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26
HasCASL: Towards Integrated Specification and Development of Functional Programs
, 2002
"... The development of programs in modern functional languages such as Haskell calls for a widespectrum specification formalism that supports the type system of such languages, in particular higher order types, type constructors, and parametric polymorphism, and contains a functional language as an exe ..."
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Cited by 24 (10 self)
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The development of programs in modern functional languages such as Haskell calls for a widespectrum specification formalism that supports the type system of such languages, in particular higher order types, type constructors, and parametric polymorphism, and contains a functional language as an executable subset in order to facilitate rapid prototyping. We lay out the design of HasCasl, a higher order extension of the algebraic specification language Casl that is geared towards precisely this purpose. Its semantics is tuned to allow program development by specification refinement, while at the same time staying close to the settheoretic semantics of first order Casl. The number of primitive concepts in the logic has been kept as small as possible; we demonstrate how various extensions to the logic, in particular general recursion, can be formulated within the language itself.
Algebraiccoalgebraic specification in CoCasl
 J. LOGIC ALGEBRAIC PROGRAMMING
, 2006
"... We introduce CoCasl as a simple coalgebraic extension of the algebraic specification language Casl. CoCasl allows the nested combination of algebraic datatypes and coalgebraic process types. We show that the wellknown coalgebraic modal logic can be expressed in CoCasl. We present sufficient criter ..."
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Cited by 20 (8 self)
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We introduce CoCasl as a simple coalgebraic extension of the algebraic specification language Casl. CoCasl allows the nested combination of algebraic datatypes and coalgebraic process types. We show that the wellknown coalgebraic modal logic can be expressed in CoCasl. We present sufficient criteria for the existence of cofree models, also for several variants of nested cofree and free specifications. Moreover, we describe an extension of the existing proof support for Casl (in the shape of an encoding into higherorder logic) to CoCasl.
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 14 (8 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.
An Intrinsic Homotopy Theory for Simplicial Complexes, with Applications to Image Analysis
"... . A simplicial complex is a set equipped with a downclosed family of distinguished finite subsets. This structure, usually viewed as codifying a triangulated space, is used here directly, to describe "spaces" whose geometric realisation can be misleading. An intrinsic homotopy theory, not ..."
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Cited by 14 (8 self)
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. A simplicial complex is a set equipped with a downclosed family of distinguished finite subsets. This structure, usually viewed as codifying a triangulated space, is used here directly, to describe "spaces" whose geometric realisation can be misleading. An intrinsic homotopy theory, not based on such realisation but agreeing with it, is introduced. The applications developed here are aimed at image analysis in metric spaces and have connections with digital topology and mathematical morphology. A metric space X has a structure t e X of simplicial complex at each resolution e > 0; the resulting homotopy group p e n (X) detects those singularities which can be captured by an ndimensional grid, with edges bound by e; this works equally well for continuous or discrete regions of euclidean spaces. Its computation is based on direct, intrinsic methods. Mathematics Subject Classifications (2000). 55U10, 55Q05, 55U35, 18G55, 18D05, 54E35, 68U10, 54G99, 05C38. Key words: Simplicial co...
Free adjunction of morphisms
 APPL. CAT. STRUCT
"... We develop a general setting for the treatment of extensions of categories by means of freely adjoined morphisms. To this end, we study what we call composition graphs, i.e. large graphs with a partial binary operation on which we impose only rudimentary requirements. The quasicategory thus obtaine ..."
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Cited by 8 (6 self)
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We develop a general setting for the treatment of extensions of categories by means of freely adjoined morphisms. To this end, we study what we call composition graphs, i.e. large graphs with a partial binary operation on which we impose only rudimentary requirements. The quasicategory thus obtained contains the quasicategory of all categories as a full reflective subquasicategory; we characterize composition graphs for which this reflexion is of a particularly simple nature. This leads to the concept of semicategory; we apply semicategories to solve characterization problems concerning absolutely initial sources, absolute monosources
Exponentiability in Categories of Lax Algebras
, 2003
"... For a complete cartesianclosed category V with coproducts, and for any pointed endofunctor T of the category of sets satisfying a suitable BeckChevalleytype condition, it is shown that the category of lax reflexive (T , V)algebras is a quasitopos. This result encompasses many known and new examp ..."
