Abstract
The paper argues that a computational constraint is one that appeals to control of computational resources in a computationalist explanation. Such constraints may arise in a theory and in its models. Instrumental use of the same concept is trivial because the constraining behavior of any function eventually reduces to its computation. Computationalism is not instrumentalism. Born-again computationalism, which is an ardent form of pancomputationalism, may need some soul searching about whether a genuinely computational explanation is necessary or needed in every domain, because the resources in a computationalist explanation are limited. Computational resources are the potential targets of computational constraints. They are representability, time, space, and, possibly, randomness, assuming that ‘BPP = BQP?’ question remains open. The first three are epitomized by the Turing machine, and manifest themselves for example in complexity theories. Randomness may be a genuine resource in quantum computing. From this perspective, some purported computational constraints may be instrumental, and some supposedly noncomputational or cognitivist constraints may be computational. Examples for both cases are provided. If pancomputationalism has instrumentalism in mind, then it may be a truism, therefore not very interesting, but born-again computationalism cannot be computationalism as conceived here.
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Notes
- 1.
Interaction of computational complexity, which looks at resource behavior of functions both at the level of theory and the model, and algorithmic information theory, which looks at resource usage at program sizes, not behaviors, in the form of Chaitin-Kolmogorov-Solomonoff complexity, is an open research area; see e.g. Chaitin (1987). There is also the notion of logical resources, i.e. number of variables and quantifiers, and size of a formula, which leads to descriptive complexity. I leave the latter complexity measures to another paper.
- 2.
In this sense, the notion of representation is not vague in a computationalist explanation, as it is sometimes claimed in dynamical approaches to cognition, e.g. Beer (2003). It is something that makes other resources of a TM accessible. Its ontology in a model is a related but different practice. Reference by an internal representation can be wrong at times, as in early child language acquisition, but representation itself is robust. Of course, not all dynamicists would agree on uselessness of a representation, but complexity results from dynamical systems, such as the one currently discussed in Blum et al. (1989), must already alert the dynamicist and embodied cognitivist that computation in cognition was never parodied as rule-following only; it’s about management of computational resources; see Mirolli (2012), Valiant (2013) and Aaronson (2013b) for some discussion.
- 3.
Also note that this comment comes from a man who had considered the mind connection of Turing unthinkable, assuming that Turing flawed by considering possible states of the mind to continue to be finite when the mind changes (Wang 1974: 325, quoting Gödel). P ≠ NP conjecture brought another perspective to the mind problem; see Aaronson (2013a,b).
- 4.
As an example from cognitive science of language, consider the word sleep. In the spectrum of sleep-like thoughts and behaviors, one ceases to use the word ‘sleep’ if it strays too far from someone’s understanding, and from desires of communication. This is self-imposed error correction, which avoids uncountably many sleep-related words. In that sense language is a digital process, i.e. it is discrete and robustly representational (Haugeland 1997), and words are its proofs.
- 5.
A cogent warning about why pancomputationalism may be a truism is given by Horsman et al. (2013: 20): “A common, and unfortunate, method of ascribing computational ability to a nonstandard system is as follows. A novel computing substrate is proposed (a stone, a soap bubble, a large interacting condensed-matter system, etc.). The physical substrate is set going and an evolution occurs. At a certain point, the end of the process is declared and measurements taken. The initial and final states of the system are compared, and then a computation and a representation picked such that if the initial state, and final states are represented in such a way then such a computation would abstractly connect them. The system is then declared to have performed such a computation: the stone has evaluated the gravitational constant, the soap bubble has solved a complex optimization problem and so on.
Such arguments, without any further testing or evidence, are rightly treated with suspicion. Given the vast range of representation for physical systems available, almost any computation can be made to fit the difference of initial and final physical states of a system. If such arguments really were correct, we would not only have to conclude that everything in the universe computes, but that everything computes every possible computation all of the time. Such extreme pancomputationalism is even less useful than the usual kind.”
- 6.
Physical possibility is crucial because every computation requires an executive medium. This is one difference between functionalism and computationalism, where in the latter the primitives of the executive substrate may shape the form of the computed function. For example, membrane computing’s in command (to permeate a local membrane) has a Turing correlate (Păun 2000), but we would expect membrane models to be programmed more naturally with in. Making a difference in complexity classes makes a difference in computationalism, but not necessarily in functionalism.
- 7.
Notice that turn itself is not a resource. It translates to bounded use of space, which is a resource.
- 8.
Roughly speaking, here is how we can get the answer algorithmically. Assume that we expect QTM M to compute M(x) = f(x), given x. Input is prepared in a superposition by for example a series of Hadamard transformations, from \(\vert 0\rangle\) to \(\vert x0\cdots 0\rangle\). M computes and ends in a state \(\vert xf(x)0\cdots 0r\rangle\), where r is the residue of quantum gates. Copy—not clone—the result to unused bits to get \(\vert xf(x)f(x)r\rangle\) by a series of quantum operations \(\vert b_{1}b_{2}\rangle \mapsto \vert b_{1}(b_{1} \oplus b_{2})\rangle\). Then run the gates in reverse to get \(\vert x0\cdots 0f(x)0\cdots 0\rangle\), and Hadamard to get \(\vert 0\cdots 0f(x)0\cdots 0\rangle\), viz. \(\vert f(x)\rangle\).
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Acknowledgements
Thanks to Mark Addis, Tarek Besold, Mark Bishop, Robin Hill, Doukas Kapantaïs, Stasinos Konstantopoulos, Giacomo Lini, Halit Oğuztüzün, Deniz Şener, Mario Verdicchio, Ioannis Votsis, Aziz Zambak for discussion and advice on the paper and related topics. I am to blame for all errors and misunderstanding.
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Bozşahin, C. (2016). What Is a Computational Constraint?. In: Müller, V.C. (eds) Computing and Philosophy. Synthese Library, vol 375. Springer, Cham. https://doi.org/10.1007/978-3-319-23291-1_1
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