4,797 research outputs found
Proper Functors and Fixed Points for Finite Behaviour
The rational fixed point of a set functor is well-known to capture the
behaviour of finite coalgebras. In this paper we consider functors on algebraic
categories. For them the rational fixed point may no longer be fully abstract,
i.e. a subcoalgebra of the final coalgebra. Inspired by \'Esik and Maletti's
notion of a proper semiring, we introduce the notion of a proper functor. We
show that for proper functors the rational fixed point is determined as the
colimit of all coalgebras with a free finitely generated algebra as carrier and
it is a subcoalgebra of the final coalgebra. Moreover, we prove that a functor
is proper if and only if that colimit is a subcoalgebra of the final coalgebra.
These results serve as technical tools for soundness and completeness proofs
for coalgebraic regular expression calculi, e.g. for weighted automata
Sound and complete axiomatizations of coalgebraic language equivalence
Coalgebras provide a uniform framework to study dynamical systems, including
several types of automata. In this paper, we make use of the coalgebraic view
on systems to investigate, in a uniform way, under which conditions calculi
that are sound and complete with respect to behavioral equivalence can be
extended to a coarser coalgebraic language equivalence, which arises from a
generalised powerset construction that determinises coalgebras. We show that
soundness and completeness are established by proving that expressions modulo
axioms of a calculus form the rational fixpoint of the given type functor. Our
main result is that the rational fixpoint of the functor , where is a
monad describing the branching of the systems (e.g. non-determinism, weights,
probability etc.), has as a quotient the rational fixpoint of the
"determinised" type functor , a lifting of to the category of
-algebras. We apply our framework to the concrete example of weighted
automata, for which we present a new sound and complete calculus for weighted
language equivalence. As a special case, we obtain non-deterministic automata,
where we recover Rabinovich's sound and complete calculus for language
equivalence.Comment: Corrected version of published journal articl
The Power of Convex Algebras
Probabilistic automata (PA) combine probability and nondeterminism. They can
be given different semantics, like strong bisimilarity, convex bisimilarity, or
(more recently) distribution bisimilarity. The latter is based on the view of
PA as transformers of probability distributions, also called belief states, and
promotes distributions to first-class citizens.
We give a coalgebraic account of the latter semantics, and explain the
genesis of the belief-state transformer from a PA. To do so, we make explicit
the convex algebraic structure present in PA and identify belief-state
transformers as transition systems with state space that carries a convex
algebra. As a consequence of our abstract approach, we can give a sound proof
technique which we call bisimulation up-to convex hull.Comment: Full (extended) version of a CONCUR 2017 paper, to be submitted to
LMC
Varieties of Cost Functions.
Regular cost functions were introduced as a quantitative generalisation of regular languages, retaining many of their equivalent characterisations and decidability properties. For instance, stabilisation monoids play the same role for cost functions as monoids do for regular languages. The purpose of this article is to further extend this algebraic approach by generalising two results on regular languages to cost functions: Eilenberg's varieties theorem and profinite equational characterisations of lattices of regular languages. This opens interesting new perspectives, but the specificities of cost functions introduce difficulties that prevent these generalisations to be straightforward. In contrast, although syntactic algebras can be defined for formal power series over a commutative ring, no such notion is known for series over semirings and in particular over the tropical semiring
Coinduction up to in a fibrational setting
Bisimulation up-to enhances the coinductive proof method for bisimilarity,
providing efficient proof techniques for checking properties of different kinds
of systems. We prove the soundness of such techniques in a fibrational setting,
building on the seminal work of Hermida and Jacobs. This allows us to
systematically obtain up-to techniques not only for bisimilarity but for a
large class of coinductive predicates modelled as coalgebras. By tuning the
parameters of our framework, we obtain novel techniques for unary predicates
and nominal automata, a variant of the GSOS rule format for similarity, and a
new categorical treatment of weak bisimilarity
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