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**Published in:** LIPIcs, Volume 178, 27th International Symposium on Temporal Representation and Reasoning (TIME 2020)

Branching Algebra is the natural branching-time generalization of Allen’s Interval Algebra. As in the linear case, the consistency problem for Branching Algebra is NP-hard. Being relatively new, however, not much is known about the computational behaviour of the consistency problem of its sub-algebras, except in the case of the recently found subset of convex branching relations, for which the consistency of a network can be tested via path consistency and it is therefore deterministic polynomial. In this paper, following Nebel and Bürckert, we define the Horn fragment of Branching Algebra, and prove that it is a sub-algebra of the latter, being closed under inverse, intersection, and composition, that it strictly contains both the convex fragment of Branching Algebra and the Horn fragment of Interval Algebra, and that its consistency problem can be decided via path consistency. Finally, we experimentally prove that the Horn fragment of Branching Algebra can be used as an heuristic for checking the consistency of a generic network with a considerable improvement over the convex subset.

Alessandro Bertagnon, Marco Gavanelli, Alessandro Passantino, Guido Sciavicco, and Stefano Trevisani. The Horn Fragment of Branching Algebra. In 27th International Symposium on Temporal Representation and Reasoning (TIME 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 178, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bertagnon_et_al:LIPIcs.TIME.2020.5, author = {Bertagnon, Alessandro and Gavanelli, Marco and Passantino, Alessandro and Sciavicco, Guido and Trevisani, Stefano}, title = {{The Horn Fragment of Branching Algebra}}, booktitle = {27th International Symposium on Temporal Representation and Reasoning (TIME 2020)}, pages = {5:1--5:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-167-2}, ISSN = {1868-8969}, year = {2020}, volume = {178}, editor = {Mu\~{n}oz-Velasco, Emilio and Ozaki, Ana and Theobald, Martin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TIME.2020.5}, URN = {urn:nbn:de:0030-drops-129736}, doi = {10.4230/LIPIcs.TIME.2020.5}, annote = {Keywords: Constraint programming, Consistency, Branching time, Horn Fragment} }

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**Published in:** LIPIcs, Volume 120, 25th International Symposium on Temporal Representation and Reasoning (TIME 2018)

Allen's Interval Algebra (IA) is one of the most prominent formalisms in the area of qualitative temporal reasoning; however, its applications are naturally restricted to linear flows of time. When dealing with nonlinear time, Allen's algebra can be extended in several ways, and, as suggested by Ragni and Wölfl [M. Ragni and S. Wölfl, 2004], a possible solution consists in defining the Branching Algebra (BA) as a set of 19 basic relations (13 basic linear relations plus 6 new basic nonlinear ones) in such a way that each basic relation between two intervals is completely defined by the relative position of the endpoints on a tree-like partial order. While the problem of deciding the consistency of a network of IA-constraints is well-studied, and every subset of the IA has been classified with respect to the tractability of its consistency problem, the fragments of the BA have received less attention. In this paper, we first define the notion of convex BA-relation, and, then, we prove that the consistency of a network of convex BA-relations can be decided via path consistency, and is therefore a polynomial problem. This is the first non-trivial tractable fragment of the BA; given the clear parallel with the linear case, our contribution poses the bases for a deeper study of fragments of BA towards their complete classification.

Marco Gavanelli, Alessandro Passantino, and Guido Sciavicco. Deciding the Consistency of Branching Time Interval Networks. In 25th International Symposium on Temporal Representation and Reasoning (TIME 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 120, pp. 12:1-12:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gavanelli_et_al:LIPIcs.TIME.2018.12, author = {Gavanelli, Marco and Passantino, Alessandro and Sciavicco, Guido}, title = {{Deciding the Consistency of Branching Time Interval Networks}}, booktitle = {25th International Symposium on Temporal Representation and Reasoning (TIME 2018)}, pages = {12:1--12:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-089-7}, ISSN = {1868-8969}, year = {2018}, volume = {120}, editor = {Alechina, Natasha and N{\o}rv\r{a}g, Kjetil and Penczek, Wojciech}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TIME.2018.12}, URN = {urn:nbn:de:0030-drops-97779}, doi = {10.4230/LIPIcs.TIME.2018.12}, annote = {Keywords: Constraint programming, Consistency, Branching time} }

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