Relational approach for a logic for order of magnitude qualitative reasoning with negligibility, non-closeness and distance
Abstract
We present a relational proof system in the style of dual tableaux for a multimodal propositional logic for order of magnitude qualitative reasoning to deal with relations of negligibility, non-closeness, and distance. This logic enables us to introduce the operation of qualitative sum for some classes of numbers. A relational formalization of the modal logic in question is introduced in this paper, i.e., we show how to construct a relational logic associated with the logic for order-of-magnitude reasoning and its dual tableau system which is a validity checker for the modal logic. For that purpose, we define a validity preserving translation of the modal language into relational language. Then we prove that the system is sound and complete with respect to the relational logic defined as well as with respect to the logic for order of magnitude reasoning. Finally, we show that in fact relational dual tableau does more. It can be used for performing the four major reasoning tasks: verification of validity, proving entailment of a formula from a finite set of formulas, model checking, and verification of satisfaction of a formula in a finite model by a given object.
Citation
Please, cite this work as:
[GM09] J. Golinska-Pilarek and E. Mu~noz-Velasco. “Relational approach for a logic for order of magnitude qualitative reasoning with negligibility, non-closeness and distance”. In: Log. J. IGPL 17.4 (2009), pp. 375-394. DOI: 10.1093/JIGPAL/JZP016. URL: https://doi.org/10.1093/jigpal/jzp016.
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Papers citing this work
The following is a non-exhaustive list of papers that cite this work:
[1] P. Balbiani. “Reasoning about negligibility and proximity in the set of all hyperreals”. In: Journal of Applied Logic 16 (Jul. 2016), p. 14–36. ISSN: 1570-8683. DOI: 10.1016/j.jal.2016.04.002. URL: http://dx.doi.org/10.1016/j.jal.2016.04.002.
[2] A. Burrieza, E. Muñoz-Velasco, and M. Ojeda-Aciego. “A flexible logic-based approach to closeness using order of magnitude qualitative reasoning”. In: Logic Journal of the IGPL 28.1 (Dec. 2019), p. 121–133. ISSN: 1368-9894. DOI: 10.1093/jigpal/jzz076. URL: http://dx.doi.org/10.1093/jigpal/jzz076.
[3] A. Burrieza, E. Muñoz-Velasco, and M. Ojeda-Aciego. “A multimodal logic for closeness”. In: Journal of Applied Non-Classical Logics 27.3–4 (Oct. 2017), p. 225–237. ISSN: 1958-5780. DOI: 10.1080/11663081.2018.1442137. URL: http://dx.doi.org/10.1080/11663081.2018.1442137.
[4] A. Burrieza, E. Muñoz-Velasco, and M. Ojeda-Aciego. “Closeness and Distance Relations in Order of Magnitude Qualitative Reasoning via PDL”. In: Current Topics in Artificial Intelligence. Springer Berlin Heidelberg, 2010, p. 71–80. ISBN: 9783642142642. DOI: 10.1007/978-3-642-14264-2_8. URL: http://dx.doi.org/10.1007/978-3-642-14264-2_8.
[5] A. Burrieza, E. Muñoz-Velasco, and M. Ojeda-Aciego. “Logics for Order-of-Magnitude Qualitative Reasoning: Formalizing Negligibility”. In: Ewa Orłowska on Relational Methods in Logic and Computer Science. Springer International Publishing, 2018, p. 203–231. ISBN: 9783319978796. DOI: 10.1007/978-3-319-97879-6_8. URL: http://dx.doi.org/10.1007/978-3-319-97879-6_8.
[6] A. BURRIEZA, E. MUÑOZ-VELASCO, and M. OJEDA-ACIEGO. “A PDL APPROACH FOR QUALITATIVE VELOCITY”. In: International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 19.01 (Feb. 2011), p. 11–26. ISSN: 1793-6411. DOI: 10.1142/s021848851100685x. URL: http://dx.doi.org/10.1142/s021848851100685x.
[7] J. Golińska-Pilarek. “On Decidability of a Logic for Order of Magnitude Qualitative Reasoning with Bidirectional Negligibility”. In: Logics in Artificial Intelligence. Springer Berlin Heidelberg, 2012, p. 255–266. ISBN: 9783642333538. DOI: 10.1007/978-3-642-33353-8_20. URL: http://dx.doi.org/10.1007/978-3-642-33353-8_20.
[8] J. Golińska-Pilarek, T. Huuskonen, and E. Muñoz-Velasco. “Relational dual tableau decision procedures and their applications to modal and intuitionistic logics”. In: Annals of Pure and Applied Logic 165.2 (Feb. 2014), p. 409–427. ISSN: 0168-0072. DOI: 10.1016/j.apal.2013.06.003. URL: http://dx.doi.org/10.1016/j.apal.2013.06.003.
[9] J. Golińska-Pilarek, A. Mora, and E. Muñoz-Velasco. “An ATP of a Relational Proof System for Order of Magnitude Reasoning with Negligibility, Non-closeness and Distance”. In: PRICAI 2008: Trends in Artificial Intelligence. Springer Berlin Heidelberg, 2008, p. 128–139. ISBN: 9783540891970. DOI: 10.1007/978-3-540-89197-0_15. URL: http://dx.doi.org/10.1007/978-3-540-89197-0_15.
[10] J. Golinska-Pilarek, E. Munoz-Velasco, and A. Mora. “A new deduction system for deciding validity in modal logic K”. In: Logic Journal of IGPL 19.2 (Jul. 2010), p. 425–434. ISSN: 1368-9894. DOI: 10.1093/jigpal/jzq033. URL: http://dx.doi.org/10.1093/jigpal/jzq033.
[11] A. Mora, E. Muñoz-Velasco, and J. Golińska-Pilarek. “Implementing a relational theorem prover for modal logic”. In: International Journal of Computer Mathematics 88.9 (Jun. 2011), p. 1869–1884. ISSN: 1029-0265. DOI: 10.1080/00207160.2010.493211. URL: http://dx.doi.org/10.1080/00207160.2010.493211.