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Types from Frames as Finite Automata

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Formal Grammar (FG 2015, FG 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9804))

Abstract

An approach to frame semantics is built on a conception of frames as finite automata, observed through the strings they accept. An institution (in the sense of Goguen and Burstall) is formed where these strings can be refined or coarsened to picture processes at various bounded granularities, with transitions given by Brzozowski derivatives.

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Notes

  1. 1.

    This introductory section presupposes some familiarity with the literature, but is followed by sections that proceed in a more careful manner, without relying on a full understanding of the Introduction.

  2. 2.

    The compatibility here becomes obvious if the moves described in footnote 5 of page 281 in Kallmeyer and Osswald (2013) are made, and a root added with attributes to the multiple base nodes.

  3. 3.

    Readers familiar with bisimulations will note that \(\sim \) is the largest bisimulation (determinism being an extreme form of image-finiteness; Hennessy and Milner 1985).

  4. 4.

    \({ Nom}_\mathsf{{A}}(\varphi )\) does the work of the scheme (Nom \(_N\)) for nominals given in Blackburn (1993), just as \({ Tml}_\mathsf{{A}}(a)\) is analogous to the scheme (Term) there for instantiating atomic information at terminal nodes.

  5. 5.

    By contrast, for a language q over \(\mathsf{{A}}\), all formulas in the set

    $$\begin{aligned} \{\Box (\langle a_q\rangle \top \supset (\langle s\rangle \top \wedge \lnot \langle s'\rangle \top )) \ | \ s\in q \text{ and } s'\in \mathsf{{A}}^*-q\} \end{aligned}$$

    must be satisfied for \(a_q\) to mark q.

  6. 6.

    That said, we might refine \(\mathbf{Sign}\), requiring of a signature \((\varSigma ,q)\) that q be a regular language. For this, it suffices to replace \(\int Q\) by \(\int R\) where \(R:{ Fin}({\mathsf{{A}}})^{op} \rightarrow \mathbf{Cat}\) is the subfunctor of Q such that \(R(\varSigma )\) is the full subcategory of \(Q(\varSigma )\) with objects regular languages. We can make this refinement without requiring that \(\varSigma \)-states in \({ Mod}(\varSigma ,q)\) be regular, forming \({ Mod}(\varSigma ,q)\) from Q (not R).

  7. 7.

    This is formulated as an institution in Fernando (2014).

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Acknowledgements

My thanks to Robin Cooper for discussions, to Glyn Morrill for help with presenting this paper at Formal Grammar 2015, and to two referees for comments and questions.

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Authors and Affiliations

  1. Trinity College Dublin, Dublin, Ireland

    Tim Fernando

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  1. Tim Fernando
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Correspondence to Tim Fernando .

Editor information

Editors and Affiliations

  1. IRISA, University of Rennes 1 , Rennes, France

    Annie Foret

  2. Department of Computer Science, Universitat Politècnica de Catalunya , Barcelona, Spain

    Glyn Morrill

  3. Tilburg University , Tilburg, The Netherlands

    Reinhard Muskens

  4. Department of General Linguistics, Heinrich-Heine-University Düsseldorf , Düsseldorf, Germany

    Rainer Osswald

  5. INRIA Nancy, Villers-lès-Nancy, France

    Sylvain Pogodalla

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Fernando, T. (2016). Types from Frames as Finite Automata. In: Foret, A., Morrill, G., Muskens, R., Osswald, R., Pogodalla, S. (eds) Formal Grammar. FG FG 2015 2016. Lecture Notes in Computer Science(), vol 9804. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53042-9_2

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  • DOI: https://doi.org/10.1007/978-3-662-53042-9_2

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