Comodels and Effects in Mathematical Operational Semantics

Abstract

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.

Presenting Distributive Laws

“…We leave as future work to find out precisely how their Theorem 31 relates to our Theorem 4.3. In [1] effects with equations are added to the syntax generated by a free monad T , using as semantic domain a suitable final B-coalgebra in the Kleisli category of T (assumed to be enriched over ωcomplete pointed partial-orders). To prove adequacy of the semantics with respect to a given operational model, the authors use a result similar to our Theorem 4.3.…”

“…We check that λ preserves the distribution axiom: . We have thus shown that λ preserves E, and it follows, in particular, that R ω , +, ×, [0], [1] is a commutative semiring. This was shown directly in [31], but the proof uses bisimulation-up-to as well as the fundamental theorem of stream calculus, which cannot be added as an equation.…”

“…First note that for F = R × Id, r 1 , t 1 F (≡ X ) r 2 , t 2 iff r 1 = r 2 and t 1 ≡ X t 2 . It is straightforward to check preservation of the axioms that only concern addition, as well as of [1]…”

Presenting Distributive Laws

“…We leave as future work to find out precisely how their Theorem 31 relates to our Theorem 4.3. In [1] effects with equations are added to the syntax generated by a free monad T , using as semantic domain a suitable final B-coalgebra in the Kleisli category of T (assumed to be enriched over ω-complete pointed partial-orders). To prove adequacy of the semantics with respect to a given operational model, the authors use a result similar to our Theorem 4.3.…”

“…We have thus shown that λ preserves E, and it follows, in particular, that R ω , +, ×, [0], [1] is a commutative semiring. This was shown directly in [31], but the proof uses bisimulation-up-to as well as the fundamental theorem of stream calculus, which cannot be added as an equation.…”