Top

Published in:

2019 | OriginalPaper | Chapter

Inspection and Selection of Representations

Authors : Daniel Raggi, Aaron Stockdill, Mateja Jamnik, Grecia Garcia Garcia, Holly E. A. Sutherland, Peter C.-H. Cheng

Published in: Intelligent Computer Mathematics

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

We present a novel framework for inspecting representations and encoding their formal properties. This enables us to assess and compare the informational and cognitive value of different representations for reasoning. The purpose of our framework is to automate the process of representation selection, taking into account the candidate representation’s match to the problem at hand and to the user’s specific cognitive profile. This requires a language for talking about representations, and methods for analysing their relative advantages. This foundational work is first to devise a computational end-to-end framework where problems, representations, and user’s profiles can be described and analysed. As AI systems become ubiquitous, it is important for them to be more compatible with human reasoning, and our framework enables just that.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Diagram Combinators in MMT

next chapter A Plugin to Export Coq Libraries to XML

As a consequence, Q tables may contain elements of multiple RSs. For reasons mentioned in Sect. 5, this is discouraged whenever it is avoidable.

Those appearing in this example are taken from a semantic net [29].

In future work, we will investigate how correspondences and their strength can be identified automatically (e.g., using machine learning).

By ‘Euler’ we mean some implementation of Euler diagrams.

and requires that both properties appear in the property table. or requires that at least one of the properties appears in the property table. If both properties appear in the table, the strength is only counted once. not requires that a specified property does not occur in the property table.

The value 0.6 for ‘instrumental’ properties is chosen arbitrarily; the only condition is that the value-importance relation is monotonic. In future work, these parameters should be tuned with experimental data.

In Euler diagrams the cardinality of sets is abstracted away; the size of zones is meaningless.

Note that the strength of the correspondences from probability to \(\Pr \) and compare sizes was set to 1 (because in principle any probability function is representable in the Bayesian or Geometric systems), but it was set to 0.5 for Contingency because not every probability function is representable in Contingency tables.

Ainsworth, S.: The functions of multiple representations. Comput. Educ. 33(2–3), 131–152 (1999)CrossRef

Barker-Plummer, D., Etchemendy, J., Liu, A., Murray, M., Swoboda, N.: Openproof - a flexible framework for heterogeneous reasoning. In: Stapleton, G., Howse, J., Lee, J. (eds.) Diagrams 2008. LNCS (LNAI), vol. 5223, pp. 347–349. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-87730-1_32CrossRef

Bengio, Y., Courville, A., Vincent, P.: Representation learning: a review and new perspectives. IEEE Trans. Pattern Anal. Mach. Intell. 35(8), 1798–1828 (2013)CrossRef

Blackwell, A., Green, T.: Notational systems-the cognitive dimensions of notations framework. In: HCI Models, Theories, and Frameworks: Toward an Interdisciplinary Science. Morgan Kaufmann (2003)

Cheng, P.C.-H.: Unlocking conceptual learning in mathematics and science with effective representational systems. Comput. Educ. 33(2–3), 109–130 (1999)CrossRef

Cheng, P.C.-H.: Probably good diagrams for learning: representational epistemic recodification of probability theory. Top. Cogn. Sci. 3(3), 475–498 (2011)CrossRef

Cheng, P.C.-H.: What constitutes an effective representation? In: Jamnik, M., Uesaka, Y., Elzer Schwartz, S. (eds.) Diagrams 2016. LNCS (LNAI), vol. 9781, pp. 17–31. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-42333-3_2CrossRef

Chomsky, N.: Three models for the description of language. IRE Trans. Inf. Theory 2(3), 113–124 (1956)CrossRef

Chomsky, N.: On certain formal properties of grammars. Inf. Control 2(2), 137–167 (1959)MathSciNetCrossRef

10.

Coquand, T.: Type theory. In: Stanford Encyclopedia of Philosophy (2006)

11.

