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

A word-to-word function is continuous for a class of languages~$\mathcal{V}$ if its inverse maps $\mathcal{V}$_languages to~$\mathcal{V}$. This notion provides a basis for an algebraic study of transducers, and was integral to the characterization of the sequential transducers computable in some circuit complexity classes. Here, we report on the decidability of continuity for functional transducers and some standard classes of regular languages. To this end, we develop a robust theory rooted in the standard profinite analysis of regular languages. Since previous algebraic studies of transducers have focused on the sole structure of the underlying input automaton, we also compare the two algebraic approaches. We focus on two questions: When are the automaton structure and the continuity properties related, and when does continuity propagate to superclasses?

We study the expressive power of polynomial recursive sequences, a nonlinear extension of the well-known class of linear recursive sequences. These sequences arise naturally in the study of nonlinear extensions of weighted automata, where (non)expressiveness results translate to class separations. A typical example of a polynomial recursive sequence is b_n=n!. Our main result is that the sequence u_n=n^n is not polynomial recursive.

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.

A word-to-word function is continuous for a class of languages $\mathcal{V}$ if its inverse maps $\mathcal{V}$-languages to $\mathcal{V}$. This notion provides a basis for an algebraic study of transducers, and was integral to the characterization of the sequential transducers computable in some circuit complexity classes. Here, we report on the decidability of continuity for functional transducers and some standard classes of regular languages. Previous algebraic studies of transducers have focused on the structure of the underlying input automaton, disregarding the output. We propose a comparison of the two algebraic approaches through two questions: When are the automaton structure and the continuity properties related, and when does continuity propagate to superclasses?

We study Monadic Second-Order Logic (MSO) over finite words, extended with (non-uniform arbitrary) monadic predicates. We show that it defines a class of languages that has algebraic, automata-theoretic and machine-independent characterizations. We consider the regularity question: given a language in this class, when is it regular? To answer this, we show a substitution property and the existence of a syntactical predicate. We give three applications. The first two are to give very simple proofs that the Straubing Conjecture holds for all fragments of MSO with monadic predicates, and that the Crane Beach Conjecture holds for MSO with monadic predicates. The third is to show that it is decidable whether a language defined by an MSO formula with morphic predicates is regular.

We study Monadic Second-Order Logic (MSO) over finite words, extended with (non-uniform arbitrary) monadic predicates. We show that it defines a class of languages that has algebraic, automata-theoretic, and machine-independent characterizations. We consider the regularity question: Given a language in this class, when is it regular? To answer this, we show a substitution property and the existence of a syntactical predicate. We give three applications. The first two are to give very simple proofs that the Straubing Conjecture holds for all fragments of MSO with monadic predicates and that the Crane Beach Conjecture holds for MSO with monadic predicates. The third is to show that it is decidable whether a language defined by an MSO formula with morphic predicates is regular.

We introduce the constrained topological sorting problem (CTS-problem): given a target language L and a directed acyclic graph (DAG) G with labeled vertices, determine if G has a topological sort which forms a word that belongs to L. This natural problem applies to several settings, including scheduling with costs or verifying concurrent programs. It also generalizes the shuffle problem of formal language theory, which asks if a list of input strings has an interleaving that achieves a target string. We accordingly call constrained shuffle problem (CSh-problem) the restriction of our CTS-problem where the input DAG consists of disjoint strings. We study the complexity of the CTS-problem and CSh-problem for regular target languages: for each fixed regular language L, we call CTS(L) and CSh(L) the corresponding problems, where the input is the DAG. Our goal is to characterize the regular languages for which these problems are tractable. We show that both problems are tractable (in NL) for unions of monomials, a useful language class for pattern matching, as well as some other cases. We extend this for the CSh-problem to unions of district group monomials. We also show NP-hardness for some other languages such as (ab)^*. These results lead to a dichotomy for a different problem phrasing, when the target is specified as a semiautomaton to enforce some closure assumptions, and where the semiautomaton is assumed to be counter-free. In this case, both problems are in NL if the transition monoid of the semiautomaton is in some class, and NP-hard otherwise. Without the counter-freeness assumption, we can extend the dichotomy to a partial result for the CSh-problem. Our proofs use a variety of tools ranging from complexity theory, combinatorics, algebraic automata theory, Ramsey's theorem, as well as a custom reduction and other new techniques.

First-order logic (FO) over words is shown to be equiexpressive with FO equipped with a restricted set of numerical predicates, namely the order, a binary predicate MSB$_0$, and the finite-degree predicates: FO[Arb] = FO[<, MSB$_0$, Fin]. The Crane Beach Property (CBP), introduced more than a decade ago, is true of a logic if all the expressible languages admitting a neutral letter are regular. Although it is known that FO[Arb] does not have the CBP, it is shown here that the (strong form of the) CBP holds for both FO[<, Fin] and FO[<, MSB$_0$]. Thus FO[<, Fin] exhibits a form of locality and the CBP, and can still express a wide variety of languages, while being one simple predicate away from the expressive power of FO[Arb]. The counting ability of FO[<, Fin] is studied as an application.

We investigate a subclass of languages recognized by vector addition systems, namely languages of nondeterministic Parikh automata. While the regularity problem (is the language of a given automaton regular?) is undecidable for this model, we show surprising decidability of the regular separability problem: given two Parikh automata, is there a regular language that contains one of them and is disjoint from the other?