6 research outputs found
Algebraic Theories over Nominal Sets
We investigate the foundations of a theory of algebraic data types with
variable binding inside classical universal algebra. In the first part, a
category-theoretic study of monads over the nominal sets of Gabbay and Pitts
leads us to introduce new notions of finitary based monads and uniform monads.
In a second part we spell out these notions in the language of universal
algebra, show how to recover the logics of Gabbay-Mathijssen and
Clouston-Pitts, and apply classical results from universal algebra.Comment: 16 page
On CSP and the Algebraic Theory of Effects
We consider CSP from the point of view of the algebraic theory of effects,
which classifies operations as effect constructors or effect deconstructors; it
also provides a link with functional programming, being a refinement of Moggi's
seminal monadic point of view. There is a natural algebraic theory of the
constructors whose free algebra functor is Moggi's monad; we illustrate this by
characterising free and initial algebras in terms of two versions of the stable
failures model of CSP, one more general than the other. Deconstructors are
dealt with as homomorphisms to (possibly non-free) algebras.
One can view CSP's action and choice operators as constructors and the rest,
such as concealment and concurrency, as deconstructors. Carrying this programme
out results in taking deterministic external choice as constructor rather than
general external choice. However, binary deconstructors, such as the CSP
concurrency operator, provide unresolved difficulties. We conclude by
presenting a combination of CSP with Moggi's computational {\lambda}-calculus,
in which the operators, including concurrency, are polymorphic. While the paper
mainly concerns CSP, it ought to be possible to carry over similar ideas to
other process calculi
Scoped effects as parameterized algebraic theories
Notions of computation can be modelled by monads. Algebraic effects offer a characterization of monads in terms of algebraic
operations and equational axioms, where operations are basic programming features, such as reading or updating the state, and axioms specify
observably equivalent expressions. However, many useful programming
features depend on additional mechanisms such as delimited scopes or
dynamically allocated resources. Such mechanisms can be supported via
extensions to algebraic effects including scoped effects and parameterized algebraic theories. We present a fresh perspective on scoped effects
by translation into a variation of parameterized algebraic theories. The
translation enables a new approach to equational reasoning for scoped
effects and gives rise to an alternative characterization of monads in
terms of generators and equations involving both scoped and algebraic
operations. We demonstrate the power of our fresh perspective by way of
equational characterizations of several known models of scoped effects
Notions of Lawvere theory
Categorical universal algebra can be developed either using Lawvere theories
(single-sorted finite product theories) or using monads, and the category of
Lawvere theories is equivalent to the category of finitary monads on Set. We
show how this equivalence, and the basic results of universal algebra, can be
generalized in three ways: replacing Set by another category, working in an
enriched setting, and by working with another class of limits than finite
products.
An important special case involves working with sifted-colimit-preserving
monads rather than filtered-colimit-preserving ones.Comment: 27 pages. v2 minor changes, final version, to appear in Applied
Categorical Structure