Sylvain Schmitz

Multiply-recursive upper bounds with Higman's Lemma

We develop a new analysis for the length of controlled bad sequences in well-quasi-orderings based on Higmans Lemma. This leads to tight multiply-recursive upper bounds that readily apply to several verification algorithms for well-structured systems.

Complexity Hierarchies beyond Elementary

We introduce a hierarchy of fast-growing complexity classes and show its suitability for completeness statements of many nonelementary problems. This hierarchy allows the classification of many decision problems with a nonelementary complexity, which occur naturally in areas such as logic, combinatorics, formal languages, and verification, with complexities ranging from simple towers of exponentials to Ackermannian and beyond.

Demystifying Reachability in Vector Addition Systems

More than 30 years after their inception, the decidability proofs for reachability in vector addition systems (VAS) still retain much of their mystery. These proofs rely crucially on a decomposition of runs successively refined by Mayr, Kosaraju, and Lambert, which appears rather magical, and for which no complexity upper bound is known. We first offer a justification for this decomposition technique, by showing that it emerges naturally in the study of the ideals of a well quasi ordering of VAS runs. In a second part, we apply recent results on the complexity of termination thanks to well quasi orders and well orders to obtain fast-growing complexity upper bounds for the decomposition algorithms, thus providing the first known upper bounds for general VAS reachability.More than 30 years after their inception, the decidability proofs for reach ability in vector addition systems (VAS) still retain much of their mystery. These proofs rely crucially on a decomposition of runs successively refined by Mayr, Kosaraju, and Lambert, which appears rather magical, and for which no complexity upper bound is known. We first offer a justification for this decomposition technique, by showing that it computes the ideal decomposition of the set of runs, using the natural embedding relation between runs as well quasi ordering. In a second part, we apply recent results on the complexity of termination thanks to well quasi orders and well orders to obtain a cubic Ackermann upper bound for the decomposition algorithms, thus providing the first known upper bounds for general VAS reach ability.

Conservative ambiguity detection in context-free grammars

The ability to detect ambiguities in context-free grammars is vital for their use in several fields, but the problem is undecidable in the general case. We present a safe, conservative approach, where the approximations cannot result in overlooked ambiguous cases. We analyze the complexity of the verification, and provide formal comparisons with several other ambiguity detection methods.

An experimental ambiguity detection tool

Although programs convey an unambiguous meaning, the grammars used in practice to describe their syntax are often ambiguous, and completed with disambiguation rules. Whether these rules achieve the removal of all the ambiguities while preserving the original intended language can be difficult to ensure. We present an experimental ambiguity detection tool for GNU Bison, and illustrate how it can assist a grammatical development for a subset of Standard ML.

Fixed-Dimensional Energy Games are in Pseudo-Polynomial Time

We generalise the hyperplane separation technique Chatterjee and Velner, 2013 from multi-dimensional mean-payoff to energy games, and achieve an algorithm for solving the latter whose running time is exponential only in the dimension, but not in the number of vertices of the game graph. This answers an open question whether energy games with arbitrary initial credit can be solved in pseudo-polynomial time for fixed dimensionsi¾?3 or larger Chaloupka, 2013. It also improves the complexity of solving multi-dimensional energy games with given initial credit from non-elementary Brazdil, Janăi¾?ar, and Kuăi¾?era, 2010 to 2EXPTIME, thus establishing their 2EXPTIME-completeness.

The Power of Priority Channel Systems

We introduce Priority Channel Systems, a new class of channel systems where messages carry a numeric priority and where higher-priority messages can supersede lower-priority messages preceding them in the fifo communication buffers. The decidability of safety and inevitability properties is shown via the introduction of a priority embedding, a well-quasi-ordering that has not previously been used in well-structured systems. We then show how Priority Channel Systems can compute Fast-Growing functions and prove that the aforementioned verification problems are

Alternating Vector Addition Systems with States

\mathbf{F}_{\varepsilon_{0}}

Coverability trees for petri nets with unordered data

-complete.

Deciding Piecewise Testable Separability for Regular Tree Languages

Alternating vector addition systems are obtained by equipping vector addition systems with states (VASS) with ‘fork’ rules, and provide a natural setting for infinite-arena games played over a VASS. Initially introduced in the study of propositional linear logic, they have more recently gathered attention in the guise of multi-dimensional energy games for quantitative verification and synthesis.