Harrison, J.: HOL light: an overview. In: Berghofer, S., Nipkow, T., Urban, C., Wenzel, M. (eds.) TPHOLs 2009. LNCS, vol. 5674, pp. 60–66. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-03359-9_4CrossRef

12.

Huffman, B., Kunčar, O.: Lifting and transfer: a modular design for quotients in Isabelle/HOL. In: Gonthier, G., Norrish, M. (eds.) CPP 2013. LNCS, vol. 8307, pp. 131–146. Springer, Cham (2013). https://doi.org/10.1007/978-3-319-03545-1_9CrossRefMATH

13.

Jamnik, M., Bundy, A., Green, I.: On automating diagrammatic proofs of arithmetic arguments. J. Log. Lang. Inf. 8(3), 297–321 (1999)MathSciNetCrossRef

14.

Jupyter. jupyter.org

15.

Kaufmann, M., Moore, J.S.: ACL2: an industrial strength version of Nqthm. In: Proceedings of 11th Annual Conference on Computer Assurance, COMPASS 1996, pp. 23–34. IEEE (1996)

16.

Kovács, L., Voronkov, A.: First-order theorem proving and Vampire. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 1–35. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39799-8_1CrossRef

17.

Kühlwein, D., Blanchette, J.C., Kaliszyk, C., Urban, J.: MaSh: machine learning for sledgehammer. In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 35–50. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-39634-2_6CrossRef

18.

Matlab. mathworks.com

19.

Mossakowski, T., Maeder, C., Lüttich, K.: The heterogeneous tool set, Hets. In: Grumberg, O., Huth, M. (eds.) TACAS 2007. LNCS, vol. 4424, pp. 519–522. Springer, Heidelberg (2007). https://doi.org/10.1007/978-3-540-71209-1_40CrossRef

20.

Newell, A.: Human Problem Solving. Prentice-Hall Inc., Upper Saddle River (1972)

21.

Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL: A Proof Assistant for Higher-Order Logic, vol. 2283. Springer, Heidelberg (2002). https://doi.org/10.1007/3-540-45949-9CrossRefMATH

22.

GNU Octave. octave.org

23.

Raggi, D., Bundy, A., Grov, G., Pease, A.: Automating change of representation for proofs in discrete mathematics (extended version). Math. Comput. Sci. 10(4), 429–457 (2016)MathSciNetCrossRef

24.

SageMath. sagemath.org

25.

Stapleton, G., Jamnik, M., Shimojima, A.: What makes an effective representation of information: a formal account of observational advantages. J. Log. Lang. Inf. 26(2), 143–177 (2017)MathSciNetCrossRef

26.

Urbas, M., Jamnik, M.: A framework for heterogeneous reasoning in formal and informal domains. In: Dwyer, T., Purchase, H., Delaney, A. (eds.) Diagrams 2014. LNCS (LNAI), vol. 8578, pp. 277–292. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44043-8_28CrossRef

27.

Urbas, M., Jamnik, M., Stapleton, G., Flower, J.: Speedith: a diagrammatic reasoner for spider diagrams. In: Cox, P., Plimmer, B., Rodgers, P. (eds.) Diagrams 2012. LNCS (LNAI), vol. 7352, pp. 163–177. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31223-6_19CrossRef

28.

Winterstein, D., Bundy, A., Gurr, C.: Dr.Doodle: a diagrammatic theorem prover. In: Basin, D., Rusinowitch, M. (eds.) IJCAR 2004. LNCS (LNAI), vol. 3097, pp. 331–335. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-25984-8_24CrossRef

29.

WordNet (2010). wordnet.princeton.edu

Title: Inspection and Selection of Representations
Authors: Daniel Raggi
Aaron Stockdill
Mateja Jamnik
Grecia Garcia Garcia
Holly E. A. Sutherland
Peter C.-H. Cheng
Publisher: Springer International Publishing
Book: Intelligent Computer Mathematics
Print ISBN: 978-3-030-23249-8

Electronic ISBN: 978-3-030-23250-4

Copyright Year: 2019
DOI: https://doi.org/10.1007/978-3-030-23250-4_16

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner