17 research outputs found
Bounding normalization time through intersection types
Non-idempotent intersection types are used in order to give a bound of the
length of the normalization beta-reduction sequence of a lambda term: namely,
the bound is expressed as a function of the size of the term.Comment: In Proceedings ITRS 2012, arXiv:1307.784
Call-by-value non-determinism in a linear logic type discipline
We consider the call-by-value lambda-calculus extended with a may-convergent
non-deterministic choice and a must-convergent parallel composition. Inspired
by recent works on the relational semantics of linear logic and non-idempotent
intersection types, we endow this calculus with a type system based on the
so-called Girard's second translation of intuitionistic logic into linear
logic. We prove that a term is typable if and only if it is converging, and
that its typing tree carries enough information to give a bound on the length
of its lazy call-by-value reduction. Moreover, when the typing tree is minimal,
such a bound becomes the exact length of the reduction
The probability of non-confluent systems
We show how to provide a structure of probability space to the set of
execution traces on a non-confluent abstract rewrite system, by defining a
variant of a Lebesgue measure on the space of traces. Then, we show how to use
this probability space to transform a non-deterministic calculus into a
probabilistic one. We use as example Lambda+, a recently introduced calculus
defined through type isomorphisms.Comment: In Proceedings DCM 2013, arXiv:1403.768
On the discriminating power of tests in resource lambda-calculus
Since its discovery, differential linear logic (DLL) inspired numerous
domains. In denotational semantics, categorical models of DLL are now commune,
and the simplest one is Rel, the category of sets and relations. In proof
theory this naturally gave birth to differential proof nets that are full and
complete for DLL. In turn, these tools can naturally be translated to their
intuitionistic counterpart. By taking the co-Kleisly category associated to the
! comonad, Rel becomes MRel, a model of the \Lcalcul that contains a notion of
differentiation. Proof nets can be used naturally to extend the \Lcalcul into
the lambda calculus with resources, a calculus that contains notions of
linearity and differentiations. Of course MRel is a model of the \Lcalcul with
resources, and it has been proved adequate, but is it fully abstract? That was
a strong conjecture of Bucciarelli, Carraro, Ehrhard and Manzonetto. However,
in this paper we exhibit a counter-example. Moreover, to give more intuition on
the essence of the counter-example and to look for more generality, we will use
an extension of the resource \Lcalcul also introduced by Bucciarelli et al for
which \Minf is fully abstract, the tests
Call-by-Value solvability, revisited
International audienceIn the call-by-value lambda-calculus solvable terms have been characterised by means of call-by-name reductions, which is disappointing and requires complex reasonings. We introduce the value substitution lambda-calculus, a simple calculus borrowing ideas from Herbelin and Zimmerman's call-by-value lambda-CBV calculus and from Accattoli and Kesner's substitution calculus lambda-sub. In this new setting, we characterise solvable terms as those terms having normal form with respect to a suitable restriction of the rewriting relation
Proof Normalisation in a Logic Identifying Isomorphic Propositions
We define a fragment of propositional logic where isomorphic propositions,
such as and , or and
are identified. We define System I, a
proof language for this logic, and prove its normalisation and consistency
Inhabitation for Non-idempotent Intersection Types
The inhabitation problem for intersection types in the lambda-calculus is
known to be undecidable. We study the problem in the case of non-idempotent
intersection, considering several type assignment systems, which characterize
the solvable or the strongly normalizing lambda-terms. We prove the
decidability of the inhabitation problem for all the systems considered, by
providing sound and complete inhabitation algorithms for them
Extensional proofs in a propositional logic modulo isomorphisms
System I is a proof language for a fragment of propositional logic where
isomorphic propositions, such as and , or
and are made
equal. System I enjoys the strong normalisation property. This is sufficient to
prove the existence of empty types, but not to prove the introduction property
(every closed term in normal form is an introduction). Moreover, a severe
restriction had to be made on the types of the variables in order to obtain the
existence of empty types. We show here that adding -expansion rules to
System I permits to drop this restriction, and yields a strongly normalising
calculus with enjoying the full introduction property.Comment: 15 pages plus references and appendi
Call-by-Value solvability, revisited
International audienceIn the call-by-value lambda-calculus solvable terms have been characterised by means of call-by-name reductions, which is disappointing and requires complex reasonings. We introduce the value substitution lambda-calculus, a simple calculus borrowing ideas from Herbelin and Zimmerman's call-by-value lambda-CBV calculus and from Accattoli and Kesner's substitution calculus lambda-sub. In this new setting, we characterise solvable terms as those terms having normal form with respect to a suitable restriction of the rewriting relation