A semantic approach to interpolation

Popescu, Andrei; erbnu, Traian Florin; Rou, Grigore

doi:10.1016/j.tcs.2008.09.038

Cited by 5 publications

(3 citation statements)

References 31 publications

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“…According to [20,58], UNIV and HCL have Craig (Sig FOL , (iii))-interpolation where (iii) is the class of injective signature morphisms (for UNIV this result is also obtained via Corollary 5.2 below), and 2. according to [58], UNIV and HCL have Craig ((ie * ), Sig FOL )-interpolation where (ie * ) is the class of the signature morphisms ϕ which are injective on the sorts and no 'new' operation symbol, i.e. outside the image of ϕ, is allowed to have the sort in the image of ϕ (i.e.…”

Section: Definition 32 ((L R)-interpolation) For Any Classes Of Simentioning

confidence: 84%

“…Finally, perhaps the most important argument is that some institutions, not having conjunctions, lack CI in the single sentences version, but have it in the sets of sentences formulation. Typical examples for this situation are equational logic [20,[58][59][60] and Horn clause logic (HCL) [20,58]. Moreover, a more careful look into this situation would reveal that in general the applications of interpolation require the set of sentences formulation.…”

Section: Craig Interpolationmentioning

confidence: 99%

“…The paper [58] gives a big list of such interpolation properties for various axiomatizable fragments of FOL.…”

Section: Definition 32 ((L R)-interpolation) For Any Classes Of Simentioning

confidence: 99%

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Borrowing interpolation

Diaconescu¹

2011

Journal of Logic and Computation

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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 so-caled '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.

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Section: Definition 32 ((L R)-interpolation) For Any Classes Of Simentioning

confidence: 84%

Section: Craig Interpolationmentioning

confidence: 99%

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Borrowing interpolation

Diaconescu¹

2011

Journal of Logic and Computation

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Craig interpolation for networks of sentences

Keisler

2012

Annals of Pure and Applied Logic

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On the algebra of structured specifications

Diaconescu

Ţuţu²

2011

Theoretical Computer Science

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We develop module algebra for structured specifications with model oriented denotations. Our work extends the existing theory with specification building operators for non-protecting importation modes and with new algebraic rules (most notably for initial semantics) and upgrades the pushout-style semantics of parameterized modules to capture the (possible) sharing between the body of the parameterized modules and the instances of the parameters. We specify a set of sufficient abstract conditions, smoothly satisfied in the actual situations, and prove the isomorphism between the parallel and the serial instantiation of multiple parameters. Our module algebra development is done at the level of abstract institutions, which means that our results are very general and directly applicable to a wide variety of specification and programming formalisms that are rigorously based upon some logical system. The structure and contributions of this paperThere are at least three levels of giving semantics or denotations to specification modules:1. The set of the (accumulated) axioms of the specification; this underlies [3] and is the most syntactic level. The paper [14] argues about several shortcomings of this viewpoint. 2. The theory of the specification, in the sense of the closure of its set of axioms under (semantic) deduction. This point of view is taken by [14] and can be considered as a middle grained approach. 3. The class of models of the specification in the style of [6,28]. This is the most semantic approach.The specification theory literature contains many arguments in favour of each of these viewpoints on denotations for specification modules, and even many more arguments against each of them. A good way to view these different perspectives on the semantics of specification modules is that these refer to slightly different aspects of the same phenomena. However, we can now understand that since the former two approaches are property oriented, in the sense that their emphasis is on the properties satisfied by the models of the specifications, they do not consider an important point of structuring which is realized only by the latter model oriented approach. Everybody sees specification or programming in-the-large as the only realistic way to build complex specifications or programs; however, many neglect another important motivation, namely that of being able to specify classes of models much beyond the capabilities of specification in-the-small. A good example illustrating this difference in specification power is given by fields. One can actually prove that the class of fields does not admit a specification in-the-small within any form of Horn clause logic (including even the use of partial functions); however, it admits (see Example 3.1 below) a rather simple CategoriesWe assume the reader is familiar with basic notions and standard notations from category theory; e.g., see [22] for an introduction to this subject. With respect to notational conventions, |C| denotes the class of objects of a category C, C(A, B) ...

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A semantic approach to interpolation

Cited by 5 publications

References 31 publications

Borrowing interpolation

Borrowing interpolation

Craig interpolation for networks of sentences

On the algebra of structured specifications

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