A Coalgebraic Semantics for Imperative Programming Languages

In the theory of programming languages, one often takes two complementary perspectives. In operational semantics, one defines and reasons about the behaviour of programs; and in denotational semantics, one abstracts away implementation details, and reasons about programs as mathematical objects or denotations. The denotational semantics should be compositional, meaning that denotations of programs are determined by the denotations of their parts. It should also be adequate with respect to operational equivalence: programs with the same denotation should be behaviourally indistinguishable. One often has to prove adequacy and compositionality independently for different languages, and the proofs are often laborious and repetitive. These proofs were provided systematically in the context of process algebras by the mathematical operational semantics framework of Turi and Plotkin – which represented transition systems as coalgebras, and program syntax by free algebras; operational specifications were given by distributive laws of syntax over behaviour. By framing the semantics on this abstract level, one derives denotational and operational semantics which are guaranteed to be adequate and compositional for a wide variety of examples. However, despite speculation on the possibility, it is hard to apply the framework to programming languages, because one obtains undesirably fine-grained behavioural equivalences, and unconventional notions of operational semantics. Moreover, the behaviour of these languages is often formalised in a different way – such as computational effects, which may be thought of as an interface between programs and external factors such as non-determinism or a variable store; and comodels, or transition systems which implement these effects. This thesis adapts the mathematical operational semantics framework to provide semantics for various classes of programming languages. After identifying the need for such an adaptation, we show how program behaviour may be characterised by final coalgebras in suitably order- enriched Kleisli categories. We define both operational and denotational semantics, first for languages with syntactic effects, and then for languages with effects and/or comodels given by a Lawvere theory. To ensure adequacy and compositionality, we define concrete and abstract operational rule-formats for these languages, based on the idea of evaluation-in-context; we give syntactic and then categorical proofs that those properties are guaranteed by operational specifications in these rule-formats.Open Acces

Coalgebraic Aspects of Bidirectional Computation

We have previously (Bx, 2014; MPC, 2015) shown that several statebased bx formalisms can be captured using monadic functional programming, using the state monad together with possibly other monadic effects, giving rise to structures we have called monadic bx (mbx). In this paper, we develop a coalgebraic theory of state-based bx, and relate the resulting coalgebraic structures (cbx) to mbx. We show that cbx support a notion of composition coherent with, but conceptually simpler than, our previous mbx definition. Coalgebraic bisimulation yields a natural notion of behavioural equivalence on cbx, which respects composition, and essentially includes symmetric lens equivalence as a special case. Finally, we speculate on the applications of this coalgebraic perspective to other bx constructions and formalisms

Entangled State Monads

We present a monadic treatment of symmetric state-based bidirectional transformations, and show how it arises naturally from the well-known asymmetric lens-based account. We introduce two presentations of a concept we dub the \"entangled\" state monad, and prove their equivalence. As a step towards a unifying account of bidirectionality in general, we exhibit existing classes of state-based approaches from the literature as instances of our new constructions. This extended abstract reports on work in progress

Reflections on monadic lenses

Bidirectional transformations (bx) have primarily been modeled as pure functions, and do not account for the possibility of the side-effects that are available in most programming languages. Recently several formulations of bx that use monads to account for effects have been proposed, both among practitioners and in academic research. The combination of bx with effects turns out to be surprisingly subtle, leading to problems with some of these proposals and increasing the complexity of others. This paper reviews the proposals for monadic lenses to date, and offers some improved definitions, paying particular attention to the obstacles to naively adding monadic effects to existing definitions of pure bx such as lenses and symmetric lenses, and the subtleties of equivalence of symmetric bidirectional transformations in the presence of effects

Towards Effects in Mathematical Operational Semantics

In this paper, we study extensions of mathematical operational semantics with algebraic effects. Our starting point is an effect-free coalgebraic operational semantics, given by a natural transformation of syntax over behaviour. The operational semantics of the extended language arises by distributing program syntax over effects, again inducing a coalgebraic operational semantics, but this time in the Kleisli category for the monad derived from the algebraic effects. The final coalgebra in this Kleisli category then serves as the denotational model. For it to exist, we ensure that the the Kleisli category is enriched over CPOs by considering the monad of possibly infinite terms, extended with a bottom element. Unlike the effectless setting, not all operational specifications give rise to adequate and compositional semantics. We give a proof of adequacy and compositionality provided the specifications can be described by evaluation-in-context. We illustrate our techniques with a simple extension of (stateless) while programs with global store, i.e. variable lookup

Comodels and Effects in Mathematical Operational Semantics

In the mid-nineties, Turi and Plotkin gave an elegant categorical treatment of denotational and operational semantics for process algebra-like languages, proving compositionality and adequacy by defining operational semantics as a distributive law of syntax over behaviour. However, its applications to stateful or effectful languages, incorporating (co)models of a countable Lawvere theory, have been elusive so far. We make some progress towards a coalgebraic treatment of such languages, proposing a congruence format related to the evaluation-in-context paradigm. We formalise the denotational semantics in suitable Kleisli categories, and prove adequacy and compositionality of the semantic theory under this congruence format