8,901 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
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
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
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
Linear-algebraic lambda-calculus
With a view towards models of quantum computation and/or the interpretation
of linear logic, we define a functional language where all functions are linear
operators by construction. A small step operational semantic (and hence an
interpreter/simulator) is provided for this language in the form of a term
rewrite system. The linear-algebraic lambda-calculus hereby constructed is
linear in a different (yet related) sense to that, say, of the linear
lambda-calculus. These various notions of linearity are discussed in the
context of quantum programming languages. KEYWORDS: quantum lambda-calculus,
linear lambda-calculus, -calculus, quantum logics.Comment: LaTeX, 23 pages, 10 figures and the LINEAL language
interpreter/simulator file (see "other formats"). See the more recent
arXiv:quant-ph/061219
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
Two linearities for quantum computing in the lambda calculus
We propose a way to unify two approaches of non-cloning in quantum
lambda-calculi: logical and algebraic linearities. The first approach is to
forbid duplicating variables, while the second is to consider all lambda-terms
as algebraic-linear functions. We illustrate this idea by defining a quantum
extension of first-order simply-typed lambda-calculus, where the type is linear
on superposition, while allows cloning base vectors. In addition, we provide an
interpretation of the calculus where superposed types are interpreted as vector
spaces and non-superposed types as their basis.Comment: Long journal version of TPNC'17 paper
(doi:10.1007/978-3-319-71069-3_22) extended with third author's
"Licenciatura"'s thesi
Polymorphic Rewriting Conserves Algebraic Strong Normalization
We study combinations of many-sorted algebraic term rewriting systems and polymorphic lambda term rewriting. Algebraic and lambda terms are mixed by adding the symbols of the algebraic signature to the polymorphic lambda calculus, as higher-order constants. We show that if a many-sorted algebraic rewrite system R is strongly normalizing (terminating, noetherian), then R + β + η + type-η rewriting of mixed terms is also strongly normalizing. The result is obtained using a technique which generalizes Girard\u27s candidats de reductibilité , introduced in the original proof of strong normalization for the polymorphic lambda calculus
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
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