16 research outputs found
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
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
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
The Vectorial -Calculus
We describe a type system for the linear-algebraic -calculus. The
type system accounts for the linear-algebraic aspects of this extension of
-calculus: it is able to statically describe the linear combinations
of terms that will be obtained when reducing the programs. This gives rise to
an original type theory where types, in the same way as terms, can be
superposed into linear combinations. We prove that the resulting typed
-calculus is strongly normalising and features weak subject reduction.
Finally, we show how to naturally encode matrices and vectors in this typed
calculus.Comment: Long and corrected version of arXiv:1012.4032 (EPTCS 88:1-15), to
appear in Information and Computatio
Lineal: A linear-algebraic Lambda-calculus
We provide a computational definition of the notions of vector space and
bilinear functions. We use this result to introduce a minimal language
combining higher-order computation and linear algebra. This language extends
the Lambda-calculus with the possibility to make arbitrary linear combinations
of terms alpha.t + beta.u. We describe how to "execute" this language in terms
of a few rewrite rules, and justify them through the two fundamental
requirements that the language be a language of linear operators, and that it
be higher-order. We mention the perspectives of this work in the field of
quantum computation, whose circuits we show can be easily encoded in the
calculus. Finally, we prove the confluence of the entire calculus.Comment: The complementary note "On the critical pairs of a rewrite system for
vector spaces" is provided in the source files. Short version :
"Linear-algebraic Lambda-calculus : higher-order and confluence", Proceedings
of RTA 08, Hagenberg, July 2008. LNCS 5117, 17, (2008). Long version : LMC
Lineal: A linear-algebraic lambda-calculus
International audienceWe provide a computational de nition of the notions of vector space and bilinear functions. We use this result to introduce a minimal language combining higher-order computation and linear algebra. This language extends the lambda-calculus with the possibility to make arbitrary linear combinations of terms : alpha t + beta u. We describe how to \execute" this language in terms of a few rewrite rules, and justify them through the two fundamental requirements that the language be a language of linear operators, and that it be higher-order. We mention the perspectives of this work in the eld of quantum computation, whose circuits we show can be easily encoded in the calculus. Finally, we prove the confluence of the entire calculus
Weighted models for higher-order computation
We study a class of quantitative models for higher-order computation: Lafont categories with (infinite) biproducts. Each of these has a complete “internal semiring” and can be enriched over its modules. We describe a semantics of nondeterministic PCF weighted over this semiring in which fixed points are obtained from the bifree algebra over its exponential structure. By characterizing them concretely as infinite sums of approximants indexed over nested finite multisets, we prove computational adequacy. We can construct examples of our semantics by weighting existing models such as categories of games over a complete semiring. This transition from qualitative to quantitative semantics is characterized as a “change of base” of enriched categories arising from a monoidal functor from coherence spaces to modules over a complete semiring. For example, the game semantics of Idealized Algol is coherence space enriched and thus gives rise to to a weighted model, which is fully abstract.</p
Call-by-value, call-by-name and the vectorial behaviour of the algebraic \lambda-calculus
We examine the relationship between the algebraic lambda-calculus, a fragment
of the differential lambda-calculus and the linear-algebraic lambda-calculus, a
candidate lambda-calculus for quantum computation. Both calculi are algebraic:
each one is equipped with an additive and a scalar-multiplicative structure,
and their set of terms is closed under linear combinations. However, the two
languages were built using different approaches: the former is a call-by-name
language whereas the latter is call-by-value; the former considers algebraic
equalities whereas the latter approaches them through rewrite rules. In this
paper, we analyse how these different approaches relate to one another. To this
end, we propose four canonical languages based on each of the possible choices:
call-by-name versus call-by-value, algebraic equality versus algebraic
rewriting. We show that the various languages simulate one another. Due to
subtle interaction between beta-reduction and algebraic rewriting, to make the
languages consistent some additional hypotheses such as confluence or
normalisation might be required. We carefully devise the required properties
for each proof, making them general enough to be valid for any sub-language
satisfying the corresponding properties
An introduction to Differential Linear Logic: proof-nets, models and antiderivatives
Differential Linear Logic enriches Linear Logic with additional logical rules
for the exponential connectives, dual to the usual rules of dereliction,
weakening and contraction. We present a proof-net syntax for Differential
Linear Logic and a categorical axiomatization of its denotational models. We
also introduce a simple categorical condition on these models under which a
general antiderivative operation becomes available. Last we briefly describe
the model of sets and relations and give a more detailed account of the model
of finiteness spaces and linear and continuous functions