80 research outputs found
Probabilistic Rewriting: Normalization, Termination, and Unique Normal Forms
While a mature body of work supports the study of rewriting systems, abstract tools for Probabilistic Rewriting are still limited. We study in this setting questions such as uniqueness of the result (unique limit distribution) and normalizing strategies (is there a strategy to find a result with greatest probability?). The goal is to have tools to analyse the operational properties of probabilistic calculi (such as probabilistic lambda-calculi) whose evaluation is also non-deterministic, in the sense that different reductions are possible
Probabilistic Rewriting: On Normalization, Termination, and Unique Normal Forms
While a mature body of work supports the study of rewriting systems, even
infinitary ones, abstract tools for Probabilistic Rewriting are still limited.
Here, we investigate questions such as uniqueness of the result (unique limit
distribution) and we develop a set of proof techniques to analyze and compare
reduction strategies. The goal is to have tools to support the operational
analysis of probabilistic calculi (such as probabilistic lambda-calculi) whose
evaluation is also non-deterministic, in the sense that different reductions
are possible.
In particular, we investigate how the behavior of different rewrite sequences
starting from the same term compare w.r.t. normal forms, and propose a robust
analogue of the notion of "unique normal form". Our approach is that of
Abstract Rewrite Systems, i.e. we search for general properties of
probabilistic rewriting, which hold independently of the specific structure of
the objects.Comment: Extended version of the paper in FSCD 2019, International Conference
on Formal Structures for Computation and Deductio
Interactive observability in Ludics: The geometry of tests
AbstractLudics [J.-Y. Girard, Locus solum, Math. Structures in Comput. Sci. 11 (2001) 301–506] is a recent proposal of analysis of interaction, developed by abstracting away from proof-theory. It provides an elegant, abstract setting in which interaction between agents (proofs/programs/processes) can be studied at a foundational level, together with a notion of equivalence from the point of view of the observer.An agent should be seen as some kind of black box. An interactive observation on an agent is obtained by testing it against other agents.In this paper we explore what can be observed interactively in this setting. In particular, we characterize the objects that can be observed in a single test: the primitive observables of the theory.Our approach builds on an analysis of the geometrical properties of the agents, and highlights a deep interleaving between two partial orders underlying the combinatorial structures: the spatial one and the temporal one
The Geometry of Synchronization (Long Version)
We graft synchronization onto Girard's Geometry of Interaction in its most
concrete form, namely token machines. This is realized by introducing
proof-nets for SMLL, an extension of multiplicative linear logic with a
specific construct modeling synchronization points, and of a multi-token
abstract machine model for it. Interestingly, the correctness criterion ensures
the absence of deadlocks along reduction and in the underlying machine, this
way linking logical and operational properties.Comment: 26 page
Factorization in Call-by-Name and Call-by-Value Calculi via Linear Logic
In each variant of the λ-calculus, factorization and normalization are two key properties that show how results are computed.Instead of proving factorization/normalization for the call-by-name (CbN) and call-by-value (CbV) variants separately, we prove them only once, for the bang calculus (an extension of the λ-calculus inspired by linear logic and subsuming CbN and CbV), and then we transfer the result via translations, obtaining factorization/normalization for CbN and CbV.The approach is robust: it still holds when extending the calculi with operators and extra rules to model some additional computational features
Factorization in Call-by-Name and Call-by-Value Calculi via Linear Logic
International audienceAbstract In each variant of the λ -calculus, factorization and normalization are two key properties that show how results are computed. Instead of proving factorization/normalization for the call-by-name (CbN) and call-by-value (CbV) variants separately, we prove them only once, for the bang calculus (an extension of the λ -calculus inspired by linear logic and subsuming CbN and CbV), and then we transfer the result via translations, obtaining factorization/normalization for CbN and CbV. The approach is robust: it still holds when extending the calculi with operators and extra rules to model some additional computational features
Probabilistic Rewriting and Asymptotic Behaviour: on Termination and Unique Normal Forms
While a mature body of work supports the study of rewriting systems, abstract
tools for Probabilistic Rewriting are still limited. In this paper we study the
question of uniqueness of the result (unique limit distribution), and develop a
set of proof techniques to analyze and compare reduction strategies. The goal
is to have tools to support the operational analysis of probabilistic calculi
(such as probabilistic lambda-calculi) where evaluation allows for different
reduction choices (hence different reduction paths)
The Sum-Product Algorithm For Quantitative Multiplicative Linear Logic
We consider an extension of multiplicative linear logic which encompasses bayesian networks and expresses samples sharing and marginalisation with the polarised rules of contraction and weakening. We introduce the necessary formalism to import exact inference algorithms from bayesian networks, giving the sum-product algorithm as an example of calculating the weighted relational semantics of a multiplicative proof-net improving runtime performance by storing intermediate results
Normal Form Bisimulations By Value
Normal form bisimilarities are a natural form of program equivalence resting
on open terms, first introduced by Sangiorgi in call-by-name. The literature
contains a normal form bisimilarity for Plotkin's call-by-value
-calculus, Lassen's \emph{enf bisimilarity}, which validates all of
Moggi's monadic laws and can be extended to validate . It does not
validate, however, other relevant principles, such as the identification of
meaningless terms -- validated instead by Sangiorgi's bisimilarity -- or the
commutation of \letexps. These shortcomings are due to issues with open terms
of Plotkin's calculus. We introduce a new call-by-value normal form
bisimilarity, deemed \emph{net bisimilarity}, closer in spirit to Sangiorgi's
and satisfying the additional principles. We develop it on top of an existing
formalism designed for dealing with open terms in call-by-value. It turns out
that enf and net bisimilarities are \emph{incomparable}, as net bisimilarity
does not validate Moggi's laws nor . Moreover, there is no easy way to
merge them. To better understand the situation, we provide an analysis of the
rich range of possible call-by-value normal form bisimilarities, relating them
to Ehrhard's relational model.Comment: Rewritten version (deleted toy similarity and explained proof method
on naive similarity) -- Submitted to POPL2
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