Zeilberger's enumeration schemes can be used to completely automate the enumeration of many permutation classes. We extend his enumeration schemes so that they apply to many more permutation classes and describe the Maple package WilfPlus, which implements this process. We also compare enumeration schemes to three other systematic enumeration techniques: generating trees, substitution decompositions, and the insertion encoding.

We consider logics on $\mathbb{Z}$ and $\mathbb{N}$ which are weaker than Presburger arithmetic and we settle the following decision problem: given a k-ary relation on $\mathbb{Z}$ and $\mathbb{N}$ which are first order definable inPresburger arithmetic, are they definable in theseweaker logics? These logics, intuitively,are obtained by considering modulo and threshold counting predicates for differences of two variables.

We consider random graphs with a fixed degree sequence. Molloy and Reed [11, 12] studied how the size of the giant component changes according to degree conditions. They showed that there is a phase transition and investigated the order of components before and after the critical phase. In this paper we study more closely the order of components at the critical phase, using singularity analysis of a generating function for a branching process which models the random graph with a given degree sequence.

Let denote the set of unrooted labelled trees of size n and let ℳ be a particular (finite, unlabelled) tree. Assuming that every tree of is equally likely, it is shown that the limiting distribution as n goes to infinity of the number of occurrences of ℳ is asymptotically normal with mean value and variance asymptotically equivalent to μn and σ2n, respectively, where the constants μ>0 and σ≥0 are computable.

Given two trees (a target T and a pattern P) and a natural number w, window embedded subtree problems consist in deciding whether P occurs as an embedded subtreeof T and/or finding the number of size (at most) w windows of T which contain pattern P as an embedded subtree. P is an embedded subtree of T if P can be obtained by deleting some nodes from T (if a node v is deleted, all edges adjacent to v are also deleted, and outgoing edges are replaced by edges going from the parent of v (if it exists) to the children of v).Deciding whether P is an embedded subtree of T is known to be NP-complete. Our algorithms run in time O(|T|22|P|) where |T| (resp. |P|) is the size of T (resp. P).

Cameron introduced the orbit algebra of a permutation group and conjectured that this algebra is an integral domain if and only if the group has no finite orbit. We prove that this conjecture holds and in fact that the age algebra of a relational structure R is an integral domain if and only if R is age-inexhaustible. We deduce these results from a combinatorial lemma asserting that if a product of two non-zero elements of a set algebra is zero then there is a finite common tranversal of their supports. The proof is built on Ramsey theorem and the integrity of a shuffle algebra.

The k-core of a graph G is the maximal subgraph of G having minimum degree at least k. In 1996, Pittel, Spencer and Wormald found the threshold λc for the emergence of a non-trivial k-core in the random graph G(n, λ/n), and the asymptotic size of the k-core above the threshold. We give a new proof of this result using a local coupling of the graph to a suitable branching process. This proof extends to a general model of inhomogeneous random graphs with independence between the edges. As an example, we study the k-core in a certain power-law or ‘scale-free’ graph with a parameter c controlling the overall density of edges. For each k ≥ 3, we find the threshold value of c at which the k-core emerges, and the fraction of vertices in the k-core when c is ϵ above the threshold. In contrast to G(n, λ/n), this fraction tends to 0 as ϵ→0.

In a balls-in-bins process with feedback, balls are sequentially thrown into bins so that the probability that a bin with n balls obtains the next ball is proportional to f(n) for some function f. A commonly studied case where there are two bins and f(n) = np for p > 0, and our goal is to study the fine behaviour of this process with two bins and a large initial number t of balls. Perhaps surprisingly, Brownian Motions are an essential part of both our proofs.

For p > 1/2, it was known that with probability 1 one of the bins will lead the process at all large enough times. We show that if the first bin starts with balls (for constant λ∈ℝ), the probability that it always or eventually leads has a non-trivial limit depending on λ.

