871 research outputs found
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
Representation and duality of the untyped lambda-calculus in nominal lattice and topological semantics, with a proof of topological completeness
We give a semantics for the lambda-calculus based on a topological duality
theorem in nominal sets. A novel interpretation of lambda is given in terms of
adjoints, and lambda-terms are interpreted absolutely as sets (no valuation is
necessary)
Algebraic totality, towards completeness
Finiteness spaces constitute a categorical model of Linear Logic (LL) whose
objects can be seen as linearly topologised spaces, (a class of topological
vector spaces introduced by Lefschetz in 1942) and morphisms as continuous
linear maps. First, we recall definitions of finiteness spaces and describe
their basic properties deduced from the general theory of linearly topologised
spaces. Then we give an interpretation of LL based on linear algebra. Second,
thanks to separation properties, we can introduce an algebraic notion of
totality candidate in the framework of linearly topologised spaces: a totality
candidate is a closed affine subspace which does not contain 0. We show that
finiteness spaces with totality candidates constitute a model of classical LL.
Finally, we give a barycentric simply typed lambda-calculus, with booleans
and a conditional operator, which can be interpreted in this
model. We prove completeness at type for
every n by an algebraic method
Deterministic stream-sampling for probabilistic programming: semantics and verification
Probabilistic programming languages rely fundamentally on some notion of
sampling, and this is doubly true for probabilistic programming languages which
perform Bayesian inference using Monte Carlo techniques. Verifying samplers -
proving that they generate samples from the correct distribution - is crucial
to the use of probabilistic programming languages for statistical modelling and
inference. However, the typical denotational semantics of probabilistic
programs is incompatible with deterministic notions of sampling. This is
problematic, considering that most statistical inference is performed using
pseudorandom number generators.
We present a higher-order probabilistic programming language centred on the
notion of samplers and sampler operations. We give this language an operational
and denotational semantics in terms of continuous maps between topological
spaces. Our language also supports discontinuous operations, such as
comparisons between reals, by using the type system to track discontinuities.
This feature might be of independent interest, for example in the context of
differentiable programming.
Using this language, we develop tools for the formal verification of sampler
correctness. We present an equational calculus to reason about equivalence of
samplers, and a sound calculus to prove semantic correctness of samplers, i.e.
that a sampler correctly targets a given measure by construction.Comment: Extended version of LiCS 2023 pape
06341 Abstracts Collection -- Computational Structures for Modelling Space, Time and Causality
From 20.08.06 to 25.08.06, the Dagstuhl Seminar 06341 ``Computational Structures for Modelling Space, Time and Causality\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Deterministic stream-sampling for probabilistic programming: semantics and verification
Probabilistic programming languages rely fundamentally on some notion of sampling, and this is doubly true for probabilistic programming languages which perform Bayesian inference using Monte Carlo techniques. Verifying samplers - proving that they generate samples from the correct distribution - is crucial to the use of probabilistic programming languages for statistical modelling and inference. However, the typical denotational semantics of probabilistic programs is incompatible with deterministic notions of sampling. This is problematic, considering that most statistical inference is performed using pseudorandom number generators.We present a higher-order probabilistic programming language centred on the notion of samplers and sampler operations. We give this language an operational and denotational semantics in terms of continuous maps between topological spaces. Our language also supports discontinuous operations, such as comparisons between reals, by using the type system to track discontinuities. This feature might be of independent interest, for example in the context of differentiable programming.Using this language, we develop tools for the formal verification of sampler correctness. We present an equational calculus to reason about equivalence of samplers, and a sound calculus to prove semantic correctness of samplers, i.e. that a sampler correctly targets a given measure by construction
Codensity Lifting of Monads and its Dual
We introduce a method to lift monads on the base category of a fibration to
its total category. This method, which we call codensity lifting, is applicable
to various fibrations which were not supported by its precursor, categorical
TT-lifting. After introducing the codensity lifting, we illustrate some
examples of codensity liftings of monads along the fibrations from the category
of preorders, topological spaces and extended pseudometric spaces to the
category of sets, and also the fibration from the category of binary relations
between measurable spaces. We also introduce the dual method called density
lifting of comonads. We next study the liftings of algebraic operations to the
codensity liftings of monads. We also give a characterisation of the class of
liftings of monads along posetal fibrations with fibred small meets as a limit
of a certain large diagram.Comment: Extended version of the paper presented at CALCO 2015, accepted for
publication in LMC
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