arXiv subjects

Arnaud Durand

Publications and source records attributed to Arnaud Durand.

At least 19 recordsLinked to original sources

arXiv2026

Recursion and proof theoretical characterizations of small circuit classes with modulo counting via discrete differential equations (long version)

The paper proposes an implicit (i.e., machine-independent) complexity approach to studying computation by polynomial-size, constant-depth circuits with gates counting modulo a constant through the lens of discrete ordinary differential equations (ODEs). So far, recursion-theoretic characterizations have been provided for functions computed by circuits of constant depth, including gates counting modulo 2 and 6 only (i.e., for the classes FAC0[2] and FAC0[6], resp.). In this paper, it is shown that considering ODE schemas, rather than bounded recursion, allows for a more fine-grained analysis, leading to (uniform) characterizations for all classes FAC0[n] (n \in N), i.e. functions computed by circuits including counting modulo n gates. Inspired by the syntactic form of the ODE schemas, we go further in this direction and present first-order bounded theories for capturing provably total functions in each of these classes.

cs.CC↗

arXiv2025

Towards New Characterizations of Small Circuit Classes via Discrete Ordinary Differential Equations

Implicit computational complexity is a lively area of theoretical computer science, which aims to provide machine-independent characterizations of relevant complexity classes. % for uniformity with subsequent uses >> 1960s (but feel free to modify it) % One of the seminal works in this field appeared in the 1960s, when Cobham introduced a function algebra closed under bounded recursion on notation to capture polynomial time computable functions ($FP$). Later on, several complexity classes have been characterized using \emph{limited} recursion schemas. In this context, an original approach has been recently introduced, showing that ordinary differential equations (ODEs) offer a natural tool for algorithmic design and providing a characterization of $FP$ by a new ODE-schema. In the present paper we generalize this approach by presenting original ODE-characterizations for the small circuit classes $AC^0$ and $FTC^0$.

cs.CC↗

arXiv2025

Characterizing Small Circuit Classes from FAC^0 to FAC^1 via Discrete Ordinary Differential Equations

In this paper, we provide a uniform framework for investigating small circuit classes and bounds through the lens of ordinary differential equations (ODEs). Following an approach recently introduced to capture the class of polynomial-time computable functions via ODE-based recursion schemas and later applied to the context of functions computed by unbounded fan-in circuits of constant depth (FAC^0), we study multiple relevant small circuit classes. In particular, we show that natural restrictions on linearity and derivation along functions with specific growth rate correspond to kinds of functions that can be proved to be in various classes, ranging from FAC^0 to FAC^1. This reveals an intriguing link between constraints over linear-length ODEs and circuit computation, providing new tools to tackle the complex challenge of establishing bounds for classes in the circuit hierarchies and possibly enhancing our understanding of the role of counters in this setting. Additionally, we establish several completeness results, in particular obtaining the first ODE-based characterizations for the classes of functions computable in constant depth with unbounded fan-in and Mod 2 gates (FACC[2]) and in logarithmic depth with bounded fan-in Boolean gates (FNC1).

cs.CC↗

arXiv2025

The flattening operator in team-based logics

We propose a systematic study of the so-called flattening operator in team semantics. This operator was first introduced by Hodges in 1997, and has not been studied in more detail since. We begin a systematic study of the expressive power this operator adds to the most well-known team-based logics, such as dependence logic, anonymity logic, inclusion logic and exclusion logic.

cs.LO↗

arXiv2022

A characterization of functions over the integers computable in polynomial time using discrete differential equations

This paper studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs), a.k.a. (Ordinary) Difference Equations. It presents a new framework using these equations as a central tool for computation and algorithm design. We present the general theory of discrete ODEs for computation theory, we illustrate this with various examples of algorithms, and we provide several implicit characterizations of complexity and computability classes. The proposed framework presents an original point of view on complexity and computation classes. It unifies several constructions that have been proposed for characterizing these classes including classical approaches in implicit complexity using restricted recursion schemes, as well as recent characterizations of computability and complexity by classes of continuous ordinary differential equations. It also helps understanding the relationships between analog computations and classical discrete models of computation theory. At a more technical point of view, this paper points out the fundamental role of linear (discrete) ODEs and classical ODE tools such as changes of variables to capture computability and complexity measures, or as a tool for programming many algorithms.

