure of formal theory looks the way it does. Instead we will focus on what this structure looks like, moving from hypocomputational, through traditional computational theories, and then beyond to hypercomputational theories.
Zeno Squeezing of Cellular Automata
 INT. JOURN. OF UNCONVENTIONAL COMPUTING, VOL. 6, PP. 399–416
, 2010
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Emergence: an algorithmic formulation
, 2005
"... When the microequations of a dynamical system generate complex macrobehaviour, there can be an explanatory gap between the smallscale and largescale descriptions of the same system. The microdynamics may be simple, but its relationship to the macrobehaviour may seem impenetrable. This phenomenon, ..."
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When the microequations of a dynamical system generate complex macrobehaviour, there can be an explanatory gap between the smallscale and largescale descriptions of the same system. The microdynamics may be simple, but its relationship to the macrobehaviour may seem impenetrable. This phenomenon, known as emergence, poses problems for the nature of scientific understanding. How do we reconcile two radically different modes of description? Emergence is formulated using the powerful tools of algorithmic information and computational theory. This provides the ground for an extension and generalisation of the phenomenon. Mathematics itself is analysed as an emergent system, linking formalist notions of mathematics as a string manipulation game with the more abstract ideas and proofs that occupy mathematicians. A philosophical problem that has plagued emergence is whether the whole can be more than the sum of its parts. This possibility, known as strong emergence, manifests when emergent macrostructures introduce brand new causal dynamics into a system. A new perspective on this
The Computational Status of Physics: A Computable Formulation of Quantum Theory ✩
, 805
"... According to the ChurchTuring Thesis, effective formal behaviours can be simulated by Turing machines; this has naturally led to speculation that physical systems can also be simulated computationally. But is this wider claim true, or do behaviours exist which are strictly hypercomputational? Sever ..."
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According to the ChurchTuring Thesis, effective formal behaviours can be simulated by Turing machines; this has naturally led to speculation that physical systems can also be simulated computationally. But is this wider claim true, or do behaviours exist which are strictly hypercomputational? Several idealised computational models are known which suggest the feasibility of physical hypercomputation – some based on cosmology; some on quantum theory; some on Newtonian physics. While the physicality of these models is debatable, they nonetheless throw into question the validity of simply extending the ChurchTuring Thesis to include all physical, as well as effective formal, systems. We propose that the physicality of hypercomputational behaviours be determined instead from first principles, and show that quantum theory can be reformulated in a way that partially resolves the question, by explaining why all physical behaviours can be regarded as ‘computing something ’ in the standard computational statemachine sense. While our approach does not rule out the possibility of hypercomputation completely, it strongly limits the form such hypercomputation must take.
Natural Computing manuscript No. (will be inserted by the editor) The Computational Status of Physics A Computable Formulation of Quantum Theory
, 805
"... the date of receipt and acceptance should be inserted later Abstract According to the ChurchTuring Thesis (CTT), effective formal behaviours can be simulated by Turing machines; this has naturally led to speculation that physical systems can also be simulated computationally. But is this wider clai ..."
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the date of receipt and acceptance should be inserted later Abstract According to the ChurchTuring Thesis (CTT), effective formal behaviours can be simulated by Turing machines; this has naturally led to speculation that physical systems can also be simulated computationally. But is this wider claim true, or do behaviours exist which are strictly hypercomputational? Several idealised computational models are known which suggest the possibility of hypercomputation, some Newtonian, some based on cosmology, some on quantum theory. While these models ’ physicality is debatable, they nonetheless throw into question the validity of extending CTT to include all physical systems. We consider the physicality of hypercomputational behaviour from first principles, by showing that quantum theory can be reformulated in a way that explains why physical behaviours can be regarded as ‘computing something ’ in the standard computational statemachine sense. While this does not rule out the physicality of hypercomputation, it strongly limits the forms it can take. Our model also has physical consequences; in particular, the continuity of motion and arrow of time become theorems within the basic model.