Abstract
Intervals and events are analyzed in terms of strings that represent points as symbols occurring uniquely. Allen interval relations, Dowty’s aspect hypothesis and inertia are understood relative to strings, compressed into canonical forms, describable in Monadic Second-Order logic. That understanding is built around a translation of strings replacing stative predicates by their borders, represented in the S-words of Schwer and Durand. Borders point to non-stative predicates, including forces that may compete, succeed to varying degrees, fail and recur.
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Notes
- 1.
Boxes are drawn instead of \(\emptyset \) and curly braces \(\{\cdot \}\) to reduce the risk of confusing, for example, the empty language \(\emptyset \) with the string \(\Box \) of length one (not to mention the null string of length 0).
- 2.
Regularity of languages is interesting here for computational reasons; for instance, since inclusions between regular languages are computable (unlike inclusions between context-free languages), so are entailments in MSO.
- 3.
The strings \(\mathfrak {s}_R(a,a')\) can be derived from strings \(\mathfrak {s}^\circ _R(a,a')\) over the alphabet \(\{a,a'\}\) by the equation
$$\mathfrak {s}_R(a,a') = b(\Box \mathfrak {s}^\circ _R(a,a')).$$For example,
A full list of \(\mathfrak {s}^\circ _R(a,a')\), for every Allen relation R, can be found in Table 7.1 in [5, p. 223].
- 4.
To see this, consider the first position where s and \(s'\) differ. Let \(\alpha \) and \(\alpha '\) be the symbols there of s and \(s'\), respectively. Then \((\alpha \cup \alpha ')-(\alpha \cap \alpha ')\) is non-empty. Let \(\gamma (a)\) be an element of that set, and \(\gamma '(a')\) belong in \((\alpha \cup \alpha ')-\{\gamma (a)\}\). Notice that \(\mathcal{{AR}}_s(a,a')\ne \mathcal{{AR}}_{s'}(a,a')\).
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Acknowledgements
This chapter is an extended version of the paper “Intervals and events with and without points” which appears in the proceedings of the Symposium on Logic and Algorithms in Computational Linguistics 2018, Stockholm University. My thanks to three referees and Roussanka Loukanova for their kind help.
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Fernando, T. (2020). Temporal Representations with and without Points. In: Loukanova, R. (eds) Logic and Algorithms in Computational Linguistics 2018 (LACompLing2018). Studies in Computational Intelligence, vol 860. Springer, Cham. https://doi.org/10.1007/978-3-030-30077-7_3
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