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

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

A Quantitative Version of Simple Types

This work introduces a quantitative version of the simple type assignment system, starting from a suitable restriction of non-idempotent intersection types. The resulting system is decidable and has the same typability power as the simple type system; thus, assigning types to terms supplies the very same qualitative information given by simple types, but at the same time can provide some interesting quantitative information. It is well known that typability for simple types is equivalent to unification; we prove a similar result for the newly introduced system. More precisely, we show that typability is equivalent to a unification problem which is a non-trivial extension of the classical one: in addition to unification rules, our typing algorithm makes use of an expansion operation that increases the cardinality of multisets whenever needed

Lazy Logical Semantics

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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

Standardization of a Call-By-Value Lambda-Calculus

We study an extension of Plotkin\u27s call-by-value lambda-calculus by means of two commutation rules (sigma-reductions). Recently, it has been proved that this extended calculus provides elegant characterizations of many semantic properties, as for example solvability. We prove a standardization theorem for this calculus by generalizing Takahashi\u27s approach of parallel reductions. The standardization property allows us to prove that our calculus is conservative with respect to the Plotkin\u27s one. In particular, we show that the notion of solvability for this calculus coincides with that for Plotkin\u27s call-by-value lambda-calculus

Lazy Strong Normalization

AbstractAmong all the reduction strategies for the untyped λ-calculus, the so called lazy β-evaluation is of particular interest due to its large applicability to functional programming languages (e.g. Haskell [Bird, R., “Introduction to Functional Programming using Haskell,” Series in Computer Science (2nd edition), Prentice Hall, (1998)]). This strategy reduces only redexes not inside a lambda abstraction.The lazy strongly β- normalizing terms are the λ-terms that don't have infinite lazy β-reduction sequences.This paper presents a logical characterization of lazy strongly β-normalizing terms using intersection types. This characterization, besides being interesting by itself, allows an interesting connection between call-by-name and call-by-value λ-calculus.In fact, it turns out that the class of lazy strongly β-normalizing terms coincides with that of call-by-value potentially valuable terms. This last class is of particular interest since it is a key notion for characterizing solvability in the call-by-value setting