A category theoretic formalism for abstract interpretation

technical reportWe 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

Fair termination revisited - with delay

AbstractA proof method for establishing the fair termination and total correctness of both nondeterministic and concurrent programs is presented. The method calls for the extension of state by auxiliary delay variables which count down to the instant in which certain action will be scheduled. It then uses well-founded ranking to prove fair termination allowing nested fair selection and loops

Semantic Domains and Denotational Semantics

The theory of domains was established in order to have appropriate spaces on which to define semantic functions for the denotational approach to programming-language semantics. There were two needs: first, there had to be spaces of several different types available to mirror both the type distinctions in the languages and also to allow for different kinds of semantical constructs - especially in dealing with languages with side effects; and second, the theory had to account for computability properties of functions - if the theory was going to be realistic. The first need is complicated by the fact that types can be both compound (or made up from other types) and recursive (or self-referential), and that a high-level language of types and a suitable semantics of types is required to explain what is going on. The second need is complicated by these complications of the semantical definitions and the fact that it has to be checked that the level of abstraction reached still allows a precise definition of computability

A Cook's Tour of Countable Nondeterminism

We provide four semantics for a small programming language involving unbounded (but countable) nondeterminism. These comprise an operational one, two denotational ones based on the Egli-Milner and Smyth orders, respectively, and a weakest precondition semantics. Their equivalence is proved. We also introduce a Hoare-like proof system for total correctness and show its soundness and completeness in an appropriate sense. Admission of countable nondeterminism results in a lack of continuity of various semantic functions; moreover some of the partial orders considered are in general not cpo's and in proofs of total correctness one has to resort to the use of (countable) ordinals. Proofs will appear in the full version of the paper