183,124 research outputs found
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
Tropical Limits of Probability Spaces, Part I: The Intrinsic Kolmogorov-Sinai Distance and the Asymptotic Equipartition Property for Configurations
The entropy of a finite probability space measures the observable
cardinality of large independent products of the probability
space. If two probability spaces and have the same entropy, there is an
almost measure-preserving bijection between large parts of and
. In this way, and are asymptotically equivalent.
It turns out to be challenging to generalize this notion of asymptotic
equivalence to configurations of probability spaces, which are collections of
probability spaces with measure-preserving maps between some of them.
In this article we introduce the intrinsic Kolmogorov-Sinai distance on the
space of configurations of probability spaces. Concentrating on the large-scale
geometry we pass to the asymptotic Kolmogorov-Sinai distance. It induces an
asymptotic equivalence relation on sequences of configurations of probability
spaces. We will call the equivalence classes \emph{tropical probability
spaces}.
In this context we prove an Asymptotic Equipartition Property for
configurations. It states that tropical configurations can always be
approximated by homogeneous configurations. In addition, we show that the
solutions to certain Information-Optimization problems are
Lipschitz-con\-tinuous with respect to the asymptotic Kolmogorov-Sinai
distance. It follows from these two statements that in order to solve an
Information-Optimization problem, it suffices to consider homogeneous
configurations.
Finally, we show that spaces of trajectories of length of certain
stochastic processes, in particular stationary Markov chains, have a tropical
limit.Comment: Comment to version 2: Fixed typos, a calculation mistake in Lemma 5.1
and its consequences in Proposition 5.2 and Theorem 6.
Belief propagation in monoidal categories
We discuss a categorical version of the celebrated belief propagation
algorithm. This provides a way to prove that some algorithms which are known or
suspected to be analogous, are actually identical when formulated generically.
It also highlights the computational point of view in monoidal categories.Comment: In Proceedings QPL 2014, arXiv:1412.810
A categorical semantics for causal structure
We present a categorical construction for modelling causal structures within
a general class of process theories that include the theory of classical
probabilistic processes as well as quantum theory. Unlike prior constructions
within categorical quantum mechanics, the objects of this theory encode
fine-grained causal relationships between subsystems and give a new method for
expressing and deriving consequences for a broad class of causal structures. We
show that this framework enables one to define families of processes which are
consistent with arbitrary acyclic causal orderings. In particular, one can
define one-way signalling (a.k.a. semi-causal) processes, non-signalling
processes, and quantum -combs. Furthermore, our framework is general enough
to accommodate recently-proposed generalisations of classical and quantum
theory where processes only need to have a fixed causal ordering locally, but
globally allow indefinite causal ordering.
To illustrate this point, we show that certain processes of this kind, such
as the quantum switch, the process matrices of Oreshkov, Costa, and Brukner,
and a classical three-party example due to Baumeler, Feix, and Wolf are all
instances of a certain family of processes we refer to as in
the appropriate category of higher-order causal processes. After defining these
families of causal structures within our framework, we give derivations of
their operational behaviour using simple, diagrammatic axioms.Comment: Extended version of a LICS 2017 paper with the same titl
Mackey-complete spaces and power series -- A topological model of Differential Linear Logic
In this paper, we have described a denotational model of Intuitionist Linear
Logic which is also a differential category. Formulas are interpreted as
Mackey-complete topological vector space and linear proofs are interpreted by
bounded linear functions. So as to interpret non-linear proofs of Linear Logic,
we have used a notion of power series between Mackey-complete spaces,
generalizing the notion of entire functions in C. Finally, we have obtained a
quantitative model of Intuitionist Differential Linear Logic, where the
syntactic differentiation correspond to the usual one and where the
interpretations of proofs satisfy a Taylor expansion decomposition
Reified valuations and adic spectra
We revisit Huber's theory of continuous valuations, which give rise to the
adic spectra used in his theory of adic spaces. We instead consider valuations
which have been reified, i.e., whose value groups have been forced to contain
the real numbers. This yields reified adic spectra which provide a framework
for an analogue of Huber's theory compatible with Berkovich's construction of
nonarchimedean analytic spaces. As an example, we extend the theory of
perfectoid spaces to this setting.Comment: v5: refereed versio
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