Polynomial-Time Algorithms for Quadratic Isomorphism of Polynomials: The Regular Case

Let $\mathbf{f}=(f\_1,\ldots,f\_m)$ and $\mathbf{g}=(g\_1,\ldots,g\_m)$ be two sets of $m\geq 1$ nonlinear polynomials over $\mathbb{K}[x\_1,\ldots,x\_n]$ ($\mathbb{K}$ being a field). We consider the computational problem of finding -- if any -- an invertible transformation on the variables mapping $\mathbf{f}$ to $\mathbf{g}$. The corresponding equivalence problem is known as {\tt Isomorphism of Polynomials with one Secret} ({\tt IP1S}) and is a fundamental problem in multivariate cryptography. The main result is a randomized polynomial-time algorithm for solving {\tt IP1S} for quadratic instances, a particular case of importance in cryptography and somewhat justifying {\it a posteriori} the fact that {\it Graph Isomorphism} reduces to only cubic instances of {\tt IP1S} (Agrawal and Saxena). To this end, we show that {\tt IP1S} for quadratic polynomials can be reduced to a variant of the classical module isomorphism problem in representation theory, which involves to test the orthogonal simultaneous conjugacy of symmetric matrices. We show that we can essentially {\it linearize} the problem by reducing quadratic-{\tt IP1S} to test the orthogonal simultaneous similarity of symmetric matrices; this latter problem was shown by Chistov, Ivanyos and Karpinski to be equivalent to finding an invertible matrix in the linear space $\mathbb{K}^{n \times n}$ of $n \times n$ matrices over $\mathbb{K}$ and to compute the square root in a matrix algebra. While computing square roots of matrices can be done efficiently using numerical methods, it seems difficult to control the bit complexity of such methods. However, we present exact and polynomial-time algorithms for computing the square root in $\mathbb{K}^{n \times n}$ for various fields (including finite fields). We then consider \\#{\tt IP1S}, the counting version of {\tt IP1S} for quadratic instances. In particular, we provide a (complete) characterization of the automorphism group of homogeneous quadratic polynomials. Finally, we also consider the more general {\it Isomorphism of Polynomials} ({\tt IP}) problem where we allow an invertible linear transformation on the variables \emph{and} on the set of polynomials. A randomized polynomial-time algorithm for solving {\tt IP} when $\mathbf{f}=(x\_1^d,\ldots,x\_n^d)$ is presented. From an algorithmic point of view, the problem boils down to factoring the determinant of a linear matrix (\emph{i.e.}\ a matrix whose components are linear polynomials). This extends to {\tt IP} a result of Kayal obtained for {\tt PolyProj}.Comment: Published in Journal of Complexity, Elsevier, 2015, pp.3

Parsing dialogue and argumentative structures

Le présent manuscrit présente de nouvelles techniques d'extraction des structures : du dialogue de groupe, d'une part; de textes argumentatifs, d'autre part. Déceler la structure de longs textes et de conversations est une étape cruciale afin de reconstruire leur signification sous-jacente. La difficulté de cette tâche est largement reconnue, sachant que le discours est une description de haut niveau du langage, et que le dialogue de groupe inclut de nombreux phénomènes linguistiques complexes. Historiquement, la représentation du discours a fortement évolué, partant de relations locales, formant des collections non-structurées, vers des arbres, puis des graphes contraints. Nos travaux utilisent ce dernier paradigme, via la Théorie de Représentation du Discours Segmenté. Notre recherche se base sur un corpus annoté de discussions en ligne en anglais, issues du jeu de société Les Colons de Catane. De par la nature stratégique des conversations, et la liberté que permet le format électronique des discussions, ces dialogues contiennent des Unités Discursives Complexes, des fils de discussion intriqués, parmi d'autres propriétés que la littérature actuelle sur l'analyse du discours ignore en général. Nous discutons de deux investigations liées à notre corpus. La première étend la définition de la contrainte de la frontière droite, une formalisation de certains principes de cohérence de la structure du discours, pour l'adapter au dialogue de groupe. La seconde fait la démonstration d'un processus d'extraction de données permettant à un joueur artificiel des Colons d'obtenir un avantage stratégique en déduisant les possessions de ses adversaires à partir de leurs négociations. Nous proposons de nouvelles méthodes d'analyse du dialogue, utilisant conjointement apprentissage automatisé, algorithmes de graphes et optimisation linéaire afin de produire des structures riches et expressives, avec une précision supérieure comparée aux efforts existants. Nous décrivons notre méthode d'analyse du discours par contraintes, d'abord sur des arbres en employant la construction d'un arbre couvrant maximal, puis sur des graphes orientés acycliques en utilisant la programmation linéaire par entiers avec une collection de contraintes originales. Nous appliquons enfin ces méthodes sur les structures de l'argumentation, avec un corpus de textes en anglais et en allemand, parallèlement annotés avec deux structures du discours et une argumentative. Nous comparons les trois couches d'annotation et expérimentons sur l'analyse de l'argumentation, obtenant de meilleurs résultats, relativement à des travaux similaires.This work presents novel techniques for parsing the structures of multi-party dialogue and argumentative texts. Finding the structure of extended texts and conversations is a critical step towards the extraction of their underlying meaning. The task is notoriously hard, as discourse is a high-level description of language, and multi-party dialogue involves many complex linguistic phenomena. Historically, representation of discourse moved from local relationships, forming unstructured collections, towards trees, then constrained graphs. Our work uses the latter framework, through Segmented Discourse Representation Theory. We base our research on a annotated corpus of English chats from the board game The Settlers of Catan. Per the strategic nature of the conversation and the freedom of online chat, these dialogues exhibit complex discourse units, interwoven threads, among other features which are mostly overlooked by the current parsing literature. We discuss two corpus-related experiments. The first expands the definition of the Right Frontier Constraint, a formalization of discourse coherence principles, to adapt it to multi-party dialogue. The second demonstrates a data extraction process giving a strategic advantage to an artificial player of Settlers by inferring its opponents' assets from chat negotiations. We propose new methods to parse dialogue, using jointly machine learning, graph algorithms and linear optimization, to produce rich and expressive structures with greater accuracy than previous attempts. We describe our method of constrained discourse parsing, first on trees using the Maximum Spanning Tree algorithm, then on directed acyclic graphs using Integer Linear Programming with a number of original constraints. We finally apply these methods to argumentative structures, on a corpus of English and German texts, jointly annotated in two discourse representation frameworks and one argumentative. We compare the three annotation layers, and experiment on argumentative parsing, achieving better performance than similar works

