Abstract
For every finitary monad T on sets and every endofunctor F on the category of T-algebras we introduce the concept of an ffg-Elgot algebra for F, that is, an algebra admitting coherent solutions for finite systems of recursive equations with effects represented by the monad T. The goal of this paper is to study the existence and construction of free ffg-Elgot algebras. To this end, we investigate the locally ffg fixed point \(\varphi F\), the colimit of all F-coalgebras with free finitely generated carrier, which is shown to be the initial ffg-Elgot algebra. This is the technical foundation for our main result: the category of ffg-Elgot algebras is monadic over the category of T-algebras.
Highlights
Terminal coalgebras yield a fully abstract domain of behavior for a given type of state-based systems whose transition type is described by an endofunctor F
In the category C of T -algebras, several notions of ’finite’ object are natural to consider, and each of those yields an ensuing notion of ’finite’ coalgebra: free objects on finitely many generators yield precisely the coalgebras that are the target of generalized determinization; finitely presentable objects are the ones that can be presented by finitely many generators and relations and yield the rational fixed point; and finitely generated objects, i.e. those presented by finitely many generators
For a functor F on a variety preserving sifted colimits, the concept of an Elgot algebra [ ] has a natural weakening obtained by working with iterative equations having ffg objects of variables
Summary
Terminal coalgebras yield a fully abstract domain of behavior for a given type of state-based systems whose transition type is described by an endofunctor F. S is a proper semiring the three fixed points coincide); rational (a.k.a. regular) Σtrees for the polynomial functor on Set associated to the signature Σ; eventually periodic and rational streams for the functor k × (−) on Set and vector spaces over the field k, respectively; the behaviors of probabilistic automata modelled as coalgebras for [0, 1] × (−)Σ on the category of positive convex algebras (that this functor is proper was recently proved by Sokolova and Woracek [ ]); (deterministic) context-free languages and constructively S-algebraic formal power-series (the weighted counterpart of context-free languages) [ ]. ΦId is non-trivial and interesting: an ffg-coalgebra T X −→γ T X may be viewed (by restricting it to its generators in X) as obtained by generalized determinization of an
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