### Dynamic relational mereotopology: Logics for stable and unstable relations

Vladislav Nenchev

DOI: http://dx.doi.org/10.12775/LLP.2013.014

#### Abstract

In this paper we present stable and unstable versions of several well-known relations from mereotopology: part-of, overlap, underlap and contact. An intuitive semantics is given for the stable and unstable relations, describing them as dynamic counterparts of the base mereotopo-logical 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 Boolean algebras and distributive lattices. Then we present some results about the first-order predicate logic of these relations and about its quantifier-free fragment. Completeness theorems for these logics are proved, the full first-order theory is proved to be hereditary undecidable and the satisfiability problem of the quantifier-free fragment is proved to be NP-complete.

#### Keywords

stable and unstable relations; mereology; mereotopology; representation theory; hereditary undecidability; quantifier-free fragment

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ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)