We introduce Distributed Memory Automata, a model of register automata suitable to capture some features of distributed algorithms designed for shared memory systems. In this model, each participant owns a local register and a shared register and has the ability to change its local value, to write it in the global memory and to test atomically the number of occurrences of its value in the shared memory, up to some threshold. We show that the control state reachability problem for Distributed Memory Automata is Pspace-complete for a fixed number of participants and is in Pspace when the number of participants is not fixed a priori.

This is a joint work with Benedikt Bollig and Fedor Ryabinin.

The first and possibly the most fundamental problem in distributed graph algorithmics is the vertex coloring problem. The nodes of a network are processes that communicate via the edges in synchronous rounds. The most basic problem is to color the network using Delta+1 colors in as few rounds a possible, where Delta is the maximum degree of the graph.

We present a new approach for randomized distributed Delta+1-coloring that is dramatically simpler than the previous best: the seminal work of Chang, Li, and Pettie (CLP) in SICOMP 2020. This approach offers numerous refinements: It matches the O(log^3 log n) round complexity in general, while on high-degree graphs (Delta » log^2 n), it uses only O(log* n) rounds. It uses small messages, i.e., works also in the CONGEST model. It also offers tradeoffs between the number of colors used, the round complexity, and the clique number of the graph.

We expect that the approach presented will lead the way towards improved results for various coloring problems in the near future.

This is based on a paper at STOC'21, joint work with Fabian Kuhn, Yannic Maus, and Tigran Tonoyan; and a more recent one on arXiv, joint with Alexandre Nolin, and Tigran Tonoyan.

We investigate the approximation rate of a typical element of the Cantor set by dyadic rationals. This is a manifestation of the times-two-times-three phenomenon, and is joint work with Demi Allen and Han Yu.

Focusing is a general technique for syntactically compartmentalising the non-deterministic choices in a proof system, which not only improves proof search but also has the representational benefit of distilling sequent proofs into synthetic normal forms. However, since focusing was traditionally specified as a restriction of the sequent calculus, the technique had not been transferred to logics that lack a (shallow) sequent presentation, as is the case for some modal or substructural logics.

With K. Chaudhuri and L. Straßburger, we extended the focusing technique to nested sequents, a generalisation of ordinary sequents which allows us to capture all the logics of the classical and intuitionistic S5 cube in a modular fashion. This relied, following the method introduced by O. Laurent, on an adequate polarisation of the syntax and an internal cut-elimination procedure for the focused system which in turn is used to show its completeness.

Recently, with A. Gheorghiu, we applied a similar method to the logic of Bunched Implications (BI), a logic that freely combines intuitionistic logic and multiplicative linear logic. For this we had first to reformulate the traditional bunched calculus for BI using nested sequents, followed by a polarised and focused variant that (again) we show is sound and complete via a cut-elimination argument.