Section: New Results

Labeled Lambda-Calculus and Variants

Participants : Tomasz Blanc, Jean-Jacques Lévy, Luc Maranget.

We introduced a new property of the labeled lambda-calculus: context irreversibility . We have if and only if . This property shows that when a (labeled) context disappears at one point of a reduction, this disappearance is irreversible: the context cannot be rebuilt in the reduction that follows. In the (unlabeled) lambda-calculus, this property is false: we have (x.xy)yyy and x.xy¬y . Lévy had put in light in his thesis that syntactic coincidences might occur in the lambda-calculus [36] . These coincidences are responsible for the loss of context irreversibility in the unlabeled calculus. Context irreversibility is similar to time irreversibility: labeled contexts can be used to time-stamp a reduction.

We examined two variants of the lambda-calculus. In the lambda-calculus by value, only redexes whose right part is a value (i.e. an abstraction) are contracted. The labels express the property of stability in the lambda-calculus but not in the lambda-calculus by value. We adapted the labels to recover this property. The lambda-calculus by value and the labeled lambda-calculus by value are confluent and verify the theorems of finite developments and standardization (although the definition of a standard reduction is adapted). To prove finite developments, we used an elegant, intuitive technique based on a notion of future redex imbrication .

We also examined the weak lambda-calculus. Although its syntactic properties did not receive great attention in the past decades, this theory is more relevant for the implementation of programming languages than the usual theory of the strong lambda-calculus. Contrary to the latter and similarly to lazy functional languages, the weak lambda-calculus does not validate the -rule i.e. the reduction under a lambda-abstraction. Without this rule, the weak lambda-calculus is not confluent. We based our labeled language on a variant of the weak lambda-calculus by the Çağman and Hindley for Combinatory Logic [30] . In this variant, a new -rule is valid: it allows reductions under lambda-abstraction in sub-terms that do not contain a bound variable. This variant enjoys confluence. We proposed a labeling of this language that expresses a confluent theory of reductions with sharing, independent of the reduction strategy. If the sub-terms of a term are initially labeled with distinct letters, then, after some steps of reduction, two sub-terms that have the same label are equal. This formal setting corresponds to the shared evaluation strategy by Wadsworth defined with dags in 1971. This work was published on the occasion of Jan Willem Klop's 60th birthday [13] . A slightly simplified version of this work, context irreversibility and the labeled lambda-calculus by value are presented in Tomasz Blanc's thesis [11] .