Arnaud Durand

First-order queries on structures of bounded degree are computable with constant delay

A relational structure is <i>d</i>-degree-bounded, for some integer <i>d</i>, if each element of the domain belongs to at most <i>d</i> tuples. In this paper, we revisit the complexity of the evaluation problem of not necessarily Boolean first-order (<b>FO</b>) queries over <i>d</i>-degree-bounded structures. Query evaluation is considered here as a dynamical process. We prove that any <b>FO</b> query on <i>d</i>-degree-bounded structures belongs to the complexity class constant-Delay<inf><i>lin</i></inf>, that is, can be computed by an algorithm that has two separate parts: it has a precomputation step of time linear in the size of the structure and then, it outputs all solutions (i.e., tuples that satisfy the formula) one by one with a constant delay (i.e., depending on the size of the formula only) between each. Seen as a global process, this implies that queries on <i>d</i>-degree-bounded structures can be evaluated in total time <i>f</i>(|ϕ|).(|<i>S</i>| + |ϕ(<i>S</i>)|) and space <i>g</i>(|ϕ|).|<i>S</i>| where <i>S</i> is the structure, ϕ is the formula, ϕ(<i>S</i>) is the result of the query and <i>f</i>, <i>g</i> are some fixed functions. Among other things, our results generalize a result of Seese on the data complexity of the model-checking problem for <i>d</i>-degree-bounded structures. Besides, the originality of our approach compared to related results is that it does not rely on the Hanfs model-theoretic technique and is simple and informative since it essentially rests on a quantifier elimination method.

On acyclic conjunctive queries and constant delay enumeration

We study the enumeration complexity of the natural extension of acyclic conjunctive queries with disequalities. In this language, a number of NP-complete problems can be expressed. We first improve a previous result of Papadimitriou and Yannakakis by proving that such queries can be computed in time c.|M|ċ|ϕ(M)| where M is the structure, ϕ(M) is the result set of the query and c is a simple exponential in the size of the formula ϕ. A consequence of our method is that, in the general case, tuples of such queries can be enumerated with a linear delay between two tuples. We then introduce a large subclass of acyclic formulas called CCQ≠ and prove that the tuples of a CCQ≠ query can be enumerated with a linear time precomputation and a constant delay between consecutive solutions. Moreover, under the hypothesis that the multiplication of two n×n boolean matrices cannot be done in time O(n2), this leads to the following dichotomy for acyclic queries: either such a query is in CCQ≠ or it cannot be enumerated with linear precomputation and constant delay. Furthermore we prove that testing whether an acyclic formula is in CCQ≠ can be performed in polynomial time. Finally, the notion of free-connex treewidth of a structure is defined. We show that for each query of free-connex treewidth bounded by some constant k, enumeration of results can be done with O(|M|k+1) precomputation steps and constant delay.

Sets with large intersection and ubiquity

A central problem motivated by Diophantine approximation is to determine the size properties of subsets of of the form where denotes an arbitrary norm, I a denumerable set, (xi,ri)i I a family of elements of × (0, 8) and a nonnegative nondecreasing function defined on [0, 8). We show that if FId, where Id denotes the identity function, has full Lebesgue measure in a given nonempty open subset V of , the set F belongs to a class Gh(V) of sets with large intersection in V with respect to a given gauge function h. We establish that this class is closed under countable intersections and that each of its members has infinite Hausdorff g-measure for every gauge function g which increases faster than h near zero. In particular, this yields a sufficient condition on a gauge function g such that a given countable intersection of sets of the form F has infinite Hausdorff g-measure. In addition, we supply several applications of our results to Diophantine approximation. For any nonincreasing sequence ? of positive real numbers converging to zero, we investigate the size and large intersection properties of the sets of all points that are ?-approximable by rationals, by rationals with restricted numerator and denominator and by real algebraic numbers. This enables us to refine the analogs of Jarniks theorem for these sets. We also study the approximation of zero by values of integer polynomials and deduce several new results concerning Mahlers and Koksmas classifications of real transcendental numbers. A central problem motivated by Diophantine approximation is to determine the size properties of subsets of of the form where denotes an arbitrary norm, I a denumerable set, (xi,ri)i I a family of elements of × (0, 8) and a nonnegative nondecreasing function defined on [0, 8). We show that if FId, where Id denotes the identity function, has full Lebesgue measure in a given nonempty open subset V of , the set F belongs to a class Gh(V) of sets with large intersection in V with respect to a given gauge function h. We establish that this class is closed under countable intersections and that each of its members has infinite Hausdorff g-measure for every gauge function g which increases faster than h near zero. In particular, this yields a sufficient condition on a gauge function g such that a given countable intersection of sets of the form F has infinite Hausdorff g-measure. In addition, we supply several applications of our results to Diophantine approximation. For any nonincreasing sequence ? of positive real numbers converging to zero, we investigate the size and large intersection properties of the sets of all points that are ?-approximable by rationals, by rationals with restricted numerator and denominator and by real algebraic numbers. This enables us to refine the analogs of Jarniks theorem for these sets. We also study the approximation of zero by values of integer polynomials and deduce several new results concerning Mahlers and Koksmas classifications of real transcendental numbers.

