Section: New Results

Resource Control

Participants : Michele Alberti, Alberto Cappai, Ugo Dal Lago, Simone Martini, Giulio Pellitta, Davide Sangiorgi, Marco Solieri, Valeria Vignudelli.

Probabilistic higher-order calculi

The first results of our efforts on probabilistic higher-order systems and languages have started to appear in 2014. In particular, we have focused our attention on the impact of probability to the classical notion of context equivalence for the lambda-calculus, showing that applicative bisimilarity continues to be a congruence [31] , and that it even coincides with context equivalence when evaluation is done in the call-by-value order [29] . The expressive power of higher-order concurrent contexts has been compared to the expressive power of lambda-calculi contexts and put in relation with other equivalences when the observed process is either an ordinary Labelled Transition Systems (LTS) or a reactive probabilistic transition system [25] . The obtained spectrum of equivalences for reactive probabilistic processes has been shown to be finer than the one for classic LTSs. We have also analysed the expressive power of different first-order testing equivalences (with nondeterministic tests, probabilistic tests, and both nondeterministic and probabilistic tests) in the spectrum for reactive probabilistic processes [26] .

Resource consumption

The main result about resource consumption has been about an open problem on the $\lambda$-calculus: we proved that the number of leftmost-outermost steps to normal form is indeed an invariant cost model in the sense of Slot and van Emde Boas' weak invariance thesis [21] . We also introduced a new recursion theoretic framework for probabilistic computation in which one is able to capture probabilistic polynomial time through Leivant's Tiering [32] .

Geometry of interaction

Novel results have been obtained for Geometry of Interaction (GoI), itself a semantics framework for linear logic introduced by Jean-Yves Girard thirty years ago. In particular, we have shown how the most concrete presentations of GoI, namely so-called token machines, can go parallel, thus exploiting the potential parallelism in functional programs (through the Curry-Howard Correspondence). This has been made concrete by studying extensions of multiplicative linear logic in which synchronization becomes an operator where tokens can indeed synchronize [30] . This has been later shown to be necessary to model quantum computation [44] . A simple, minimalistic GoI model of the resource $\lambda$-calculus has also been introduced [43] .