Strong normalization of lambda-bar-mu-mu-tilde-calculus with explicit substitutions

International audienceThe lambda-bar-mu-mu-tilde-calculus, defined by Curien and Herbelin, is a variant of the lambda-mu-calculus that exhibits symmetries such as terms/contexts and call-by-name/call-by-value. Since it is a symmetric, and hence a non-deterministic calculus, usual proof techniques of normalization needs some adjustments to work in this setting. Here we prove the strong normalization (SN) of simply typed lambda-bar-mu-mu-tilde-calculus with explicit substitutions. For that purpose, we first prove SN of simply typed lambda-bar-mu-mu-tilde-calculus (by a variant of the reducibility technique from Barbanera and Berardi), then we formalize a proof technique of SN via PSN (preservation of strong normalization), and we prove PSN by the perpetuality technique, as formalized by Bonelli

THE FREQUENCIES OF HAPTOGLOBIN TYPES IN FIVE POPULATIONS *

Haptoglobin types have been determined by starch gel electrophoresis of blood from five populations. The gene frequencies obtained for allele Hp 1 were as follows: American whites, 043; American Negroes, 0.59; African Negroes, 0.72; Apaches, 0.59; and Asiatic Indians, 0.18. In tribes of the Ivory Coast and Liberia, there was a suggestion of a cline which parallels that for haemoglobin S. Evidence is presented that the condition of ahaptoglobinemia is under genetic control but not by a gene allelic to the Hp 1 -Hp 2 series. The importance of the ahaptoglobinemic individuals for genetic studies and the possibility of selection in the maintenance of the genetic polymorphism are discussed. The authors wish to acknowledge the excellent assistance of Alojzia Sandor, who carried out the electrophoretic separations.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/66263/1/j.1469-1809.1958.tb01460.x.pd

Towards a canonical classical natural deduction system

Preprint submitted to Elsevier, 6 July 2012This paper studies a new classical natural deduction system, presented as a typed calculus named lambda-mu- let. It is designed to be isomorphic to Curien and Herbelin's lambda-mu-mu~-calculus, both at the level of proofs and reduction, and the isomorphism is based on the correct correspondence between cut (resp. left-introduction) in sequent calculus, and substitution (resp. elimination) in natural deduction. It is a combination of Parigot's lambda-mu -calculus with the idea of "coercion calculus" due to Cervesato and Pfenning, accommodating let-expressions in a surprising way: they expand Parigot's syntactic class of named terms. This calculus and the mentioned isomorphism Theta offer three missing components of the proof theory of classical logic: a canonical natural deduction system; a robust process of "read-back" of calculi in the sequent calculus format into natural deduction syntax; a formalization of the usual semantics of the lambda-mu-mu~-calculus, that explains co-terms and cuts as, respectively, contexts and hole- filling instructions. lambda-mu-let is not yet another classical calculus, but rather a canonical reflection in natural deduction of the impeccable treatment of classical logic by sequent calculus; and provides the "read-back" map and the formalized semantics, based on the precise notions of context and "hole-expression" provided by lambda-mu-let. We use "read-back" to achieve a precise connection with Parigot's lambda-mu , and to derive lambda-calculi for call-by-value combining control and let-expressions in a logically founded way. Finally, the semantics , when fully developed, can be inverted at each syntactic category. This development gives us license to see sequent calculus as the semantics of natural deduction; and uncovers a new syntactic concept in lambda-mu-mu~ ("co-context"), with which one can give a new de nition of eta-reduction

