Kleisli, Parikh and Peleg Compositions and Liftings for Multirelations

Multirelations provide a semantic domain for computing systems that involve two dual kinds of nondeterminism. This paper presents relational formalisations of Kleisli, Parikh and Peleg compositions and liftings of multirelations. These liftings are similar to those that arise in the Kleisli category of the powerset monad. We show that Kleisli composition of multirelations is associative, but need not have units. Parikh composition may neither be associative nor have units, but yields a category on the subclass of up-closed multirelations. Finally, Peleg composition has units, but need not be associative; a category is obtained when multirelations are union-closed.Comment: 20 page

Towards Patterns for Heaps and Imperative Lambdas

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 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