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A modal logic framework for reasoning about comparative distances and topology
, 2009
"... We propose and investigate a uniform modal logic framework for reasoning about topology and relative distance in metric and more general distance spaces, thus enabling the comparison and combination of logics from distinct research traditions such as Tarski’s S4 for topological closure and interior, ..."
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We propose and investigate a uniform modal logic framework for reasoning about topology and relative distance in metric and more general distance spaces, thus enabling the comparison and combination of logics from distinct research traditions such as Tarski’s S4 for topological closure and interior, conditional logics, and logics of comparative similarity. This framework is obtained by decomposing the underlying modallike operators into firstorder quantifier patterns. We then show that quite a powerful and natural fragment of the resulting firstorder logic can be captured by one binary operator comparing distances between sets and one unary operator distinguishing between realised and limit distances (i.e., between minimum and infimum). Due to its greater expressive power, this logic turns out to behave quite differently from both S4 and conditional logics. We provide finite (Hilbertstyle) axiomatisations and ExpTimecompleteness proofs for the logics of various classes of distance spaces, in particular metric spaces. But we also show that the logic of the real line (and various other important metric spaces) is not recursively enumerable. This result is proved by an encoding of Diophantine equations.
Graphtheoretic fibring of logics
 Part II  Completeness preservation. Preprint, SQIG  IT and IST  TU Lisbon
, 2008
"... A graphtheoretic account of fibring of logics is developed, capitalizing on the interleaving characteristics of fibring at the linguistic, semantic and proof levels. Fibring of two signatures is seen as an mgraph where the nodes and the medges include the sorts and the constructors of the signatu ..."
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A graphtheoretic account of fibring of logics is developed, capitalizing on the interleaving characteristics of fibring at the linguistic, semantic and proof levels. Fibring of two signatures is seen as an mgraph where the nodes and the medges include the sorts and the constructors of the signatures at hand. Fibring of two models is an mgraph where the nodes and the medges are the values and the operations in the models, respectively. Fibring of two deductive systems is an mgraph whose nodes are language expressions and the medges represent the inference rules of the two original systems. The sobriety of the approach is confirmed by proving that all the fibring notions are universal constructions. This graphtheoretic view is general enough to accommodate very different fibrings of propositional based logics encompassing logics with nondeterministic semantics, logics with an algebraic semantics, logics with partial semantics, and substructural logics, among others. Soundness and weak completeness are proved to be preserved under very general conditions. Strong completeness is also shown to be preserved under tighter conditions. In this setting, the collapsing problem appearing in several combinations of logic systems can be avoided. 1
Nonfinitely axiomatisable twodimensional modal logics
, 2011
"... We show the first examples of recursively enumerable (even decidable) twodimensional products of finitely axiomatisable modal logics that are not finitely axiomatisable. In particular, we show that any axiomatisation of some bimodal logics that are determined by classes of product frames with linea ..."
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We show the first examples of recursively enumerable (even decidable) twodimensional products of finitely axiomatisable modal logics that are not finitely axiomatisable. In particular, we show that any axiomatisation of some bimodal logics that are determined by classes of product frames with linearly ordered first components must be infinite in two senses: It should contain infinitely many propositional variables, and formulas of arbitrarily large modal nestingdepth. 1
Dynamic Mereotopology II: Axiomatizing some Whiteheadean Type Spacetime Logics
"... In this paper we present an Whiteheadean style pointfree theory of space and time. Here ”pointfree ” means that neither space points, nor time moments are assumed as primitives. The algebraic formulation of the theory, called dynamic contact algebra (DCA), is a Boolean algebra whose elements symbo ..."
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In this paper we present an Whiteheadean style pointfree theory of space and time. Here ”pointfree ” means that neither space points, nor time moments are assumed as primitives. The algebraic formulation of the theory, called dynamic contact algebra (DCA), is a Boolean algebra whose elements symbolize dynamic regions changing in time. It has three spatiotemporal relations between dynamic regions: space contact, time contact and preceding. We prove a representation theorem for DCAs of topological type, reflecting the dynamic nature of regions, which is a reason to call DCAs dynamic mereotopoly. We also present several complete quantifierfree logics based on the language of DCAs.
DOI: 10.12775/LLP.2013.014
"... Logics for stable and unstable relations Abstract. In this paper we present stable and unstable versions of several wellknown relations from mereotopology: partof, overlap, underlap and contact. An intuitive semantics is given for the stable and unstable relations, describing them as dynamic coun ..."
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Logics for stable and unstable relations Abstract. In this paper we present stable and unstable versions of several wellknown relations from mereotopology: partof, overlap, underlap and contact. An intuitive semantics is given for the stable and unstable relations, describing them as dynamic counterparts of the base mereotopological relations. Stable relations are described as ones that always hold, while unstable relations hold sometimes. A set of firstorder 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 firstorder predicate logic of these relations and about its quantifierfree fragment. Completeness theorems for these logics are proved, the full firstorder theory is proved to be hereditary undecidable and the satisfiability problem of the quantifierfree fragment is proved to be NPcomplete.
Graphtheoretic Fibring of Logics Part II Completeness Preservation
, 2008
"... A graphtheoretic account of fibring of logics is developed, capitalizing on the interleaving characteristics of fibring. Signatures, interpretation structures and deductive systems are defined as enriched graphs. This graphtheoretic view is general enough to accommodate very different propositiona ..."
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A graphtheoretic account of fibring of logics is developed, capitalizing on the interleaving characteristics of fibring. Signatures, interpretation structures and deductive systems are defined as enriched graphs. This graphtheoretic view is general enough to accommodate very different propositional based logics encompassing logics with nondeterministic semantics, logics with an algebraic semantics, logics with partial semantics, substructural logics, among others. Fibring is seen as a universal construction in the category of logic systems. Graphtheoretic fibring allows the explicit construction of the interpretation structure resulting from the fibring of a pair of interpretation structures. Soundness and weak completeness are proved to be preserved under very general conditions. Strong completeness is also shown to be preserved under tighter conditions. In this setting, the collapsing problem appearing in several combinations of logic systems is avoided. 1