1,544 research outputs found
Healthiness from Duality
Healthiness is a good old question in program logics that dates back to
Dijkstra. It asks for an intrinsic characterization of those predicate
transformers which arise as the (backward) interpretation of a certain class of
programs. There are several results known for healthiness conditions: for
deterministic programs, nondeterministic ones, probabilistic ones, etc.
Building upon our previous works on so-called state-and-effect triangles, we
contribute a unified categorical framework for investigating healthiness
conditions. We find the framework to be centered around a dual adjunction
induced by a dualizing object, together with our notion of relative
Eilenberg-Moore algebra playing fundamental roles too. The latter notion seems
interesting in its own right in the context of monads, Lawvere theories and
enriched categories.Comment: 13 pages, Extended version with appendices of a paper accepted to
LICS 201
Resource modalities in game semantics
The description of resources in game semantics has never achieved the
simplicity and precision of linear logic, because of a misleading conception:
the belief that linear logic is more primitive than game semantics. We advocate
instead the contrary: that game semantics is conceptually more primitive than
linear logic. Starting from this revised point of view, we design a categorical
model of resources in game semantics, and construct an arena game model where
the usual notion of bracketing is extended to multi- bracketing in order to
capture various resource policies: linear, affine and exponential
Tangent Categories from the Coalgebras of Differential Categories
Following the pattern from linear logic, the coKleisli category of a differential category is a Cartesian differential category. What then is the coEilenberg-Moore category of a differential category? The answer is a tangent category! A key example arises from the opposite of the category of Abelian groups with the free exponential modality. The coEilenberg-Moore category, in this case, is the opposite of the category of commutative rings. That the latter is a tangent category captures a fundamental aspect of both algebraic geometry and Synthetic Differential Geometry. The general result applies when there are no negatives and thus encompasses examples arising from combinatorics and computer science
Guard Your Daggers and Traces: On The Equational Properties of Guarded (Co-)recursion
Motivated by the recent interest in models of guarded (co-)recursion we study
its equational properties. We formulate axioms for guarded fixpoint operators
generalizing the axioms of iteration theories of Bloom and Esik. Models of
these axioms include both standard (e.g., cpo-based) models of iteration
theories and models of guarded recursion such as complete metric spaces or the
topos of trees studied by Birkedal et al. We show that the standard result on
the satisfaction of all Conway axioms by a unique dagger operation generalizes
to the guarded setting. We also introduce the notion of guarded trace operator
on a category, and we prove that guarded trace and guarded fixpoint operators
are in one-to-one correspondence. Our results are intended as first steps
leading to the description of classifying theories for guarded recursion and
hence completeness results involving our axioms of guarded fixpoint operators
in future work.Comment: In Proceedings FICS 2013, arXiv:1308.589
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