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Cited by 7 (3 self)
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For a complete cartesianclosed category V with coproducts, and for any pointed endofunctor T of the category of sets satisfying a suitable BeckChevalleytype condition, it is shown that the category of lax reflexive (T , V)algebras is a quasitopos. This result encompasses many known and new examples of quasitopoi. 1.
The HasCasl prologue: categorical syntax and semantics of the partial λcalculus
 COMPUT. SCI
, 2006
"... We develop the semantic foundations of the specification language HasCasl, which combines algebraic specification and functional programming on the basis of Moggi’s partial λcalculus. Generalizing Lambek’s classical equivalence between the simply typed λcalculus and cartesian closed categories, we ..."
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Cited by 6 (4 self)
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We develop the semantic foundations of the specification language HasCasl, which combines algebraic specification and functional programming on the basis of Moggi’s partial λcalculus. Generalizing Lambek’s classical equivalence between the simply typed λcalculus and cartesian closed categories, we establish an equivalence between partial cartesian closed categories (pccc’s) and partial λtheories. Building on these results, we define (settheoretic) notions of intensional Henkin model and syntactic λalgebra for Moggi’s partial λcalculus. These models are shown to be equivalent to the originally described categorical models in pccc’s via the global element construction. The semantics of HasCasl is defined in terms of syntactic λalgebras. Correlations between logics and classes of categories facilitate reasoning both on the logical and on the categorical side; as an application, we pinpoint unique choice as the distinctive feature of topos logic (in comparison to intuitionistic higherorder logic of partial functions, which by our results is the logic of pccc’s with equality). Finally, we give some applications of the modeltheoretic equivalence result to the semantics of HasCasl and its relation to firstorder Casl.
Classifying Categories for Partial Equational Logic
, 2002
"... Along the lines of classical categorical type theory for total functions, we establish equivalence results between certain classes of partial equational theories on the one hand and corresponding classes of categories on the other hand, staying close to standard categorical notions. ..."
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Cited by 6 (3 self)
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Along the lines of classical categorical type theory for total functions, we establish equivalence results between certain classes of partial equational theories on the one hand and corresponding classes of categories on the other hand, staying close to standard categorical notions.
The Logic of the Partial λCalculus With Equality
"... We investigate the logical aspects of the partial λcalculus with equality, exploiting an equivalence between partial λtheories and partial cartesian closed categories (pcccs) established here. The partial λcalculus with equality provides a fullblown intuitionistic higher order logic, which in a ..."
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Cited by 5 (1 self)
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We investigate the logical aspects of the partial λcalculus with equality, exploiting an equivalence between partial λtheories and partial cartesian closed categories (pcccs) established here. The partial λcalculus with equality provides a fullblown intuitionistic higher order logic, which in a precise sense turns out to be almost the logic of toposes, the distinctive feature of the latter being unique choice. We give a linguistic proof of the generalization of the fundamental theorem of toposes to pcccs with equality; type theoretically, one thus obtains that the partial λcalculus with equality encompasses a MartinLöfstyle dependent type theory. This work forms part of the semantical foundations for the higher order algebraic specification language HasCasl.
Combinatorial Trees in Priestley Spaces
 Comment. Math. Univ. Carolinae
, 2004
"... We show that prohibiting a combinatorial tree in the Priestley duals determines an axomatizable class of distributive lattices. ..."
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Cited by 4 (4 self)
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We show that prohibiting a combinatorial tree in the Priestley duals determines an axomatizable class of distributive lattices.