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Hybridization of Institutions
"... Abstract. Modal logics are successfully used as specification logics for reactive systems. However, they are not expressive enough to refer to individual states and reason about the local behaviour of such systems. This limitation is overcome in hybrid logics which introduce special symbols for nami ..."
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Abstract. Modal logics are successfully used as specification logics for reactive systems. However, they are not expressive enough to refer to individual states and reason about the local behaviour of such systems. This limitation is overcome in hybrid logics which introduce special symbols for naming states in models. Actually, hybrid logics have recently regained interest, resulting in a number of new results and techniques as well as applications to software specification. In this context, the first contribution of this paper is an attempt to ‘universalize ’ the hybridization idea. Following the lines of [16], where a method to modalize arbitrary institutions is presented, the paper introduces a method to hybridize logics at the same institutionindependent level. The method extends arbitrary institutions with Kripke semantics (for multimodalities with arbitrary arities) and hybrid features. This paves the ground for a general result: any encoding (expressed as comorphism) from an arbitrary institution to first order logic (FOL) determines a comorphism from its hybridization to FOL. This second contribution opens the possibility of effective tool support to specification languages based upon logics with hybrid features. 1
Quasivarieties and Initial Semantics for Hybridized Institutions
"... We define and develop the concept of quasivariety for models of hybrid logics and we apply this for determining initial semantics for classes of hybrid logics theories. The hybrid logic is considered here in a very general sense, internal to abstract institutions (in the sense of the socalled inst ..."
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We define and develop the concept of quasivariety for models of hybrid logics and we apply this for determining initial semantics for classes of hybrid logics theories. The hybrid logic is considered here in a very general sense, internal to abstract institutions (in the sense of the socalled institution theory of Goguen and Burstall). This means our result is applicable to a wide variety of hybrid logics including for example those resulting from the various kinds of combinations between conventional hybrid logics and various other logical systems. 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.
Under consideration for publication in Math. Struct. in Comp. Science Encoding Hybridised Institutions into First Order Logic
, 2013
"... A ‘hybridisation ’ of a logic, referred to as the base logic, consists of developing the characteristic features of hybrid logic on top of the respective base logic, both at the level of syntax (i.e. modalities, nominals, etc.) and of the semantics (i.e. possible worlds). By ‘hybridised institutions ..."
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A ‘hybridisation ’ of a logic, referred to as the base logic, consists of developing the characteristic features of hybrid logic on top of the respective base logic, both at the level of syntax (i.e. modalities, nominals, etc.) and of the semantics (i.e. possible worlds). By ‘hybridised institutions’ we mean the result of this process when logics are treated abstractly as institutions (in the sense of the institution theory of Goguen and Burstall). This work develops encodings of hybridised institutions into (manysorted) first order logic (abbreviated FOL) as a ‘hybridisation ’ process of abstract encodings of institutions into FOL, which may be seen as an abstraction of the well known standard translation of modal logic into first order logic. The concept of encoding employed by our work is that of comorphism from institution theory, which is a rather comprehensive concept of encoding as it features encodings both of the syntax and of the semantics of logics/institutions. Moreover we consider the socalled theoroidal version of comorphisms that encode signatures to theories, a feature that accommodates a wide range of concrete applications. Our theory is also general enough to accommodate various constraints on the possible worlds semantics as well a wide variety of quantifications. We also provide pragmatic sufficient conditions for the conservativity of the encodings to be preserved through the hybridisation process, which provides the possibility to shift a formal verification process from the hybridised institution to FOL. 1.
From Universal Logic to Computer Science, and back
"... Abstract. Computer Science has been long viewed as a consumer of mathematics in general, and of logic in particular, with few and minor contributions back. In this article we are challenging this view with the case of the relationship between specification theory and the universal trend in logic. 1 ..."
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Abstract. Computer Science has been long viewed as a consumer of mathematics in general, and of logic in particular, with few and minor contributions back. In this article we are challenging this view with the case of the relationship between specification theory and the universal trend in logic. 1 From Universal Logic... Although universal logic has been clearly recognised as a trend in mathematical logic since about one decade only, mainly due to the efforts of JeanYves Béziau and his colleagues, it had a presence here and there since much longer. For example the anthology [9] traces universal logic ideas back to the work of Paul Herz in 1922. In fact there is a whole string of famous names in logic that have been involved with universal logic in the last century, including Paul Bernays,