52 research outputs found
Tukey reducibility for categories -- In search of the strongest statement in finite Ramsey theory
Every statement of the Ramsey theory of finite structures corresponds to the
fact that a particular category has the Ramsey property. We can, then, compare
the strength of Ramsey statements by comparing the ``Ramsey strength'' of the
corresponding categories. The main thesis of this paper is that establishing
pre-adjunctions between pairs of categories is an appropriate way of comparing
their ``Ramsey strength''. What comes as a pleasant surprise is that
pre-adjunctions generalize the Tukey reducibility in the same way categories
generalize preorders. In this paper we set forth a classification program of
statements of finite Ramsey theory based on their relationship with respect to
this generalized notion of Tukey reducibility for categories. After identifying
the ``weakest'' Ramsey category, we prove that the Finite Dual Ramsey Theorem
is as powerful as the full-blown version of the Graham-Rothschild Theorem, and
conclude the paper with the hypothesis that the Finite Dual Ramsey Theorem is
the ``strongest'' of all finite Ramsey statements
The Expectation Monad in Quantum Foundations
The expectation monad is introduced abstractly via two composable
adjunctions, but concretely captures measures. It turns out to sit in between
known monads: on the one hand the distribution and ultrafilter monad, and on
the other hand the continuation monad. This expectation monad is used in two
probabilistic analogues of fundamental results of Manes and Gelfand for the
ultrafilter monad: algebras of the expectation monad are convex compact
Hausdorff spaces, and are dually equivalent to so-called Banach effect
algebras. These structures capture states and effects in quantum foundations,
and also the duality between them. Moreover, the approach leads to a new
re-formulation of Gleason's theorem, expressing that effects on a Hilbert space
are free effect modules on projections, obtained via tensoring with the unit
interval.Comment: In Proceedings QPL 2011, arXiv:1210.029
Dual Ramsey properties for classes of algebras
Almost any reasonable class of finite relational structures has the Ramsey
property or a precompact Ramsey expansion. In contrast to that, the list of
classes of finite algebras with the precompact Ramsey expansion is surprisingly
short. In this paper we show that any nontrivial variety (that is, equationally
defined class of algebras) enjoys various \emph{dual} Ramsey properties. We
develop a completely new set of strategies that rely on the fact that left
adjoints preserve the dual Ramsey property, and then treat classes of algebras
as Eilenberg-Moore categories for a monad. We show that finite algebras in any
nontrivial variety have finite dual small Ramsey degrees, and that every finite
algebra has finite dual big Ramsey degree in the free algebra on countably many
free generators. As usual, these come as consequences of ordered versions of
the statements
Logical Relations for Monadic Types
Logical relations and their generalizations are a fundamental tool in proving
properties of lambda-calculi, e.g., yielding sound principles for observational
equivalence. We propose a natural notion of logical relations able to deal with
the monadic types of Moggi's computational lambda-calculus. The treatment is
categorical, and is based on notions of subsconing, mono factorization systems,
and monad morphisms. Our approach has a number of interesting applications,
including cases for lambda-calculi with non-determinism (where being in logical
relation means being bisimilar), dynamic name creation, and probabilistic
systems.Comment: 83 page
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