Abstract
This paper discusses the empirical question concerning the physical realization (or implementation) of a computation. We give a precise definition of the realization of a Turing-computable algorithm into a physical situation. This definition is not based, as usual, on an interpretation function of physical states, but on an implementation function from machine states to physical states (as suggested by Piccinini G, Computation in physical systems. The Stanford encyclopedia of philosophy. http://plato.stanford.edu/archives/fall2012/entries/computation-physicalsystems. Accessed 5 Dec 2013, 2012). We show that our definition avoids difficulties posed by Putnam’s theorem (Putnam H, Representation and reality. MIT Press, Cambridge, 1988) and Kripke’s objections (Stabler EP Jr, Kripke on functionalism and automata. Synthese 70(1):1–22, 1987; Scheutz M, What is not to implement a computation: a critical analysis of Chalmers’ notion of implementation. http://hrilab.tufts.edu/publications/scheutzcogsci12chalmers.pdf. Accessed 5 Dec 2013, 2001). Using our notion of representation, we analyse Gandy machines, intended in a physical sense, as a case study and show an inaccuracy in Gandy’s analysis with respect to the locality notion. This shows the epistemological relevance of our realization concept. We also discuss Gandy machines in quantum context. In fact, it is well known that in quantum mechanics, locality is seriously questioned, therefore it is worthwhile to analyse briefly, whether quantum machines are Gandy machines.
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- 1.
Let us consider only those functions whose arguments and values are natural numbers.
- 2.
From now on we will use the term “computable”.
- 3.
The term explication appears for the first time in The Two Concepts of Probability, but a more complete discussion of this concept can be found in the first chapter of Logical Foundations of Probability in which Carnap writes: “By an explication we understand the transformation of an inexact, prescientific concept, the explicandum, into an exact concept, the explicatum” (cap.1, p.1). […] “The term ‘explicatum’ has been suggested by the following two usages. Kant calls a judgment explicative if the predicate is obtained by analysis of the subject. Husserl, in speaking about the synthesis of identification between a confused, nonarticulated sense and a subsequently intended distinct, articulated sense, calls the latter an ‘Explicat’ of the former. What I mean by ‘explicandum’ and ‘explicatum’ is to some extent similar to what C.H. Langford calls ‘analysandum’ and ‘analysans’: ‘the analysis that states an appropriate relation between the analysandum and the analysans’; he says that the motive of an analysis ‘is usually that of supplanting a relative vague idea by a more precise one (cap. 1, § 2)’”. See Carnap (1945, 1950).
- 4.
- 5.
- 6.
The term ‘empirical’ is to be understood in a purely epistemological way, that is, as a comparison with experimental data. In spite of this no applications are involved.
- 7.
The identification of rigid designators in natural languages is a very complex issue and it is still a matter of controversy. In our paper, the term ‘rigid designator’ is to be understood in the sense that we refer to parts of the world in a manner as neutral as possible. The use of rigid designators is not essential for the purposes of our argument, it helps us to simplify our discourse, and it is a useful expedient to avoid a conceptual connotation of physical objects of which we are speaking.
- 8.
It is impossible to stick to an excessive rigor in defining the delicate and difficult concept of realization. So we have chosen the practical way of modern natural science, which consists of bringing the object of our investigation to a partially ideal situation that can find approximate concrete examples in the world. In our laboratories, for example, it is trivial that gravity can never be screened, so our real system is necessarily not isolated. However it is not difficult to find situations in which gravity does not have any relevant physical effect from the computational point of view. For example on the computer that we use to write this paper.
- 9.
Characteristics could be non observable as well.
- 10.
From now on we move in a classical physics context, where space is represented in Euclidean terms.
- 11.
- 12.
Such functions always exist, but in general they are not simple. Physics can reduce their complexity.
- 13.
It is true that our current computers are all based on quantum-mechanical effects due to the doping of semiconductors, but in general a doped semiconductor could stay in a continuous set of possible physical states (velocity, temperature, positions etc) and our intermediation function chooses exactly the one we are interested in, that is the two levels of potential.
- 14.
We use the term ‘state’ (distinguishing it from the term ‘internal state’) because we wish to highlight the connection between the concepts of machine states and physical states, but our reasoning could easily be reformulated using the more familiar notion of ‘configuration’.
- 15.
Something very similar is maintained in Horsman et al. (2014). The latter’s approach is comparable to ours in the sense that it emphasizes the importance of the relation between computational and physical space. They express something very similar to condition 2. of our definition of realization (see infra), speaking of commuting diagrams. In spite of this our feeling is that their approach, though interesting, is naive from an epistemological point of view.
- 16.
Remember that in the pools case, TL is classical mechanics.
- 17.
There need not necessarily be a different quadruple for each couple skql.
- 18.
On the notion of ‘realization’ it is necessary to move from Giunti’s important study, 1997, especially par. 16. Our approach is partially similar from a formal point of view, but it is conceptually different, since it involves the laws of physics.
- 19.
In general, C will not be surjective, since not all the RTL space is used. Being C injective, it will be invertible as well.
- 20.
We will explain this condition in a subsequent comment.
- 21.
