Relational type-checking for MELL proof-structures. Part 1: Multiplicatives

Relational semantics for linear logic is a form of non-idempotent intersection type system, from which several informations on the execution of a proof-structure can be recovered. An element of the relational interpretation of a proof-structure R with conclusion $\Gamma$ acts thus as a type (refining $\Gamma$) having R as an inhabitant. We are interested in the following type-checking question: given a proof-structure R, a list of formulae $\Gamma$, and a point x in the relational interpretation of $\Gamma$, is x in the interpretation of R? This question is decidable. We present here an algorithm that decides it in time linear in the size of R, if R is a proof-structure in the multiplicative fragment of linear logic. This algorithm can be extended to larger fragments of multiplicative-exponential linear logic containing $\lambda$-calculus

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

Light Logics and the Call-by-Value Lambda Calculus

The so-called light logics have been introduced as logical systems enjoying quite remarkable normalization properties. Designing a type assignment system for pure lambda calculus from these logics, however, is problematic. In this paper we show that shifting from usual call-by-name to call-by-value lambda calculus allows regaining strong connections with the underlying logic. This will be done in the context of Elementary Affine Logic (EAL), designing a type system in natural deduction style assigning EAL formulae to lambda terms.Comment: 28 page

Linear Rank Intersection Types

Non-idempotent intersection types provide quantitative information about typed programs, and have been used to obtain time and space complexity measures. Intersection type systems characterize termination, so restrictions need to be made in order to make typability decidable. One such restriction consists in using a notion of finite rank for the idempotent intersection types. In this work, we define a new notion of rank for the non-idempotent intersection types. We then define a novel type system and a type inference algorithm for the ?-calculus, using the new notion of rank 2. In the second part of this work, we extend the type system and the type inference algorithm to use the quantitative properties of the non-idempotent intersection types to infer quantitative information related to resource usage

Codimension-3 Singularities and Yukawa Couplings in F-theory

F-theory is one of the frameworks where all the Yukawa couplings of grand unified theories are generated and their computation is possible. The Yukawa couplings of charged matter multiplets are supposed to be generated around codimension-3 singularity points of a base complex 3-fold, and that has been confirmed for the simplest type of codimension-3 singularities in recent studies. However, the geometry of F-theory compactifications is much more complicated. For a generic F-theory compactification, such issues as flux configuration around the codimension-3 singularities, field-theory formulation of the local geometry and behavior of zero-mode wavefunctions have virtually never been addressed before. We address all these issues in this article, and further discuss nature of Yukawa couplings generated at such singularities. In order to calculate the Yukawa couplings of low-energy effective theory, however, the local descriptions of wavefunctions on complex surfaces and a global characterization of zero-modes over a complex curve have to be combined together. We found the relation between them by re-examining how chiral charged matters are characterized in F-theory compactification. An intrinsic definition of spectral surfaces in F-theory turns out to be the key concept. As a biproduct, we found a new way to understand the Heterotic--F theory duality, which improves the precision of existing duality map associated with codimension-3 singularities.Comment: 91 pages; minor clarification, typos corrected and a reference added (v3

The involutions-as-principal types/ application-as-unification analogy

In 2005, S. Abramsky introduced various universal models of computation based on Affine Combinatory Logic, consisting of partial involutions over a suitable formal language of moves, in order to discuss reversible computation in a game-theoretic setting. We investigate Abramsky\u2019s models from the point of view of the model theory of \u3bb-calculus, focusing on the purely linear and affine fragments of Abramsky\u2019s Combinatory Algebras. Our approach stems from realizing a structural analogy, which had not been hitherto pointed out in the literature, between the partial involution interpreting a combinator and the principal type of that term, with respect to a simple types discipline for \u3bb-calculus. This analogy allows for explaining as unification between principal types the somewhat awkward linear application of involutions arising from Geometry of Interaction (GoI). Our approach provides immediately an answer to the open problem, raised by Abramsky, of characterising those finitely describable partial involutions which are denotations of combinators, in the purely affine fragment. We prove also that the (purely) linear combinatory algebra of partial involutions is a (purely) linear \u3bb-algebra, albeit not a combinatory model, while the (purely) affine combinatory algebra is not. In order to check the complex equations involved in the definition of affine \u3bb-algebra, we implement in Erlang the compilation of \u3bb-terms as involutions, and their execution

Set-Theoretic Types for Polymorphic Variants

Polymorphic variants are a useful feature of the OCaml language whose current definition and implementation rely on kinding constraints to simulate a subtyping relation via unification. This yields an awkward formalization and results in a type system whose behaviour is in some cases unintuitive and/or unduly restrictive. In this work, we present an alternative formalization of poly-morphic variants, based on set-theoretic types and subtyping, that yields a cleaner and more streamlined system. Our formalization is more expressive than the current one (it types more programs while preserving type safety), it can internalize some meta-theoretic properties, and it removes some pathological cases of the current implementation resulting in a more intuitive and, thus, predictable type system. More generally, this work shows how to add full-fledged union types to functional languages of the ML family that usually rely on the Hindley-Milner type system. As an aside, our system also improves the theory of semantic subtyping, notably by proving completeness for the type reconstruction algorithm.Comment: ACM SIGPLAN International Conference on Functional Programming, Sep 2016, Nara, Japan. ICFP 16, 21st ACM SIGPLAN International Conference on Functional Programming, 201