cs.LO↗

arXiv2022

Enumerating Answers to First-Order Queries over Databases of Low Degree

A class of relational databases has low degree if for all $δ>0$, all but finitely many databases in the class have degree at most $n^δ$, where $n$ is the size of the database. Typical examples are databases of bounded degree or of degree bounded by $\log n$. It is known that over a class of databases having low degree, first-order boolean queries can be checked in pseudo-linear time, i.e.\ for all $ε>0$ in time bounded by $n^{1+ε}$. We generalize this result by considering query evaluation. We show that counting the number of answers to a query can be done in pseudo-linear time and that after a pseudo-linear time preprocessing we can test in constant time whether a given tuple is a solution to a query or enumerate the answers to a query with constant delay.

cs.DB↗

arXiv2022

Enumeration Classes Defined by Circuits

We refine the complexity landscape for enumeration problems by introducing very low classes defined by using Boolean circuits as enumerators. We locate well-known enumeration problems, e.g., from graph theory, Gray code enumeration, and propositional satisfiability in our classes. In this way we obtain a framework to distinguish between the complexity of different problems known to be in $\mathbf{DelayP}$, for which a formal way of comparison was not possible to this day.

cs.CC↗

arXiv2022

Modular SAT-based techniques for reasoning tasks in team semantics

We study the complexity of reasoning tasks for logics in team semantics. Our main focus is on the data complexity of model checking but we also derive new results for logically defined counting and enumeration problems. Our approach is based on modular reductions of these problems into the corresponding problems of various classes of Boolean formulas. We illustrate our approach via several new tractability/intractability results.

cs.LO↗

arXiv2021

Tractability Frontier of Data Complexity in Team Semantics

We study the data complexity of model-checking for logics with team semantics. We focus on dependence, inclusion, and independence logic formulas under both strict and lax team semantics. Our results delineate a clear tractability/intractability frontiers in data complexity of both quantifier-free and quantified formulas for each of the logics. For inclusion logic under the lax semantics, we reduce the model-checking problem to the satisfiability problem of so-called dual-Horn Boolean formulas. Via this reduction, we give an alternative proof for the known result that the data complexity of inclusion logic is in PTIME.

cs.LO↗

arXiv2018

Recursion schemes, discrete differential equations and characterization of polynomial time computation

This papers studies the expressive and computational power of discrete Ordinary Differential Equations (ODEs). It presents a new framework using discrete ODEs as a central tool for computation and provides several implicit characterizations of complexity and computability classes. The proposed framework presents an original point of view on complexity and computability classes. It also unifies in an elegant settings various constructions that have been proposed for characterizing these classes. This includes Cobham's and, Bellantoni and Cook's definition of polynomial time and later extensions on the approach, as well as recent characterizations of computability and complexity by classes of ordinary differential equations. It also helps understanding the relationships between analog computations and classical discrete models of computation theory. At a more technical point of view, this paper points out the fundamental role of linear (discrete) ordinary differential equations and classical ODE tools such as changes of variables to capture computability and complexity measures, or as a tool for programming various algorithms.

cs.LO↗

arXiv2018

Probabilistic team semantics

Team semantics is a semantical framework for the study of dependence and independence concepts ubiquitous in many areas such as databases and statistics. In recent works team semantics has been generalised to accommodate also multisets and probabilistic dependencies. In this article we study a variant of probabilistic team semantics and relate this framework to a Tarskian two-sorted logic. We also show that very simple quantifier-free formulae of our logic give rise to NP-hard model checking problems.

cs.LO↗

arXiv2017

Model-Theoretic Characterizations of Boolean and Arithmetic Circuit Classes of Small Depth

