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Structural Operational Semantics and Modal Logic, Revisited
"... A previously introduced combination of the bialgebraic approach to structural operational semantics with coalgebraic modal logic is reexamined and improved in some aspects. Firstly, a more abstract, conceptual proof of the main compositionality theorem is given, based on an understanding of modal l ..."
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A previously introduced combination of the bialgebraic approach to structural operational semantics with coalgebraic modal logic is reexamined and improved in some aspects. Firstly, a more abstract, conceptual proof of the main compositionality theorem is given, based on an understanding of modal logic as a study of coalgebras in slice categories of adjunctions. Secondly, a more concrete understanding of the assumptions of the theorem is provided, where proving compositionality amounts to finding a syntactic distributive law between two collections of predicate liftings. Keywords: structural operational semantics, modal logic, coalgebra 1
Pointwise Extensions of GSOSDefined Operations
"... Distributive laws of syntax over behaviour (cf. [1, 3]) are, among other things, a wellstructured way of defining algebraic operations on final coalgebras. For a simple example, consider the set B ω of infinite streams of elements of B; this carries a final coalgebra w = 〈hd,tl〉: B ω → B × B ω for ..."
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Cited by 2 (2 self)
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Distributive laws of syntax over behaviour (cf. [1, 3]) are, among other things, a wellstructured way of defining algebraic operations on final coalgebras. For a simple example, consider the set B ω of infinite streams of elements of B; this carries a final coalgebra w = 〈hd,tl〉: B ω → B × B ω for the endofunctor F = B × − on Set. If B comes with a binary operation +, one can define an addition operation ⊕ on streams coinductively: hd(σ ⊕ τ) = hd(σ) + hd(τ) tl(σ ⊕ τ) = tl(σ) ⊕ tl(τ). It is easy to see that these equations define a distributive law, i.e., a natural transformation λ: ΣF ⇒ FΣ, where ΣX = X 2 is the signature endofunctor corresponding to a single binary operation. The operation ⊕: B ω × B ω → B ω is now defined as the unique morphism to the final coalgebra as in: ΣB ω B ω
Traces for Coalgebraic Components
 MATH. STRUCT. IN COMP. SCIENCE
, 2010
"... This paper contributes a feedback operator, in the form of a monoidal trace, to the theory of coalgebraic, statebased modelling of components. The feedback operator on components is shown to satisfy the trace axioms of Joyal, Street and Verity. We employ McCurdy’s tube diagrams, an extension of sta ..."
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This paper contributes a feedback operator, in the form of a monoidal trace, to the theory of coalgebraic, statebased modelling of components. The feedback operator on components is shown to satisfy the trace axioms of Joyal, Street and Verity. We employ McCurdy’s tube diagrams, an extension of standard string diagrams for monoidal categories, for representing and manipulating component diagrams. The microcosm principle then yields a canonical “inner” traced monoidal structure on the category of resumptions (elements of final coalgebras / components). This generalises an observation by Abramsky, Haghverdi and Scott.
Towards Effects in Mathematical Operational Semantics
"... In this paper, we study extensions of mathematical operational semantics with algebraic effects. Our starting point is an effectfree coalgebraic operational semantics, given by a natural transformation of syntax over behaviour. The operational semantics of the extended language arises by distributi ..."
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In this paper, we study extensions of mathematical operational semantics with algebraic effects. Our starting point is an effectfree 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 evaluationincontext. We illustrate our techniques with a simple extension of (stateless) while programs with global store, i.e. variable lookup.
Towards Bialgebraic Semantics for the Linear Time – Branching Time Spectrum ∗
"... Abstract. Process algebra, e.g. CSP, offers different semantical observations (e.g. traces, failures, divergences) on a single syntactical system description. These observations are either computed algebraically from the process syntax, or “extracted ” from a single operational model. Bialgebras cap ..."
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Abstract. Process algebra, e.g. CSP, offers different semantical observations (e.g. traces, failures, divergences) on a single syntactical system description. These observations are either computed algebraically from the process syntax, or “extracted ” from a single operational model. Bialgebras capture both approaches in one framework and characterize their equivalence; however, due to use of finality, lack the capability to simultaneously cater for various semantics. We suggest to relax finality to quasifinality. This allows for several semantics, which also can be coarser than bisimulation. As a case study, we show that our approach works out in the case of the CSP failures model. 1
Probabilistic Simulation ⋆
, 2010
"... Abstract. Jonsson and Larsen’s notion of probabilistic simulation is studied from a coalgebraic perspective. The notion is compared with two generic coalgebraic definitions of simulation: Hughes and Jacobs ’ one, and the one introduced previously by the author. We show that the first almost coincide ..."
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Abstract. Jonsson and Larsen’s notion of probabilistic simulation is studied from a coalgebraic perspective. The notion is compared with two generic coalgebraic definitions of simulation: Hughes and Jacobs ’ one, and the one introduced previously by the author. We show that the first almost coincides with the second, and that the second is a special case of the last. We investigate implications of this characterization; notably the JonssonLarsen simulation is shown to be sound, i.e. its existence implies trace inclusion. 1