J. Meseguer. General logics. In H.-D. E. et al., editor, Logic Colloquium'87, pages 275--329. North-Holland, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....9] it will be used as the current paper addresses the related issue of composing specifications. Section 2 ends with the presentation of a new, quantifier free, sound and strongly complete logic for multialgebraic specifications. We thus obtain a general logic of multialgebras (in the sense of [11]) The central focus of the paper, however, is neither nondeterminism nor reasoning but, instead, the possibilities o#ered by to combine di#erent algebraic specification formalisms in one framework (where the reasoning system can, of course, faciliate proving consequences of the combined ....

....strategies into PA specifications. For the reason of space limitations we have to assume the reader to be familiar not only with the general background on algebraic specifications and category theory, but also with the institutions and their mappings. We use the definitions and notation from [11]. 2 The institution of multialgebras We summarize the relevant notions about multialgebras (see e.g. 21] 6] contains the proofs concerning the institution of multialgebras) The algebraic signature, # = S,# ) and terms over # with variables from a set X , T#,X , are defined in the usual ....

Jose Meseguer. General logics. In H.-D. Ebbinghaus et al., editors, Logic colloquium '87, pages 275--329. Elsevier Science Publisher B.V.(North-Holland), 1989.

....describing the functionality of the system. Assuming that this global description would have also included some axioms would not pose any significant technical problem. The module specifications are bound to the global description, on one hand, by means of institution (semi )representations [9], 19] and on the other hand, by signature morphisms. The semantics of this kind of system is defined as the class of all models of the global signature where the corresponding retracts (up to model translation) are models of the given modules. This can be seen as a generalization of the approach ....

....i.e. Ax Sen(S) and whose morphisms h : S;Ax) S ; Ax ) are signature morphisms h : S S satisfying Sen(h) Ax) Ax . A. Arrows between institutions In this subsection we recall the notions of institution representation [18] also called plain map of institutions by Meseguer [9]) and of institution semi representation defined by Tarlecki in [19] which will be needed in the paper. The notion of an institution representation represents the idea that an institution is encoded in terms of another one. In the case of semi representations this encoding is only at the level ....

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J. Meseguer. General logic. In Logic Colloq.'87, pages 279--329. H.-D. Ebbinghaus et al. eds., North Holland 1998.

....In our study of the problem of integrating heterogeneous formal notations with emphasis on the integration of the axiomatic (logical) semantics, we have blended together concepts and methods from formal logic, categorical algebra and institution theory. We use Meseguer s General Logics [33] as a unifying presentation of the logical (axiomatic) and the denotational semantics of a formal notation, related via a soundness condition. We also use from [14] the concept of a non plain mapping of Logics (which is a slight adaptation of Meseguer s (simple) map of logics in [33] as the ....

....Logics [33] as a unifying presentation of the logical (axiomatic) and the denotational semantics of a formal notation, related via a soundness condition. We also use from [14] the concept of a non plain mapping of Logics (which is a slight adaptation of Meseguer s (simple) map of logics in [33]) as the basic correctness preserving means of relating Logics. Using this framework, we have modelled the interpretation of LPF into classical (infinitary) logic introduced by Jones and Middelburg in [27] to provide an indicative example of an interesting non plain mapping of Logics in [17] ....

Jose Meseguer. General logics. In H.D. Ebbinghaus, editor, Logic Colloquium'87, pages 275--329, 1989.

....checked that diagrams and diagram morphisms form a category. Colimits in this category can be computed using left Kan extensions and colimits in SPEC. In the sequel we will generally use the term refinement to mean a diagram morphism. 2. 5 Logic Morphisms and Code Generation Inter logic morphisms [3] are used to translate specifications from the specification logic to the logic of a programming language. See [8] for more details. They are also useful for translating between the specification logic and the logic supported byvarious theorem provers and analysis tools. They are also useful for ....

Meseguer, J. General logics. In Logic Colloquium 87, H. Ebbinghaus, Ed. North Holland, Amsterdam, 1989, pp. 275--329.

....The fact that the colimit calculation constructs a re nement of a given diagram (here the sorting speci cation) with respect to an abstract re nement (here BtoS) is a key tool in our approach to mechanizing the development process. 2. 5 Logic Morphisms and Code Generation Inter logic morphisms [9] are used to translate speci cations from the speci cation logic to the logic of a programming language. See [18] for more details. They are also useful for translating between the speci cation logic and the logic supported by various theorem provers and analysis tools. They are also useful for ....

Meseguer, J. General logics. In Logic Colloquium 87, H. Ebbinghaus, Ed. North Holland, Amsterdam, 1989, pp. 275-329.