In this paper we give a characterization of both Boolean and arithmetic circuit classes of logarithmic depth in the vein of descriptive complexity theory, i.e., the Boolean classes $\textrm{NC}^1$, $\textrm{SAC}^1$ and $\textrm{AC}^1$ as well as their arithmetic counterparts $\#\textrm{NC}^1$, $\#\textrm{SAC}^1$ and $\#\textrm{AC}^1$. We build on Immerman's characterization of constant-depth polynomial-size circuits by formulas of first-order logic, i.e., $\textrm{AC}^0 = \textrm{FO}$, and augment the logical language with an operator for defining relations in an inductive way. Considering slight variations of the new operator, we obtain uniform characterizations of the three just mentioned Boolean classes. The arithmetic classes can then be characterized by functions counting winning strategies in semantic games for formulas characterizing languages in the corresponding Boolean class.

cs.CC↗

arXiv2016

Descriptive Complexity of $\#\textrm{AC}^0$ Functions

We introduce a new framework for a descriptive complexity approach to arithmetic computations. We define a hierarchy of classes based on the idea of counting assignments to free function variables in first-order formulae. We completely determine the inclusion structure and show that #P and #AC^0 appear as classes of this hierarchy. In this way, we unconditionally place #AC^0 properly in a strict hierarchy of arithmetic classes within #P. We compare our classes with a hierarchy within #P defined in a model-theoretic way by Saluja et al. We argue that our approach is better suited to study arithmetic circuit classes such as #AC^0 which can be descriptively characterized as a class in our framework.

cs.CC↗

arXiv2015

Approximation and Dependence via Multiteam Semantics

We define a variant of team semantics called multiteam semantics based on multisets and study the properties of various logics in this framework. In particular, we define natural probabilistic versions of inclusion and independence atoms and certain approximation operators motivated by approximate dependence atoms of Väänänen.

cs.LO↗

arXiv2015

Describability via ubiquity and eutaxy in Diophantine approximation

We present a comprehensive framework for the study of the size and large intersection properties of sets of limsup type that arise naturally in Diophantine approximation and multifractal analysis. This setting encompasses the classical ubiquity techniques, as well as the mass and the large intersection transference principles, thereby leading to a thorough description of the properties in terms of Hausdorff measures and large intersection classes associated with general gauge functions. The sets issued from eutaxic sequences of points and optimal regular systems may naturally be described within this framework. The discussed applications include the classical homogeneous and inhomogeneous approximation, the approximation by algebraic numbers, the approximation by fractional parts, the study of uniform and Poisson random coverings, and the multifractal analysis of L{é}vy processes.

math.CA↗

arXiv2014

Hypergraph Acyclicity and Propositional Model Counting

We show that the propositional model counting problem #SAT for CNF- formulas with hypergraphs that allow a disjoint branches decomposition can be solved in polynomial time. We show that this class of hypergraphs is incomparable to hypergraphs of bounded incidence cliquewidth which were the biggest class of hypergraphs for which #SAT was known to be solvable in polynomial time so far. Furthermore, we present a polynomial time algorithm that computes a disjoint branches decomposition of a given hypergraph if it exists and rejects otherwise. Finally, we show that some slight extensions of the class of hypergraphs with disjoint branches decompositions lead to intractable #SAT, leaving open how to generalize the counting result of this paper.

cs.CC↗

arXiv2013

Metric Diophantine approximation on the middle-third Cantor set

Let $μ\geq 2$ be a real number and let $\Mcal(μ)$ denote the set of real numbers approximable at order at least $μ$ by rational numbers. More than eighty years ago, Jarní k and, independently, Besicovitch established that the Hausdorff dimension of $\Mcal(μ)$ is equal to $2/μ$. We investigate the size of the intersection of $\Mcal(μ)$ with Ahlfors regular compact subsets of the interval $[0, 1]$. In particular, we propose a conjecture for the exact value of the dimension of $\Mcal(μ)$ intersected with the middle-third Cantor set and give several results supporting this conjecture. We especially show that the conjecture holds for a natural probabilistic model that is intended to mimic the distribution of the rationals. The core of our study relies heavily on dimension estimates concerning the set of points lying in an Ahlfors regular set and approximated at a given rate by a system of random points.

math.NT↗

arXiv2013

Dependence logic with a majority quantifier

We study the extension of dependence logic D by a majority quantifier M over finite structures. We show that the resulting logic is equi-expressive with the extension of second-order logic by second-order majority quantifiers of all arities. Our results imply that, from the point of view of descriptive complexity theory, D(M) captures the complexity class counting hierarchy.

cs.LO↗