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From Total Equational to Partial First Order Logic
, 1998
"... The focus of this chapter is the incremental presentation of partial firstorder logic, seen as a powerful framework where the specification of most data types can be directly represented in the most natural way. Both model theory and logical deduction are described in full detail. Alternatives to pa ..."
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Cited by 19 (8 self)
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The focus of this chapter is the incremental presentation of partial firstorder logic, seen as a powerful framework where the specification of most data types can be directly represented in the most natural way. Both model theory and logical deduction are described in full detail. Alternatives to partiality, like (variants of) error algebras and ordersortedness are also discussed, showing their uses and limitations. Moreover, both the total and the partial (positive) conditional fragment are investigated in detail, and in particular the existence of initial (free) models for such restricted logical paradigms is proved. Some more powerful algebraic frameworks are sketched at the end. Equational specifications introduced in last chapter, are a powerful tool to represent the most common data types used in programming languages and their semantics. Indeed, Bergstra and Tucker have shown in a series of papers (see [BT87] for a complete exposition of results) that a data type is semicompu...
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
Colimits of OrderSorted Specifications
 In Recent Trends in Algebraic Development Techniques, Proc. 12th International Workshop, WADT '97
"... . We prove cocompleteness of the category of CASL signatures, of monotone signatures, of strongly regular signatures and of strongly locally filtered signatures. This shows that using these signature categories is compatible with a pushout or colimit based module system. 1 Introduction "Given ..."
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Cited by 4 (1 self)
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. We prove cocompleteness of the category of CASL signatures, of monotone signatures, of strongly regular signatures and of strongly locally filtered signatures. This shows that using these signature categories is compatible with a pushout or colimit based module system. 1 Introduction "Given a species of structure, say widgets, then the result of interconnecting a system of widgets to form a superwidget corresponds to taking the colimit of the diagram of widgets in which the morphisms show how they are interconnected." J. Goguen [8] An important application of this is the slogan "Putting theories together to make specifications" [3]. That is, specifications should be developed in a modular way, using colimits to combine different modules properly. An orthogonal question is that of the logic that is used to specify the individual modules. Ordersorted algebra is a logic that has been proposed as a means to deal with exceptions, partiality and inheritance. See, among others, Goguen an...
What is a Logic Translation?
, 2009
"... We study logic translations from an abstract perspective, without any commitment to the structure of sentences and the nature of logical entailment, which also means that we cover both prooftheoretic and modeltheoretic entailment. We show how logic translations induce notions of logical expressive ..."
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Cited by 3 (3 self)
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We study logic translations from an abstract perspective, without any commitment to the structure of sentences and the nature of logical entailment, which also means that we cover both prooftheoretic and modeltheoretic entailment. We show how logic translations induce notions of logical expressiveness, consistency strength and sublogic, leading to an explanation of paradoxes that have been described in the literature. Connectives and quantifiers, although not present in the definition of logic and logic translation, can be recovered by their abstract properties and are preserved and reflected by translations under suitable conditions.