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BTM Models of Algorithmic Skills

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Extended Cognition and the Dynamics of Algorithmic Skills

Part of the book series: Studies in Applied Philosophy, Epistemology and Rational Ethics ((SAPERE,volume 35))

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Abstract

In this chapter I first sketch an overview on the principal contemporary approaches to cognitive arithmetic, showing that these approaches somewhat underestimate the role of online symbolic transformations like those performed in the execution of an algorithm (Rumelhart et al. 1986). Second, I propose to inspect arithmetical skills from an algorithmic stance. Assuming that the Bidimensional Turing machine is a reliable model of the various elements at stake in algorithmic performances, I formulate a set of hypotheses about the development of algorithmic skills and the related algorithmic performances, which may in principle be empirically verified. An eventual confirmation of those hypotheses may also help answering the question about the role of external devices, like paper and pencil, in algorithmic performances. Last, I describe some experiments made on a feed-forward neural network in order to test a developmental hypothesis on the acquisition of a set of basic number facts (in this case, the set of all possible results of single-digit additions).

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Notes

  1. 1.

    The term “constructivist” is here used in a peculiar sense, as pointing to a specific position in the field of the psychology of arithmetical development, without any reference to the way this term is used in philosophy of education by, e.g., von Glasersfeld (1989).

  2. 2.

    See Ashcraft (1992) for an overview of the main issues of this field.

  3. 3.

    As I understand the argument, the set of numerlogs should be intended as a subset of that of numerons.

  4. 4.

    This kind of tasks typically involves counting sets of objects variously arranged, like arrays of dots, sets of figures, cards, and so on.

  5. 5.

    \(\mathrm{{BTM}_{4}}\) formalizes an algorithm which employs a strategy of this kind (Sect. 5.3.3, pp. 116–118).

  6. 6.

    A more recent development of these ideas is in Leslie et al. (2008).

  7. 7.

    Feigenson et al. (2004) refers to these systems with the apt expression: Core systems of number. See also Piazza (2010) for an exhaustive explanation of the concepts treated in this paragraph.

  8. 8.

    This topic is currently debated. A thorough study involving 16 right brain-damaged subjects shows a dissociation between deviations in physical and number line-bisection tasks, suggesting that the navigation along physical space and number lines is governed by different brain networks (Doricchi et al. 2005).

  9. 9.

    Longo (2011) proposes that the cognitive notion of a MNL should be also considered as a source of robust evidence in advanced mathematical reasoning. See also Sect. 1.2 (note 8) of this book.

  10. 10.

    In the following example, an important feature of the notation used for BTMs will be evident, namely that complex internal states are identified by their simple internal state symbol in conjunction with the number of non empty registers they include. This means, for instance, that an internal state \(\langle q_{1} [r_{1}] \rangle \) is distinct from an internal state \(\langle q_{1} [r_{1}], [r_{2}] \rangle \). Furthermore, when an internal state containing 2 variables, for instance, \(r_{1}\) and \(r_{2}\), triggers as output an internal state in which both variables are used as argument for a function (e.g, \(\langle q_{1}, [r_{1}], [r_{2}] \rangle \longmapsto \langle q_{1}, [f (r_{1}, r_{2})] \rangle \)), the output internal state will include only one register, which will take the name of the first non-empty register of that internal state (in the example, \(\langle q_{1}, [f (r_{1}, r_{2})] \rangle \) is equivalent to an input internal state \(\langle q_{1}, [r_{1}] \rangle \)).

  11. 11.

    More precisely, the implication derives from this comparison in conjunction with the assumption of empirical adequacy of the models (see point 3 of the method sketched in Sect. 5.3.1).

  12. 12.

    See also Hutchins (1995) for an early and very influential treatment of this issue.

  13. 13.

    In the original 1974 model, the WM is composed of a Central Executive that interacts with two slave systems: the Phonological Loop and the Visuo-spatial Sketchpad. In 2000 Baddeley introduced a third slave system, the Episodic Buffer (Baddeley 2000). See also Baddeley (1987, 1996, 2003, 2012) for further elaborations and discussions on the current validity of the model.

  14. 14.

    Lisa Feigenson suggests that, although humans can show impressive quantitative feats, as when we count objects in arrays containing hundreds, or estimate big approximate quantities, we cannot overcome the capacity of our WM, which can simultaneously hold only three/four items. So, how can we accomplish such computational tasks? The answer is that our WM makes up for its strict limits with a fair amount of flexibility, for it can represent items as either objects, or sets, or ensembles (Feigenson 2011).

  15. 15.

    This is a slightly modified version of a bidimensional Turing machine described in Giunti (2009).

  16. 16.

    The domain of this function is also infinite; however, it simplifies the procedure in the sense that it is no more necessary to be able to sum an arbitrary 1-digit number to any natural number, but only to know the immediate successor of any number.

  17. 17.

    The construction of the net and all the experiments carried out on it have been made in collaboration with Giorgio Fumera, Associate Professor at the Università di Cagliari, Dipartimento di Ingegneria Elettrica ed Elettronica, who has provided software simulations, result plots, and full technical support. Experiments have been carried out with a software written in Python language, by using a wrapper of the C library for neural networks FANN (version 2.1.0, http://www.leenissen.dk/fann).

  18. 18.

    This way of number encoding is in accordance with Butterworth’s Numerosity Coding Structure. See Sect. 5.1.1 of this book.

  19. 19.

    See Reed and Marks (1998).

  20. 20.

    In addition to the numerosity coding structure, which has been used in this simulation, three more schemes for encoding numbers have been, to date, used in connectionist models (Zorzi et al. 2005):

    1. 1.

      “Barcode” magnitude representation (Anderson 1998, 2002). Numbers are encoded as sets of adjacent active units of an ordered set of nodes, where each node is labeled with a particular number. In this scheme, e.g., the number 7 is encoded via the activation of the nodes labeled “6”, “7” and “8”.

    2. 2.

      Compressed number line (Dehaene 2001). Each number is represented by a pattern of activation of the input neurons, where a neuron set to 1 is surrounded by noisy neurons activated according to a gaussian distribution with fixed variance. In this scheme, the number series is generated by following a logarithmic scale, in such a way that representations of larger numbers share more active neurons than representations of smaller numbers. This method of representing numbers has the purpose of mirroring empirical evidences which indicate that smaller numerosities are more easily discriminated than larger ones.

    3. 3.

      Number line with scalar variability (Gallistel and Gelman 1992; Dehaene 2001). This scheme differs from the compressed number line, for the number series is linear. For any number n the total activation is constant while the variance is proportional to n itself.

    It would be interesting to see whether the effect of learning strategy is verified even if numbers are encoded according to any of these other schemes.

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  1. Dipartimento di Pedagogia, Psicologia, Filosofia. ALOPHIS—Applied LOgic, Philosophy, and HIstory of Science, Università di Cagliari, Cagliari, Italy

    Simone Pinna

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Pinna, S. (2017). BTM Models of Algorithmic Skills. In: Extended Cognition and the Dynamics of Algorithmic Skills. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 35. Springer, Cham. https://doi.org/10.1007/978-3-319-51841-1_5

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