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Logic and Complexity in Cognitive Science

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Book cover Johan van Benthem on Logic and Information Dynamics

Part of the book series: Outstanding Contributions to Logic ((OCTR,volume 5))

Abstract

This chapter surveys the use of logic and computational complexity theory in cognitive science. We emphasize in particular the role played by logic in bridging the gaps between Marr’s three levels: representation theorems for non-monotonic logics resolve algorithmic/implementation debates, while complexity theory probes the relationship between computational task analysis and algorithms. We argue that the computational perspective allows feedback from empirical results to guide the development of increasingly subtle computational models. We defend this perspective via a survey of the role of logic in several classic problems in cognitive science (the Wason selection task, the frame problem, the connectionism/symbolic systems debate) before looking in more detail at case studies involving quantifier processing and social cognition. In these examples, models developed by Johan van Benthem have been supplemented with complexity analysis to drive successful programs of empirical research.

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Notes

  1. 1.

    For more references on the interface between logic and cognition, see also the 2007 special issue of Topoi on “Logic and Cognition”, ed. J. van Benthem, H. Hodges, and W. Hodges; the 2008 special issue of Journal of Logic, Language and Information on “Formal Models for Real People”, ed. M. Counihan; the 2008 special issue of Studia Logica on “Psychologism in Logic?”, ed. H. Leitgeb, including [8]; and the 2013 special issue of Journal of Logic, Language and Information on “Logic and Cognition” ed. J. Szymanik and R. Verbrugge.

  2. 2.

    Siegelmann repeatedly appeals to a result in Siegelmann and Sontag [107] when arguing in later papers that analog neural networks do not require arbitrary precision (and are thus physically realizable). In particular, their Lemma 4.1 shows that for every neural network which computes over real numbers, there exists a neural network which computes over truncated reals (i.e. reals precise only to a finite number of digits). However, the length of truncation required is a function of the length of the computation—longer computations require longer truncated strings. Consequently, if length of computation is allowed to grow arbitrarily, so must the length of the strings of digits over which the computation is performed in a truncated network. Thus, one still must allow computation over arbitrarily precise reals if one is considering the computational properties of analog neural networks in general, i.e. over arbitrarily long computation times.

  3. 3.

    For an overview, see the 2006 special issue of Trends in Cognitive Sciences (vol. 10, no. 7) on probabilistic models of cognition, or [129].

  4. 4.

    However, they do not indicate how such implementational networks avoid their general critique, see [18].

  5. 5.

    The intuitive connection between efficiency and PTIME-computability depends crucially on considering efficiency over arbitrarily large input size \(n\). For example, an algorithm bounded by \(n^{\frac{1}{5}\log \log n}\) could be used practically even though it is not polynomial (since \(n^{\frac{1}{5}\log \log n} > n^2\) only when \(n>e^{e^{10}}\), [61]). Conversely, an algorithm bounded by \(n^{98466506514687}\) is PTIME-computable, but even for small \(n\) it is not practical to implement. We return to these considerations in the following sections.

  6. 6.

    See also [89].

  7. 7.

    However, see [102].

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Acknowledgments

We would like to thank Alexandru Baltag, Johan van Benthem, Peter Gärdenfors, Iris van Rooij, and Keith Stenning for many comments and suggestions. The first author would also like to thank Douwe Kiela and Thomas Icard for helpful discussions of this material; he was supported by NSF grant 1028130. The second author was supported by NWO Vici grant 277-80-001 and NWO Veni grant 639-021-232. The third author was supported by NWO Vici grant 277-80-001.

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  1. Department of Philosophy, University of Edinburgh, Edinburgh, Scotland

    Alistair M. C. Isaac

  2. Institute for Logic, Language, and Computation, University of Amsterdam, Amsterdam, The Netherlands

    Jakub Szymanik

  3. Institute of Artificial Intelligence, University of Groningen, Groningen, The Netherlands

    Rineke Verbrugge

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  1. University of Amsterdam Inst Logic,Language & Computation, Amsterdam, The Netherlands

    Alexandru Baltag

  2. Amsterdam, The Netherlands

    Sonja Smets

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Isaac, A.M.C., Szymanik, J., Verbrugge, R. (2014). Logic and Complexity in Cognitive Science. In: Baltag, A., Smets, S. (eds) Johan van Benthem on Logic and Information Dynamics. Outstanding Contributions to Logic, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-06025-5_30

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