49 research outputs found
Normalizing the Taylor expansion of non-deterministic {\lambda}-terms, via parallel reduction of resource vectors
It has been known since Ehrhard and Regnier's seminal work on the Taylor
expansion of -terms that this operation commutes with normalization:
the expansion of a -term is always normalizable and its normal form is
the expansion of the B\"ohm tree of the term. We generalize this result to the
non-uniform setting of the algebraic -calculus, i.e.
-calculus extended with linear combinations of terms. This requires us
to tackle two difficulties: foremost is the fact that Ehrhard and Regnier's
techniques rely heavily on the uniform, deterministic nature of the ordinary
-calculus, and thus cannot be adapted; second is the absence of any
satisfactory generic extension of the notion of B\"ohm tree in presence of
quantitative non-determinism, which is reflected by the fact that the Taylor
expansion of an algebraic -term is not always normalizable. Our
solution is to provide a fine grained study of the dynamics of
-reduction under Taylor expansion, by introducing a notion of reduction
on resource vectors, i.e. infinite linear combinations of resource
-terms. The latter form the multilinear fragment of the differential
-calculus, and resource vectors are the target of the Taylor expansion
of -terms. We show the reduction of resource vectors contains the
image of any -reduction step, from which we deduce that Taylor expansion
and normalization commute on the nose. We moreover identify a class of
algebraic -terms, encompassing both normalizable algebraic
-terms and arbitrary ordinary -terms: the expansion of these
is always normalizable, which guides the definition of a generalization of
B\"ohm trees to this setting
On linear combinations of lambda-terms
International audienceWe define an extension of lambda-calculus with linear combinations, endowing the set of terms with a structure of R-module, where R is a fixed set of scalars. Terms are moreover subject to identities similar to usual pointwise definition of linear combinations of functions with values in a vector space. We then extend beta-reduction on those algebraic lambda-terms as follows: at + u reduces to at + u as soon as term t reduces to t and a is a non-zero scalar. We prove that reduction is confluent. Under the assumption that the set R of scalars is positive (i.e. a sum of scalars is zero iff all of them are zero), we show that this algebraic lambda-calculus is a conservative extension of ordinary lambda-calculus. On the other hand, we show that if R admits negative elements, then every term reduces to every other term. We investigate the causes of that collapse, and discuss some possible fixes
The algebraic -calculus is a conservative extension of the ordinary -calculus
The algebraic -calculus is an extension of the ordinary
-calculus with linear combinations of terms. We establish that two
ordinary -terms are equivalent in the algebraic -calculus iff
they are -equal. Although this result was originally stated in the early
2000's (in the setting of Ehrhard and Regnier's differential
-calculus), the previously proposed proofs were wrong: we explain why
previous approaches failed and develop a new proof technique to establish
conservativity
Strategies as Resource Terms, and Their Categorical Semantics
As shown by Tsukada and Ong, simply-typed, normal and η-long resource terms correspond to plays in Hyland-Ong games, quotiented by Melliès' homotopy equivalence. Though inspiring, their proof is indirect, relying on the injectivity of the relational model {w.r.t.} both sides of the correspondence - in particular, the dynamics of the resource calculus is taken into account only via the compatibility of the relational model with the composition of normal terms defined by normalization. In the present paper, we revisit and extend these results. Our first contribution is to restate the correspondence by considering causal structures we call augmentations, which are canonical representatives of Hyland-Ong plays up to homotopy. This allows us to give a direct and explicit account of the connection with normal resource terms. As a second contribution, we extend this account to the reduction of resource terms: building on a notion of strategies as weighted sums of augmentations, we provide a denotational model of the resource calculus, invariant under reduction. A key step - and our third contribution - is a categorical model we call a resource category, which is to the resource calculus what differential categories are to the differential λ-calculus
Completeness of algebraic CPS simulations
The algebraic lambda calculus and the linear algebraic lambda calculus are
two extensions of the classical lambda calculus with linear combinations of
terms. They arise independently in distinct contexts: the former is a fragment
of the differential lambda calculus, the latter is a candidate lambda calculus
for quantum computation. They differ in the handling of application arguments
and algebraic rules. The two languages can simulate each other using an
algebraic extension of the well-known call-by-value and call-by-name CPS
translations. These simulations are sound, in that they preserve reductions. In
this paper, we prove that the simulations are actually complete, strengthening
the connection between the two languages.Comment: In Proceedings DCM 2011, arXiv:1207.682
Confluence via strong normalisation in an algebraic \lambda-calculus with rewriting
The linear-algebraic lambda-calculus and the algebraic lambda-calculus are
untyped lambda-calculi extended with arbitrary linear combinations of terms.
