Abstract
Hypercomputation—the hypothesis that Turing-incomputable objects can be computed through infinitary means—is ineffective, as the unsolvability of the halting problem for Turing machines depends just on the absence of a definite value for some paradoxical construction; nature and quantity of computing resources are immaterial. The assumption that the halting problem is solved by oracles of higher Turing degree amounts just to postulation; infinite-time oracles are not actually solving paradoxes, but simply assigning them conventional values. Special values for non-terminating processes are likewise irrelevant, since diagonalization can cover any amount of value assignments. This should not be construed as a restriction of computing power: Turing’s uncomputability is not a ‘barrier’ to be broken, but simply an effect of the expressive power of consistent programming systems.
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Notes
The hypothesis currently belongs to the field of non-conventional computation, but should be clearly distinguished from the approaches based on physical, chemical, or biological implementations of novel computing strategies; see Toffoli (1998) for an extensive introduction. Non-conventional computers may renounce the classical Von Neumann scheme, and explore alternative architectures; in general, they aim at overcoming some instance of intractability (e.g., see Sipser 2006, Chap. 9), without compromising with speculations on super-Turing computability.
For instance, relativistic hypercomputation, as theorized by Pitowsky, Hogarth, Németi, and others, faces substantial objections by Davis (2004), Silagadze (2005). Flaws in Kieu’s solution of Hilbert’s Tenth Problem through quantum adiabatic processes are discussed by Hagar and Korolev (2006), Smith (2006). The Calude-Pavlov approach to the solution of the Halting Problem through quantum measurement is criticized by Davis (2006). Objections to Xia’s Newtonian super-task can be found in Barrow (2005, Chap. 10).
The ‘conceptual’ treatment within category theory is not the only way of producing a unified account of diagonal theorems. A comparably high level of generality, and equivalent conclusions on the nature of uncomputability, has been eventually obtained also from the ‘formal’ viewpoint, initiated by Carnap’s remarks. A full elaboration can be found in Gaifman (2006).
Physical plausibility remains out of question. According to McLaughlin (1998), also the logical rejection of super-tasks could be vindicated, by renouncing standard theoretical approaches; Thomson’s lamp would turn out to be ‘dysfunctional’ on the grounds of internal set theory, a version of non-standard analysis.
Cooper (2006) admits the presence of this petitio principii, dubbing it ‘Davis’ Paradox’. Nevertheless, he appears confident that hypercomputation can be realized through ‘the set of all real numbers, within which the scientist commonly describes the material universe’. As we saw in Cotogno (2003), this is not the case: if computable reals are taken in approximated form, then they do not exceed the boundaries of the Church-Turing Thesis; otherwise, if taken in infinite precision, they assume the realization of super-tasks, and face the related objections.
As Wells (2004) reports, Tarski himself was implicitly entertaining the idea that classical definitions may involve some form of hypercomputational decidability. The discussion initiated by Wells has the potential of shedding new light on the relationship between classical and constructive foundations; we may consider it as the most interesting achievement of the hypercomputation area.
If one really intends to theorize computability without being vexed by diagonalization, renouncing bivalence is not enough: one should leave classical logic altogether, as done in the programme of synthetic computability, proposed by Bauer (2006). In this approach, all functions are assumed to be computable a priori, with no concern for algorithmic definitions, but logical principles are intuitionistic: diagonal uncomputability is, in a way, turned into a built-in feature.
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Acknowledgments
I am grateful to Martin Davis for letting me know about the corpus of his writings on hypercomputation. I would like to thank Amit Hagar for sharing first-hand information about his criticism of adiabatic hypercomputation. Thanks are also due to the anonymous referees for their stimulating comments.
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Cotogno, P. A Brief Critique of Pure Hypercomputation. Minds & Machines 19, 391–405 (2009). https://doi.org/10.1007/s11023-009-9161-7
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DOI: https://doi.org/10.1007/s11023-009-9161-7