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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 modal-like operators into first-order quantifier patterns. We then show that quite a powerful and natural fragment of the resulting first-order 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 (Hilbert-style) axiomatisations and ExpTime-completeness 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.
Graph-theoretic fibring of logics
- Part II - Completeness preservation. Preprint, SQIG - IT and IST - TU Lisbon
, 2008
"... A graph-theoretic 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 m-graph where the nodes and the m-edges include the sorts and the constructors of the signatu ..."
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Cited by 4 (3 self)
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A graph-theoretic 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 m-graph where the nodes and the m-edges include the sorts and the constructors of the signatures at hand. Fibring of two models is an m-graph where the nodes and the m-edges are the values and the operations in the models, respectively. Fibring of two deductive systems is an m-graph whose nodes are language expressions and the m-edges 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 graph-theoretic view is general enough to accommodate very different fibrings of propositional based logics encompassing logics with non-deterministic 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
Non-finitely axiomatisable two-dimensional modal logics
, 2011
"... We show the first examples of recursively enumerable (even decidable) two-dimensional 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) two-dimensional 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 nesting-depth. 1
Dynamic Mereotopology II: Axiomatizing some Whiteheadean Type Space-time Logics
"... In this paper we present an Whiteheadean style point-free theory of space and time. Here ”point-free ” 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 point-free theory of space and time. Here ”point-free ” 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 spatio-temporal relations between dynamic regions: space contact, time contact and preceding. We prove a representation theorem for DCA-s of topological type, reflecting the dynamic nature of regions, which is a reason to call DCA-s dynamic mereotopoly. We also present several complete quantifier-free logics based on the language of DCA-s.
DOI: 10.12775/LLP.2013.014
"... Logics for stable and unstable relations Abstract. In this paper we present stable and unstable versions of sev-eral 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 coun ..."
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Logics for stable and unstable relations Abstract. In this paper we present stable and unstable versions of sev-eral 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.
Graph-theoretic Fibring of Logics Part II- Completeness Preservation
, 2008
"... A graph-theoretic 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 graph-theoretic view is general enough to accommodate very different propositiona ..."
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A graph-theoretic 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 graph-theoretic view is general enough to accommodate very different propositional based logics encompassing logics with non-deterministic 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. Graph-theoretic 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
