15,375 research outputs found
Relational Graph Models at Work
We study the relational graph models that constitute a natural subclass of
relational models of lambda-calculus. We prove that among the lambda-theories
induced by such models there exists a minimal one, and that the corresponding
relational graph model is very natural and easy to construct. We then study
relational graph models that are fully abstract, in the sense that they capture
some observational equivalence between lambda-terms. We focus on the two main
observational equivalences in the lambda-calculus, the theory H+ generated by
taking as observables the beta-normal forms, and H* generated by considering as
observables the head normal forms. On the one hand we introduce a notion of
lambda-K\"onig model and prove that a relational graph model is fully abstract
for H+ if and only if it is extensional and lambda-K\"onig. On the other hand
we show that the dual notion of hyperimmune model, together with
extensionality, captures the full abstraction for H*
No solvable lambda-value term left behind
In the lambda calculus a term is solvable iff it is operationally relevant.
Solvable terms are a superset of the terms that convert to a final result
called normal form. Unsolvable terms are operationally irrelevant and can be
equated without loss of consistency. There is a definition of solvability for
the lambda-value calculus, called v-solvability, but it is not synonymous with
operational relevance, some lambda-value normal forms are unsolvable, and
unsolvables cannot be consistently equated. We provide a definition of
solvability for the lambda-value calculus that does capture operational
relevance and such that a consistent proof-theory can be constructed where
unsolvables are equated attending to the number of arguments they take (their
"order" in the jargon). The intuition is that in lambda-value the different
sequentialisations of a computation can be distinguished operationally. We
prove a version of the Genericity Lemma stating that unsolvable terms are
generic and can be replaced by arbitrary terms of equal or greater order.Comment: 43 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
Principal Typings in a Restricted Intersection Type System for Beta Normal Forms with De Bruijn Indices
The lambda-calculus with de Bruijn indices assembles each alpha-class of
lambda-terms in a unique term, using indices instead of variable names.
Intersection types provide finitary type polymorphism and can characterise
normalisable lambda-terms through the property that a term is normalisable if
and only if it is typeable. To be closer to computations and to simplify the
formalisation of the atomic operations involved in beta-contractions, several
calculi of explicit substitution were developed mostly with de Bruijn indices.
Versions of explicit substitutions calculi without types and with simple type
systems are well investigated in contrast to versions with more elaborate type
systems such as intersection types. In previous work, we introduced a de Bruijn
version of the lambda-calculus with an intersection type system and proved that
it preserves subject reduction, a basic property of type systems. In this paper
a version with de Bruijn indices of an intersection type system originally
introduced to characterise principal typings for beta-normal forms is
presented. We present the characterisation in this new system and the
corresponding versions for the type inference and the reconstruction of normal
forms from principal typings algorithms. We briefly discuss the failure of the
subject reduction property and some possible solutions for it
Resource control and intersection types: an intrinsic connection
In this paper we investigate the -calculus, a -calculus
enriched with resource control. Explicit control of resources is enabled by the
presence of erasure and duplication operators, which correspond to thinning and
con-traction rules in the type assignment system. We introduce directly the
class of -terms and we provide a new treatment of substitution by its
decompo-sition into atomic steps. We propose an intersection type assignment
system for -calculus which makes a clear correspondence between three
roles of variables and three kinds of intersection types. Finally, we provide
the characterisation of strong normalisation in -calculus by means of
an in-tersection type assignment system. This process uses typeability of
normal forms, redex subject expansion and reducibility method.Comment: arXiv admin note: substantial text overlap with arXiv:1306.228
Strong normalisation for applied lambda calculi
We consider the untyped lambda calculus with constructors and recursively
defined constants. We construct a domain-theoretic model such that any term not
denoting bottom is strongly normalising provided all its `stratified
approximations' are. From this we derive a general normalisation theorem for
applied typed lambda-calculi: If all constants have a total value, then all
typeable terms are strongly normalising. We apply this result to extensions of
G\"odel's system T and system F extended by various forms of bar recursion for
which strong normalisation was hitherto unknown.Comment: 14 pages, paper acceptet at electronic journal LMC
Normalization by Evaluation for Call-by-Push-Value and Polarized Lambda-Calculus
We observe that normalization by evaluation for simply-typed lambda-calculus
with weak coproducts can be carried out in a weak bi-cartesian closed category
of presheaves equipped with a monad that allows us to perform case distinction
on neutral terms of sum type. The placement of the monad influences the normal
forms we obtain: for instance, placing the monad on coproducts gives us
eta-long beta-pi normal forms where pi refers to permutation of case
distinctions out of elimination positions. We further observe that placing the
monad on every coproduct is rather wasteful, and an optimal placement of the
monad can be determined by considering polarized simple types inspired by
focalization. Polarization classifies types into positive and negative, and it
is sufficient to place the monad at the embedding of positive types into
negative ones. We consider two calculi based on polarized types: pure
call-by-push-value (CBPV) and polarized lambda-calculus, the natural deduction
calculus corresponding to focalized sequent calculus. For these two calculi, we
present algorithms for normalization by evaluation. We further discuss
different implementations of the monad and their relation to existing
normalization proofs for lambda-calculus with sums. Our developments have been
partially formalized in the Agda proof assistant
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