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Combining and Representing Logical Systems Using ModelTheoretic Parchments
 In Recent Trends in Algebraic Development Techniques, volume 1376 of LNCS
, 1997
"... . The paper addresses important problems of building complex logical systems and their representations in universal logics in a systematic way. We adopt the modeltheoretic view of logic as captured in the notions of institution and of parchment (an algebraic way of presenting institutions). We prop ..."
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Cited by 15 (4 self)
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. The paper addresses important problems of building complex logical systems and their representations in universal logics in a systematic way. We adopt the modeltheoretic view of logic as captured in the notions of institution and of parchment (an algebraic way of presenting institutions). We propose a new, modified notion of parchment together with parchment morphisms and representations. In contrast to the original parchment definition and our earlier work, in modeltheoretic parchments introduced here the universal semantic structure is distributed over individual signatures and models. We lift formal properties of the categories of institutions and their representations to this level: the category of modeltheoretic parchments is complete, and their representations may be put together using categorical limits as well. However, modeltheoretic parchments provide a more adequate framework for systematic combination of logical systems than institutions. We indicate how the necessar...
Completeness of CategoryBased Equational Deduction
 Mathematical Structures in Computer Science
, 1995
"... Equational deduction is generalised within a categorybased abstract model theory framework, and proved complete under a hypothesis of quantifier projectivity, using a semantic treatment that regards quantifiers as models rather than variables, and regards valuations as model morphisms rather tha ..."
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Cited by 13 (7 self)
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Equational deduction is generalised within a categorybased abstract model theory framework, and proved complete under a hypothesis of quantifier projectivity, using a semantic treatment that regards quantifiers as models rather than variables, and regards valuations as model morphisms rather than functions. Applications include many and order sorted [conditional] equational logics, Horn clause logic, equational deduction modulo a theory, constraint logics, and more, as well as any possible combination among them. In the cases of equational deduction modulo a theory and of constraint logic the completeness result is new. One important consequence is an abstract version of Herbrand's Theorem, which provides an abstract model theoretic foundations for equational and constraint logic programming. 1 Introduction A uniform treatment of the model theory of classical equational logic is now possible due to the comprehensive development of categorical universal algebra; without any c...
Combining and Representing Logical Systems
, 1997
"... The paper addresses important problems of building complex logical systems and their representations in universal logics in a systematic way. Following Goguen and Burstall, we adopt the modeltheoretic view of logic as captured in the notion of institution and of parchment (a certain algebraic ..."
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Cited by 12 (3 self)
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The paper addresses important problems of building complex logical systems and their representations in universal logics in a systematic way. Following Goguen and Burstall, we adopt the modeltheoretic view of logic as captured in the notion of institution and of parchment (a certain algebraic way of presenting institutions). We propose a modified notion of parchment together with a notion of parchment morphism and representation, respectively. We lift formal properties of the categories of institutions and their representations to this level: the category of parchments is complete, and parchment representations may be put together using categorical limits as well. However, parchments provide a more adequate framework for systematic combination of logical systems than institutions. We indicate how the necessary invention for proper combination of various logical features may be introduced either on an ad hoc basis (when putting parchments together using limits in the cat...
Type class polymorphism in an institutional framework
 IN JOSÉ FIADEIRO, EDITOR, 17TH WADT, LECTURE NOTES IN COMPUTER SCIENCE
, 2005
"... Higherorder logic with shallow type class polymorphism is widely used as a specification formalism. Its polymorphic entities (types, operators, axioms) can easily be equipped with a ‘naive ’ semantics defined in terms of collections of instances. However, this semantics has the unpleasant property ..."
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Cited by 12 (7 self)
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Higherorder logic with shallow type class polymorphism is widely used as a specification formalism. Its polymorphic entities (types, operators, axioms) can easily be equipped with a ‘naive ’ semantics defined in terms of collections of instances. However, this semantics has the unpleasant property that while model reduction preserves satisfaction of sentences, model expansion generally does not. In other words, unless further measures are taken, type class polymorphism fails to constitute a proper institution, being only a socalled rps preinstitution; this is unfortunate, as it means that one cannot use institutionindependent or heterogeneous structuring languages, proof calculi, and tools with it. Here, we suggest to remedy this problem by modifying the notion of model to include information also about its potential future extensions. Our construction works at a high level of generality in the sense that it provides, for any preinstitution, an institution in which the original preinstitution can be represented. The semantics of polymorphism used in the specification language HasCasl makes use of this result. In fact, HasCasl’s polymorphism is a special case of a general notion of polymorphism in institutions introduced here, and our construction leads to the right notion of semantic consequence when applied to this generic polymorphism. The appropriateness of the construction for other frameworks that share the same problem depends on methodological questions to be decided case by case. In particular, it turns out that our method is apparently unsuitable for observational logics, while it works well with abstract state machine formalisms such as statebased Casl.
Interpolation, Preservation, and Pebble Games
 Journal of Symbolic Logic
, 1996
"... Preservation and interpolation results are obtained for L1! and sublogics L ` L1! such that equivalence in L can be characterized by suitable backandforth conditions on sets of partial isomorphisms. 1 Introduction In the heyday of infinitary logic in the 1960's and 70's, most attention was focuse ..."