For p ≤ 1/2, it was known that with probability 1 the bins will alternate in leadership. We show, however, that if the initial fraction of balls in one of the bins is > 1/2, the time until it is overtaken by the remaining bin scales like Θ(t1+1/(1-2p)) for p < 1/2 and exp(Θ(t)) for p = 1/2. In fact, the overtaking time has a non-trivial distribution around the scaling factor, which we determine explicitly.

Our proofs use a continuous-time embedding of the balls-in-bins process (due to Rubin) and a non-standard approximation of the process by Brownian Motion. The techniques presented also extend to more general functions f.


 (1) Shepherdson proved that a discrete unitary commutative semi-ring A + satisfies IE 0 (induction scheme restricted to quantifierfree formulas) iff A is integral part of a real closed field; and Berarducci asked about extensions of this criterion when exponentiation is added to the language of rings. Let T range over axiom systems for ordered fields with exponentiation; for three values of T we provide a theory $_{\llcorner} T _{\lrcorner}$ in the language of rings plus exponentiation such that the models (A, expA ) of $_{\llcorner} T _{\lrcorner}$ are all integral parts A of models M of T with A + closed under expM and expA = expM | A+. Namely T = EXP, the basic theory of real exponential fields; T = EXP+ the Rolle and the intermediate value properties for all 2x-polynomials; and T = Texp , the complete theory of the field of reals with exponentiation.(2) $_{\llcorner}$ Texp $_{\lrcorner}$ is recursively axiomatizable iff Texp is decidable.  $_{\llcorner}$ Texp $_{\lrcorner}$ implies LE0(xy) (least element principle for open formulas in the language <,+,x,-1,xy ) but the reciprocal is an open question. $_{\llcorner}$ Texp $_{\lrcorner}$ satisfies “provable polytime witnessing”: if $_{\llcorner} $ Texp $_{\lrcorner}$ proves ∀x∃y : |y| < |x|k)R(x,y) (where $|y|:=_{\llcorner}$ log(y) $_{\lrcorner}$ , k < ω and R is an NP relation), then it proves ∀x R(x,ƒ(x)) for some polynomial time function f.(3) We introduce “blunt” axioms for Arithmetics: axioms which do as if every real number was a fraction (or even a dyadic number). The falsity of such a contention in the standard model of the integers does not mean inconsistency; and bluntness has both a heuristic interest and a simplifying effect on many questions – in particular we prove that the blunt version of $_{\llcorner}$ Texp $_{\lrcorner}$ is a conservative extension of $_{\llcorner} $ Texp $_{\lrcorner}$ for sentences in ∀Δ0(xy ) (universal quantifications of bounded formulas in the language of rings plus xy ). Blunt Arithmetics – which can be extended to a much richer language – could become a useful tool in the non standard approach to discrete geometry, to modelization and to approximate computation with reals.


We give a complete characterization of the class of functions that arethe intensional behaviours of primitive recursive (PR) algorithms. This class is the set of primitive recursive functions that have a null basic caseof recursion. This result is obtained using the property of ultimate unarity and a geometrical approach of sequential functions on N the set of positive integers.

Lorsqu'on observe des orbites de certains automates cellulaires, on peut penser qu'elles apparaissent comme des mélanges d'orbites d'autres automates (composants). Dans cet article, nous tentons de comprendre ce phénomène en construisant un hybride de deux automates au moyen d'un troisième. Deux types d'automates cellulaires sont introduits : les captifs et les foulards. Nous comparons des propriétés de ces hybrides dans le cadre des classifications algébriques introduites par [B. Martin (2001) ; N. Ollinger (2002) ; I. Rapaport (1998) ; G. Teyssier (2005) : PhD. Thesis, École Normale Supérieure de Lyon].

We prove that there exists a structure M whose monadic second order theory is decidable, and such that the first-order theory of every expansion of M by a constant is undecidable. 


