Galois correspondence for counting quantifiers

We introduce a new type of closure operator on the set of relations, max-implementation, and its weaker analog max-quantification. Then we show that approximation preserving reductions between counting constraint satisfaction problems (#CSPs) are preserved by these two types of closure operators. Together with some previous results this means that the approximation complexity of counting CSPs is determined by partial clones of relations that additionally closed under these new types of closure operators. Galois correspondence of various kind have proved to be quite helpful in the study of the complexity of the CSP. While we were unable to identify a Galois correspondence for partial clones closed under max-implementation and max-quantification, we obtain such results for slightly different type of closure operators, k-existential quantification. This type of quantifiers are known as counting quantifiers in model theory, and often used to enhance first order logic languages. We characterize partial clones of relations closed under k-existential quantification as sets of relations invariant under a set of partial functions that satisfy the condition of k-subset surjectivity. Finally, we give a description of Boolean max-co-clones, that is, sets of relations on {0,1} closed under max-implementations.Comment: 28 pages, 2 figure

Definable sets, motives and p-adic integrals

We associate canonical virtual motives to definable sets over a field of characteristic zero. We use this construction to show that very general p-adic integrals are canonically interpolated by motivic ones.Comment: 45 page

Quantified Constraints in Twenty Seventeen

I present a survey of recent advances in the algorithmic and computational complexity theory of non-Boolean Quantified Constraint Satisfaction Problems, incorporating some more modern research directions

Do Hard SAT-Related Reasoning Tasks Become Easier in the Krom Fragment?

Many reasoning problems are based on the problem of satisfiability (SAT). While SAT itself becomes easy when restricting the structure of the formulas in a certain way, the situation is more opaque for more involved decision problems. We consider here the CardMinSat problem which asks, given a propositional formula $\phi$ and an atom $x$, whether $x$ is true in some cardinality-minimal model of $\phi$. This problem is easy for the Horn fragment, but, as we will show in this paper, remains $\Theta_2$-complete (and thus $\mathrm{NP}$-hard) for the Krom fragment (which is given by formulas in CNF where clauses have at most two literals). We will make use of this fact to study the complexity of reasoning tasks in belief revision and logic-based abduction and show that, while in some cases the restriction to Krom formulas leads to a decrease of complexity, in others it does not. We thus also consider the CardMinSat problem with respect to additional restrictions to Krom formulas towards a better understanding of the tractability frontier of such problems

Ax's theorem with an additive character

Motivated by Emmanuel Kowalski's exponential sums over definable sets in finite fields, we generalize Ax's theorem on pseudo-finite fields to a continuous-logic setting allowing for an additive character. The role played by Weil's Riemann hypothesis for curves over finite fields is taken by the `Weil bound' on exponential sums. Subsequent model-theoretic developments, including simplicity and the Chatzidakis-Van den Dries-Macintyre definable measures, also generalize. Analytically, we have the following consequence: consider the algebra of functions \Ff_p^n \to \Cc obtained from the additive characters and the characteristic functions of subvarieties by pre- or post-composing with polynomials, applying min and sup operators to the real part, and averaging over subvarieties. Then any element of this class can be approximated, uniformly in the variables and in the prime $p$, by a polynomial expression in $\Psi_p(\xi)$ at certain algebraic functions $\xi$ of the variables, where $\Psi(n \mod p) = exp(2 \pi i n/p)$ is the standard additive character.Comment: Version 3: various local changes, with some material reorganized for clarity. The main mathematical difference is in section 5, where the connection to Duke-Friedlander-Iwaniec is considerably tightene

A presentation theorem for continuous logic and Metric Abstract Elementary Classes

We give a presentation theorem for continuous first-order logic and Metric Abstract Elementary classes in terms of $L_{\omega_1, \omega}$ and Abstract Elementary Classes, respectively. This presentation is accomplished by analyzing dense subsets that are closed under functions. We extend this correspondence to types and saturation

Definable equivalence relations and zeta functions of groups

We prove that the theory of the $p$-adics ${\mathbb Q}_p$ admits elimination of imaginaries provided we add a sort for ${\rm GL}_n({\mathbb Q}_p)/{\rm GL}_n({\mathbb Z}_p)$ for each $n$. We also prove that the elimination of imaginaries is uniform in $p$. Using $p$-adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed $p$) of these formal zeta functions that extends to the subanalytic context. As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups.Comment: 89 pages. Various corrections and changes. To appear in J. Eur. Math. So

Asymptotic behaviour of rational curves

We investigate the asympotic behaviour of the moduli space of morphisms from the rational curve to a given variety when the degree becomes large. One of the crucial tools is the homogeneous coordinate ring of the variey. First we explain in details what happens in the toric case. Then we examine the general case.Comment: This is a revised and slightly expanded version of notes for a course delivered during the summer school on rational curves held in June 2010 at Institut Fourier, Grenobl