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Institution Morphisms
, 2001
"... Institutions formalize the intuitive notion of logical system, including syntax, semantics, and the relation of satisfaction between them. Our exposition emphasizes the natural way that institutions can support deduction on sentences, and inclusions of signatures, theories, etc.; it also introduces ..."
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Cited by 58 (18 self)
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Institutions formalize the intuitive notion of logical system, including syntax, semantics, and the relation of satisfaction between them. Our exposition emphasizes the natural way that institutions can support deduction on sentences, and inclusions of signatures, theories, etc.; it also introduces terminology to clearly distinguish several levels of generality of the institution concept. A surprising number of different notions of morphism have been suggested for forming categories with institutions as objects, and an amazing variety of names have been proposed for them. One goal of this paper is to suggest a terminology that is uniform and informative to replace the current chaotic nomenclature; another goal is to investigate the properties and interrelations of these notions in a systematic way. Following brief expositions of indexed categories, diagram categories, twisted relations, and Kan extensions, we demonstrate and then exploit the duality between institution morphisms in the original sense of Goguen and Burstall, and the "plain maps" of Meseguer, obtaining simple uniform proofs of completeness and cocompleteness for both resulting categories. Because of this duality, we prefer the name "comorphism" over "plain map;" moreover, we argue that morphisms are more natural than comorphisms in many cases. We also consider "theoroidal" morphisms and comorphisms, which generalize signatures to theories, based on a theoroidal institution construction, finding that the "maps" of Meseguer are theoroidal comorphisms, while theoroidal morphisms are a new concept. We introduce "forward" and "seminatural" morphisms, and develop some of their properties. Appendices discuss institutions for partial algebra, a variant of order sorted algebra, two versions of hidden algebra, and...
Static Semantic Analysis and Theorem Proving for CASL
 In F. ParisiPresicce (Ed.): Recent Trends in Algebraic Development Techniques
, 1998
"... . This paper presents a static semantic analysis for CASL, the Common Algebraic Specification Language. Abstract syntax trees are generated including subsorts and overloaded functions and predicates. The static semantic analysis, through the implementation of an overload resolution algorithm, checks ..."
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Cited by 22 (12 self)
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. This paper presents a static semantic analysis for CASL, the Common Algebraic Specification Language. Abstract syntax trees are generated including subsorts and overloaded functions and predicates. The static semantic analysis, through the implementation of an overload resolution algorithm, checks and qualifies these abstract syntax trees. The result is a fully qualified CASL abstract syntax tree where the overloading has been resolved. This abstract syntax tree corresponds to a theory in the institution underlying CASL, subsorted partial firstorder logic with sort generation constraints (SubPCFOL). Two ways of embedding SubPCFOL in higherorder logic (HOL) of the logical framework Isabelle are discussed: the first one from SubPFOL to HOL via PFOL (partial firstorder logic) first drops subsorting and then partiality, and the second one is the counterpart via SubFOL (subsorted firstorder logic). The C in SubPCFOL stands for sort generation constraints, which are translated separat...
Permissive Subsorted Partial Logic in CASL
, 1997
"... . This paper presents a permissive subsorted partial logic used in the CoFI Algebraic Specification Language. In contrast to other ordersorted logics, subsorting is not modeled by set inclusions, but by injective embeddings allowing for more general models in which subtypes can have different data t ..."
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Cited by 13 (8 self)
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. This paper presents a permissive subsorted partial logic used in the CoFI Algebraic Specification Language. In contrast to other ordersorted logics, subsorting is not modeled by set inclusions, but by injective embeddings allowing for more general models in which subtypes can have different data type representations. Furthermore, there are no restrictions like monotonicity, regularity or local filtration on signatures at all. Instead, the use of overloaded functions and predicates in formulae is required to be sufficiently disambiguated, such that all parses have the same semantics. An overload resolution algorithm is sketched. 1 Introduction During the past decades a large number of algebraic specification languages have been developed. The presence of so many similar specification languages with no common framework hinders the dissemination and application of research results in algebraic specification. In particular, it makes it difficult to produce educational material, to reus...
The Institution of Multialgebras  a general framework for algebraic software development
, 2002
"... this technicality ..."
Representations, Hierarchies, and Graphs of Institutions
, 1996
"... For the specification of abstract data types, quite a number of logical systems have been developed. In this work, we will try to give an overview over this variety. As a prerequisite, we first study notions of {\em representation} and embedding between logical systems, which are formalized as {\em ..."
