Sergey Goncharov

Sergey Goncharov, Stefan Milius, Lutz Schröder et al.

Compositionality proofs in higher-order languages are notoriously involved, and general semantic frameworks guaranteeing compositionality are hard to come by. In particular, Turi and Plotkin's bialgebraic abstract GSOS framework, which has been successfully applied to obtain off-the-shelf compositionality results for first-order languages, so far does not apply to higher-order languages. In the present work, we develop a theory of abstract GSOS specifications for higher-order languages, in effect transferring the core principles of Turi and Plotkin's framework to a higher-order setting. In our theory, the operational semantics of higher-order languages is represented by certain dinatural transformations that we term pointed higher-order GSOS laws. We give a general compositionality result that applies to all systems specified in this way and discuss how compositionality of the SKI calculus and the $λ$-calculus w.r.t. a strong variant of Abramsky's applicative bisimilarity are obtained as instances.

Sergey Goncharov

Notions of iteration range from the arguably most general Elgot iteration to a very specific Kleene iteration. The fundamental nature of Elgot iteration has been extensively explored by Bloom and Esik in the form of iteration theories, while Kleene iteration became extremely popular as an integral part of (untyped) formalisms, such as automata theory, regular expressions and Kleene algebra. Here, we establish a formal connection between Elgot iteration and Kleene iteration in the form of Elgot monads and Kleene monads, respectively. We also introduce a novel class of while-monads, which like Kleene monads admit a relatively simple description in algebraic terms. Like Elgot monads, while-monads cover a large variety of models that meaningfully support while-loops, but may fail the Kleene algebra laws, or even fail to support a Kleen iteration operator altogether.

Sergey Goncharov, Stefan Milius, Lutz Schröder et al.

Compositionality proofs in higher-order languages are notoriously involved, and general semantic frameworks guaranteeing compositionality are hard to come by. In particular, Turi and Plotkin's bialgebraic abstract GSOS framework, which provides off-the-shelf compositionality results for first-order languages, so far does not apply to higher-order languages. In the present work, we develop a theory of abstract GSOS specifications for higher-order languages, in effect transferring the core principles of Turi and Plotkin's framework to a higher-order setting. In our theory, the operational semantics of higher-order languages is represented by certain dinatural transformations that we term \emph{(pointed) higher-order GSOS laws}. We give a general compositionality result that applies to all systems specified in this way and discuss how compositionality of combinatory logics and the $λ$-calculus w.r.t.\ a strong variant of Abramsky's applicative bisimilarity are obtained as instances.

Robin Jourde, Henning Urbat, Sergey Goncharov et al.

A key requirement on any well-behaved process language is its compositionality: behavioural equivalence of processes should be respected by the constructors of the language. Turi and Plotkin's abstract GSOS provides an elegant bialgebraic framework for modelling rule formats that guarantee compositionality from the outset. Their original results, however, are restricted to compositionality of strong bisimilarity, a rather fine-grained notion of process equivalence. In the present paper, we demonstrate that Turi and Plotkin's approach also applies to trace equivalence, which only observes external actions of processes. To this end, we revisit the general compositionality result of their original theory and present it in a refined form with regard to the required naturality conditions. This step makes abstract GSOS applicable over Kleisli categories and thereby enables reasoning about compositionality in the setting of coalgebraic trace semantics. As our main contribution, we introduce De Simone laws, a type of GSOS laws over Kleisli categories, and prove that their operational models are compositional for coalgebraic trace equivalence. This result recovers and explains compositionality of the well-known De Simone rule format for labelled transition systems in a natural categorical setting. As a further application, we derive from our general framework a novel De Simone-type format for probabilistic systems, compositional for probabilistic trace equivalence.