175 research outputs found
An ordered framework for partial multivalued functors
The category Rel of sets and relations intimately ties the notions of
function, partial multivalued function, and direct image under a function
through the description of Rel as the Kleisli category of the covariant power
set functor on Set. We present a suitable framework to obtain a similar
relationship between the concepts of functor, partial multivalued functor, and
the direct image under a functor.Comment: Accepted for presentation at the Asia-Pacific World Congress on
Computer Science and Engineering 2015, Fij
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
A categorical foundation for Bayesian probability
Given two measurable spaces and with countably generated
-algebras, a perfect prior probability measure on and a
sampling distribution , there is a corresponding inference
map which is unique up to a set of measure zero. Thus,
given a data measurement , a posterior probability
can be computed. This procedure is iterative: with
each updated probability , we obtain a new joint distribution which in
turn yields a new inference map and the process repeats with each
additional measurement. The main result uses an existence theorem for regular
conditional probabilities by Faden, which holds in more generality than the
setting of Polish spaces. This less stringent setting then allows for
non-trivial decision rules (Eilenberg--Moore algebras) on finite (as well as
non finite) spaces, and also provides for a common framework for decision
theory and Bayesian probability.Comment: 15 pages; revised setting to more clearly explain how to incorporate
perfect measures and the Giry monad; to appear in Applied Categorical
Structure
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
Kleisli morphisms and randomized congruences for the Giry monad
AbstractStochastic relations are the Kleisli morphisms for the Giry monad. This paper proposes the study of the associated morphisms and congruences. The relationship between kernels of these morphisms and congruences is studied, and a unique factorization of a morphism through this kernel is shown to exist. This study is based on an investigation into countably generated equivalence relations on the space of all subprobabilities. Operations on these relations are investigated quite closely. This utilizes positive convex structures and indicates cross-connections to Eilenberg–Moore algebras for the Giry monad. Hennessy–Milner logic serves as an illustration for randomized morphisms and congruences
Relating Operator Spaces via Adjunctions
This chapter uses categorical techniques to describe relations between
various sets of operators on a Hilbert space, such as self-adjoint, positive,
density, effect and projection operators. These relations, including various
Hilbert-Schmidt isomorphisms of the form tr(A-), are expressed in terms of dual
adjunctions, and maps between them. Of particular interest is the connection
with quantum structures, via a dual adjunction between convex sets and effect
modules. The approach systematically uses categories of modules, via their
description as Eilenberg-Moore algebras of a monad
Cumulants, free cumulants and half-shuffles
Free cumulants were introduced as the proper analog of classical cumulants in
the theory of free probability. There is a mix of similarities and differences,
when one considers the two families of cumulants. Whereas the combinatorics of
classical cumulants is well expressed in terms of set partitions, the one of
free cumulants is described, and often introduced in terms of non-crossing set
partitions. The formal series approach to classical and free cumulants also
largely differ. It is the purpose of the present article to put forward a
different approach to these phenomena. Namely, we show that cumulants, whether
classical or free, can be understood in terms of the algebra and combinatorics
underlying commutative as well as non-commutative (half-)shuffles and
(half-)unshuffles. As a corollary, cumulants and free cumulants can be
characterized through linear fixed point equations. We study the exponential
solutions of these linear fixed point equations, which display well the
commutative, respectively non-commutative, character of classical, respectively
free, cumulants.Comment: updated and revised version; accepted for publication in PRS
Trace semantics via determinization
This paper takes a fresh look at the topic of trace semantics in the theory of coalgebras. The first development of coalgebraic trace semantics used final coalgebras in Kleisli categories, stemming from an initial algebra in the underlying category (see notably~\cite{HasuoJS07}). This approach requires some non-trivial assumptions, like dcpo enrichment, which do not always hold, even in cases where one can reasonably speak of traces (like for weighted automata). More recently, it has been noticed (see~\cite{SBBR10}) that trace semantics can also arise by first performing a determinization construction. In this paper, we develop a systematic approach, in which the two approaches correspond to different orders of composing a functor and a monad, and accordingly, to different distributive laws. The relevant final coalgebra that gives rise to trace semantics does not live in a Kleisli category, but more generally, in a category of Eilenberg-Moore algebras. In order to exploit its finality, we identify an extension operation, that changes the state space of a coalgebra into a free algebra, which abstractly captures determinization of automata. Notably, we show that the two different views on trace semantics are equivalent, in the examples where both approaches are applicable.We are grateful to the anonymous referees for valuable comments. The work of Alexandra Silva is partially funded by the ERDF through the Programme COMPETE and by the Portuguese Foundation for Science and Technology, project Ref. FCOMP-01-0124-FEDER-020537 and SFRH/BPD/71956/2010
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