14 research outputs found
An Explicit Framework for Interaction Nets
Interaction nets are a graphical formalism inspired by Linear Logic
proof-nets often used for studying higher order rewriting e.g. \Beta-reduction.
Traditional presentations of interaction nets are based on graph theory and
rely on elementary properties of graph theory. We give here a more explicit
presentation based on notions borrowed from Girard's Geometry of Interaction:
interaction nets are presented as partial permutations and a composition of
nets, the gluing, is derived from the execution formula. We then define
contexts and reduction as the context closure of rules. We prove strong
confluence of the reduction within our framework and show how interaction nets
can be viewed as the quotient of some generalized proof-nets
On the functor l^2
We study the functor l^2 from the category of partial injections to the
category of Hilbert spaces. The former category is finitely accessible, and its
homsets are algebraic domains; the latter category has conditionally algebraic
domains for homsets. The functor preserves daggers, monoidal structures,
enrichment, and various (co)limits, but has no adjoints. Up to unitaries, its
direct image consists precisely of the partial isometries, but its essential
image consists of all continuous linear maps between Hilbert spaces.Comment: 13 pages; updated Proposition 2.1
Observational Equivalence and Full Abstraction in the Symmetric Interaction Combinators
The symmetric interaction combinators are an equally expressive variant of
Lafont's interaction combinators. They are a graph-rewriting model of
deterministic computation. We define two notions of observational equivalence
for them, analogous to normal form and head normal form equivalence in the
lambda-calculus. Then, we prove a full abstraction result for each of the two
equivalences. This is obtained by interpreting nets as certain subsets of the
Cantor space, called edifices, which play the same role as Boehm trees in the
theory of the lambda-calculus
A Correspondence between Maximal Abelian Sub-Algebras and Linear Logic Fragments
We show a correspondence between a classification of maximal abelian
sub-algebras (MASAs) proposed by Jacques Dixmier and fragments of linear logic.
We expose for this purpose a modified construction of Girard's hyperfinite
geometry of interaction which interprets proofs as operators in a von Neumann
algebra. The expressivity of the logic soundly interpreted in this model is
dependent on properties of a MASA which is a parameter of the interpretation.
We also unveil the essential role played by MASAs in previous geometry of
interaction constructions
On the Resolution Semiring
In this thesis, we study a semiring structure with a product based on theresolution rule of logic programming. This mathematical object was introducedinitially in the setting of the geometry of interaction program in order to modelthe cut-elimination procedure of linear logic. It provides us with an algebraicand abstract setting, while being presented in a syntactic and concrete way, inwhich a theoretical study of computation can be carried on.We will review first the interactive interpretation of proof theory withinthis semiring via the categorical axiomatization of the geometry of interactionapproach. This interpretation establishes a way to translate functional programsinto a very simple form of logic programs.Secondly, complexity theory problematics will be considered: while thenilpotency problem in the semiring we study is undecidable in general, it willappear that certain restrictions allow for characterizations of (deterministicand non-deterministic) logarithmic space and (deterministic) polynomial timecomputation
Denotational validation of higher-order Bayesian inference
We present a modular semantic account of Bayesian inference algorithms for
probabilistic programming languages, as used in data science and machine
learning. Sophisticated inference algorithms are often explained in terms of
composition of smaller parts. However, neither their theoretical justification
nor their implementation reflects this modularity. We show how to conceptualise
and analyse such inference algorithms as manipulating intermediate
representations of probabilistic programs using higher-order functions and
inductive types, and their denotational semantics. Semantic accounts of
continuous distributions use measurable spaces. However, our use of
higher-order functions presents a substantial technical difficulty: it is
impossible to define a measurable space structure over the collection of
measurable functions between arbitrary measurable spaces that is compatible
with standard operations on those functions, such as function application. We
overcome this difficulty using quasi-Borel spaces, a recently proposed
mathematical structure that supports both function spaces and continuous
distributions. We define a class of semantic structures for representing
probabilistic programs, and semantic validity criteria for transformations of
these representations in terms of distribution preservation. We develop a
collection of building blocks for composing representations. We use these
building blocks to validate common inference algorithms such as Sequential
Monte Carlo and Markov Chain Monte Carlo. To emphasize the connection between
the semantic manipulation and its traditional measure theoretic origins, we use
Kock's synthetic measure theory. We demonstrate its usefulness by proving a
quasi-Borel counterpart to the Metropolis-Hastings-Green theorem