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Preservation of interpolation features by fibring
 Journal of Logic and Computation
"... Fibring is a metalogical constructor that permits to combine different logics by operating on their deductive systems under certain natural restrictions, as for example that the two given logics are presented by deductive systems of the same type. Under such circumstances, fibring will produce a new ..."
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Fibring is a metalogical constructor that permits to combine different logics by operating on their deductive systems under certain natural restrictions, as for example that the two given logics are presented by deductive systems of the same type. Under such circumstances, fibring will produce a new deductive system by means of the free use of inference rules from both deductive systems, provided the rules are schematic, in the sense of using variables that are open for application to formulas with new linguistic symbols (from the point of view of each logic component). Fibring is a generalization of fusion, a less general but wider developed mechanism which permits results of the following kind: if each logic component is decidable (or sound, or complete with respect to a certain semantics) then the resulting logic heirs such a property. The interest for such preservation results for combining logics is evident, and they have been achieved in the more general setting of fibring in several cases. The Craig interpolation property and the Maehara interpolation have a special significance when combining logics, being related to certain problems of complexity theory, some properties of model theory and to the usual (global) metatheorem of deduction. When the peculiarities of the distinction between local and global deduction interfere, justifying what we call careful reasoning, the question of preservation of interpolation becomes more subtle and other forms of interpolation can be distinguished. These questions are investigated and several (global and local) preservation results for interpolation are obtained for fibring logics that fulfill mild requirements. AMS Classification: 03C40, 03B22, 03B45 1
NEIGHBOURHOOD STRUCTURES: BISIMILARITY AND BASIC MODEL THEORY
, 2009
"... Neighbourhood structures are the standard semantic tool used to reason about nonnormal modal logics. The logic of all neighbourhood models is called classical modal logic. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denote ..."
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Neighbourhood structures are the standard semantic tool used to reason about nonnormal modal logics. The logic of all neighbourhood models is called classical modal logic. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denoted by 2 2. We use this coalgebraic modelling to derive notions of equivalence between neighbourhood structures. 2 2bisimilarity and behavioural equivalence are well known coalgebraic concepts, and they are distinct, since 2 2 does not preserve weak pullbacks. We introduce a third, intermediate notion whose witnessing relations we call precocongruences (based on pushouts). We give backandforth style characterisations for 2 2bisimulations and precocongruences, we show that on a single coalgebra, precocongruences capture behavioural equivalence, and that between neighbourhood structures, precocongruences are a better approximation of behavioural equivalence than 2 2bisimulations. We also introduce a notion of modal saturation for neighbourhood models, and investigate its relationship with definability and imagefiniteness. We prove a HennessyMilner theorem for modally saturated and for imagefinite neighbourhood models. Our main results are an analogue of Van Benthem’s characterisation theorem and a modeltheoretic proof of Craig interpolation for classical modal logic.
Modules in Transition Conservativity, Composition, and Colimits
"... Abstract. Several modularity concepts for ontologies have been studied in the literature. Can they be brought to a common basis? We propose to use the language of category theory, in particular diagrams and their colimits, for answering this question. We outline a general approach for representing c ..."
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Abstract. Several modularity concepts for ontologies have been studied in the literature. Can they be brought to a common basis? We propose to use the language of category theory, in particular diagrams and their colimits, for answering this question. We outline a general approach for representing combinations of logical theories, or ontologies, through interfaces of various kinds, based on diagrams and the theory of institutions. In particular, we consider theory interpretations, language extensions, symbol identification, and conservative extensions. We study the problem of inheriting conservativity between subtheories in a diagram to its colimit ontology. Finally, we apply this to the problem of conservativity when composing DDLs or Econnections. 1
Interpolation via translations
"... A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties betwe ..."
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A new technique is presented for proving that a consequence system enjoys Craig interpolation or Maehara interpolation based on the fact that these properties hold in another consequence system. This technique is based on the existence of a back and forth translation satisfying some properties between the consequence systems. Some examples of translations satisfying those properties are described. Namely a translation between the global/local consequence systems induced by fragments of linear logic, a KolmogorovGentzenGödel style translation, and a new translation between the global consequence systems induced by full Lambek calculus and linear logic, mixing features of a KiriyamaOno style translation with features of a KolmogorovGentzenGödel style translation. These translations establish a strong relationship between the logics involved and are used to obtain new results about whether Craig interpolation and Maehara interpolation hold in that logics. AMS Classification: 03C40, 03F03, 03B22
Some Model Theoretical Results over Horn Formula
"... Abstract. Horn clause plays a basic role in logic programming and are important for many constructive logics. Interpolation theorem is also an important topic for many monotonic or nonmonotonic logic systems. The interpolation theorem for Horn clauses under the derivability from their proof theory ..."
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Abstract. Horn clause plays a basic role in logic programming and are important for many constructive logics. Interpolation theorem is also an important topic for many monotonic or nonmonotonic logic systems. The interpolation theorem for Horn clauses under the derivability from their proof theory was formulated by Gabby and Maksimova. Using model theoretical methods, this paper demonstrates the following results for Horn clauses under the classical propositional derivability. First we formulate that the restrict over a sublanguage of a close set of Horn formulas is logically equivalent to a set of Horn formulas. Next the interpolation theorem and the parallel interpolation theorem over Horn formulas under classical propositional derivability was formulated. The paper answers partially the open problem proposed by Kourousias and Makinson in 2007. Key words: Horn Formula, Interpolation Theorem, Model Theory. 1
Borrowing Interpolation
"... We present a generic method for establishing interpolation properties by ‘borrowing ’ across logical systems. The framework used is that of the socaled ‘institution theory ’ which is a categorical abstract model theory providing a formal definition for the informal concept of ‘logical system ’ and ..."
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We present a generic method for establishing interpolation properties by ‘borrowing ’ across logical systems. The framework used is that of the socaled ‘institution theory ’ which is a categorical abstract model theory providing a formal definition for the informal concept of ‘logical system ’ and a mathematical concept of ‘homomorphism ’ between logical systems. We develop three different styles or patterns to apply the proposed borrowing interpolation method. These three ways are illustrated by the development of a series of concrete interpolation results for logical systems that are used in mathematical logic or in computing science, most of these interpolation properties apparently being new results. These logical systems include fragments of (classical many sorted) first order logic with equality, preordered algebra and its Horn fragment, partial algebra, higher order logic. Applications are also expected for many other logical systems, including membership algebra, various types of order sorted algebra, the logic of predefined types, etc., and various combinations of the logical systems discussed here. 1.