The syntax and semantics of quantitative type theory

We present Quantitative Type Theory, a Type Theory that records usage information for each variable in a judgement, based on a previous system by McBride. The usage information is used to give a realizability semantics using a variant of Linear Combinatory Algebras, refining the usual realizability semantics of Type Theory by accurately tracking resource behaviour. We define the semantics in terms of Quantitative Categories with Families, a novel extension of Categories with Families for modelling resource sensitive type theories

SARS-CoV-2 B.1.617.2 Delta variant replication and immune evasion

Abstract: The B.1.617.2 (Delta) variant of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) was first identified in the state of Maharashtra in late 2020 and spread throughout India, outcompeting pre-existing lineages including B.1.617.1 (Kappa) and B.1.1.7 (Alpha)1. In vitro, B.1.617.2 is sixfold less sensitive to serum neutralizing antibodies from recovered individuals, and eightfold less sensitive to vaccine-elicited antibodies, compared with wild-type Wuhan-1 bearing D614G. Serum neutralizing titres against B.1.617.2 were lower in ChAdOx1 vaccinees than in BNT162b2 vaccinees. B.1.617.2 spike pseudotyped viruses exhibited compromised sensitivity to monoclonal antibodies to the receptor-binding domain and the amino-terminal domain. B.1.617.2 demonstrated higher replication efficiency than B.1.1.7 in both airway organoid and human airway epithelial systems, associated with B.1.617.2 spike being in a predominantly cleaved state compared with B.1.1.7 spike. The B.1.617.2 spike protein was able to mediate highly efficient syncytium formation that was less sensitive to inhibition by neutralizing antibody, compared with that of wild-type spike. We also observed that B.1.617.2 had higher replication and spike-mediated entry than B.1.617.1, potentially explaining the B.1.617.2 dominance. In an analysis of more than 130 SARS-CoV-2-infected health care workers across three centres in India during a period of mixed lineage circulation, we observed reduced ChAdOx1 vaccine effectiveness against B.1.617.2 relative to non-B.1.617.2, with the caveat of possible residual confounding. Compromised vaccine efficacy against the highly fit and immune-evasive B.1.617.2 Delta variant warrants continued infection control measures in the post-vaccination era

Irrelevance, Polymorphism, and Erasure in Type Theory

Dependent type theory is a proven technology for verified functional programming in which programs and their correctness proofs may be developed using the same rules in a single formal system. In practice, large portions of programs developed in this way have no computational relevance to the ultimate result of the program and should therefore be removed prior to program execution. In previous work on identifying and removing irrelevant portions of programs, computational irrelevance is usually treated as an intrinsic property of program expressions. We find that such an approach forces programmers to maintain two copies of commonly used datatypes: a computationally relevant one and a computationally irrelevant one. We instead develop an extrinsic notion of computational irrelevance and find that it yields several benefits including (1) avoidance of the above mentioned code duplication problem; (2) an identification of computational irrelevance with a highly general form of parametric polymorphism; and (3) an elective (i.e., user-2 directed) notion of proof irrelevance. We also develop a program analysis for identifying irrelevant expressions and show how previously studied types embodying computational irrelevance (including subset types and squash types) are expressible in the extension of type theory developed herein

Typed closure conversion for the calculus of constructions

International audienceDependently typed languages such as Coq are used to specify and verify the full functional correctness of source programs. Type-preserving compilation can be used to preserve these specifications and proofs of correctness through compilation into the generated target-language programs. Unfortunately, type-preserving compilation of dependent types is hard. In essence, the problem is that dependent type systems are designed around high-level compositional abstractions to decide type checking, but compilation interferes with the type-system rules for reasoning about run-time terms.We develop a type-preserving closure-conversion translation from the Calculus of Constructions (CC) with strong dependent pairs (Σ types)—a subset of the core language of Coq—to a type-safe, dependently typed compiler intermediate language named CC-CC. The central challenge in this work is how to translate the source type-system rules for reasoning about functions into target type-system rules for reasoning about closures. To justify these rules, we prove soundness of CC-CC by giving a model in CC. In addition to type preservation, we prove correctness of separate compilation

Irrelevance, Heterogeneous Equality, and Call-by-value Dependent Type Systems

We present a full-spectrum dependently typed core language which includes both nontermination and computational irrelevance (a.k.a. erasure), a combination which has not been studied before. The two features interact: to protect type safety we must be careful to only erase terminating expressions. Our language design is strongly influenced by the choice of CBV evaluation, and by our novel treatment of propositional equality which has a heterogeneous, completely erased elimination form