The first-value operator assigns to any sequence of partial functions of the same type a new such function. Its domain is the union of the domains of the sequence functions, and its value atany point is just the value of the first function in the sequence which is defined at that point. In this paper, the first-value operator is applied to establish hierarchies of classes of functions under various settings. For effective sequences of computable discrete functions, we obtain ahierarchy connected with Ershov's one within $\Delta^{0}_2$ . The non-effective version over real functions is connected with the degrees of discontinuity and yields a hierarchy related to Hausdorff's difference hierarchy in the Borel class  $\Delta^{B}_2$ . Finally, the effective version over approximately computable real functions forms a hierarchy which provides a useful tool in computable analysis.

We revisit the problem of deciding whether a finitely generated subgroup H is a free factor of a given free group F. Known algorithms solve this problem in time polynomial in the sum of the lengths of the generators of H and exponential in the rank of F. We show that the latter dependency can be made exponential in the rank difference rank(F) - rank(H), which often makes a significant change.

Inf-Datalog extends the usual least fixpoint semantics of Datalog with greatest fixpoint semantics: we defined inf-Datalog and characterized theexpressive power of various fragments of inf-Datalog in [CITE].In the present paper, we study the complexity of query evaluation on finite modelsfor (various fragments of) inf-Datalog.We deduce a unified and elementary proof that global model-checking (i.e. computing all nodes satisfying a formula in a given structure) has1. quadratic data complexity in timeand linear program complexity in spacefor CTL and alternation-free modal μ-calculus, and2. linear-space (data and program) complexities, linear-time program complexityand polynomial-time data complexity for Lµk (modal μ-calculus with fixed alternation-depth at most k).

It is well-known that some of the most basic properties of words, like thecommutativity (xy = yx) and the conjugacy (xz = zy), can be expressedas solutions of word equations. An important problem is to decide whetheror not a given equation on words has a solution. For instance,the equation xMyN = zP has only periodic solutions in a freemonoid, that is, if xMyN = zP holds with integers m,n,p ≥ 2,then there exists a word w such that x, y, z are powers of w.This result, which received a lot of attention, was first provedby Lyndon and Schützenberger for free groups.In this paper, we investigate equations on partial words.Partial words are sequences over a finite alphabet that may containa number of “do not know” symbols. When we speak about equationson partial words, we replace the notion of equality(=) with compatibility (↑).Among other equations, we solve xy ↑ yx,xz ↑ zy, and special cases of xmyn ↑ zp for integers m,n,p ≥ 2. 


Let w be an infinite fixed point of a binary k-uniform morphism f, and let E w bethe critical exponent of w. We give necessary and sufficient conditions for E w to bebounded, and an explicit formula to compute it when it is. In particular, we show that E wis always rational. We also sketch an extension of our method to non-uniform morphisms over generalalphabets.

We present the first (polynomial-time) algorithm for reducinga given deterministic finite state automaton (DFA) intoa hyper-minimized DFA, which may have fewer states thanthe classically minimized DFA. The price we pay is that thelanguage recognized by the new machine can differ from theoriginal on a finite number of inputs. These hyper-minimizedautomata are optimal, in the sense that every DFA with fewerstates must disagree on infinitely many inputs. With smallmodifications, the construction works also for finite statetransducers producing outputs. Within a class of finitely differing languages, thehyper-minimized automaton is not necessarily unique. There mayexist several non-isomorphic machines using the minimum number ofstates, each accepting a separate language finitely-differentfrom the original one. We will show that there are largestructural similarities among all these smallest automata.

A new algorithm is presented for the D0L sequence equivalence problem which, when the alphabets are fixed, works in time polynomial in the rest of the input data. The algorithm uses a polynomial encoding of words and certain well-known properties of $\mathbb{Z}$ -rational sequences.

Real functions on the domain [0,1)n – often used to describe digitalimages – allow for different well-known types of binary operations. In thisnote, we recapitulate how weighted finite automata can be used in order torepresent those functions and how certain binary operations are reflected inthe theory of these automata. Different types of products of automata are employed, includingthe seldomly-used full Cartesian product. We show, however, the infeasibilityof functional composition; simple examples yield that the class of automaticfunctions (i.e., functions computable by automata) is not closed under thisoperation.