The former presents the axioms of linear algebra in the form of a rewrite
system, while the latter uses equalities. When given by rewrites, algebraic
lambda-calculi are not confluent unless further restrictions are added. We
provide a type system for the linear-algebraic lambda-calculus enforcing strong
normalisation, which gives back confluence. The type system allows an abstract
interpretation in System F.Comment: In Proceedings LSFA 2011, arXiv:1203.542
Semantics of a Typed Algebraic Lambda-Calculus
Algebraic lambda-calculi have been studied in various ways, but their
semantics remain mostly untouched. In this paper we propose a semantic analysis
of a general simply-typed lambda-calculus endowed with a structure of vector
space. We sketch the relation with two established vectorial lambda-calculi.
Then we study the problems arising from the addition of a fixed point
combinator and how to modify the equational theory to solve them. We sketch an
algebraic vectorial PCF and its possible denotational interpretations
A System F accounting for scalars
The Algebraic lambda-calculus and the Linear-Algebraic lambda-calculus extend
the lambda-calculus with the possibility of making arbitrary linear
combinations of terms. In this paper we provide a fine-grained, System F-like
type system for the linear-algebraic lambda-calculus. We show that this
"scalar" type system enjoys both the subject-reduction property and the
strong-normalisation property, our main technical results. The latter yields a
significant simplification of the linear-algebraic lambda-calculus itself, by
removing the need for some restrictions in its reduction rules. But the more
important, original feature of this scalar type system is that it keeps track
of 'the amount of a type' that is present in each term. As an example of its
use, we shown that it can serve as a guarantee that the normal form of a term
is barycentric, i.e that its scalars are summing to one
An application of parallel cut elimination in multiplicative linear logic to the Taylor expansion of proof nets
We examine some combinatorial properties of parallel cut elimination in
multiplicative linear logic (MLL) proof nets. We show that, provided we impose
a constraint on some paths, we can bound the size of all the nets satisfying
this constraint and reducing to a fixed resultant net. This result gives a
sufficient condition for an infinite weighted sum of nets to reduce into
another sum of nets, while keeping coefficients finite. We moreover show that
our constraints are stable under reduction.
Our approach is motivated by the quantitative semantics of linear logic: many
models have been proposed, whose structure reflect the Taylor expansion of
multiplicative exponential linear logic (MELL) proof nets into infinite sums of
differential nets. In order to simulate one cut elimination step in MELL, it is
necessary to reduce an arbitrary number of cuts in the differential nets of its
Taylor expansion. It turns out our results apply to differential nets, because
their cut elimination is essentially multiplicative. We moreover show that the
set of differential nets that occur in the Taylor expansion of an MELL net
automatically satisfies our constraints.
Interestingly, our nets are untyped: we only rely on the sequentiality of
linear logic nets and the dynamics of cut elimination. The paths on which we
impose bounds are the switching paths involved in the Danos--Regnier criterion
for sequentiality. In order to accommodate multiplicative units and weakenings,
our nets come equipped with jumps: each weakening node is connected to some
other node. Our constraint can then be summed up as a bound on both the length
of switching paths, and the number of weakenings that jump to a common node
Sur la syntaxe de la sémantique quantitative
Le développement de Taylor des λ-termes et des preuves de la logique linéaire est le fruit d’une relecture syntaxique par Ehrhard et Regnier de la sémantique quantitative de Girard : il associe à chaque terme ou preuve une combinaison linéaire infinie d’approximations multilinéaires et finies de l’objet de départ. Il matérialise une correspondance étroite entre le comportement calculatoire des termes, défini par la β-réduction, et leur interprétation dans certains modèles dénotationnels : le développement de Taylor d’un terme est toujours normalisable, et sa forme normale correspond exactement à l’arbre de Böhm du terme. Cette correspondance se retrouve dans le fait que, pour de nombreux modèles de la logique linéaire, la promotion d’un morphisme s’obtient comme une superposition d’opérations multilinéaires, faisant du développement de Taylor des preuves une structure sous-jacente de ces modèles.Ce mémoire présente quelques avancées récentes visant à raffiner l’analyse de la normalisation (qui est un processus potentiellement infini) offerte par le développement de Taylor pour la ramener au niveau de la β-réduction ou de l’élimination des coupures (qui correspond à un calcul fini).On démontre que cette approche permet d’étendre l’analyse à un cadre non-uniforme, susceptible de prendre en compte par exemple une forme de non-déterminisme calculatoire — alors que la normalisation peut échouer dans ce cadre. On démontre également que la même approche peut être appliquée aux réseaux de démonstration de la logique linéaire. Enfin les techniques développées précédemment permettent de revisiter et simplifier le résultat originel d’Ehrhard et Regnier pour la normalisation dans le cas uniforme, tout en l’adaptant à une forme restreinte de non-déterminisme