A Behavioural Model for Klop's Calculus

Dezani‐Ciancaglini, Mariangiola; Tatsuta, Makoto

doi:10.1016/j.entcs.2006.07.026

Cited by 3 publications

(2 citation statements)

References 6 publications

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“…Intersection types in the framework of Church-Klop's calculus have been studied in e.g. [DCT07], but, to our knowledge, they were always considered idempotent. Hence, the quantitative analysis provided by non-idempotency was not carried out.…”

Section: Introductionmentioning

confidence: 99%

Complexity of Strongly Normalising λ-Terms via Non-idempotent Intersection Types

Bernadet

Lengrand

2011

Lecture Notes in Computer Science

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We present a typing system for the λ-calculus, with nonidempotent intersection types. As it is the case in (some) systems with idempotent intersections, a λ-term is typable if and only if it is strongly normalising. Non-idempotency brings some further information into typing trees, such as a bound on the longest β-reduction sequence reducing a term to its normal form.We actually present these results in Klop's extension of λ-calculus, where the bound that is read in the typing tree of a term is refined into an exact measure of the longest reduction sequence.This complexity result is, for longest reduction sequences, the counterpart of de Carvalho's result for linear head-reduction sequences.

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Section: Introductionmentioning

confidence: 99%

Complexity of Strongly Normalising λ-Terms via Non-idempotent Intersection Types

Bernadet

Lengrand

2011

Lecture Notes in Computer Science

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“…In particular, the equivalence between the first condition and the third one has solved an open problem in the conclusion of [4]. After this submission, we found we could extend the discussion in this paper to Λ * , and we proved the above equivalencies for Λ * in [5].…”

Section: [[M ]mentioning

confidence: 68%

Normalisation is Insensible to \lambda-Term Identity or Difference

Tatsuta

Dezani‐Ciancaglini

21st Annual IEEE Symposium on Logic in Computer Science (LICS'06)

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This paper analyses the computational behaviour of λ-term applications. The properties we are interested in are weak normalisation (i.e. there is a terminating reduction) and strong normalisation (i.e. all reductions are terminating).One can prove that the application of a λ-term M to a fixed number n of copies of the same arbitrary strongly normalising λ-term is strongly normalising if and only if the application of M to n different arbitrary strongly normalising λ-terms is strongly normalising. I.e. one has that M X . . . X n is strongly normalising, for an arbitrary strongly normalising X, if and only if MX 1 . . . X n is strongly normalising for arbitrary strongly normalising X 1 , . . . , X n . The analogous property holds when replacing strongly normalising by weakly normalising.As an application of the result on strong normalisation the λ-terms whose interpretation is the top element (in the environment which associates the top element to all variables) of the Honsell-Lenisa model turn out to be exactly the λ-terms which, applied to an arbitrary number of strongly normalising λ-terms, always produces strongly normalising λ-terms. This proof uses a finitary logical description of the model by means of intersection types. This answers an open question stated by Dezani, Honsell and Motohama.

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Calculi, types and applications

Berardi

de’Liguoro

2008

Theoretical Computer Science

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PrefaceThe 60th birthdays of Mario Coppo, Mariariangiola Dezani-Ciancaglini, and Simona Ronchi Della Rocca all fall between 2006 and 2007. These three researchers are widely known in the world community of theoretical computer scientists, and especially among the researchers in λ-calculus, type theory and semantics of programming languages. Owing to the significance of their contributions, the impact of their ideas and the large number of coworkers and students they have, we wish to honour them with the present collection of essays, covering most of their favourite topics, also showing how far the concepts and results originating from their works have gone.Our aim has been to illustrate both the past of the type assignment, the intersection types and the λ-calculus by presenting old results with new proofs in a new light; and to have a look at what is going on in these fields, since the original nucleus of the types for the λ-calculus has been widened encompassing object and process calculi, linear logic and implicit complexity.All of the present contributions have been invited, staring with a small group of coauthors of Coppo, Dezani and Ronchi. The number of papers has grown rapidly, as the news of our collection circulated, and other relevant themes were also suggested especially with regards to the most recent developments. The resulting collection is therefore the merit of the contributors, whose work has been submitted to a rigorous process of refereeing, and of the referees themselves, who provided criticism as well as suggestions to improve the papers. The absence of many researchers in the field is mostly casual, only due to the lack of space and to the poorness of our forces.Torino

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A Behavioural Model for Klop's Calculus

Abstract: A model characterising strong normalisation for Klop's extension of λ-calculus is presented. The main technical tools for this result are an inductive definition of strongly normalising terms of Klop's calculus and an intersection type system for terms of Klop's calculus.

Cited by 3 publications

References 6 publications

Complexity of Strongly Normalising λ-Terms via Non-idempotent Intersection Types

Complexity of Strongly Normalising λ-Terms via Non-idempotent Intersection Types

Normalisation is Insensible to \lambda-Term Identity or Difference

Calculi, types and applications

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