267 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
Models of Non-Well-Founded Sets via an Indexed Final Coalgebra Theorem
The paper uses the formalism of indexed categories to recover the proof of a
standard final coalgebra theorem, thus showing existence of final coalgebras
for a special class of functors on categories with finite limits and colimits.
As an instance of this result, we build the final coalgebra for the powerclass
functor, in the context of a Heyting pretopos with a class of small maps. This
is then proved to provide a model for various non-well-founded set theories,
depending on the chosen axiomatisation for the class of small maps
Generic Trace Semantics via Coinduction
Trace semantics has been defined for various kinds of state-based systems,
notably with different forms of branching such as non-determinism vs.
probability. In this paper we claim to identify one underlying mathematical
structure behind these "trace semantics," namely coinduction in a Kleisli
category. This claim is based on our technical result that, under a suitably
order-enriched setting, a final coalgebra in a Kleisli category is given by an
initial algebra in the category Sets. Formerly the theory of coalgebras has
been employed mostly in Sets where coinduction yields a finer process semantics
of bisimilarity. Therefore this paper extends the application field of
coalgebras, providing a new instance of the principle "process semantics via
coinduction."Comment: To appear in Logical Methods in Computer Science. 36 page
Non-wellfounded trees in Homotopy Type Theory
We prove a conjecture about the constructibility of coinductive types - in
the principled form of indexed M-types - in Homotopy Type Theory. The
conjecture says that in the presence of inductive types, coinductive types are
derivable. Indeed, in this work, we construct coinductive types in a subsystem
of Homotopy Type Theory; this subsystem is given by Intensional Martin-L\"of
type theory with natural numbers and Voevodsky's Univalence Axiom. Our results
are mechanized in the computer proof assistant Agda.Comment: 14 pages, to be published in proceedings of TLCA 2015; ancillary
files contain Agda files with formalized proof
Generalized Vietoris Bisimulations
We introduce and study bisimulations for coalgebras on Stone spaces [14]. Our
notion of bisimulation is sound and complete for behavioural equivalence, and
generalizes Vietoris bisimulations [4]. The main result of our paper is that
bisimulation for a coalgebra is the topological closure of
bisimulation for the underlying coalgebra
Dependent Inductive and Coinductive Types are Fibrational Dialgebras
In this paper, I establish the categorical structure necessary to interpret
dependent inductive and coinductive types. It is well-known that dependent type
theories \`a la Martin-L\"of can be interpreted using fibrations. Modern
theorem provers, however, are based on more sophisticated type systems that
allow the definition of powerful inductive dependent types (known as inductive
families) and, somewhat limited, coinductive dependent types. I define a class
of functors on fibrations and show how data type definitions correspond to
initial and final dialgebras for these functors. This description is also a
proposal of how coinductive types should be treated in type theories, as they
appear here simply as dual of inductive types. Finally, I show how dependent
data types correspond to algebras and coalgebras, and give the correspondence
to dependent polynomial functors.Comment: In Proceedings FICS 2015, arXiv:1509.0282
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