As suggested by Piccinini (2012, p. 9).
- 22.
See Scheutz (1999).
- 23.
See also Horsman et al. (2014).
- 24.
This definition is different from what Piccinini dubs a ‘simple mapping account’. The latter turns out to be mathematically more vague. As a result, the perspective added is that we restrict ourselves to mappings that are acceptable. Nevertheless the decision to change the mapping from functional states to states of the machine is not the significant part of our argument. Rather, this choice depends on the fact that we want to tackle the relationship between the term ‘computational’, on the one hand, and ‘physical world’, on the other, at the greatest possible distance between them. For this reason we speak of states of the machine. Our work differs from the standard philosophical literature, not only in the use of the state of the machine, but also in the definition of the realization, which takes into account all the epistemological problems, to our knowledge, raised till now upon this concept.
- 25.
Using the words of Chalmers: “The ambitions of artificial intelligence rest on a related claim of computational sufficiency” (Chalmers 1996, pp. 309), that is, “the right kind of computational structure suffices for the possession of a mind” (Chalmers 2011, p. 325). So, “computation will provide a powerful formalism for the replication and explanation of mentality” (Chalmers 1996, pp. 309–310). Hilary Putnam’s theorem says that “every ordinary open system is a realization of every abstract finite automaton” (Putnam 1988, p. 121), and its proof requires two physical principles; a principle of continuity, and the principle of Noncyclical behaviour (for more details see Putnam 1988, pp. 120–125). “Together with the thesis of computational sufficiency, this [theorem] would imply that a rock has a mind.” […] “We must either embrace an extreme form of panpsychism or reject the principle on which the hopes of artificial intelligence rest. Putnam himself takes the result to show that computational functionalism cannot provide a foundation for a theory of mind” (Chalmers 1996, pp. 309–310). See also Piccinini (2012).
- 26.
For a interesting discussion of these issues see Tamburrini (2002).
- 27.
We attributed this view to Robin Gandy, because it is very similar to what he calls “Thesis M” (Gandy 1980, p. 124). So do Copeland and Shagrir (2007), although they are convinced that there are discrete and deterministic physical machines that can compute hyper-computable functions. This is not our topic. However Cotogno (2003) is rather sceptical about hyper-computation. Pitowsky and Shagrir (2003) show that a physical digital hypercomputer, whose existence is compatible with General Relativity, is a Gandy machine and it can compute a function that is non Turing-computable. See also Kieu (2002), Syropoulus (2008). Moreover consider that originally Gandy has in mind not only classical mechanics, but classical electromagnetism as well.
- 28.
Later we will give a sketchy presentation of this notion.
- 29.
Remember that we have specified the notion of ‘realization’ making explicit the fact that it is not a relationship between the mathematical concept of a Turing machine and a part of the world, but between the former and the physical theory true for that part of the world. Therefore, in this case TL will be a physical theory endowed with various principles.
- 30.
- 31.
Nonetheless many physical theories based on an atomistic conception of matter and space have been proposed; one of the most recent and interesting is Rovelli (2004).
- 32.
Gandy also shows that several forms of weakening on the conditions of function F would allow computing functions not computable in the sense of Turing.
- 33.
- 34.
- 35.
See Shagrir (2002) for various interpretations of Gandy Machines. Shagrir (2002) emphasizes that in Gandy’s paper there is an ambiguity, since it is not clear whether he intends his thesis as a physical one – as in our paper – or as a mathematical one. That is, is Gandy’s machine a mathematical concept inspired by physical models, or is it an actual physical system? In the first case, Gandy shows an important theorem according to which Gandy’s machines (parallel) computes only Turing computable functions. But we are interested in the physical interpretation of Gandy’s point, which is present in many passages of this splendid paper. We also think that the ambiguity noted by Shagrir has contributed to hiding the inaccuracy we are considering here. See also Shagrir (1997).
- 36.
In Gandy’s formalism there is no distinction between labels of regions and labels of bodies.
- 37.
Gandy dubs this hierarchical structure “stereotype”, because isomorphic physical situations have the same computational relevance.
- 38.
We do not discuss the Gandy’s formalism but, for our purpose, we refer the reader to the simpler and clearer formalism of Sieg (2008). In our opinion, the use of Sieg’s formalism does not affect our contention that there is an inaccuracy in Gandy.
- 39.
- 40.
- 41.
Note that this is an inaccuracy in Gandy’s analyses. A very interesting second question is: can we find the same inaccuracy also in other formalizations of Gandy Machines? We will analyze this problem in a new paper.
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Acknowledgments
We thank Claudio Calosi, Maurizio Colucci, Paola Gentili, Marco Giunti, Rossella Lupacchini, Wilfried Sieg and Guglielmo Tamburrini for their comments and suggestions. We thank the reviewers who have commented our paper and the scholars attending the IACAP 2014 Conference for their questions.
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Fano, V., Graziani, P., Macrelli, R., Tarozzi, G. (2016). Are Gandy Machines Really Local?. In: Müller, V.C. (eds) Computing and Philosophy. Synthese Library, vol 375. Springer, Cham. https://doi.org/10.1007/978-3-319-23291-1_3
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