56 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 eta-long resource terms
correspond to plays in Hyland-Ong games, quotiented by Melli\`es' homotopy
equivalence. The original proof of this inspiring result 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
lambda-calculus.Comment: extended versio
Extensional Taylor Expansion
We introduce a calculus of extensional resource terms. These are resource
terms \`a la Ehrhard-Regnier, but in infinite -long form, while retaining
a finite syntax and dynamics: in particular, we prove strong confluence and
normalization.
Then we define an extensional version of Taylor expansion, mapping ordinary
-terms to sets (or infinite linear combinations) of extensional
resource terms: just like for ordinary Taylor expansion, the dynamics of our
resource calculus allows to simulate the -reduction of -terms;
the extensional nature of expansion shows in that we are also able to simulate
-reduction.
In a sense, extensional resource terms form a language of (non-necessarily
normal) finite approximants of Nakajima trees, much like ordinary resource
terms are approximants of B\"ohm-trees. Indeed, we show that the equivalence
induced on -terms by the normalization of extensional Taylor-expansion
is nothing but , the greatest consistent sensible -theory.
Taylor expansion has profoundly renewed the approximation theory of the
-calculus by providing a quantitative alternative to order-based
approximation techniques, such as Scott continuity and B\"ohm trees.
Extensional Taylor expansion enjoys similar advantages: e.g., to exhibit models
of , it is now sufficient to provide a model of the extensional resource
calculus. We apply this strategy to give a new, elementary proof of a result by
Manzonetto: is the -theory induced by a well-chosen reflexive
object in the relational model of the -calculus
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
Finitary Simulation of Infinitary -Reduction via Taylor Expansion, and Applications
Originating in Girard's Linear logic, Ehrhard and Regnier's Taylor expansion
of -terms has been broadly used as a tool to approximate the terms of
several variants of the -calculus. Many results arise from a
Commutation theorem relating the normal form of the Taylor expansion of a term
to its B\"ohm tree. This led us to consider extending this formalism to the
infinitary -calculus, since the version of
this calculus has B\"ohm trees as normal forms and seems to be the ideal
framework to reformulate the Commutation theorem.
We give a (co-)inductive presentation of . We define
a Taylor expansion on this calculus, and state that the infinitary
-reduction can be simulated through this Taylor expansion. The target
language is the usual resource calculus, and in particular the resource
reduction remains finite, confluent and terminating. Finally, we state the
generalised Commutation theorem and use our results to provide simple proofs of
some normalisation and confluence properties in the infinitary
-calculus
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
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
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