Non-strict don’t care algebras and specifications

Abstract

Non-strict don't care functions, whose foremost representative is the ubiquitous if_then_else, play an essential role in computer science. As for what concerns semantics, they can be modelled by their totalizations with the appropriate use of elements representing undefinedness, as D. Scott has shown in his denotational approach. The situation is not so straighforward when we consider non-strict functions in the context of an algebraic framework. In this paper, after presenting the basic properties of the category of non-strict algebras, we explore the relationship between non-strict don't care and total algebras. Then the conditional algebraic specifications are investigated; it is shown that non-strict conditional specifications are equivalent to disjunctive specifications and necessary and sufficient conditions for the existence of initial models are given. Since non-strict don't care specifications generalize both the total and the partial case, it is shown how the results about initiality can be obtained as specializations.