Section: New Results

Models and theories of lambda calculus

Participant : Giulio Manzonetto.

The lambda calculus is a paradigmatic programming language that is at the basis of all functional programming languages. Lambda theories (equational extensions of lambda calculus) allow to study all the meaningful notions of program equivalence, and constitute a very rich and complex mathematical structure. In our research we use the denotational models of lambda calculus to study computational properties of programs, and to understand the structure of the set of lambda theories.

In 2009 we have proved that there is an inverse relationship between the effectivity of the models of lambda calculus and the effectivity of the corresponding theories. More precisely, we have shown that no effective model of lambda calculus can have lambda-beta or lambda-beta-eta as equational theory [11] . This can be seen as a partial answer of an open problem posed by Honsell 25 years ago. Moreover, we have defined and studied a new model of lambda calculus, living in a category of sets and (multi-)relations and provided a characterization of its equational theory [14] . We have shown that this model has a rich and powerful structure that allows to interpret a non-deterministic and parallel extension of lambda calculus [13] . Finally, we have generalized the notion of model of lambda calculus obtaining the class of Church algebras [15] , and used them to study the lattices of equational theories. This allows to prove a meta-stone representation theorem whose scope is much bigger than the one of the other representation theorems: it is indeed applicable to all varieties of algebras.

This work was achieved in collaboration with Chantal Berline, Antonio Bucciarelli, Thomas Ehrhard (PPS, Paris), Antonino Salibra (Ca'Foscari, Venice).