Externalized and Internalized Notions of Behavioral Refinement

Abstract

Abstract.Many different behavioral refinement notions for algebraic specifications have been proposed in the literature but the relationship between the various concepts is still unclear. In this paper we provide a classification and a comparative study of behavioral refinements according to two directions, the externalized approach which uses an explicit behavioral abstraction operator that is applied to the specification to be implemented, and the internalized approach which uses a built-in behavioral semantics of specifications. We show that both concepts are equivalent under suitable conditions. The formal basis of our study is provided by the COL institution (constructor-based observational logic). Hence, as a side-effect of our study on internalized behavioral refinements, we introduce also a novel concept of behavioral refinement for COL-specifications.

Proving Behavioral Refinements of COL-specifications

“…Using the notion of a COL-implementation constructor we can generalize Definition 3 to the case where the abstract and implementing specifications have different signatures. As discussed in [7] an important question is, of course, which implementation constructors are appropriate for behavioral refinements. As a first approach one could simply consider COL-signature morphisms σ COL : Σ COL → ΣI COL .…”

“…, s n → s, and for all observable sorts s ∈ S Obs such that the new sort Cont[s→s ] exists, the equations: The main idea underlying the above definitions is that, to any Σ-algebra A (where Σ is the standard signature underlying the COL-signature Σ COL ), corresponds a unique (up to isomorphism) "lifted" AL(Σ COL )-algebra AL(A) which extends A (i.e., AL(A)| Σ = A) and satisfies the lifting axioms Ax(Σ COL ). 7 Moreover, this lifted algebra AL(A) is defined in a way which ensures a one to one correspondence between:…”

“…As discussed in [7], behavioral refinement concepts can be classified into two principal trends. The first one, pursued e.g.…”

“…This sentence is finite, since for any state sort s ∈ SState, there is only a finite number of sorts Cont[s→s ], where s ∈ S Obs is an observable sort 7. In other words, the lifting axioms Ax(ΣCOL) induce a strongly persistent free functor from Σ-algebras to AL(ΣCOL)-algebras 8.…”