Sylvain Schmitz

Assistant professor

Complexity Hierarchies Beyond Elementary

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Don't Panic!

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:


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


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.
  1. 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.
  2. 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
Best Paper Award at RTA-TLCA 2014Building 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...


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!).

More talks...

Other activities


MPRI 2015–2016
First half of the logical and computational structures for linguistic modeling course.
First half of the initiation to verification course.
Computer science option of the French "agrégation de mathématiques"
Courses on formal languages and automata.
Bachelor of computer science
Labs for the second half of the computability and complexity course.
Labs for the formal languages course.

Further documents and course pages related to my older teaching activities can be found in my teaching activities page.

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Sylvain Schmitz
LSV, CNRS & ENS de Cachan
61, avenue du Président Wilson
94235 CACHAN Cedex, France
+33 (0)1 47 40 75 42
+33 (0)1 47 40 75 21
+33 (0)1 47 40 75 20