271 research outputs found
Well-Pointed Coalgebras
For endofunctors of varieties preserving intersections, a new description of
the final coalgebra and the initial algebra is presented: the former consists
of all well-pointed coalgebras. These are the pointed coalgebras having no
proper subobject and no proper quotient. The initial algebra consists of all
well-pointed coalgebras that are well-founded in the sense of Osius and Taylor.
And initial algebras are precisely the final well-founded coalgebras. Finally,
the initial iterative algebra consists of all finite well-pointed coalgebras.
Numerous examples are discussed e.g. automata, graphs, and labeled transition
systems
Extensive amenability and an application to interval exchanges
Extensive amenability is a property of group actions which has recently been
used as a tool to prove amenability of groups. We study this property and prove
that it is preserved under a very general construction of semidirect products.
As an application, we establish the amenability of all subgroups of the group
IET of interval exchange transformations that have angular components of
rational rank~.
In addition, we obtain a reformulation of extensive amenability in terms of
inverted orbits and use it to present a purely probabilistic proof that
recurrent actions are extensively amenable. Finally, we study the triviality of
the Poisson boundary for random walks on IET and show that there are subgroups
admitting no finitely supported measure with trivial boundary.Comment: 28 page
Applications of Metric Coinduction
Metric coinduction is a form of coinduction that can be used to establish
properties of objects constructed as a limit of finite approximations. One can
prove a coinduction step showing that some property is preserved by one step of
the approximation process, then automatically infer by the coinduction
principle that the property holds of the limit object. This can often be used
to avoid complicated analytic arguments involving limits and convergence,
replacing them with simpler algebraic arguments. This paper examines the
application of this principle in a variety of areas, including infinite
streams, Markov chains, Markov decision processes, and non-well-founded sets.
These results point to the usefulness of coinduction as a general proof
technique
Bisimilarity, Hypersets, and Stable Partitioning: a Survey
Since Hopcroft proposed his celebrated algorithm for minimizing states in a finite automaton, the race for efficient partition refinement methods has inspired much research in algorithmics. In parallel, the notion of bisimulation has gained ground in theoretical investigations not less than in applications, till it even pervaded the axioms of a variant Zermelo-Fraenkel set theory. As is well-known, the coarsest stable partitioning problem and the determination of bisimilarity (i.e., the largest partition stable relative to finitely many dyadic relations) are two faces of the same coin. While there is a tendency to refer these topics to varying frameworks, we will contend that the set-theoretic view not only offers a clear conceptual background (provided stability is referred to a non-well-founded membership), but is leading to new insights on the algorithmic complexity issues
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