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Deriving generic bounds for time-series constraints based on regular expressions characteristics

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Abstract

We introduce the concept of regular expression characteristics as a unified way to concisely express bounds on time-series constraints. This allows us not only to define time-series constraints in a compositional way, but also to deal with their combinatorial aspect in a compositional way, without developing ad-hoc bounds for each time-series constraint separately.

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Notes

  1. The ‘b’ of b σ stands for ‘before’, while the ‘a’ of a σ stands for ‘after’.

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Acknowledgements

We thank Pierre Flener for his feedback on the notation system for regular expression characteristics in Section 3.1.

Notation for Regular Expression Characteristics ω σ the size of a regular expression σ (see Definition 4) \({{\Omega }_{\sigma }^{\left \langle \ell , u \right \rangle }(v)}\) the set of support time series of two words v and w in \({\mathcal {L}_{\sigma }}\) wrt \(\left \langle \ell , u \right \rangle \) (see Definition 5) η σ (v) the height of a word v in \({\mathcal {L}_{\sigma }}\) (see Definition 6) η σ the height of a regular expression σ (see Definition 7) \(\phi _{\sigma }^{\left \langle n \right \rangle }\) the range of a regular expression σ wrt \(\left \langle n \right \rangle \) (see Definition 8) Θ σ the set of inducing words of a regular expression σ (see Definition 10, Table 1) \({{\Gamma }_{\sigma }^{\left \langle \ell , u \right \rangle }(v, w)}\) the set of superpositions of two words v and w in \({\mathcal {L}_{\sigma }}\) wrt \(\left \langle \ell , u \right \rangle \) (see Definition 11) \({o_{\sigma }^{\left \langle \ell , u \right \rangle }(v, w)}\) the overlap of two words v and w in \({\mathcal {L}_{\sigma }}\) wrt \(\left \langle \ell , u \right \rangle \) (see Definition 12) \(o_{\sigma }^{\left \langle \ell , u \right \rangle }\) the overlap of a regular expression σ wrt \(\left \langle \ell , u \right \rangle \) (see Definition 13) \({\overline {\nu }_{\sigma }^{\left \langle \ell , u \right \rangle }(v, w, i)}\) the shift of a subword w within a word v in \({\mathcal {L}_{\sigma }}\) wrt \(\left \langle \ell , u \right \rangle \) (see Definition 14) \(\delta _{\sigma }^{\left \langle \ell , u \right \rangle }(v, w)\) the smallest variation of maxima of two words w and v in \({\mathcal {L}_{\sigma }}\) wrt \(\left \langle \ell , u \right \rangle \) (see Definition 15) \(\delta _{\sigma }^{\left \langle \ell , u \right \rangle }\) the smallest variation of maxima of a regular expression σ wrt \(\left \langle \ell , u \right \rangle \) (see Definition 16)

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Correspondence to Ekaterina Arafailova.

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This is an extended version of parts of the CP 2016 paper [3] which involves asubset of the original authors. This paper introduces 6new characteristics of regular expressions to express generic bounds on time-series constraints, which were not discussed in the original paper [3]. Ekaterina Arafailova is supported by the EU H2020 programme under grant 640954 for project GRACeFUL. Nicolas Beldiceanu is partially supported by the GRACeFUL project and by the Gaspard Monge Program for Optimization and Operations Research (PGMO). Helmut Simonis is supported by Science Foundation Ireland (SFI) under grant SFI/10/IN.1/I3032; the Insight Centre for Data Analytics is supported by SFI under grant SFI/12/RC/2289.

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Arafailova, E., Beldiceanu, N. & Simonis, H. Deriving generic bounds for time-series constraints based on regular expressions characteristics. Constraints 23, 44–86 (2018). https://doi.org/10.1007/s10601-017-9276-z

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