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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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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.
More on Orthogonality in Locally Presentable Categories
"... Introduction Many "everyday" categories have the following type of presentation: a general locally finitely presentable (LFP) category L, representing the signature in some sense, is given, together with a set \Sigma of morphisms having finitely presentable domains and codomains. And our category K ..."
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Introduction Many "everyday" categories have the following type of presentation: a general locally finitely presentable (LFP) category L, representing the signature in some sense, is given, together with a set \Sigma of morphisms having finitely presentable domains and codomains. And our category K is the full subcategory of L on all objects K orthogonal to ) Supported by the Grant Agency of the Czech Republic under the grant No. 201/99/0310. 1 each s : X ! X 0 in \Sigma (notation: K ? s), which means that every morphism f : X ! K uniquely factors through s; notation: K = \Sigma ? . Such subcategories K of L are called in [AR] the !orthogonality c
Amalgamation in the Semantics of CASL
"... We present a semantics for architectural specifications in Casl, including an extended static analysis compatible with modeltheoretic requirements. The main obstacle here is the lack of amalgamation for Casl models. To circumvent this problem, we extend the Casl logic by introducing enriched signat ..."
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We present a semantics for architectural specifications in Casl, including an extended static analysis compatible with modeltheoretic requirements. The main obstacle here is the lack of amalgamation for Casl models. To circumvent this problem, we extend the Casl logic by introducing enriched signatures, where subsort embeddings form a category rather than just a preorder. The extended model functor satisfies the amalgamation property as well as its converse, which makes it possible to express the amalgamability conditions in the semantic rules in static terms. Using these concepts, we develop the semantics at various levels in an institutionindependent fashion. Moreover, amalgamation for enriched Casl means that a variety of results for institutions with amalgamation, such as computation of normal forms and theorem proving for structured specifications, can now be used for Casl.