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Cited by 12 (5 self)
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Preservation and interpolation results are obtained for L1! and sublogics L ` L1! such that equivalence in L can be characterized by suitable backandforth conditions on sets of partial isomorphisms. 1 Introduction In the heyday of infinitary logic in the 1960's and 70's, most attention was focused on L!1! and its fragments (see e.g. Keisler [19]), since countable formulas seemed best behaved. The past decade has seen a renewed interest in L1! and its finite variable fragments L (k) 1! (for 2 k ! !) and the modal fragment L \Pi 1! (see e.g. Ebbinghaus and Flum [17] on the former and Barwise and Moss [9] on the latter), due to various connections with topics in computer science. These logics form a hierarchy of increasingly powerful logics L \Pi 1! ae L (2) 1! ae L (3) 1! ae : : : ae L (k) 1! ae : : : ae L1! ; with each of these inclusions being proper. Moreover, there is a useful and elegant algebraic characterization of equivalence in L in each of these logics L, from b...
Synchronization of Logics
 Studia Logica
, 1996
"... Motivated by applications in software engineering, we propose two forms of combination of logics: synchronization on formulae and synchronization on models. We start by reviewing satisfaction systems, consequence systems, onestep derivation systems and theory spaces, as well as their functorial ..."
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Cited by 12 (9 self)
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Motivated by applications in software engineering, we propose two forms of combination of logics: synchronization on formulae and synchronization on models. We start by reviewing satisfaction systems, consequence systems, onestep derivation systems and theory spaces, as well as their functorial relationships. We define the synchronization on formulae of two consequence systems and provide a categorial characterization of the construction. For illustration we consider the synchronization of linear temporal logic and equational logic. We define the synchronization on models of two satisfaction systems and provide a categorial characterization of the construction. We illustrate the technique in two cases: linear temporal logic versus equational logic; and linear temporal logic versus branching temporal logic. Finally, we lift the synchronization on formulae to the category of logics over consequence systems. Key words: combination of logics, synchronization on formulae, sync...
First Order Linear Logic in Symmetric Monoidal Closed Categories
, 1991
"... There has recently been considerable interest in the development of `logical frameworks ' which can represent many of the logics arising in computer science in a uniform way. Within the Edinburgh LF project, this concept is split into two components; the first being a general proof theoretic encodin ..."
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Cited by 11 (0 self)
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There has recently been considerable interest in the development of `logical frameworks ' which can represent many of the logics arising in computer science in a uniform way. Within the Edinburgh LF project, this concept is split into two components; the first being a general proof theoretic encoding of logics, and the second a uniform treatment of their model theory. This thesis forms a case study for the work on model theory. The models of many first and higher order logics can be represented as fibred or indexed categories with certain extra structure, and this has been suggested as a general paradigm. The aim of the thesis is to test the strength and flexibility of this paradigm by studying the specific case of Girard's linear logic. It should be noted that the exact form of this logic in the first order case is not entirely certain, and the system treated here is significantly different to that considered by Girard.
Parameterisation of Logics
 Recent trends in algebraic development techniques Selected papers
, 1999
"... . Combined logics have recently deserved much attention. In this paper we develop a detailed study of a form of combination that generalises the temporalisation construction proposed in [9]. It consists of replacing an atomic part (formal parameter) of one (parameterised) logic by another (actual pa ..."
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Cited by 10 (6 self)
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. Combined logics have recently deserved much attention. In this paper we develop a detailed study of a form of combination that generalises the temporalisation construction proposed in [9]. It consists of replacing an atomic part (formal parameter) of one (parameterised) logic by another (actual parameter) logic. We provide a categorial characterisation of parameterisation and illustrate it with an example. Under reasonable assumptions, we show that the result logic is a conservative extension of both the parameterised and parameter logics and also that soundness, completeness and decidability are transferred. 1 Introduction We need to work with evermore complex systems. The challenge is to identify abstractions that may lead to a modular and integrated management of this complexity. One such approach is the combination of logics. In practice, it is geared by the need for integrating heterogeneous platforms and tools. Theoretically, the study of logics for combined structures has bee...
General logics
 In Logic Colloquium 87
, 1989
"... theory, categorical logic. model theory that emerged in computer science studies of software specification and semantics. To handle proof theory, our institutions use an extension of traditional categorical logic with sets of sentences as objects instead of single sentences, and with morphisms repre ..."
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Cited by 9 (3 self)
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theory, categorical logic. model theory that emerged in computer science studies of software specification and semantics. To handle proof theory, our institutions use an extension of traditional categorical logic with sets of sentences as objects instead of single sentences, and with morphisms representing proofs as usual. A natural equivalence relation on institutions is defined such that its equivalence classes are logics. Several invariants are defined for this equivalence, including a Lindenbaum
A graphtheoretic account of logics
, 2009
"... A graphtheoretic account of logics is explored based on the general notion of mgraph (that is, a graph where each edge can have a finite sequence of nodes as source). Signatures, interpretation structures and deduction systems are seen as mgraphs. After defining a category freely generated by a m ..."
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Cited by 9 (9 self)
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A graphtheoretic account of logics is explored based on the general notion of mgraph (that is, a graph where each edge can have a finite sequence of nodes as source). Signatures, interpretation structures and deduction systems are seen as mgraphs. After defining a category freely generated by a mgraph, formulas and expressions in general can be seen as morphisms. Moreover, derivations involving rule instantiation are also morphisms. Soundness and completeness theorems are proved. As a consequence of the generality of the approach our results apply to very different logics encompassing, among others, substructural logics as well as logics with nondeterministic semantics, and subsume all logics endowed with an algebraic semantics. 1