....quite obvious (note that for semi (co)morphisms, none is required) There are also local and hiding theorem links, which are omitted here for simplicity. Finally, each type of morphism also comes in a simple theoroidal variant [20] meaning that signatures may be mapped to theories. Following [26], the category Th of theories has as objects theories, i.e. signatures plus sets of axioms. A theory morphism is a signature morphism mapping axioms to logical consequences. Let Sig : Th Sign be the functor forgetting axioms, and : Sign Th denote the obvious inclusion, which is a right inverse ....

....has as objects theories, i.e. signatures plus sets of axioms. A theory morphism is a signature morphism mapping axioms to logical consequences. Let Sig : Th Sign be the functor forgetting axioms, and : Sign Th denote the obvious inclusion, which is a right inverse to Sig . Again following [26], a theoroidal comorphism = I J is said to be a subinstitution comorphism (and I is said to be a subinstitution of J) if is an embedding of categories, is a pointwise injection, and is a natural isomorphism. In the literature, a whole bunch of di erent types of translations ....

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J. Meseguer. General logics. In Logic Colloquium 87, pages 275-329. North Holland, 1989.

....semantics applies: the proof calculi for structured and architectural speci cations are independent of the framework that is used for basic speci cations. The semantics of basic speci cation de nes an institution [GB92] while the proof calculus of basic speci cations extends this to a logic [Mes89] institution plus proof theoretic entailment relation) Hence, the proof calculi for structured and architectural speci cations are parameterized over an arbitrary logic actually, the logic is required to ful l some mild technical conditions. The actual subdivision is not entirely as clean ....

....full subcategory of Mod( induced by the class of those models M satisfying . The category Pres of presentations (also called at speci cations) is just the full subcategory of theories having nite sets of axioms. Given institutions I and J , a simple theoroidal institution comorphism [GR01, Mes89, Tar96] also called simple map of institutions [Mes89] R = I J consists of a functor : Sign Pres J1 a natural transformation : Sen Sen , a natural transformation : Mod op Mod such that the following comorphism condition is satis ed for ....

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J. Meseguer. General logics. In Logic Colloquium 87, pages 275{ 329. North Holland, 1989.

....to wit in knowledge representation and formal specification. Due to its intuitive simplicity and theoretical interest, the fibring mechanism for combining logics has deserved close attention [9, 3, 20, 25] In [6] c parchments were proposed for bringing fibring to the realm of institutions [10, 15, 11, 24], as an alternative to other approaches for combining institutions [16 18] A major strength of fibring is the possibility to establish general transfer results from the logics being combined to the resulting fibred logic. Soundness and completeness preservation for propositional based logics was ....

.... putting in context the notion of fullness and the role that it plays in the completeness results, bringing us closer to the rich field of algebraic logic [4, 1] We are also interested in studying the representation of fibring in logical frameworks, by capitalizing on the theory of general logics [15]. Finally, future work must also cover transfer results for other relevant properties, like decidability, complexity or interpolation. ....

J. Meseguer. General logics. In Proceedings of the Logic Colloquium'87, pages 275--329. North-Holland, 1989.

....and veri cation in the small, while we discuss its implementation in MAYA and related work in sections 5 and 5. 2 A Formal Notion of Development Graphs In order to de ne development graphs we start with a short recapitulation of the basics of logics as they are given, for instance, in [10]. Thereby the notion of a logic is based on the notions of an institution and an entailment system. An institution I = Sign; Sen; Mod; j= consists of a category of signatures Sign, two functors Sen and Mod giving respectively the set of valid sentences Sen( and the models Mod( for some ....

J. Meseguer. General logics, In Logic Colloquium 87, pages 275-329, North Holland, 1989.

....It is easily checked that diagrams and diagram morphisms form a category. Colimits in this category can be computed using colimits in SPEC. In the sequel we will generally use the term refinement to mean a diagram morphism. 9 2.5. Logic Morphisms and Code Generation Inter logic morphisms [2] are used to translate specifications from the specification logic to the logic of a programming language. See [10] for more details. They are also useful for translating between the specification logic and the logic supported by various theorem provers and analysis tools. They are also useful for ....

....b into an element a and the remainder of the bag b ; i.e. an inverse of insert bag(a; b ) If the bags are ultimately implemented as lists, then it is natural to use first and rest to perform the decomposition. However consider two equal bags f1,1,2g= f2,1,1g. If f1,1,2g is represented by [1,1,2] then it will decompose into 1 and f1,2g. If f2,1,1g is represented by [2,1,1] then it will decompose into 2 and f1,1g. Hence the decomposition is not functional on bags, even though it is functional on lists. There are various approaches to this problem. We observe however that any extra work ....

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Meseguer, J. General logics. In Logic Colloquium 87, H. Ehrig et al., Ed. North Holland, Amsterdam, 1989, pp. 275--329.

