Charles Paperman’s research while affiliated with University of Lille and other places

Given two families of sets $\mathcal{F}$ and $\mathcal{G}$, the $\mathcal{F}$ separability problem for $\mathcal{G}$ asks whether for two given sets $U, V \in \mathcal{G}$ there exists a set $S \in \mathcal{F}$, such that U is included in S and V is disjoint with S. We consider two families of sets $\mathcal{F}$: modular sets $S \subseteq \mathbb{N}^d$, defined as unions of equivalence classes modulo some natural number $n \in \mathbb{N}$, and unary sets. Our main result is decidability of modular and unary separability for the class $\mathcal{G}$ of reachability sets of Vector Addition Systems, Petri Nets, Vector Addition Systems with States, and for sections thereof.

XML schema validation can be performed in constant memory in the streaming model if and only if the schema admits only trees of bounded depth - an acceptable assumption from the practical view-point. In this paper we refine this analysis by taking into account that data can be streamed block-by-block, rather then letter-by-letter, which provides opportunities to speed up the computation by parallelizing the processing of each block. For this purpose we introduce the model of streaming circuits, which process words of arbitrary length in blocks of fixed size, passing constant amount of information between blocks. This model allows us to transfer fundamental results about the circuit complexity of regular languages to the setting of streaming schema validation, which leads to effective constructions of streaming circuits of depth logarithmic in the block size, or even constant under certain assumptions on the input schema. For nested-relational DTDs, a practically motivated class of bounded-depth XML schemas, we provide an efficient construction yielding constant-depth streaming circuits with particularly good parameters.

In this paper, we study the lattice and the Boolean algebra, possibly closed under quotient, generated by the languages of the form u∗, where u is a word. We provide effective equational characterisations of these classes, i.e. one can decide using our descriptions whether a given regular language belongs or not to each of them.

Low circuit complexity classes and regular languages exhibit very tight interactions that shade light on their respective expressiveness. We propose to study these interactions at a functional level, by investigating the deterministic rational transductions computable by constant-depth, polysize circuits. To this end, a circuit framework of independent interest that allows variable output length is introduced. Relying on it, there is a general characterization of the set of transductions realizable by circuits. It is then decidable whether a transduction is definable in $\mathrm{AC}^0$ and, assuming a well-established conjecture, the same for $\mathrm{ACC}^0$.

We investigate the decidability of the definability problem for fragments of first order logic over finite words enriched with regular numerical predicates. In this paper, we focus on the quantifier alternation hierarchies of first order logic. We obtain that deciding this problem for each level of the alternation hierarchy of both first order logic and its two-variable fragment when equipped with all regular numerical predicates is not harder than deciding it for the corresponding level equipped with only the linear order. Relying on some recent results, this proves the decidability for each level of the alternation hierarchy of the two-variable first order fragment while in the case of the first order logic the question remains open for levels greater than two. The main ingredients of the proofs are syntactic transformations of first-order formulas as well as the infinitely testable property, a new algebraic notion on varieties that we define.

We consider two-variable first-order logic on finite words with a fixed number of quantifier alternations. We show that all languages with a neutral letter definable using the order and finite-degree predicates are also definable with the order predicate only. From this result we derive the separation of the alternation hierarchy of two-variable logic on this signature.

This paper is a contribution to the study of regular languages defined by fragments of first order or even monadic second order logic. More specifically, we consider the operation of enriching a given fragment by adding modular predicates. Our first result gives a simple algebraic counterpart to this operation in terms of semidirect products of varieties together with a combinatorial description based on elementary operations on languages. Now, a difficult question is to know whether the decidability of a given fragment is preserved under this enrichment. We first prove that this is always the case for so-called local varieties. The problem remains open in the nonlocal case but our main results also gives several sufficient conditions to preserve decidability. We use these latter results to establish the decidability of three fragments of the first order logic with two variables.