533 research outputs found
Probabilistic Monads, Domains and Classical Information
Shannon's classical information theory uses probability theory to analyze
channels as mechanisms for information flow. In this paper, we generalize
results of Martin, Allwein and Moskowitz for binary channels to show how some
more modern tools - probabilistic monads and domain theory in particular - can
be used to model classical channels. As initiated Martin, et al., the point of
departure is to consider the family of channels with fixed inputs and outputs,
rather than trying to analyze channels one at a time. The results show that
domain theory has a role to play in the capacity of channels; in particular,
the (n x n)-stochastic matrices, which are the classical channels having the
same sized input as output, admit a quotient compact ordered space which is a
domain, and the capacity map factors through this quotient via a
Scott-continuous map that measures the quotient domain. We also comment on how
some of our results relate to recent discoveries about quantum channels and
free affine monoids.Comment: In Proceedings DCM 2011, arXiv:1207.682
A Recipe for State-and-Effect Triangles
In the semantics of programming languages one can view programs as state
transformers, or as predicate transformers. Recently the author has introduced
state-and-effect triangles which capture this situation categorically,
involving an adjunction between state- and predicate-transformers. The current
paper exploits a classical result in category theory, part of Jon Beck's
monadicity theorem, to systematically construct such a state-and-effect
triangle from an adjunction. The power of this construction is illustrated in
many examples, covering many monads occurring in program semantics, including
(probabilistic) power domains
The semantics of jitter in anticipating time itself within nano-technology
The development of nano-technology calls for a careful examination of anticipatory systems at this small scale. For the characteristics of time at the boundary between classical and quantum domains are quite critical for the advancement of the new technology. It has long been well recognised that time is not absolute even in classical subjects like navigation and dynamics where idealised concepts like mean solar time, International Atomic Time and Newtonâs dynamical time have had to be used. Time is the data of the Universe and belongs in the semantics of its extensional form. At the boundary between classical and quantum behaviour the uncertainty of time data becomes a significant effect and this is why it is of great importance in nanotechnology, in areas such as the interoperability of different time domains in hardware, where noise in the form of jitter causes a system to behave in an unpredictable fashion, a severe and expensive problem for anticipating how time is to be handled. A fundamental difficulty is that jitter is represented using numbers, giving rise to undecidability and incompleteness according to Gödelâs theorems. To escape the clutches of Gödel undecidability it is necessary to advance to cartesian closed categories beyond the category of sets to represent the relationship between different times as adjoint endofunctors in monad and comonad constructions
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
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
Abstract Hidden Markov Models: a monadic account of quantitative information flow
Hidden Markov Models, HMM's, are mathematical models of Markov processes with
state that is hidden, but from which information can leak. They are typically
represented as 3-way joint-probability distributions.
We use HMM's as denotations of probabilistic hidden-state sequential
programs: for that, we recast them as `abstract' HMM's, computations in the
Giry monad , and we equip them with a partial order of increasing
security. However to encode the monadic type with hiding over some state
we use rather
than the conventional that suffices for
Markov models whose state is not hidden. We illustrate the
construction with a small
Haskell prototype.
We then present uncertainty measures as a generalisation of the extant
diversity of probabilistic entropies, with characteristic analytic properties
for them, and show how the new entropies interact with the order of increasing
security. Furthermore, we give a `backwards' uncertainty-transformer semantics
for HMM's that is dual to the `forwards' abstract HMM's - it is an analogue of
the duality between forwards, relational semantics and backwards,
predicate-transformer semantics for imperative programs with demonic choice.
Finally, we argue that, from this new denotational-semantic viewpoint, one
can see that the Dalenius desideratum for statistical databases is actually an
issue in compositionality. We propose a means for taking it into account
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