....i.e. id,Bao(A) The satisfaction condition then implies that o,BaA. Hence, o,B Mod(T) 1 Corollary 3. 8: If the satisfaction relation has the p property, Mod= SpecO, 1 It is well known that (SpecO, is a pullback preserving functor and, hence, the p property leads to an exact institution [Meseguer 89] This is a particularly important property on the modularity allowed by the formalism because it tells us that the models of a composite specification (taken as a colimit) are all the possible compositions (taken as limits) of the models of the components. That is, in order to develop a system ....

....of models. The existence of this adjointness implies that the model functor Mod: THEOP CAT can be factorised as Beha; BEHA ) which characterises the O institution as follows: Corollary 4. 6: A O institution with the p property has a terminal semantics iff it is categorical, in the sense of [Meseguer 89] on BEHA. 1 That is, categoricity characterises programmability . As an example of a O institution which has terminal semantics we have linear temporal logic with trace based semantics as defined in [Fiadeiro and Costa 93] The construction and proof of the existence of the adjunction can also ....

J.Meseguer, "General Logics", in H.-D.Ebbinghaus et al (eds) Logic Colloquium 87, North-Holland 1989.

.... for object specifications, and show how object specifications are themselves related to process specifications by an adjunction (section 4) Finally (section 5) we bring institutions into the picture and show how the temporal logic defined in section 2 is categorical over Proc in the sense of [Meseguer 89] We should state that the results that we present in this paper were first developed in the context of topological categories [Fiadeiro and Costa 93] Although topological categories provide a more adequate framework for putting in evidence the structural properties of both domains ....

....binary relation Z in I Mod(Z) I xPROP(Z) inductively defined as in 2.4. I Notice that, as antecipated, the satisfaction condition is a consequence of property 2.8. Relationships between institutions and semantic domains, namely adjoint situations as illustrated in section 3, have been studied in [Meseguer 89] within a generalisation of the notion of categorical logic. Adapting Meseguer s definition to our context, the institution will be categorical on ProcP provided that: a)there are functors J:ThP Proc and H:Proc ThP with H left adjoint to J; b) the functor Mod:ThP Cat is naturally isomorphic to ....

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J.Meseguer, "General Logics", in H.-D.Ebbinghaus et al (eds) Logic Colloquium 87, North-Holland 1989.

....i.e. whether each InsertSort model is also a Sorting model. 3 Preliminaries When studying development graphs with hiding, we want to focus on the structuring and want to abstract from the details of the underlying logical system. Therefore, we recall the abstract notion of logic from Meseguer [Mes89]. Logics consist of model theory and proof theory. Model theory is captured by the notion of institution, providing an abstract framework for talking about signatures, models, sentences and satisfaction. Proof theory is captured by the notion of entailment system, providing an abstract framework ....

J. Meseguer. General logics, In Logic Colloquium 87, pages 275-329, North Holland, 1989.

....) op ) is complete, so INS is complete. 3. 2 Institution Representations A slightly different notion of mapping between institutions, namely institution representation, was introduced by Tarlecki in [Tar87] see also [Tar96a, Tar96b] This is a special case of Meseguer s map of institutions [Mes89]: Definition 19. Given two institutions I = Sign; Mod;Sen; j= and I 0 = Sign 0 ; Mod 0 ; Sen 0 ; j= 0 ) an institution representation from I to I 0 consists of Phi : Sign Sign 0 , a natural transformation fi : Phi; Mod 0 ) Mod, and a natural transformation ff : Sen ) Phi; ....

Jos'e Meseguer. General logics. In H.-D. Ebbinghaus et al., editors, Proceedings, Logic Colloquium

....be shared by quite disparate languages and systems. Furthermore, using foundations of this kind it becomes possible to adopt a systematically logical approach to programming languages that connects intimately logic and computation. From this perspective, emphasized in the theory of general logics [9], a declarative programming language is a logic that has good representational capabilities for the applications that we want to express in it and for which deduction can be mechanized with reasonable efficiency for programming purposes. Therefore, this formalism independent foundations for ....

....logic and Maude. 2 Reflection in General Logics and in Rewriting Logic In this section we propose metalogical foundations for the notions of reflective logic and reflective declarative language, and the related notion of internal strategy. We use concepts from the theory of general logics [9]. However, our definition of proof subcalculus [2, Definition 1.1.6] and declarative language [2, Definition 1.1.7] are new, more general versions of earlier definitions. Of course, reflective phenomena admit of degrees, in that some languages may choose to represent only certain metalevel ....

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J. Meseguer. General logics. In H.-D. Ebbinghaus et al., editor, Logic Colloquium'87, pages 275--329. North-Holland, 1989.

