7,910 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
Full Abstraction for the Resource Lambda Calculus with Tests, through Taylor Expansion
We study the semantics of a resource-sensitive extension of the lambda
calculus in a canonical reflexive object of a category of sets and relations, a
relational version of Scott's original model of the pure lambda calculus. This
calculus is related to Boudol's resource calculus and is derived from Ehrhard
and Regnier's differential extension of Linear Logic and of the lambda
calculus. We extend it with new constructions, to be understood as implementing
a very simple exception mechanism, and with a "must" parallel composition.
These new operations allow to associate a context of this calculus with any
point of the model and to prove full abstraction for the finite sub-calculus
where ordinary lambda calculus application is not allowed. The result is then
extended to the full calculus by means of a Taylor Expansion formula. As an
intermediate result we prove that the exception mechanism is not essential in
the finite sub-calculus
On Differential Rota-Baxter Algebras
A Rota-Baxter operator of weight is an abstraction of both the
integral operator (when ) and the summation operator (when
). We similarly define a differential operator of weight
that includes both the differential operator (when ) and the
difference operator (when ). We further consider an algebraic
structure with both a differential operator of weight and a
Rota-Baxter operator of weight that are related in the same way that
the differential operator and the integral operator are related by the First
Fundamental Theorem of Calculus. We construct free objects in the corresponding
categories. In the commutative case, the free objects are given in terms of
generalized shuffles, called mixable shuffles. In the noncommutative case, the
free objects are given in terms of angularly decorated rooted forests. As a
byproduct, we obtain structures of a differential algebra on decorated and
undecorated planar rooted forests.Comment: 21 page
Noncommutative differential calculus for Moyal subalgebras
We build a differential calculus for subalgebras of the Moyal algebra on R^4
starting from a redundant differential calculus on the Moyal algebra, which is
suitable for reduction. In some cases we find a frame of 1-forms which allows
to realize the complex of forms as a tensor product of the noncommutative
subalgebras with the external algebra Lambda^*.Comment: 13 pages, no figures. One reference added, minor correction
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
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 coherent differential PCF
The categorical models of the differential lambda-calculus are additive
categories because of the Leibniz rule which requires the summation of two
expressions. This means that, as far as the differential lambda-calculus and
differential linear logic are concerned, these models feature finite
non-determinism and indeed these languages are essentially non-deterministic.
In a previous paper we introduced a categorical framework for differentiation
which does not require additivity and is compatible with deterministic models
such as coherence spaces and probabilistic models such as probabilistic
coherence spaces. Based on this semantics we develop a syntax of a
deterministic version of the differential lambda-calculus. One nice feature of
this new approach to differentiation is that it is compatible with general
fixpoints of terms, so our language is actually a differential extension of PCF
for which we provide a fully deterministic operational semantics
Modules over monads and operational semantics
This paper is a contribution to the search for efficient and high-level
mathematical tools to specify and reason about (abstract) programming languages
or calculi. Generalising the reduction monads of Ahrens et al., we introduce
transition monads, thus covering new applications such as
lambda-bar-mu-calculus, pi-calculus, Positive GSOS specifications, differential
lambda-calculus, and the big-step, simply-typed, call-by-value lambda-calculus.
Moreover, we design a suitable notion of signature for transition monads
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