A category theoretic formalism for abstract interpretation

We present a formal theory of abstract interpretation based on a new category theoretic formalism. This formalism allows one to derive a collecting semantics which preserves continuity of lifted functions and for which the lifting functon is itself continuous. The theory of abstract interpretation is then presented as an approximation of this collecting semantics. The use of categories rather than compete partial orders eliminates the need for introducing two distinct partial orders and for introducing any closure operation on the allowable elements, as is necessary with powerdomains. Furthermore, our construction can be applied to any situation for which the underlying domains are complete partial orders, since the domains are not further restricted in any way. This formalism can be applied to first order languages.