....Then, a way to obtain proof support is to encode the logic into another logic that has good tool support. For encoding logics, we use the notion of institution representation. De nition 5. Given institutions I and J , a simple institution representation [20] also called simple map of institutions [11]) I J consists of a functor : Sign I Pres J 2 , a natural transformation : Sen I Sen J , a natural transformation : Mod J op Mod I such that the following representation condition is satis ed for all 2 Sign I , M 0 2 Mod J ( and 2 ....

J. Meseguer. General logics. In Logic Colloquium 87, pages 275-329. North Holland, 1989.

....proof calculi. It turns out that it is possible to simultaneously simplify matters. 2 Institutions, logics, morphisms and representations We now recall formalizations of several notions mentioned in the introduction. Institutions [14] capture the model theory of a logic, entailment systems [15] capture proof theory, while logics [15] combine both. Institution morphisms [14] capture the intuition that one logic is built upon, or projected onto another one, while institution representations [23] also called maps of institutions [15] or institution comorphisms [13] capture the intuition ....

....is possible to simultaneously simplify matters. 2 Institutions, logics, morphisms and representations We now recall formalizations of several notions mentioned in the introduction. Institutions [14] capture the model theory of a logic, entailment systems [15] capture proof theory, while logics [15] combine both. Institution morphisms [14] capture the intuition that one logic is built upon, or projected onto another one, while institution representations [23] also called maps of institutions [15] or institution comorphisms [13] capture the intuition that one logic is encoded into another ....

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J. Meseguer. General logics. In Logic Colloquium 87, pages 275-329. North Holland, 1989.

....only the operational and proof theoretic aspects of provers. We need to add a semantic component to the framework and to provide a notion of model for RThs. We need this to be able to reason about how semantics compose when we connect together heterogeneous provers. The work on general logics [29] provides the starting point for this work. ....

Jose Meseguer. General logics. In H.-D. Ebbinghaus et al., editor, Logic Colloquium'87, pages 275-329. North-Holland, 1989.

....interrelations with projections, subsort memberships and overloaded operations. Then Casl # models coincide with many sorted models of the resulting theory, and Casl # sentences may be directly replaced by the corresponding # # sentences. It can be easily verified that this defines a simple map [42] between the two Casl institutions considered (with and without subsorting, respectively) see [49] Semantic Functions: In the Casl institution, applications of predicates and operations in atomic formulae and terms are fully qualified by their profiles, so there is no overloading at that level. ....

Jose Meseguer. General Logics. In H.-D. Ebbinghaus, J. Fernandez-Prida, M. Garrido, D. Lascar, and M. Rodrguez Artalejo (eds.) Logic Colloquium '87, pages 275--329. North-Holland, 1989.

....of the goal with the chosen simpli er set. After the proof of a goal is nished, it turns into a theorem. You can then use it in the proof of other theorems, or, if it has the form of a rewrite rule, add it to a simpli er set. 1 Formally, they are institution representations in the sense of [13, 24] 8 Fig. 5. The Hol Casl instantiation of the IsaWin system 4 Integrating Hol Casl and IsaWin into Maya The development graph manager Maya has a connection to the speci cation language Casl (see [1] Hence, it provides tool support for the administration of formal software developments with ....

J. Meseguer. General logics. In Logic Colloquium 87, pages 275-329. North Holland, 1989.

....is still a lot of research work to be done in order to extend the usual algebraic techniques of logic to cope with the possible absence of congruence. We are also interested in studying the representation of bring in logical frameworks. Namely, capitalizing on Meseguer s theory of general logics [18], we aim at characterizing the mechanism of bring of logics within rewriting logic [17] In particular, general representation preservation results are envisaged, that may determine the exact extent to which representations of bred logics can be obtained out of representations of the logics ....

J. Meseguer. General logics. In H.-D. Ebbinghaus et al , editor, Proceedings of the Logic Colloquium'87, pages 275-329. North-Holland, 1989.

....situation when truth values are ordered, we require a whole Tarskian closure operation as in [3] On the other hand, we shall also extend these, following the ideas in [4] to cope with possible non truth functionality of constructors. In the spirit of the theory of institutions and general logics [8, 9], we consider a logic to consist of an indexing functor to a suitable category of logic systems. In our case, the logic systems of interest are non truth functional (ntf) rooms . For simplicity, we shall only work at this level of abstraction. As shown in [3] everything can be smoothly lifted to ....

J. Meseguer. General logics. In H.-D. Ebbinghaus et al, editor, Procs. of the Logic Colloquium'87, pages 275-329. North-Holland, 1989.

