Non-Strict Don't Care Algebras and Specifications

Non-strict don't care functions, whose foremost representative is the ubiquitous if then else, play an essential role in computer science. As far as the semantics is concerned, they can be modelled by their totalizations with the appropriate use of elements representing undeenedness, as D. Scott has shown in his denotational approach. The situation is not so straightforward when we consider non-strict functions in the context of an algebraic framework; this point is discussed in the last section, where we explore the relationship between non-strict don't care and total algebras. The central part of this paper, after presenting the basic properties of the category of non-strict algebras, is an investigation of conditional algebraic speciications. It is shown that non-strict conditional speciications are equivalent to disjunctive speciications, and necessary and suucient conditions for the existence of initial models are given. Since non-strict don't care speciications generalize both the total and the partial case, it is shown how the results for initiality can be obtained as specializations.