# Logics for stable and unstable mereological relations

Open Mathematics (2011)

- Volume: 9, Issue: 6, page 1354-1379
- ISSN: 2391-5455

## Access Full Article

top## Abstract

top## How to cite

topVladislav Nenchev. "Logics for stable and unstable mereological relations." Open Mathematics 9.6 (2011): 1354-1379. <http://eudml.org/doc/269055>.

@article{VladislavNenchev2011,

abstract = {In this paper we present logics about stable and unstable versions of several well-known relations from mereology: part-of, overlap and underlap. An intuitive semantics is given for the stable and unstable relations, describing them as dynamic counterparts of the base mereological relations. Stable relations are described as ones that always hold, while unstable relations hold sometimes. A set of first-order sentences is provided to serve as axioms for the stable and unstable relations, and representation theory is developed in similar fashion to Stone’s representation theory for distributive lattices. First-order predicate logic and modal logic are presented with semantics based on structures with stable and unstable mereological relations. Completeness theorems for these logics are proved, as well as decidability in the case of the modal logic, hereditary undecidability in the case of the first-order logic, and NP-completeness for the satisfiability problem of the quantifier-free fragment of the first-order logic.},

author = {Vladislav Nenchev},

journal = {Open Mathematics},

keywords = {Stable and unstable relations; Mereology; Representation theory; First-order logic; Hereditary undecidability; Quantifier-free fragment; Modal logic; stable relation; unstable relation; mereology; representation theory; first-order logic; hereditary undecidability; quantifier-free fragment; modal logic},

language = {eng},

number = {6},

pages = {1354-1379},

title = {Logics for stable and unstable mereological relations},

url = {http://eudml.org/doc/269055},

volume = {9},

year = {2011},

}

TY - JOUR

AU - Vladislav Nenchev

TI - Logics for stable and unstable mereological relations

JO - Open Mathematics

PY - 2011

VL - 9

IS - 6

SP - 1354

EP - 1379

AB - In this paper we present logics about stable and unstable versions of several well-known relations from mereology: part-of, overlap and underlap. An intuitive semantics is given for the stable and unstable relations, describing them as dynamic counterparts of the base mereological relations. Stable relations are described as ones that always hold, while unstable relations hold sometimes. A set of first-order sentences is provided to serve as axioms for the stable and unstable relations, and representation theory is developed in similar fashion to Stone’s representation theory for distributive lattices. First-order predicate logic and modal logic are presented with semantics based on structures with stable and unstable mereological relations. Completeness theorems for these logics are proved, as well as decidability in the case of the modal logic, hereditary undecidability in the case of the first-order logic, and NP-completeness for the satisfiability problem of the quantifier-free fragment of the first-order logic.

LA - eng

KW - Stable and unstable relations; Mereology; Representation theory; First-order logic; Hereditary undecidability; Quantifier-free fragment; Modal logic; stable relation; unstable relation; mereology; representation theory; first-order logic; hereditary undecidability; quantifier-free fragment; modal logic

UR - http://eudml.org/doc/269055

ER -

## References

top- [1] Balbes R., Dwinger Ph., Distributive Lattices, University of Missouri Press, Columbia, 1974 Zbl0321.06012
- [2] Balbiani Ph., Tinchev T., Vakarelov D., Modal logics for region-based theory of space, Fund. Inform., 2007, 81(1–3), 29–82 Zbl1142.03012
- [3] Blackburn P., de Rijke M., Venema Y., Modal Logic, Cambridge Tracts Theoret. Comput. Sci., 53, Cambridge University Press, Cambridge, 2001 Zbl0988.03006
- [4] Chagrov A., Zakharyaschev M., Modal Logic, Oxford Logic Guides, Oxford University Press, New York, 1997
- [5] Egenhofer M.J., Franzosa R., Point-set topological spatial relations, International Journal of Geographic Information Systems, 1991, 5(2), 161–174 http://dx.doi.org/10.1080/02693799108927841
- [6] Ershov Yu.L., Problems of Decidability and Constructive Models, Nauka, Moscow, 1980 (in Russian)
- [7] Fine K., Essence and modality, Philosophical Perspectives, 1994, 8, 1–16 http://dx.doi.org/10.2307/2214160
- [8] Finger M., Gabbay D.M., Adding a temporal dimension to a logic system, J. Logic Lang. Inform., 1992, 1(3), 203–233 http://dx.doi.org/10.1007/BF00156915 Zbl0798.03031
- [9] Jonsson P., Drakengren T., A complete classification of tractability in RCC-5, J. Artificial Intelligence Res., 1997, 6, 211–221 Zbl0894.68096
- [10] Kontchakov R., Kurucz A., Wolter F., Zakharyaschev M., Spatial logic + temporal logic = ?, In: Handbook of Spatial Logics, Springer, Dordrecht, 2007, 497–564 http://dx.doi.org/10.1007/978-1-4020-5587-4_9
- [11] de Laguna T., Point, line, and surface, as sets of solids, J. Philos., 1922, 19(17), 449–461 http://dx.doi.org/10.2307/2939504
- [12] Lutz C., Wolter F., Modal logics of topological relations, Log. Methods Comput. Sci., 2006, 2(2), #5 Zbl1126.03026
- [13] Nenchev V., Vakarelov D., An axiomatization of dynamic ontology of stable and unstable mereological relations, In: 7th Panhellenic Logic Symposium, Patras, July 15–19, 2009, 137–141
- [14] Nenov Y., Vakarelov D., Modal Llogics for mereotopological relations, In: Advances in Modal Logic, 7, Nancy, September 9–12, 2008, College Publications, London, 2008, 249–272 Zbl1244.03069
- [15] Randell D.A., Cui Z., Cohn A.G., A spatial logic based on regions and connection, In: 3rd International Conference on Knowledge Representation and Reasoning, 1992, Morgan Kaufmann, 1992, 165–176
- [16] Segerberg K., An Essay in Classical Modal Logic, Filosofiska Studier, 13, Filosofiska Föreningen och Filosofiska Institutionen vid Uppsala Universitet, Uppsala, 1971 Zbl0311.02028
- [17] Simons P., Parts. A Study in Ontology, Clarendon Press, Oxford, 1987
- [18] Vakarelov D., Logical analysis of positive and negative similarity relations in property systems, In: 1st World Conference on the Fundamentals of Artificial Intelligence, Paris, July 1–5, 1991, 491–499
- [19] Vakarelov D., A modal logic for set relations, In: 10th International Congress of Logic, Methodology and Philosophy of Science, Florence, August 19–25, 1995, 183
- [20] Vakarelov D., A modal approach to dynamic ontology: modal mereotopology, Logic Log. Philos., 2008, 17(1–2), 163–183 Zbl1156.03028
- [21] Vakarelov D., Dynamic mereotopology: a point-free theory of changing regions. I. Stable and unstable mereotopological relations, Fund. Inform., 2010, 100, 159–180 Zbl1225.03027
- [22] Whitehead A.N., Process and Reality, Cambridge University Press, Cambridge, 1929
- [23] Wolter F., Zakharyaschev M., Spatio-temporal representation and reasoning based on RCC-8, In: 7th International Conference on Knowledge Representation and Reasoning, 2000, Morgan Kaufmann, 2000, 3–14

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.