....Object Medium Agent Fig. 1. Medium as Sphere for Communities of Agents 2.2 Formalization of the Media Concept In formalization, the media concept with its three main components is re ned to a media structure with ten components. The frameworks for formalization are General Logic [2,11] and Labelled Deductive Systems (LDS) 5] See App. A) General Logic is the framework from which languages, theories and models of components are being selected. With LDS, those components are combined and structured to form a medium. Let us give the media structure rst and explain it ....

J. Meseguer. General logics. In Logic Colloquium, pages 275-329. North-Holland, 1989.

....the logics as institutions in the sense of Goguen and Burstall [25] In some cases, the recognition of the underlying logic is not obvious, and the formalization as an institution is a non trivial task. Once this has been done, the institutions can be related using institution representations [38,62]. In some cases, there is an obvious subinstitution of the Casl institution that closely corresponds to the institution underlying the speci cation language in question. We therefore single out a number of subinstitutions of Casl and develop a uniform naming scheme for them. In other cases, the ....

.... : 0 2 D 2 and any (D 1 ; D 2 ) structured 0 speci cation SP 0 . 13 2.4 Institution Representations In order to relate sublanguages of Casl, we relate their underlying institutions. We therefore use the notion of institution representation (also called simple map of institution) [38,62]. The idea behind an institution representation is to encode an institution I within an institution J . A simple institution representation from an institution I to an institution J consists of the following components: a translation of I signatures to J presentations. Given an I signature , ....

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J. Meseguer. General logics. In Logic Colloquium 87, pages 275-329. North Holland, 1989.

.... language L 0 and all n ary (atomic) formulae n in L 0 : Analogously 9 n is read as for some linguistic expansion L 0 of the specificationAE language L and some n ary (atomic) formulae n i in L 0 : Although, the abstract framework of General Logics [27] has been employed for this presentation, as in [12] emphasis is given to the particular instance for (finitary) propositional and first order languages. The choice of (finitary) propositional and first order languages has been made on the basis of familiarity; as is explained in [12] similar ....

....instance for (finitary) propositional and first order languages. The choice of (finitary) propositional and first order languages has been made on the basis of familiarity; as is explained in [12] similar methods are applicable for a larger class of logics. 2 Preliminaries A General Logic [27] L = hSign; gram; L , sem; j= L i provides a abstract presentation of a formal logic, consisting of (1) a category Sign of signatures; 2) a functor gram:Sign Set, that assigns to each signature Sigma the set of sentences wellformed formulae built over Sigma ; 3) a Sign indexed family of ....

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Jose Meseguer. General logics. In H.D. Ebbinghaus, editor, Logic Colloquium'87, pages 275--329, 1989.

....context of this paper the focus is on extending (finitary) propositional and first order languages. The choice of these languages has been made on the basis of familiarity; as is explained in [13] similar methods are applicable for larger classes of logics. 2 Preliminaries An Entailment System [30] E = hSign; gram; E i (also called a Institution in the chronologically earlier paper [18] and in the more recent revision of [20] provides an abstract presentation of logical consequence, consisting of (1) a category Sign of signatures; 2) a functor gram:Sign Set, that assigns to each ....

....under translation 1 ; The tuple G[L] hSign; grami is the Grammar G[E ] which presents the (category of) languages in E . In addition, E is called compact iff whenever Gamma E Sigma , there exists a finite A Gamma such that A E Sigma . As it is explained in [13] also noted in [30]) this presentation of logical consequence is independent of the means by which it has been defined (e.g. proofcalculus, satisfaction system, forcing, etc. The usefulness of the Entailment Systems framework stems from its power of abstraction. By analysing properties and describing development ....

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Jose Meseguer. General logics. In H.D. Ebbinghaus, editor, Logic Colloquium'87, pages 275--329, 1989.

....some fundamental meta logical statements (such as abstraction and generalisation of expressions in an object language) that associate derivability with linguistic transformations. The paper is structured as follows: Uniform interpolants, uniform schemata and the employed General Logics framework [25] are reviewed in section 2. For more on the significance of uniform interpolants and uniform schemata for manipulating specification modules see [11, 10] and [13, 10] respectively. The skeleton of a generic (i.e. notation semantics independent) 1 method to extend conservatively a specification ....

....is presented in section 3. For a detailed analysis of this extension with emphasis on the salient characteristics of the resulting entailment, proofs and indicative examples see Chapter 6 of the first author s Ph.D. thesis [10] 2 Preliminaries Definition 2. 1 [General Logic] A General Logic [25] L = hSign; gram; E , sem; j= I i provides an abstract presentation of a formal logic, consisting of 1. a category Sign of signatures; 2. a functor gram:Sign Set, that assigns to each signature Sigma the set of sentences wellformed formulae built over Sigma ; 3. a Sign indexed family of ....

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Jose Meseguer. General logics. In H.D. Ebbinghaus, editor, Logic Colloquium'87, pages 275--329, 1989.

