Weak bisimilarity coalgebraically

We argue that weak bisimilarity of processes can be conveniently captured in a semantic domain by a combination of traces and coalgebraic finality, in such a way that important process algebra aspects such as parallel composition and recursion can be represented compositionally. We illustrate the usefulness of our approach by providing a fully-abstract denotational semantics for CCS under weak bisimilarity

Weak bisimilarity coalgebraically

We argue that weak bisimilarity of processes can be conveniently captured in a semantic domain by a combination of traces and coalgebraic finality, in such a way that important process algebra aspects such as parallel composition and recursion can be represented compositionally. We illustrate the usefulness of our approach by providing a fully-abstract denotational semantics for CCS under weak bisimilarity

A coalgebraic treatment of conditional transition systems with upgrades

We consider conditional transition systems, that model software product lines with upgrades, in a coalgebraic setting. By using Birkhoff's duality for distributive lattices, we derive two equivalent Kleisli categories in which these coalgebras live: Kleisli categories based on the reader and on the so-called lattice monad over Poset. We study two different functors describing the branching type of the coalgebra and investigate the resulting behavioural equivalence. Furthermore we show how an existing algorithm for coalgebra minimisation can be instantiated to derive behavioural equivalences in this setting

Global semantic typing for inductive and coinductive computing

Inductive and coinductive types are commonly construed as ontological (Church-style) types, denoting canonical data-sets such as natural numbers, lists, and streams. For various purposes, notably the study of programs in the context of global semantics, it is preferable to think of types as semantical properties (Curry-style). Intrinsic theories were introduced in the late 1990s to provide a purely logical framework for reasoning about programs and their semantic types. We extend them here to data given by any combination of inductive and coinductive definitions. This approach is of interest because it fits tightly with syntactic, semantic, and proof theoretic fundamentals of formal logic, with potential applications in implicit computational complexity as well as extraction of programs from proofs. We prove a Canonicity Theorem, showing that the global definition of program typing, via the usual (Tarskian) semantics of first-order logic, agrees with their operational semantics in the intended model. Finally, we show that every intrinsic theory is interpretable in a conservative extension of first-order arithmetic. This means that quantification over infinite data objects does not lead, on its own, to proof-theoretic strength beyond that of Peano Arithmetic. Intrinsic theories are perfectly amenable to formulas-as-types Curry-Howard morphisms, and were used to characterize major computational complexity classes Their extensions described here have similar potential which has already been applied

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

The Theory of Traces for Systems with Nondeterminism, Probability, and Termination

This paper studies trace-based equivalences for systems combining nondeterministic and probabilistic choices. We show how trace semantics for such processes can be recovered by instantiating a coalgebraic construction known as the generalised powerset construction. We characterise and compare the resulting semantics to known definitions of trace equivalences appearing in the literature. Most of our results are based on the exciting interplay between monads and their presentations via algebraic theories.Comment: This paper is an extended version of a LICS 2019 paper "The Theory of Traces for Systems with Nondeterminism and Probability". It contains all the proofs, additional explanations, material, and example

The theory of traces for systems with nondeterminism and probability

This paper studies trace-based equivalences for systems combining nondeterministic and probabilistic choices. We show how trace semantics for such processes can be recovered by instantiating a coalgebraic construction known as the generalised powerset construction. We characterise and compare the resulting semantics to known definitions of trace equivalences appearing in the literature. Most of our results are based on the exciting interplay between monads and their presentations via algebraic theories