Towards patterns for heaps and imperative lambdas

Abstract

In functional programming, point-free relation calculi have been fruitful for general theories of program construction, but for specific applications pointwise expressions can be more convenient and comprehensible. In imperative programming, refinement calculi have been tied to pointwise expression in terms of state variables, with the curious exception of the ubiquitous but invisible heap. To integrate pointwise with point-free, de Moor and Gibbons [dMG00] extended lambda calculus with non-injective pattern matching interpreted using relations. This article gives a semantics of that language using "ideal relations" between partial orders, and a second semantics using predicate transformers. The second semantics is motivated by its potential use with separation algebra, for pattern matching in programs acting on the heap. Laws including lax beta and eta are proved in these models and a number of open problems are posed.Dedicated to José Nuno Oliveira on the occasion of his 60th birthday.

Programming from metaphorisms

“…Can f g be generalized to some R S and still retain metaphors' ability to equate objects of incompatible orders [38]? Facts (26), (27) and (33) point towards such a generalization. This relates to another direction for possible genericity: metaphorisms as given in this paper call for a division allegory [20] such as that of binary relations.…”

“…. Note how these laws include what are normally regarded as the two key benefits of the calculus of relations: converse functions as specifications and intersection of specifications [25,26] .…”

“…where pairing is extended to arbitrary relations in the expected way: (y, z ) (R △ S ) x ⇔ (y R x ) ∧ (z S x ). Note how these laws include what are normally regarded as the two key benefits of the calculus of relations: converse functions as specifications and intersection of specifications [25,26] .…”

Whither Specifications as Programs

“…Upward closed sets of predicates play an important role in that algebra [33], which should be explored in connection with the present investigation and its potential application to higher order programs. The extension of functional programming calculus to imperative refinement is one setting in which strong laws (Cartesian closure) for a well-behaved subset are expressed as implications with inequational antecedents [32,34]. These works use backward predicate transformers in order to model general specifications and in particular the combination of angelic and demonic nondeterminacy.…”