Overall Objectives
New Software and Platforms
Bilateral Contracts and Grants with Industry
Partnerships and Cooperations
Bibliography
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## Section: New Results

### Software specification and verification

#### Formal reasoning about asymptotic complexity

Participants : Armaël Guéneau, Arthur Charguéraud [team Camus] , François Pottier.

For a couple years, Armaël Guéneau, Arthur Charguéraud, François Pottier have been investigating the use of Separation Logic, extended with Time Credits, as an approach to the formal verification of the time complexity of OCaml programs. In particular, Armaël has developed in Coq a theory and a set of tactics that allow working with asymptotic complexity bounds. He has presented the main aspects of this work at the conference ESOP 2018 [21]. Furthermore, a key part of the machinery for working with asymptotic complexity bounds has been released as a standalone, reusable Coq library, procrastination. Armaël presented this library at the Coq Workshop in July 2018 [29].

In 2018, Armaël has worked on a more ambitious case study, namely a recent incremental cycle detection algorithm, whose amortized complexity analysis is nontrivial. A machine-checked proof has been completed; a paper is in preparation.

#### Time Credits and Time Receipts in Iris

Participants : Glen Mével, Jacques-Henri Jourdan [CNRS] , François Pottier.

From March to August 2018, Glen Mével did an M2 internship at Gallium, where he was co-advised by Jacques-Henri Jourdan (CNRS) and François Pottier. Glen extended the program logic Iris with time credits and time receipts.

Time credits are a well-understood concept, and have been used in several papers already by Armaël Guéneau, Arthur Charguéraud, and François Pottier. However, because Iris is implemented and proved sound inside Coq, extending Iris with time credits requires a nontrivial proof, which Glen carried out, based on a program transformation which inserts “tick” instructions into the code. As an application of time credits, Glen verified inside Iris the correctness of Okasaki's notion of “debits”, which allows reasoning about the time complexity of programs that use thunks.

Time receipts are a new concept, which (we showed) allows proving that certain undesirable events, such as integer overflows, cannot occur until a very long time has elapsed. Glen extended Iris with time receipts and proved the soundness of this extension. As an application of time credits and receipts together, Jacques-Henri Jourdan updated Charguéraud and Pottier's earlier verification of the Union-Find data structure [3] and proved that integer ranks cannot realistically overflow, even if they are stored using only $logW$ bits, where $W$ is the number of bits in a machine word.

This work has been first submitted to POPL 2019, then (after significant revision) re-submitted to ESOP 2019.

#### Verified Interval Maps

Participant : François Pottier.

In the setting of ANR project Vocal, which aims to build a library of verified data structures for OCaml, François Pottier carried out a formal reconstruction of “interval maps”. An interval map, a data structure proposed by Bonichon and Cuoq in 2010, represents a set of possible heaps, that is, a set of mappings of integer addresses to abstract values. Interval maps are used in the Frama-C program analysis tool. François Pottier re-implemented this data structure in Coq and carried out a formal verification of its main operations. This work, which represents about 4 months of work, remains unpublished at this time. It would be desirable to publish it and to envision its integration in Frama-C; this however requires further effort.

#### Chunked Sequences

Participants : Émilie Guermeur, Arthur Charguéraud, François Pottier.

In June and July 2018, Émilie Guermeur, an undergraduate student at Carnegie Mellon University (Pittsburgh, USA) did a 6-week internship, co-advised by Arthur Charguéraud and François Pottier. She wrote a full-fledged OCaml implementation of “chunked sequences”, a data structure which offers an efficient representation of sequences of elements. This data structure exists in two forms, a persistent form and an ephemeral (mutable) form; efficient conversion operations are offered. François Pottier subsequently implemented a test harness, based on afl-fuzz, which allowed us to submit Émilie's code to intensive testing and detect and fix a few bugs. This work is not yet published; we intend to pursue it in 2019, to publish the library and perhaps to verify it.

#### TLA+

Participants : Damien Doligez, Leslie Lamport [Microsoft Research] , Ioannis Filippidis, Martin Riener [team VeriDis] , Stephan Merz [team VeriDis] .

Damien Doligez is head of the “Tools for Proofs” team in the Microsoft-Inria Joint Centre. The aim of this project is to extend the TLA+ language with a formal language for hierarchical proofs, formalizing Lamport's ideas [36]. This requires building tools to help write TLA+ specifications and mechanically check proofs.

Since October 2018, Ioannis Filippidis has been working on extending the TLAPS tool to deal with proofs of temporal properties. Under some well-defined circumstances, an occurrence of the enabled operator applied to a formula $f$ can be replaced by a version of $f$ where the primed variables are replaced by new existentially-quantified variables. The result is a first-order formula that can be sent to one of TLAPS's first-order back-ends. This rewriting of enabled suffices to prove a large class of liveness properties. Ioannis has started implementing this in TLAPS.