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Quantifierfree logic for multialgebraic theories
, 2002
"... We develop a new logic for deriving consequences of multialgebraic theories (specifications). Multilagebras are used as models for nondeterminism in the context of algebraic specifications. They are many sorted algebras with set valued operations. Atomic formulae are set inclusion t ≺ t ′ –the inter ..."
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We develop a new logic for deriving consequences of multialgebraic theories (specifications). Multilagebras are used as models for nondeterminism in the context of algebraic specifications. They are many sorted algebras with set valued operations. Atomic formulae are set inclusion t ≺ t ′ –the interpretation of t is included in the interpretation of t ′ , and element equality t. = t ′ – t and t ′ denote the same element of the carrier. We introduce the RasiowaSikorski logic RS for proving multilagebraic tautologies and show its soundness and completeness. We then extend this system for proving consequences of specifications based on translation of theories into logical formulae. Finally, we show how such a translation may be avoided –introduction of specific cut rules leads to a sound and complete Gentzen system for proving directly consequences of specifications.
Combining specification formalisms in the 'general logic' of multialgebras
, 2003
"... We recall basic facts about the institution of multialgebras, and introduce a new, quantifierfree reasoning system for deriving consequences of multialgebraic specifications. We then show how can be used for combining specifications developed in other algebraic frameworks. We spell out the defi ..."
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We recall basic facts about the institution of multialgebras, and introduce a new, quantifierfree reasoning system for deriving consequences of multialgebraic specifications. We then show how can be used for combining specifications developed in other algebraic frameworks. We spell out the definitions of embeddings of institution of partial algebras, and membership algebras, MA.
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.
Institutional semantics for manyvalued logics
"... We develop manyvalued logic, including a generic abstract model theory, over a fully abstract syntax. We show that important manyvalued logic model theories, such as traditional firstorder manyvalued logic and fuzzy multialgebras, may be conservatively embedded into our abstract framework. Our ..."
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We develop manyvalued logic, including a generic abstract model theory, over a fully abstract syntax. We show that important manyvalued logic model theories, such as traditional firstorder manyvalued logic and fuzzy multialgebras, may be conservatively embedded into our abstract framework. Our development is technically based upon the socalled theory of institutions of Goguen and Burstall and may serve as a template for defining at hand manyvalued logic model theories over various concrete syntaxes or, from another perspective, to combine manyvalued logic with other logical systems. We also show that our generic manyvalued logic abstract model theory enjoys a couple of important institutional model theory properties that support the development of deep model theory methods. Key words: institutions, manyvalued logic 1.