Sylvain Schmitz

Assistant professor

Complexity Hierarchies Beyond Elementary

I've been busy these last few years trying to get a clear picture of the complexity of problems that arise from the use of well-quasi-orders and well-structured transition systems. Here is some of the material I co-authored on the subject:

A write-up on non-elementary complexity classes and problems: Complexity Hierarchies Beyond Elementary. Ultimately, I think the main point I try to make in the technical developments is that people should forget all the low-level details found in this paper—which are shown to have little to no impact at such high complexities—, and just use the classes without worrying.
The slides of a talk on the subject. (Warning: those are heavy HTML5 slides.)
An invited paper presented by Ph. Schnoebelen at Concur 2013 on the The Power of Well-Structured Systems, which is meant to provide a gentle introduction to some advanced topics covered in the ESSLLI 2012 lecture notes (see last point).
Another invited paper I presented at RP 2014 on the bounds in the particular case of ordinals: Complexity Bounds for Ordinal-Based Termination.
The teaching material of a course on Algorithmic Aspects of WQO Theory taught together with Ph. Schnoebelen at ESSLLI 2012.

Research

Keywords: Verification, infinite systems, well quasi orders, formal languages, parsing, computational linguistics

Publications

Most recent work:

A Sequent Calculus for a Modal Logic on Finite Data Trees

Joint work with David Baelde from LSV and Simon Lunel and based on Simon's Master Thesis. As a first dive into proof systems for data-aware logics, in the context of the ANR Prodaq project, we propose a sequent calculus for a modal fragment of DataXPath. We further show that proof search in this calculus can be performed in PSPACE, which is optimal for the logic.

The Ideal View on Rackoff's Coverability Technique

Joint work with R. Lazić from U. Warwick. We re-prove Bozzelli & Ganty's 2EXP upper bound on the backward coverability algorithm for VAS using ideal representations of downwards-closed sets. This yields a generic means of proving complexity upper bounds for coverability in WSTS. Presented at RP 2015.

Fixed-dimensional Energy Games are in Pseudo-polynomial Time

Joint work with M. Jurdziński and R. Lazić from U. Warwick. We show that multi-dimensional energy games can be solved in pseudo-polynomial time when the dimension is fixed, answering an open question of Chaloupka (2013). This entails a 2EXP upper bound for multidimensional energy games when both the dimension and the initial credit are part of the input, answering an open question of Brázdil et al. (2010), and closes the gap with 2EXP hardness proven last year with J.-B. Courtois. Presented at ICALP 2015.

Demystifying Reachability in Vector Addition Systems

Joint work with J. Leroux from LaBRI. The decidability of reachability in vector addition systems (VAS), or equivalently in Petri nets, is arguably one of the landmark results in theoretical computer science. The first proof by Mayr (1981) and its further simplifications by Kosaraju (1982) and Lambert (1992) are extremely complex on two accounts.

First, on a conceptual side, they rely on a somewhat magical decomposition technique. We expose the tricks involved in this construction, by showing that this decomposition technique can be recast as the computation of an ideal decomposition of the set of runs, for an appropriately chosen well-quasi-ordering on runs.
Second, on a complexity side, no upper bounds were known until now for the decomposition algorithms. Using recent results on the complexity of algorithms that terminate thanks to well-quasi-orderings, we provide the first known upper bound on the VAS reachability problem: it belongs to $F_{ω^{3}}$ , a non primitive-recursive complexity class, among the lowest multiply-recursive complexites.

Presented at LICS 2015.

Implicational Relevance Logic is 2EXPTIME-Complete

Building on the complexity of root coverability in branching VASSs, which was shown 2EXP-complete by Demri et al., I show that the same complexity holds for provability in the implicational fragment of relevance logic, a problem open for almost 25 years. The result extends all the way up to intuitionistic contractive multiplicative exponential linear logic (IMELLC). Was presented at RTA-TLCA 2014 where it received the Best Paper Award during the Vienna Summer of Logic. Extended version accepted for publication in JSL.

Non-Elementary Complexities for Branching VASS, MELL, and Extensions

Joint work with R. Lazić from U. Warwick. We investigate the complexity of decision problems in substructural logics and branching variants of vector addition systems. Concretely, we define alternating branching vector addition systems with states (ABVASS) and provide reductions in both directions between their reachability problem and provability in propositional linear logic. These reductions carry over for fragments of linear logic and restricted ABVASS, and furthermore carry over in presence of structural weakening when considering lossy ABVASS and in presence of structural contraction when considering expansive ABVASS. For these two relaxations, we provide optimal complexity bounds on reachability, and thus on provability on the associated substructural logics: Tower-complete in the lossy case, and Ackermann-complete in the expansive case. The former entails a new Tower lower bound on BVASS reachability and MELL provability. Was presented at CSL-LICS 2014 during the Vienna Summer of Logic. Extended version published in ToCL.

Full BibTeX

More publications...

Talks

LICS 2015, 11:15am, July 6th, 2015, Kyoto, Japan.: I presented the article Demystifying Reachability in Vector Addition Systems written with J. Leroux. Here are the slides.
DIMAP Logic Day 2015, 9am, June 1st, 2015, University of Warwick, Warwick, UK.: I presented the recent joint work with J. Leroux on deriving complexity upper bounds for the reachability problem in vector addition systems.
DIMAP Seminar, 4pm, May 26th, 2015, University of Warwick, Warwick, UK.: I gave a lecture (on the blackboard) on complexity classes beyond Elementary. Using the case of reachability in lossy counter machines as a running example, I sketched the proofs of the complexity lower and upper bounds, and motivated the need for fast-growing complexity classes. The lecture was based mainly on the complexity classes paper.
Oxford Verification Seminar, 11am, May 6th, 2015, University of Oxford, Oxford, UK.: I gave a lecture on ideal decompositions of downward-closed sets, and in particular on how they provide a new understanding of the structures and computations defined in the reachability algorithms for VAS developped by Mayr (1981), Kosaraju (1982), and Lambert (1992). The lecture is based on a joint paper with J. Leroux. Here are the slides.
Theory Seminar, 11am, April 28th, 2015, Queen Mary University of London, London, UK.: I gave a lecture on the complexity of provability in systems of substructural logic, more precisely affine or contractive variants of linear logic. The main message is that a lot of insights into algorithmic complexity can be gained through inter-reductions with reachability problems in extensions of vector addition systems. The lecture is based on a joint paper with R. Lazić. Here are the slides.
ACTS 2015, 11:45am, February 11, 2015, CMI, Chennai, India.: I gave a chalk talk on the blackboard on the recent complexity upper bounds obtained with J. Leroux for the reachability problem in vector addition systems. Here are some mostly unreadable (guess you had to be there) slides compiled from photos (many thanks to A. Sangnier for taking these!).