Diagrammatic logic applied to a parameterisation process

Abstract

This paper provides an abstract definition of a class of logics, called diagrammatic logics, together with a definition of morphisms and 2-morphisms between them. The definition of the 2-category of diagrammatic logics relies on category theory, mainly on adjunction, categories of fractions and limit sketches. This framework is applied to the formalisation of a parameterisation process. This process, which consists of adding a formal parameter to some operations in a given specification, is presented as a morphism of logics. Then the parameter passing process for recovering a model of the given specification from a model of the parameterised specification and an actual parameter is shown to be a 2-morphism of logics.Downloaded640 types, like the type of integers, while the second layer deals with algebraic structures, like the structure of groups, which are implemented using the abstract data types in the first layer. In addition, this analysis made it clear that in EAT, groups (for instance) are not implemented individually, but are implemented through parameterised families of groups. Lambán et al. (2003) defined an operation called the imp construction -it was given this name because of its role in the implementation process in the system EAT. In fact, the imp construction is a parameterisation process: starting from a specification Σ in which some operations are labelled as 'pure ' (Domínguez et al. 2006), the imp construction builds a new specification Σ A with a distinguished sort A added to the domain of each non-pure operation. Then the parameter passing process derives an implementation of Σ from each implementation of Σ A and each value in the interpretation of A. Moreover, the exact parameterisation property was proved in Lambán et al. (2003): the implementations of EAT algebraic structures are as general as possible in the sense that they are ingredients of terminal objects in some categories of models. The parameter set in the Kenzo and EAT systems is encoded by means of a record of Common Lisp functions, which has a field for each operation in the algebraic structure to be implemented. The pure terms correspond to functions, which can be obtained from the fixed data and do not require any explicit storage, so each particular instance of the record gives rise to an algebraic structure; more details can be found in Lambán et al. (2003). Lambán et al. (2003) then reinterpreted these results for the EAT algebraic structures in terms of object-oriented techniques, like hidden algebras (Goguen and Malcolm 2000) or coalgebras (Rutten 2000). Domínguez et al. (2007) then extended these results in the algebraic framework of institutions (Goguen and Burstall 1984): the imp construction is represented through institution encodings in the sense of Tarlecki (2000). Goguen and Roşu (2002) provided a survey of the different notions of morphisms between institutions,where institution encodings are called forward institution morphisms. In fact, institution encodings are not that common in institution theories, where ...

A Parameterization Process: From a Functorial Point of View

“…Thanks to our results, now some data structures in both systems can be interpreted in more generic (and categorical) setting. In particular, their two different layers of data structures can be understood using an additional level of abstraction which allows seeing the parameterization process as a morphism of logics and the parameter passing process as a 2-morphism of logics, in a relevant 2-category of logics [5]. This provides a better understanding of the internal organization of the Kenzo and EAT systems that should prove useful in future versions.…”

“…A first version of this approach can be found in [3], and a more abstract point of view, relying on diagrammatic logic, is presented in [5]. With respect to the previous papers like [13], we provide a new interpretation of the parameterization process and in addition an interpretation of the parameter passing process.…”

Decorated proofs for computational effects: States

“…Definition 2.5. For each propagator g (1) : X → Y and each catcher k (2) (2) . The rules in Figure 3 are the rules for the construction of ▽k and for the decorated coproduct X ∼ = X + 0.…”

“…As for other inference systems, we may define theories and specifications (or presentations of theories) with respect to L. They are called decorated specifications and decorated theories, respectively. This approach is based on the general framework for diagrammatic theories and specifications [5,1], but no knowledge of this framework is assumed in this paper. A decorated theory is made of types, expressions, equations and coproducts which satisfy the decorated rules for exceptions.…”

“…It follows that the handling of exceptions is quite difficult to formalize in this framework; several solutions are proposed in [15,10,14]. In this paper we rather use the categorical approach of diagrammatic logics, as introduced in [5] and developed in [1].…”