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 non-computable 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 (one-dimensional) 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.

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 Turing-incomputable 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

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.

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When the microequations of a dynamical system generate complex macrobehaviour, there can be an explanatory gap between the small-scale and large-scale 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