....systems. One fundamental characteristic of the proposed method is that the integration takes place only at the level of the entailment: the means of proof and the (denotational) semantics are left local to each component. The paper is structured as follows. We review the concepts of a Logic [23] and a non plain mapping of Logics in Section 2. A Logic is used as a unifying presentation of the axiomatic and denotational semantics of a formal notation, related via a soundness condition. A nonplain mapping of Logics is used as the basic correctness preserving means of relating Logics. LPF ....

....every theorem that is proved using the denotational semantics is also entailed by the corresponding axiomatic semantics. In this case, the interrelation of formal notations is reduced to the interrelation of the logics describing their semantics. In this paper, we will use the concept of a Logic [23] as a generic presentation of a (sound) logic. A Logic consists of (i) an Institution [17] which encodes models and satisfaction, and (ii) an Entailment System [23] also called Institution earlier in [12] and later in [14] which encodes the (logical) consequence. A soundness condition is used ....

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Jose Meseguer. General logics. In H.D. Ebbinghaus, editor, Logic Colloquium'87, pages 275--329, 1989.

....from (x; x) by replacing some, but not necessarily all, free occurrences of x by x 0 , with the proviso that x 0 is free for x in (x; x) Thus, x; x 0 ) may or may not contain free occurrences of x. 8 An institution with semicomposable signatures is called an exact institution in [Mes89] and semiexact in [DGS93] This prerequisite is equivalent to the amalgamation lemma, see Chapter 4. Michel Bidoit, Mar a Victoria Cengarle, and Rolf Hennicker 0 1 1 2 2 po 1 ( 1 ; 2 ) po 2 ( 1 ; 2 ) PO( 1 ; 2 ) Fig. 11.1. Pushout diagram 0 1 2 inl inr 1 0 2 Fig. ....

Jose Meseguer. General Logics. In H.-D. Ebbinghaus, J. FernandezPrida, M. Garrido, D. Lascar, and M. Rodrguez Artalejo, editors, Logic Colloquium '87, pages 275-329. North-Holland, 1989.

.... information hiding operation, modularisation and (uniform) interpolation, blended together in a Development Workspace (DW) in order to generate E dev proof obligations that assist in the design, synthesis and management of E spec specification modules.A DW [9, 10] is a Subentailment System [25] hE spec ; J : E spec E dev i that presents a conservative extension J of an Entailment System E spec , called the specification formalism , to an Entailment System E dev , called the development formalism . A DW possesses the following characteristics (among others) 1) E dev is ....

...., and (2) to provide insights and inspiration for extending concrete calculi and satisfaction systems. Some of the potential applications of this framework in information technology and directions for further research are highlighted in Section 5. 2 Preliminaries Definition 1. A General Logic [25] L = hSign; gram; E , sem; j= I i provides an abstract presentation of a formal logic, consisting of 1. a category Sign of signatures; 2. a functor gram:Sign Set, that assigns to each signature Sigma the set of sentences well formed formulae built over Sigma ; 3. a Sign indexed family of ....

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Jose Meseguer. General logics. In H.D. Ebbinghaus, editor, Logic Colloquium'87, pages 275--329, 1989.

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J. Meseguer. General logics. In H.-D. E. et al., editor, Logic Colloquium'87, pages 275--329. North-Holland, 1989.

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Jose Meseguer. General logics. In H.-D. Ebbinghaus et al., editors, Proceedings, Logic Colloquium 1987.

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Jos'e Meseguer. General logics. In H.-D. Ebbinghaus et al., editor, Logic Colloquium'87, pages 275--329. North-Holland, 1989.

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Jos'e Meseguer. General logics. In H.-D. Ebbinghaus et al., editor, Logic Colloquium'87, pages 275--329. North-Holland, 1989.

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J. Meseguer, General logics, in: H.-D. Ebbinghaus et al., eds., Logic Colloquium'87 (North-Holland, 1989) 275--329.

....notion of logic . The possibilities for confusion and obscurantism are indeed endless; but such verbal games are for the most part a waste of time and will not occupy us in these lectures. For a general mathematical notion of logic and general axiomatic requirements for declarative languages see [32]. For program reasoning and veri cation purposes, declarative programs have the important advantage of being already a piece of mathematics. Speci cally: a declarative program P in a language based on a given logic is typically a logical theory in that logic; the properties of P that we ....

J. Meseguer. General logics. In H.-D. E. et al., editor, Logic Colloquium'87, pages 275-329. NorthHolland, 1989.

.... and object oriented programming from reflective equational logic as in FOOPS [30] and all three paradigms together from a reflective Horn clause logic with equality as in FOOPlog [30] Logical programming can be given a precise grounding using the notions of institution [22] and logical system [54], and this is in part responsible for the cleanliness and simplicity of the various languages that we have designed. A logical programming language wears its semantics on its sleeve and does not need the complex machinery of Scott Strachey style denotational semantics [70, 74] or of ....