Towards a canonical classical natural deduction system

This paper studies a new classical natural deduction system, presented as a typed calculus named \lml. It is designed to be isomorphic to Curien-Herbelin's calculus, both at the level of proofs and reduction, and the isomorphism is based on the correct correspondence between cut (resp. left-introduction) in sequent calculus, and substitution (resp. elimination) in natural deduction. It is a combination of Parigot's $\lambda\mu$-calculus with the idea of ``coercion calculus'' due to Cervesato-Pfenning, accommodating let-expressions in a surprising way: they expand Parigot's syntactic class of named terms. This calculus aims to be the simultaneous answer to three problems. The first problem is the lack of a canonical natural deduction system for classical logic. \lml is not yet another classical calculus, but rather a canonical reflection in natural deduction of the impeccable treatment of classical logic by sequent calculus. The second problem is the lack of a formalization of the usual semantics of Curien-Herbelin's calculus, that explains co-terms and cuts as, respectively, contexts and hole-filling instructions. The mentioned isomorphism is the required formalization, based on the precise notions of context and hole-expression offered by \lml. The third problem is the lack of a robust process of ``read-back'' into natural deduction syntax of calculi in the sequent calculus format, that affects mainly the recent proof-theoretic efforts of derivation of $\lambda$-calculi for call-by-value. An isomorphic counterpart to the $Q$-subsystem of Curien-Herbelin's-calculus is derived, obtaining a new $\lambda$-calculus for call-by-value, combining control and let-expressions.Fundação para a Ciência e a Tecnologia (FCT

Realizability Interpretation and Normalization of Typed Call-by-Need λ\lambda-calculus With Control

We define a variant of realizability where realizers are pairs of a term and a substitution. This variant allows us to prove the normalization of a simply-typed call-by-need \lambda$-$calculus with control due to Ariola et al. Indeed, in such call-by-need calculus, substitutions have to be delayed until knowing if an argument is really needed. In a second step, we extend the proof to a call-by-need \lambda$$-calculus equipped with a type system equivalent to classical second-order predicate logic, representing one step towards proving the normalization of the call-by-need classical second-order arithmetic introduced by the second author to provide a proof-as-program interpretation of the axiom of dependent choice

Proteomics Mapping of Cord Blood Identifies Haptoglobin “Switch-On” Pattern as Biomarker of Early-Onset Neonatal Sepsis in Preterm Newborns

Intra-amniotic infection and/or inflammation (IAI) are important causes of preterm birth and early-onset neonatal sepsis (EONS). A prompt and accurate diagnosis of EONS is critical for improved neonatal outcomes. We sought to explore the cord blood proteome and identify biomarkers and functional protein networks characterizing EONS in preterm newborns.We studied a prospective cohort of 180 premature newborns delivered May 2004-September 2009. A proteomics discovery phase employing two-dimensional differential gel electrophoresis (2D-DIGE) and mass spectrometry identified 19 differentially-expressed proteins in cord blood of newborns with culture-confirmed EONS (n = 3) versus GA-matched controls (n = 3). Ontological classifications of the proteins included transfer/carrier, immunity/defense, protease/extracellular matrix. The 1(st)-level external validation conducted in the remaining 174 samples confirmed elevated haptoglobin and haptoglobin-related protein immunoreactivity (Hp&HpRP) in newborns with EONS (presumed and culture-confirmed) independent of GA at birth and birthweight (P<0.001). Western blot concurred in determining that EONS babies had conspicuous Hp&HpRP bands in cord blood ("switch-on pattern") as opposed to non-EONS newborns who had near-absent "switch-off pattern" (P<0.001). Fetal Hp phenotype independently impacted Hp&HpRP. A bayesian latent-class analysis (LCA) was further used for unbiased classification of all 180 cases based on probability of "antenatal IAI exposure" as latent variable. This was then subjected to 2(nd)-level validation against indicators of adverse short-term neonatal outcome. The optimal LCA algorithm combined Hp&HpRP switch pattern (most input), interleukin-6 and neonatal hematological indices yielding two non-overlapping newborn clusters with low (≤20%) versus high (≥70%) probability of IAI exposure. This approach reclassified ∼30% of clinical EONS diagnoses lowering the number needed to harm and increasing the odds ratios for several adverse outcomes including intra-ventricular hemorrhage.Antenatal exposure to IAI results in precocious switch-on of Hp&HpRP expression. As EONS biomarker, cord blood Hp&HpRP has potential to improve the selection of newborns for prompt and targeted treatment at birth