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Register a new userNon-idempotent intersection types in logical form
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Added by
Thomas Ehrhard
Publication date
2019
fields
Informatics Engineering
and research's language is
English
Authors
Thomas Ehrhard
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Intersection types are an essential tool in the analysis of operational and denotational properties of lambda-terms and functional programs. Among them, non-idempotent intersection types provide precise quantitative information about the evaluation of terms and programs. However, unlike simple or second-order types, intersection types cannot be considered as a logical system because the application rule (or the intersection rule, depending on the presentation of the system) involves a condition expressing that the proofs of premises satisfy a very strong uniformity condition: the underlying lambda-terms must be the same. Using earlier work introducing an indexed version of Linear Logic, we show that non-idempotent typing can be given a logical form in a system where formulas represent hereditarily indexed families of intersection types.
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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.
This paper investigates type isomorphism in a lambda-calculus with intersection and union types. It is known that in lambda-calculus, the isomorphism between two types is realised by a pair of terms inverse one each other. Notably, invertible terms are linear terms of a particular shape, called finite hereditary permutators. Typing properties of finite hereditary permutators are then studied in a relevant type inference system with intersection and union types for linear terms. In particular, an isomorphism preserving reduction between types is defined. Type reduction is confluent and terminating, and induces a notion of normal form of types. The properties of normal types are a crucial step toward the complete characterisation of type isomorphism. The main results of this paper are, on one hand, the fact that two types with the same normal form are isomorphic, on the other hand, the characterisation of the isomorphism between types in normal form, modulo isomorphism of arrow types.
We study polymorphic type assignment systems for untyped lambda-calculi with effects, based on Moggis monadic approach. Moving from the abstract definition of monads, we introduce a version of the call-by-value computational lambda-calculus based on Wadlers variant with unit and bind combinators, and without let. We define a notion of reduction for the calculus and prove it confluent, and also we relate our calculus to the original work by Moggi showing that his untyped metalanguage can be interpreted and simulated in our calculus. We then introduce an intersection type system inspired to Barendregt, Coppo and Dezani system for ordinary untyped lambda-calculus, establishing type invariance under conversion, and provide models of the calculus via inverse limit and filter model constructions and relate them. We prove soundness and completeness of the type system, together with subject reduction and expansion properties. Finally, we introduce a notion of convergence, which is precisely related to reduction, and characterize convergent terms via their types.
In this paper we investigate the $lambda$ -calculus, a $lambda$-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 $lambda$ -terms and we provide a new treatment of substitution by its decompo-sition into atomic steps. We propose an intersection type assignment system for $lambda$ -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 $lambda$ -calculus by means of an in-tersection type assignment system. This process uses typeability of normal forms, redex subject expansion and reducibility method.
This volume contains a final and revised selection of papers presented at the Seventh Workshop on Intersection Types and Related Systems (ITRS 2014), held in Vienna (Austria) on July 18th, affiliated with TLCA 2014, Typed Lambda Calculi and Applications (held jointly with RTA, Rewriting Techniques and Applications) as part of FLoC and the Vienna Summer of Logic (VSL) 2014. Intersection types have been introduced in the late 1970s as a language for describing properties of lambda calculus which were not captured by all previous type systems. They provided the first characterisation of strongly normalising lambda terms and have become a powerful syntactic and semantic tool for analysing various normalisation properties as well as lambda models. Over the years the scope of research on intersection types has broadened. Recently, there have been a number of breakthroughs in the use of intersection types and similar technology for practical purposes such as program analysis, verification and concurrency, and program synthesis. The aim of the ITRS workshop series is to bring together researchers working on both the theory and practical applications of systems based on intersection types and related approaches (e.g., union types, refinement types, behavioral types).
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