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245
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...
From Action Calculi to Linear Logic
, 1998
"... . Milner introduced action calculi as a framework for investigating models of interactive behaviour. We present a typetheoretic account of action calculi using the propositionsastypes paradigm; the type theory has a sound and complete interpretation in Power's categorical models. We go on to ..."
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. Milner introduced action calculi as a framework for investigating models of interactive behaviour. We present a typetheoretic account of action calculi using the propositionsastypes paradigm; the type theory has a sound and complete interpretation in Power's categorical models. We go on to give a sound translation of our type theory in the (type theory of) intuitionistic linear logic, corresponding to the relation between Benton's models of linear logic and models of action calculi. The conservativity of the syntactic translation is proved by a modelembedding construction using the Yoneda lemma. Finally, we briefly discuss how these techniques can also be used to give conservative translations between various extensions of action calculi. 1 Introduction Action calculi arose directly from the ßcalculus [MPW92]. They were introduced by Milner [Mil96], to provide a uniform notation for capturing many calculi of interaction such as the ßcalculus, the calculus, models of distribut...
Equivalences among Various Logical Frameworks of Partial Algebras
 Computer Science Logic. 9th Workshop, CSL'95. Paderborn
, 1996
"... We examine a variety of liberal logical frameworks of partial algebras. Therefore we use simple, conjunctive and weak embeddings of institutions which preserve model categories and may map sentences to sentences, finite sets of sentences, or theory extensions using unique existential quantifiers, re ..."
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Cited by 17 (7 self)
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We examine a variety of liberal logical frameworks of partial algebras. Therefore we use simple, conjunctive and weak embeddings of institutions which preserve model categories and may map sentences to sentences, finite sets of sentences, or theory extensions using unique existential quantifiers, respectively. They faithfully represent theories, model categories, theory morphisms, colimit of theories, reducts etc. Moreover, along simple and conjunctive embeddings, theorem provers can be reused in a way that soundness and completeness is preserved. Our main result states the equivalence of all the logical frameworks with respect to weak embeddability. This gives us compilers between all frameworks. Thus it is a chance to unify the different branches of specification using liberal partial logics. This is important for reaching the goal of formal interoperability of different specification languages for software development. With formal interoperability, a specification can contain part...
Complete Cuboidal Sets in Axiomatic Domain Theory (Extended Abstract)
 In Proceedings of 12th Annual Symposium on Logic in Computer Science
, 1997
"... ) Marcelo Fiore !mf@dcs.ed.ac.uk? Gordon Plotkin y !gdp@dcs.ed.ac.uk? John Power !ajp@dcs.ed.ac.uk? Department of Computer Science Laboratory for Foundations of Computer Science University of Edinburgh, The King's Buildings Edinburgh EH9 3JZ, Scotland Abstract We study the enrichme ..."
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Cited by 17 (4 self)
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) Marcelo Fiore !mf@dcs.ed.ac.uk? Gordon Plotkin y !gdp@dcs.ed.ac.uk? John Power !ajp@dcs.ed.ac.uk? Department of Computer Science Laboratory for Foundations of Computer Science University of Edinburgh, The King's Buildings Edinburgh EH9 3JZ, Scotland Abstract We study the enrichment of models of axiomatic domain theory. To this end, we introduce a new and broader notion of domain, viz. that of complete cuboidal set, that complies with the axiomatic requirements. We show that the category of complete cuboidal sets provides a general notion of enrichment for a wide class of axiomatic domaintheoretic structures. Introduction The aim of Axiomatic Domain Theory (ADT) is to provide a conceptual understanding of why domains are adequate as mathematical models of computation. (For a discussion see [12, x Axiomatic Domain Theory ].) The approach taken is to axiomatise the structure needed on a category so that its objects can be considered as domains, and its maps as continuous...
An Extension of Models of Axiomatic Domain Theory to Models of Synthetic Domain Theory
 In Proceedings of CSL 96
, 1997
"... . We relate certain models of Axiomatic Domain Theory (ADT) and Synthetic Domain Theory (SDT). On the one hand, we introduce a class of nonelementary models of SDT and show that the domains in them yield models of ADT. On the other hand, for each model of ADT in a wide class we construct a model of ..."
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. We relate certain models of Axiomatic Domain Theory (ADT) and Synthetic Domain Theory (SDT). On the one hand, we introduce a class of nonelementary models of SDT and show that the domains in them yield models of ADT. On the other hand, for each model of ADT in a wide class we construct a model of SDT such that the domains in it provide a model of ADT which conservatively extends the original model. Introduction The aim of Axiomatic Domain Theory (ADT) is to axiomatise the structure needed on a category so that its objects can be considered to be domains (see [11, x Axiomatic Domain Theory]). Models of axiomatic domain theory are given with respect to an enrichment base provided by a model of intuitionistic linear type theory [2, 3]. These enrichment structures consist of a monoidal adjunction C \Gamma! ? /\Gamma D between a cartesian closed category C and a symmetric monoidal closed category with finite products D, as well as with an !inductive fixedpoint object (Definition 1...
Countable Lawvere Theories and Computational Effects
, 2006
"... Lawvere theories have been one of the two main category theoretic formulations of universal algebra, the other being monads. Monads have appeared extensively over the past fifteen years in the theoretical computer science literature, specifically in connection with computational effects, but Lawvere ..."
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Cited by 16 (3 self)
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Lawvere theories have been one of the two main category theoretic formulations of universal algebra, the other being monads. Monads have appeared extensively over the past fifteen years in the theoretical computer science literature, specifically in connection with computational effects, but Lawvere theories have not. So we define the notion of (countable) Lawvere theory and give a precise statement of its relationship with the notion of monad on the category Set. We illustrate with examples arising from the study of computational effects, explaining how the notion of Lawvere theory keeps one closer to computational practice. We then describe constructions that one can make with Lawvere theories, notably sum, tensor, and distributive tensor, reflecting the ways in which the various computational effects are usually combined, thus giving denotational semantics for the combinations.
LeftDetermined Model Categories and Universal Homotopy Theories
"... We say that a model category is leftdetermined if the weak equivalences are generated (in a sense specified below) by the cofibrations. While the model category of simplicial sets is not leftdetermined, we show that its nonoriented variant, the category of symmetric simplicial sets (in the sense ..."
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Cited by 16 (2 self)
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We say that a model category is leftdetermined if the weak equivalences are generated (in a sense specified below) by the cofibrations. While the model category of simplicial sets is not leftdetermined, we show that its nonoriented variant, the category of symmetric simplicial sets (in the sense of Lawvere and Grandis) carries a natural leftdetermined model category structure. This is used to give another and, as we believe simpler, proof of a recent result of D. Dugger about universal homotopy theories. 1
Extending Development Graphs With Hiding
, 2001
"... Development graphs are a tool for dealing with structured specifications in a formal program development in order to ease the management of change and reusing proofs. In this work, we extend development graphs with hiding (e.g. hidden operations). Hiding is a particularly difficult to realize operat ..."
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Cited by 15 (10 self)
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Development graphs are a tool for dealing with structured specifications in a formal program development in order to ease the management of change and reusing proofs. In this work, we extend development graphs with hiding (e.g. hidden operations). Hiding is a particularly difficult to realize operation, since it does not admit such a good decomposition of the involved specifications as other structuring operations do. We develop both a semantics and proof rules for development graphs with hiding. The rules are proven to be sound, and also complete relative to an oracle for conservative extensions. We also show that an absolute complete set of rules cannot exist. The whole framework is developed in a way independent of the underlying logical system (and thus also does not prescribe the nature of the parts of a specification that may be hidden).