Strongly Complete Logics for Coalgebras

Abstract

Abstract. Coalgebras for a functor model different types of transition systems in a uniform way. This paper focuses on a uniform account of finitary logics for set-based coalgebras. In particular, a general construction of a logic from an arbitrary set-functor is given and proven to be strongly complete under additional assumptions. We proceed in three parts.Part I argues that sifted colimit preserving functors are those functors that preserve universal algebraic structure. Our main theorem here states that a functor preserves sifted colimits if and only if it has a finitary presentation by operations and equations. Moreover, the presentation of the category of algebras for the functor is obtained compositionally from the presentations of the underlying category and of the functor.Part II investigates algebras for a functor over ind-completions and extends the theorem of Jónsson and Tarski on canonical extensions of Boolean algebras with operators to this setting.Part III shows, based on Part I, how to associate a finitary logic to any finite-sets preserving functor T . Based on Part II we prove the logic to be strongly complete under a reasonable condition on T .

Bisimulation of Labeled State-to-Function Transition Systems of Stochastic Process Languages

“…However, following a familiar argument, we show that the functor associated with a FuTS does possess a final coalgebra and therefore has an associated notion of behavioral equivalence indeed. It is noted, in the presence of a final coalgebra for FuTS a more general definition of behavioral equivalence based on cospans coincides [20]. A correspondence result is proven in this paper that shows that the concrete bisimulation of a FuTS, coincides with behavioral equivalence of its functor.…”

Smooth coalgebra: testing vector analysis

“…Both approaches were aligned in [46] 1 , generalizing the earlier application in [55]. Such loose semantical connections of algebras and coalgebras, as two independent dimensions of semantical descriptions, distinguish the testing frameworks used in the present paper, and previously applied in [55,53,54,46,51,52], from the tight semantical connections of algebras and coalgebras, as arising on the two sides of a duality, and used in algebraic semantics of coalgebraic logic [29,30,31]. The idea of testing is echoed more closely in the testing approach to the equivalence of concurrent processes [13,7], and the two dimensional approach is implemented in terms of algebras and coalgebras in the categorical approach to Structured Operational Semantics, which was developed as an extension of Denotational Semantics of monads [61,24,26].…”

On universal algebra over nominal sets

“…At this point, we are encouraged by a result of (Kurz and Rosický, 2006), stating that all 'universal-algebraic' endofunctors L have a presentation. In particular, it follows that Alg(L) is an equationally defined class (variety) of many-sorted algebras in the standard sense of universal algebra.…”

“…Central to our approach is the flexibility provided by the notion of a presentation of functor (introduced in (Bonsangue and Kurz, 2006) and further developed in (Kurz and Rosický, 2006;Kurz and Petrişan, 2008)). It allows us to identify the abstract notion of Lalgebras with the concrete notion of algebras for a signature and equations.…”