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Considerations in Representation Selection for Problem Solving: A Review

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Abstract

Choosing how to represent knowledge effectively is a long-standing open problem. Cognitive science has shed light on the taxonomisation of representational systems from the perspective of cognitive processes, but a similar analysis is absent from the perspective ofproblem solving, where the representations are employed. In this paper we review how representation choices are made for solving problems in the context of theorem proving from three perspectives: cognition, heterogeneity, and computational demands. We contrast the different factors that are most important for each perspective in the context of problem solving to produce a list of considerations for developers of problem solving tools regarding representations that are appropriate for particular users and effective for specific problem domains.

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Notes

  1. 1.

    Famously, seven plus or minus twochunks [22].

  2. 2.

    By analogy to computers, we devote ‘registers’ that would otherwise be used on the problem to maintaining the ‘call stack’, but the human brain’s ‘call stack’ capacity is small, and – due to the nature of graph search – easy to overflow.

  3. 3.

    Although work that builds upon these dimensions (e.g., [4]) often includes concepts very close to those we are about to discuss.

  4. 4.

    In this case, the hierarchy is that the box is restricted to tacks, and there is no hierarchical relationship to the candle; thenecessary hierarchy has box restricted toobjects, which includes tacks and candles.

  5. 5.

    We consider onlyvisual representations; representations that are audial or tactile, for example, are beyond the scope of this review.

  6. 6.

    [32] identified variations on this divide, but all are sufficiently similar for our discussion.

  7. 7.

    Note again that, visually,a is to theleft ofb.

  8. 8.

    HOL stands for higher-order logic, an extension of predicate/first-order logic. LCF stands for the logic for computable functions, a theorem prover based on the logicof computable functions.

  9. 9.

    Isabelle (without ‘HOL’) is ameta-logic system: a developer tailors Isabelle to work in their particular system. For example, Isabelle/ZF allows people to use Zermelo-Fraenkel (ZF) set theory rather than higher-order logic. This is an interesting step towards heterogeneity, but the different logics are inaccessible to each other.

  10. 10.

    For example,\((x = \square ) \rightarrow (\text {shape}(x) = \texttt {square})\) places the square in the statement.

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Acknowledgements

Aaron Stockdill is supported by the Hamilton Cambridge International Scholarship. This work was supported by the EPSRC grants EP/R030650/1, EP/T019603/1, EP/R030642/1, and EP/T019034/1.

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Authors and Affiliations

  1. University of Cambridge, Cambridge, UK

    Aaron Stockdill, Daniel Raggi & Mateja Jamnik

  2. University of Sussex, Brighton, UK

    Grecia Garcia Garcia & Peter C.-H. Cheng

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  1. Aaron Stockdill

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  2. Daniel Raggi

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  3. Mateja Jamnik

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  4. Grecia Garcia Garcia

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  5. Peter C.-H. Cheng

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Correspondence toAaron Stockdill.

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Editors and Affiliations

  1. Jadavpur University, Kolkata, India

    Amrita Basu

  2. University of Cambridge, Cambridge, UK

    Gem Stapleton

  3. Lancaster University in Leipzig, Leipzig, Germany

    Sven Linker

  4. Deakin University, Burwood, VIC, Australia

    Catherine Legg

  5. Kyoto University, Kyoto, Japan

    Emmanuel Manalo

  6. Universidade Federal Fluminense, Niterói, Brazil

    Petrucio Viana

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Stockdill, A., Raggi, D., Jamnik, M., Garcia Garcia, G., Cheng, P.CH. (2021). Considerations in Representation Selection for Problem Solving: A Review. In: Basu, A., Stapleton, G., Linker, S., Legg, C., Manalo, E., Viana, P. (eds) Diagrammatic Representation and Inference. Diagrams 2021. Lecture Notes in Computer Science(), vol 12909. Springer, Cham. https://doi.org/10.1007/978-3-030-86062-2_4

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