.... [43] In fact, we would claim that a language that can only be given a semantics in one of these styles, and 2 The basic intuitions for this view were expressed in [26] and formalized using institutions in [20] The definition below is an informal exposition of the more recent formalization in [54]. 3 In particular, there should be reasonably simple notions of sentence, deduction, model and satisfaction, preferably with a completeness theorem, saying that the notion of deduction is fully adequate for the notion of model, in the sense that given any set P of sentences, another sentence s ....

Jos'e Meseguer. General logics. In H.-D. Ebbinghaus et al., editors, Proceedings, Logic Colloquium,

.... logic L with a finitary syntax and inference system within rewriting logic by means of a representation map Phi : L Gamma RWLogic: The map Phi should preserve and reflect theoremhood, that is, it should be a conservative map of entailment systems in the sense of the theory of general logics [111]. The reason why rewriting logic is a good framework that is, why it is easy to define maps Phi of this kind for many logics is that the formulas of a logic L can typically be equationally axiomatized by an equational theory, and the rules of inference can be typically understood as rewrite ....

....to general maps Phi : L Gamma Q between two logics L and Q, that can then be reified as equationally defined functions Phi : ModuleL Gamma ModuleQ within rewriting logic. That is, rewriting logic provides an executable framework to implement key concepts in the theory of general logics [111]. There is, in addition, a very fruitful relationship between rewriting logic and the theory of reasoning theories proposed by Giunchiglia, Pecchiari and Talcott [64] Reasoning theories provide a logic independent architecture for combining and interoperating different mechanized formal systems. ....

J. Meseguer. General logics. In H.-D. E. et al., editor, Logic Colloquium'87, pages 275--329. North-Holland, 1989.

....namely, that they denote an entire class of colored nets instead of just a single one. In fact, there is a general concept of CNSs that is parameterized by an underlying logic. A logic has a deductive system and a model theoretic semantics, a concept that can be formalized by general logics [47] which contain institutions [35] as the model theoretic component. We denote by CNSL the class of colored net speci cations over the underlying logic L. Possible candidates for L include equational logics such as many sorted equational logic (MSA) order sorted equational logic (OSA) or ....

J. Meseguer. General logics. In H.-D. Ebbinghaus et al., editors, Proceedings, Logic Colloquium,

....It seems natural, then, to ask about the relationships between deduction in these logics, and to extend the question so as to encompass whether the corresponding models are also related. A suitable framework in which to carry out this study is the theory of general logics developed by Meseguer [10]. There, a logic is described in a very abstract manner and two separated components are distinguished in it: an entailment system and an institution, corresponding with the syntactic and the semantic parts of the logic, respectively. We will begin by studying derivability and, for that, we will ....

....We would like to relate the institutions ICRWL and IRL by means of a map of institutions ( Phi; ff; fi) ICRWL IRL having nice properties, in such a way that it indicated that ICRWL could be considered as a subinstitution of IRL . The formal definition of subinstitution appeared originally in [10] and has been further generalized in subsequent articles. One of those extensions was introduced by Meseguer in [13] where it is called an embedding. The only requirement imposed on a map of institutions ( Phi; ff; fi) I I 0 to be an embedding is that for each T 2 Th I , the functor fi T : ....

J. Meseguer. General logics. In H.-D. Ebbinghaus, J. Fern'andez-Prida, M. Garrido, D. Lascar, and M. Rodr'iguez-Artalejo, editors, Logic Colloquium'87, pages 275--329. North-Holland, 1989.

....But where will the metatheory supporting such meta tools come from To make such tools mathematically rigorous, the first thing obviously needed is to have a mathematical metatheory of logics and of translations between logics. We have been investigating the theory of general logics [47, 44, 52, 11, 16] for this purpose. This theory axiomatizes the proof theoretic and model theoretic facets of logics and their translations, includes the theory of institutions as its modeltheoretic component [30] and is related to other similar metatheories (see the survey [52] But meta tools need more than a ....

....say Java, in which the implementation itself is not a suitable formal axiomatization of the tool being implemented. This leads us to the need for a metatheory of logics, as a necessary foundation for the design of formal meta tools. In our work we have used the theory of general logics proposed in [47], which provides an axiomatic framework to formalize the proof theory and model theory of a logic, and which also provides adequate notions of mapping between logics, that is, of logic translations. This theory contains Goguen and Burstall s theory of institutions [30] as its model theoretic ....

