×

About the unification type of \(\mathbf{K}+\square \square \bot\). (English) Zbl 07517438

Summary: The unification problem in a propositional logic is to determine, given a formula \(\phi\), whether there exists a substitution \(\sigma\) such that \(\sigma(\phi)\) is in that logic. In that case, \(\sigma\) is a unifier of \(\phi\). When a unifiable formula has minimal complete sets of unifiers, it is either infinitary, finitary, or unitary, depending on the cardinality of its minimal complete sets of unifiers. Otherwise, it is nullary. In this paper, we prove that in modal logic \(\mathbf{K}+\square \square \bot\), unifiable formulas are either finitary, or unitary.

MSC:

03B45 Modal logic (including the logic of norms)
03B70 Logic in computer science
68T27 Logic in artificial intelligence
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Baader, F., Fernández Gil, O., Rostamigiv, M.: Restricted unification in the description logic \(\mathcal{F}{\mathcal L}_0 \). In: 35th International Workshop on Unification, Informal proceedings, pp 8-14 (2021) · Zbl 07497917
[2] Baader, F.; Ghilardi, S., Unification in modal and description logics, Logic Journal of the IGPL, 19, 705-730 (2011) · Zbl 1258.03018 · doi:10.1093/jigpal/jzq008
[3] Baader, F.; Narendran, P., Unification of concept terms in description logics, J. Symb. Comput., 31, 277-305 (2001) · Zbl 0970.68166 · doi:10.1006/jsco.2000.0426
[4] Balbiani, P., Remarks about the unification type of several non-symmetric non-transitive modal logics, Logic Journal of the IGPL, 27, 639-658 (2019) · Zbl 1494.03046 · doi:10.1093/jigpal/jzy078
[5] Balbiani, P.; Gencer, Ç., KD is nullary, J. Appl. Non-Class. Log., 27, 196-205 (2017) · Zbl 1398.03086 · doi:10.1080/11663081.2018.1428000
[6] Balbiani, P.; Gencer, Ç., Unification in epistemic logics, J. Appl. Non-Class. Log., 27, 91-105 (2017) · Zbl 1398.03070 · doi:10.1080/11663081.2017.1368845
[7] Balbiani, P.; Gencer, Ç., About the unification type of modal logics between KB and KTB, Stud. Logica., 108, 941-966 (2020) · Zbl 1486.03038 · doi:10.1007/s11225-019-09883-0
[8] Balbiani, P., Gencer, Ç., Rostamigiv, M., Tinchev, T.: About the unification types of the modal logics determined by classes of deterministic frames, arXiv:2004.07904v1[cs.LO] (2020)
[9] Balbiani, P., Tinchev, T.: Unification in modal logic Alt_1. In: Advances in Modal Logic, College Publications, pp 117-134 (2016) · Zbl 1400.03037
[10] Balbiani, P.; Tinchev, T., Elementary unification in modal logic KD45, J. Appl. Logics, 5, 301-317 (2018) · Zbl 1349.03024
[11] Blackburn, P.; de Rijke, M.; Venema, Y., Modal Logic (2001), Cambridge: Cambridge University Press, Cambridge · Zbl 0988.03006 · doi:10.1017/CBO9781107050884
[12] Chagrov, A.; Zakharyaschev, M., Modal Logic (1997), Oxford: Oxford University Press, Oxford · Zbl 0871.03007
[13] Dzik, W., Unitary unification of S5 modal logics and its extensions, Bull. Sect. Log., 32, 19-26 (2003) · Zbl 1039.03009
[14] Dzik, W.: Unification Types in Logic. Wydawnicto Uniwersytetu Slaskiego (2007) · Zbl 1148.03003
[15] Dzik, W.; Wojtylak, P., Projective unification in modal logic, Logic Journal of the IGPL, 20, 121-153 (2012) · Zbl 1260.03041 · doi:10.1093/jigpal/jzr028
[16] Gabbay, D.; Shehtman, V., Products of modal logics, part 1, Logic Journal of the IGPL, 6, 73-146 (1998) · Zbl 0902.03008 · doi:10.1093/jigpal/6.1.73
[17] Ghilardi, S., Best solving modal equations, Ann. Pure Appl. Log., 102, 183-198 (2000) · Zbl 0949.03010 · doi:10.1016/S0168-0072(99)00032-9
[18] Ghilardi, S., Unification, finite duality and projectivity in varieties of Heyting algebras, Ann. Pure Appl. Log., 127, 99-115 (2004) · Zbl 1058.03020 · doi:10.1016/j.apal.2003.11.010
[19] Ghilardi, S.; Sacchetti, L., Filtering unification and most general unifiers in modal logic, J. Symb. Log., 69, 879-906 (2004) · Zbl 1069.03011 · doi:10.2178/jsl/1096901773
[20] Goranko, V. , Otto, M.: Model theory of modal logic. In: Handbook of Modal Logic, pp 249-329. Elsevier (2007)
[21] Iemhoff, R., A syntactic approach to unification in transitive reflexive modal logics, Notre Dame Journal of Formal Logic, 57, 233-247 (2016) · Zbl 1436.03133
[22] Jer̆ábek, E.: Logics with directed unification. In: Algebra and Coalgebra meet Proof Theory, Workshop at Utrecht University (2013)
[23] Jer̆ábek, E., Blending margins: the modal logic K has nullary unification type, J. Log. Comput., 25, 1231-1240 (2015) · Zbl 1328.03019 · doi:10.1093/logcom/ext055
[24] Kost, S., Projective unification in transitive modal logics, Logic Journal of the IGPL, 26, 548-566 (2018) · Zbl 1492.03007 · doi:10.1093/jigpal/jzy013
[25] Kracht, M.: Tools and techniques in modal logic. Elsevier (1999) · Zbl 0927.03002
[26] Miyazaki, Y.: Normal modal logics containing KTB with some finiteness conditions. In: Advances in Modal Logic, pp 171-190. College Publications (2004) · Zbl 1102.03019
[27] Nagle, M.; Thomason, S., The extensions of the modal logic K5, J. Symb. Log., 50, 102-109 (1985) · Zbl 0586.03012 · doi:10.2307/2273793
[28] Shapirovsky, I., Shehtman, V.: Local tabularity without transitivity. In: Advances in Modal Logic, pp 520-534. College Publications (2016) · Zbl 1400.03051
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.