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Using Signatures in Type Theory to Represent Situations

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New Frontiers in Artificial Intelligence (JSAI-isAI 2014)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9067))

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Abstract

Signatures have been introduced to represent situations in formal semantics based on modern type theories. In this paper, we study the notion of signature in more details, presenting it formally and discussing its use in representations of situations. In particular, the new forms of signature entries, the subtyping entries and the manifest entries, are formally presented and studied. Besides being signature entries, these two forms of entries may be introduced to form contextual entries as well and this may have interesting implications in applications of the notion of context to, for example, belief contexts.

This work is partially supported by the research grants from Leverhulme, Royal Academy of Engineering and the CAS/SAFEA International Partnership Program for Creative Research Teams.

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Notes

  1. 1.

    Contextual manifest entries were first proposed by the second author in [12], where they are studied in a different context, focussing on its intensional nature, as compared with traditional extensional definition entries in proof assistants.

  2. 2.

    Local coercions are useful in formal semantics based on MTTs. See, for example, [1, 14] for discussions.

  3. 3.

    In LF, we use the notation [x : K]b for \(\uplambda x:K.b\) and \((x:K)K'\) for \(\varPi x:K.K'\).

  4. 4.

    The word ‘coercion’ has been used for related but maybe different things including coercions in programming languages and coercions in linguistics. See Asher and Luo [1] for a use of coercive subtyping in modelling linguistic coercions and Retoré et al. [2] for another proposal of using coercions to deal with some linguistic coercions in lexical semantics.

  5. 5.

    It is important that the condition is not stated for the whole system of coercive subtyping, for otherwise it would become trivial. Here we do not detail the description of the subsystem because we would then have to make explicit some technical details we feel unnecessary for this paper. An interested reader may look at [17] for details how coherence is defined in a global case.

  6. 6.

    At the moment, this is only a conjecture: although the authors do not see any real problems in doing so, tedious and careful work is needed to carry such a proof out (work in progress).

  7. 7.

    Similar ideas have been put forth in [5] and [4] to deal with intensional adjectives and adverbs.

  8. 8.

    Here, we do not discuss the issue whether such a proposal is adequate to represent intensional beliefs. For instance, one might argue against such proposals simply by arguing that ordinary logical inference does not capture the intended inference concerning beliefs. We are simply take this as an example to show that the usefulness of subtyping/manifest entries in contexts.

  9. 9.

    For example, using the heterogenous equality Eq, this belief can be expressed as \(\forall x:G\forall y:W.Eq(G,W,x,y)\). We do not get into the formal details here.

  10. 10.

    A similar problem due to Kripke is the Pierre problem, according to which Pierre thinks that Londres is beautiful but London is not [9]. It is obvious how this can be handled given what we have said.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Royal Holloway, University of London, Egham, UK

    Stergios Chatzikyriakidis & Zhaohui Luo

Authors
  1. Stergios Chatzikyriakidis
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  2. Zhaohui Luo
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Correspondence to Stergios Chatzikyriakidis .

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Editors and Affiliations

  1. Tokyo Institute of Technology, Tokyo, Japan

    Tsuyoshi Murata

  2. Ochanomizu University, Tokyo, Japan

    Koji Mineshima

  3. Department Information Science, Faculty of Science, Ochanomizu University , Tokyo, Japan

    Daisuke Bekki

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Chatzikyriakidis, S., Luo, Z. (2015). Using Signatures in Type Theory to Represent Situations. In: Murata, T., Mineshima, K., Bekki, D. (eds) New Frontiers in Artificial Intelligence. JSAI-isAI 2014. Lecture Notes in Computer Science(), vol 9067. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48119-6_13

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  • DOI: https://doi.org/10.1007/978-3-662-48119-6_13

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