[Article contains additional citation context not shown here]

J. Meseguer. General logics. In H.-D. E. et al., editor, Logic Colloquium'87, pages 275--329. North-Holland, 1989.

....With the above definition of sentences and satisfaction, membership equational logic is indeed an institution 6 [13] that we shall denote MEqtl. Furthermore, it is also quite clear that membership equational logic is a special type of Horn formalism and should therefore be viewed as a sublogic [24] of many sorted Horn logic with equality, denoted MSHorn = That is, there is a sublogic inclusion I : MEqtl MSHorn = Indeed, recall that signatures in MSHorn = are triples (L; Sigma ; Pi ) with L a set of sorts, Sigma = f Sigma w;l g (w;l)2L ThetaL a family of function symbols, ....

....not only isomorphic, but in fact identical, that is, Alg Omega = Mod I( Omega ) and similarly Alg Omega ; Gamma = Mod I( Omega ) Gamma . Conversely, we can regard many sorted Horn logic with equality as a special case of membership equational logic by defining a map of institutions 7 [24] J : MSHorn = MEqtl . The map J assigns to each signature (L; Sigma ; Pi ) the theory whose signature 1. has kind set K = L ] fp(w) j w 2 L Gamma L and Pi w 6= g; 2. for each kind k 2 K the set of sorts S k is if k 2 L then S k = Pi k , if k = p(w) then S p(w) Pi w ; 3. ....

[Article contains additional citation context not shown here]

J. Meseguer. General logics. In H.-D. E. et al., editor, Logic Colloquium'87, pages 275--329. North-Holland, 1989.

....as follows. M j= Sigma Gamma iff M j= Sigma for each 2 Gamma: Then, the relation between sets of sentences and sentences given by Gamma j= Sigma iff M j= Sigma for each M 2 jMod I ( Sigma; Gamma)j allows us to associate to an institution an entailment system in the sense of [22]. For any signature Sigma, the closure of a set Gamma of Sigma sentences is Gamma ffl = f j Gamma j= Sigma g. The Sigma theory presented by ( Sigma; Gamma) is then given by ( Sigma; Gamma ffl ) Given presentations of theories ( Sigma; Gamma) and ( Sigma 0 ; Gamma 0 ) a ....

....We show in this section that, if C is cocomplete, then Dg(C) is also cocomplete. This is probably a folklore result. Since we are not aware of a 1 Note that in the above definition the objects of ThI are presentations of theories. We follow here the terminology of Meseguer s general logics [22], instead of Goguen and Burstall s original definition [18] In what follows, when we talk about a theory ( Sigma; Gamma) we shall mean a theory presentation. 4 suitable textbook exposition to give as a reference, we include it here to make the paper self contained. Given a cocomplete category ....

[Article contains additional citation context not shown here]

J. Meseguer. General logics. In H.-D. Ebbinghaus et al., editors, Logic Colloquium'87, pages 275--329. North-Holland, 1989. 15

....But where will the meta theory supporting such meta tools come from To make such tools mathematically rigorous, the first thing obviously needed is to have a mathematical meta theory of logics and of translations between logics. We have been investigating the theory of general logics [39, 36, 44, 9, 14] for this purpose. This theory axiomatizes the proof theoretic and model theoretic facets of logics and their translations, includes the theory of institutions as its modeltheoretic component [25] and is related to other similar meta theories (see the survey [44] But meta tools need more than ....

....Java, in which the implementation itself is not a suitable formal axiomatization of the tool being implemented. This leads us to the need for a meta theory of logics, as a necessary foundation for the design of formal meta tools. In our work we have used the theory of general logics proposed in [39], which provides an axiomatic framework to formalize the proof theory and model theory of a logic, and which also provides adequate notions of mapping between logics, that is, of logic translations. This theory contains Goguen and Burstall s theory of institutions [25] as its modeltheoretic ....

J. Meseguer. General logics. In H.-D. E. et al., editor, Logic Colloquium'87, pages 275--329. North-Holland, 1989.

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J. Meseguer. General logics. In Logic Colloquium 87, pages 275-329. North Holland, 1989. 38

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J. Meseguer. General logics. In H.-D. Ebbinghaus et al., editor, Logic Colloquium'87, pages 275--329. Elsevier Science Publishers B. V. (NorthHolland) , 1989.

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J. Meseguer. General logics. In Logic Colloquium 87, pages 275-329. North Holland, 1989.

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J. Meseguer. General logics. In H.-D. Ebbinghaus et al., editor, Logic Colloquium'87, pages 275-329. North-Holland, 1989.

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J. Meseguer. General logics. In Logic Colloquium 87, p. 275-329. North Holland, 1989.

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Jos'e Meseguer. General logics. In H.-D. Ebbinghaus et al., editor, Logic Colloquium '87, pages 275--329. North-Holland,

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