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Cited by 5 (4 self)
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For the specification of abstract data types, quite a number of logical systems have been developed. In this work, we will try to give an overview over this variety. As a prerequisite, we first study notions of {\em representation} and embedding between logical systems, which are formalized as {\em institutions} here. Different kinds of representations will lead to a looser or tighter connection of the institutions, with more or less good possibilities of faithfully embedding the semantics and of reusing proof support. In the second part, we then perform a detailed ``empirical'' study of the relations among various wellknown institutions of total, ordersorted and partial algebras and firstorder structures (all with Horn style, i.e.\ universally quantified conditional, axioms). We thus obtain a {\em graph} of institutions, with different kinds of edges according to the different kinds of representations between institutions studied in the first part. We also prove some separation results, leading to a {\em hierarchy} of institutions, which in turn naturally leads to five subgraphs of the above graph of institutions. They correspond to five different levels of expressiveness in the hierarchy, which can be characterized by different kinds of conditional generation principles. We introduce a systematic notation for institutions of total, ordersorted and partial algebras and firstorder structures. The notation closely follows the combination of features that are present in the respective institution. This raises the question whether these combinations of features can be made mathematically precise in some way. In the third part, we therefore study the combination of institutions with the help of socalled parchments (which are certain algebraic presentations of institutions) and parchment morphisms. The present book is a revised version of the author's thesis, where a number of mathematical problems (pointed out by Andrzej Tarlecki) and a number of misuses of the English language (pointed out by Bernd KriegBr\"uckner) have been corrected. Also, the syntax of specifications has been adopted to that of the recently developed Common Algebraic Specification Language {\sc Casl} \cite{CASL/Summary,Mosses97TAPSOFT}.
Different Types of Arrow Between Logical Frameworks
 Proc. ICALP 96, LNCS 1099, 158169
, 1996
"... this paper we argue that these different types of arrow can be generated by one basic type of arrow and monadic constructions on categories of logical frameworks, with the effect of automatically having functors relating the new categories of logical frameworks with the old ones. The paper is organi ..."
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Cited by 5 (2 self)
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this paper we argue that these different types of arrow can be generated by one basic type of arrow and monadic constructions on categories of logical frameworks, with the effect of automatically having functors relating the new categories of logical frameworks with the old ones. The paper is organized as follows: in Sect. 2, some types of logical framework and some categorical notions are recalled. Section 3 then introduces, using monads and adjunctions, one wellknown and three new notions of maps between institutions, which vary in the strictness of keeping the signaturesentence distinction. In each case, we briefly show the application to different logical frameworks. Section 4 concludes the paper. Due to lack of space, we omit proofs, which will appear elsewhere. 2 Preliminaries
Translating OBJ3 into CASL: the Institution Level
 In Recent Trends in Algebraic Development Techniques, Proc. 13th International Workshop, WADT '98
, 1998
"... We translate OBJ3 to CASL. At the level of basic specifications, we set up several institution representations between the underlying institutions. They correspond to different methodological views of OBJ3. The translations can be the basis for automated tools translating OBJ3 to CASL. ..."
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Cited by 3 (0 self)
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We translate OBJ3 to CASL. At the level of basic specifications, we set up several institution representations between the underlying institutions. They correspond to different methodological views of OBJ3. The translations can be the basis for automated tools translating OBJ3 to CASL.
C.A.: Inversive meadows and divisive meadows
, 2009
"... Abstract. An inversive meadow is a commutative ring with identity and a total multiplicative inverse operation satisfying 0 −1 = 0. Previously, inversive meadows were shortly called meadows. In this paper, we introduce divisive meadows, which are inversive meadows with the multiplicative inverse ope ..."
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Cited by 2 (2 self)
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Abstract. An inversive meadow is a commutative ring with identity and a total multiplicative inverse operation satisfying 0 −1 = 0. Previously, inversive meadows were shortly called meadows. In this paper, we introduce divisive meadows, which are inversive meadows with the multiplicative inverse operation replaced by a division operation. We introduce a translation from the terms over the signature of divisive meadows into the terms over the signature of inversive meadows and a translation the other way round to show that it depends on the angle from which they are viewed whether inversive meadows or divisive meadows must be considered more basic. Divisive meadows are more basic if variants with a partial multiplicative inverse or division operation are considered as well. We also take a survey of firstorder logics that are appropriate to handle those partial variants of inversive and divisive meadows.
An institutional view on categorical logic and the CurryHowardTaitisomorphism
"... We introduce a generic notion of propositional categorical logic and provide a construction of an institution with proofs out of such a logic, following the CurryHowardTait paradigm. We then prove logicindependent soundness and completeness theorems. The framework is instantiated with a number ..."
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Cited by 1 (1 self)
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We introduce a generic notion of propositional categorical logic and provide a construction of an institution with proofs out of such a logic, following the CurryHowardTait paradigm. We then prove logicindependent soundness and completeness theorems. The framework is instantiated with a number of examples: classical, intuitionistic, linear and modal propositional logics. Finally, we speculate how this framework may be extended beyond the propositional case.
The Common Framework Initiative for algebraic specification and development of software: recent progress
, 2001
"... The Common Framework Initiative (CoFI)isanopeninternational collaboration which aims to provide a common framework for algebraic specification and development of software. The central element of the Common Framework is a specification language called Casl for formal specification of functional r ..."
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The Common Framework Initiative (CoFI)isanopeninternational collaboration which aims to provide a common framework for algebraic specification and development of software. The central element of the Common Framework is a specification language called Casl for formal specification of functional requirements and modular software design which subsumes many previous algebraic specification languages.