Random Wavelet Series Based on a Tree-Indexed Markov Chain

We study the global and local regularity properties of random wavelet series whose coefficients exhibit correlations given by a tree-indexed Markov chain. We determine the law of the spectrum of singularities of these series, thereby performing their multifractal analysis. We also show that almost every sample path displays an oscillating singularity at almost every point and that the points at which a sample path has at most a given Hölder exponent form a set with large intersection.

On Randomly Placed Arcs on the Circle

We completely describe in terms of Hausdorff measures the size of the set of points of the circle that are covered infinitely often by a sequence of random arcs with given lengths. We also show that this set is a set with large intersection.

On the complexity of recognizing the Hilbert basis of a linear diophantine system

The problem of computing the Hilbert basis of a homogeneous linear Diophantine system over nonnegative integers is often considered in automated deduction and integer programming. In automated deduction, the Hilbert basis of a corresponding system serves to compute the minimal complete set of associative-commutative unifiers, whereas in integer programming the Hilbert bases are tightly connected to integer polyhedra and to the notion of total dual integrality. In this paper, we sharpen the previously known result that the problem, asking whether a given solution belongs to the Hilbert basis of a given system, is coNP-complete. We show that the problem has a pseudopolynomial algorithm if the number of equations in the system is fixed, but it is coNP-complete in the strong sense if the given system is unbounded. This result is important in the scope of automated deduction, where the input is given in unary and therefore the previously known coNP-completeness result was unusable. Moreover, we show that, from the complexity standpoint, it is not important to know the underlying homogeneous linear Diophantine system when we ask whether a given set of vectors constitutes a Hilbert basis.

Enumeration Complexity of Logical Query Problems with Second-order Variables

We consider query problems defined by first order formulas of the form F(x,T) with free first order and second order variables and study the data complexity of enumerating results of such queries. By considering the number of alternations in the quantifier prefixes of formulas, we show that such query problems either admit a constant delay or a polynomial delay enumeration algorithm or are hard to enumerate. We also exhibit syntactically defined fragments inside the hard cases that still admit good enumeration algorithms and discuss the case of some restricted classes of database structures as inputs.

Large intersection properties in Diophantine approximation and dynamical systems

We investigate the large intersection properties of the set of points that are approximated at a certain rate by a family of affine subspaces. We then apply our results to various sets arising in the metric theory of Diophantine approximation, in the study of the homeomorphisms of the circle, and in the perturbation theory for Hamiltonian systems.

Nonerasing, Counting, and Majority over the Linear Time Hierarchy

In this paper, we investigate several extensions of the linear time hierarchy (denoted by LTH). We first prove that it is not necessary to erase the oracle tape between two successive oracle calls, thereby lifting a common restriction on LTH machines. We also define a natural counting extension of LTH and show that it corresponds to a robust notion of counting bounded arithmetic predicates. Finally, we show that the computational power of the majority operator is equivalent to that of the exact counting operator in both contexts.

Random fractals and tree-indexed Markov chains

We study the size properties of a general model of fractal sets that are based on a tree-indexed family of random compacts and a tree-indexed Markov chain. These fractals may be regarded as a generalization of those resulting from the Moran-like deterministic or random recursive constructions considered by various authors. Among other applications, we consider various extensions of Mandelbrots fractal percolation process.