Section: New Results

Probabilistic Systems and Resource Control

Participants : Martin Avanzini, Mario Bravetti, Raphaelle Crubillé, Ugo Dal Lago, Francesco Gavazzo, Gabriele Vanoni, Akira Yoshimizu.

Probabilistic Programming and Static Analysis

In FoCUS, we are interested in studying probabilistic higher-order programming languages and, more generally, the fundamental properties of probabilistic computation when placed in an interactive scenario, for instance concurrency. One of the most basic but nevertheless desirable properties of programs is of course termination. Termination can be seen as a minimal guarantee about the time complexity of the underlying program. When probabilistic choice comes into play, termination can be defined by stipulating that a program is terminating if its probability of convergence is 1, this way giving rise to the notion of almost sure termination. Alternatively, a probabilistic program is said to be positively almost surely terminating if its average runtime is finite. The latter condition easily implies the former. Termination, already undecidable for deterministic (universal) programming languages, remains so in the presence of probabilistic choice, even becoming provably harder.

The FoCUS team has been the first in advocating the use of types to guarantee probabilistic termination, in the form of a monadic sized-type system [17]. Developed in collaboration with Grellois by Dal Lago, this system substantially generalises usual sized-types, and allows this way to capture probabilistic, higher-order programs which terminate almost surely. Complementary, in collaboration with Ghyselen, Avanzini and Dal Lago have recently defined a formal system for reasoning about the expected runtime of higher-order probabilistic programs, through a refinement type system capable of modeling probabilistic effects with exceptional accuracy [26]. To the best of our knowledge, this provides the first formal methodology for average case complexity analysis of higher-order programs. Remarkably, the system is also extensionally complete.

In 2018, we have started to investigate the foundations for probabilistic abstract reduction systems (probabilistic ARSs), which constitute a general framework to study fundamental properties of probabilistic computations, such as termination or confluence. In 2019, we have significantly revised this initial development [11]. Particularly, we have refined Lyapunov ranking functions by conceiving them as probabilistic embeddings. The ramifications of this work are two-fold. First, we obtain a sound and complete method for reasoning about strong positive almost sure termination. Second, this method has been instantiated in the setting of (first-order) probabilistic rewrite systems, giving rise to the notion of barycentric algebras, generalising the well-known interpretation method. Barycentric algebras have been integrated in the termination prover NaTT (See https://www.trs.css.i.nagoya-u.ac.jp/NaTT/.), confirming the feasibility of the approach.

We have also worked on higher-order model checking as a way to prove termination of probabilsitic variations on higher-order recursion schemes [36], obtaining encouraging results. More specifically, an algorithm for approximating the probability of convergence of any such scheme has been designed and proved sound, although the problem of precisely computing the probability of convergence is shown to be undecidable at order 2 or higher. Finally, we have published a new version of a contribution we wrote in 2017 about how implicit computational complexity could help in proving that certain cryptographic constructions have the desired complexity-theoretic properties [12].

Higher-Order end Effectful Programs: Relational Reasoning

In FoCUS, we are also interested in relational reasoning about programs written in higher-order programming languages. In the recent years, this research has been directed to effectful programs, namely programs whose behaviour is not purely functional. Moreover, there has recently been a shift in our interests, driven by the projects REPAS and DIAPASoN, towards quantitative kinds of relational reasoning, in which programs are not necessarily dubbed equivalent (or not), but rather put at a certain distance.

The first contribution we had in this direction is due to Dal Lago and Gavazzo [31], who generalized the so-called open normal-form bisimilarity technique to higher-order programs exhibiting any kind of monadic effect. The key ingredient here is that of a relator, and allows to lift relations on a set to relations on monadic extensions to the same set. This allows to define open normal-form bisimilarity, and to prove it correct. This, together, with other contributions, have also appeared in Gavazzo's PhD Thesis, which has been successfully defended in April 2019 [10], and which has been awarder the Prize for the Best PhD Thesis in Theoretical Computer Science by the Italian Chapter of the EATCS.

We have also given the notion of differential logical relations [33], a generalization of Plotkin's logical relations in which programs are dubbed being at a certain distance rather than being just equivalent. Noticeably, this distance is not necessarily numeric, but is itself functional if the compared programs have a non-ground type. This allows to evaluate the distance between programs taking into account the possible actions the environment can make on the compared programs.

Alternative Probabilistic Models

We are also interested in exploring probabilistic models going beyond the usual ones, in which determinisitic programming languages are endowed with discrete probabilistic choice.

We have first of all studied bayesian $\lambda$-calculi, namely $\lambda$-calculi in which not only an operator for probabilistic choice is available, but also one for scoring, which serves as the basis to model conditioning in probabilistic programming. We give a geometry of interaction model for such a typed $\lambda$-calculus [34], namely a paradigmatic calculus for higher-order Bayesian programming in the style of PCF. The model is based on the category of measurable spaces and partial measurable functions, and is proved adequate with respect to both a distribution-based and a sampling-based operational semantics.

We have also introduced a probabilistic extension of a framework to specify and analyze software product lines [15]. We define a syntax of the language including probabilistic operators and define operational and denotational semantics for it. We prove that the expected equivalence between these two semantic frameworks holds. Our probabilistic framework is supported by a set of scripts to show the model behavior.