Abstract

Turi and Plotkin introduced an elegant approach to structural operational semantics based on universal coalgebra, parametric in the type of syntax and the type of behaviour. Their framework includes abstract GSOS, a categorical generalisation of the classical GSOS rule format, as well as its categorical dual, coGSOS. Both formats are well behaved, in the sense that each specification has a unique model on which behavioural equivalence is a congruence. Unfortunately, the combination of the two formats does not feature these desirable properties. We show that monotone specifications—that disallow negative premises—do induce a canonical distributive law of a monad over a comonad, and therefore a unique, compositional interpretation.

Highlights

  • Structural operational semantics (SOS) is an expressive and popular framework for defining the operational semantics of programming languages and calculi

  • There is a wide variety of specification formats that syntactically restrict the full power of SOS, but guarantee certain desirable properties to hold [1]

  • In their seminal paper [30], Turi and Plotkin introduced an elegant mathematical approach to structural operational semantics, where the type of syntax is modeled by an endofunctor Σ and the type of behaviour is modeled by an endofunctor B

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Summary

Introduction

Structural operational semantics (SOS) is an expressive and popular framework for defining the operational semantics of programming languages and calculi. The symbol x −→ /a represents a negative premise, which is satisfied whenever x does not make an a-transition Both GSOS and coGSOS specifications induce distributive laws, and as a consequence they induce unique supported models on which behavioural equivalence is a congruence. It is straightforward to combine GSOS and coGSOS as a natural transformation of the form Σ B∞ ⇒ BΣ∗, called a biGSOS specification, generalising both formats Such specifications are, in some sense, too expressive: they do not induce unique supported models, as already observed in [30]. This allows us to relax the requirement of DCPO structure only to countable sets, given that the functor B is countably accessible (this is weaker than being finitary, a standard condition in the theory of coalgebras) and the syntax consists only of countably many operations each with finite arity This applies to labelled transition systems (with countable branching) and certain kinds of weighted transition systems.

Preliminaries
Algebras and monads
Coalgebras and comonads
Distributive laws
Abstract rule formats
Similarity on cofree coalgebras
Monotone biGSOS specifications
Models of monotone specifications
Distributive laws for biGSOS specifications
Liftings for countably accessible functors
Future work
A Naturality of biGSOS induced from ntree
B Proofs of Section 7
Tκ Tκ αTκ α T I ηκ
Full Text
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