Discourse parsing for multi-party chat dialogues

In this paper we present the first ever, to the best of our knowledge, discourse parser for multi-party chat dialogues. Discourse in multi-party dialogues dramatically differs from monologues since threaded conversations are commonplace rendering prediction of the discourse structure compelling. Moreover, the fact that our data come from chats renders the use of syntactic and lexical information useless since people take great liberties in expressing themselves lexically and syntactically. We use the dependency parsing paradigm as has been done in the past (Muller et al., 2012; Li et al., 2014). We learn local probability distributions and then use MST for decoding. We achieve 0.680 F 1 on unlabelled structures and 0.516 F 1 on fully labeled structures which is better than many state of the art systems for monologues, despite the inherent difficulties that multi-party chat dialogues have

Structure of high Reynolds number boundary layers over cube canopies

The influence of a cube-based canopy on coherent structures of the flow was investigated in a high Reynolds number boundary layer (thickness δ ∼ 30 000 wall units). Wind tunnel experiments were conducted considering wall configurations that represent three idealised urban terrains. Stereoscopic Particle Image Velocimetry was employed using a large field of view in a streamwise-spanwise plane combined to two-point hot-wire measurements. The analysis of the flow within the inertial layer highlights the independence of its characteristics from the wall configuration. The population of coherent structures is in agreement with that of smooth-wall boundary layers, i.e. consisting of large and very-large scale motions, sweeps and ejections, as well as smaller-scale vortical structures. The characteristics of vortices appear to be independent of the roughness configuration while their spatial distribution is closely linked to large meandering motions of the boundary layer. The canopy geometry only significantly impacts the wall-normal exchanges within the roughness sublayer. Bi-dimensional spectral analysis demonstrates that wall-normal velocity fluctuations are constrained by the presence of the canopy for the densest investigated configurations. This threshold in plan density above which large scales from the overlying boundary layer can penetrate the roughness sublayer is consistent with the change of the flow regime reported in the literature and constitutes a major difference with flows over vegetation canopies

Influence of the plan density of a cube-based canopy on the structure of the lower atmospheric boundary layer

An experimental investigation of the structure of the boundary-layer developing over an urban-like rough wall is performed considering wall configurations that represent three idealised urban terrains. Both stereoscopic particle image velocimetry and two-point hot-wire data are investigated through one- and two-dimensionnal spectral analysis. It is shown that the flow characteristics within the inertial layer appear to be independent from the wall configuration. This region is populated with coherent structures of the same type as those found in smooth-wall flows, namely very large-scale motions and large-scale motions, the later showing self-similar features. These coherent structures, whose footprint is mainly visible on the streamwise and spanwise velocity components, also appear to be present in the roughness sublayer. Within the roughness sublayer, the influence of the canopy density shows in the characteristics of the wall-normal velocity component w. The most energetic scales in the auto-spectra of w and in the co-spectra between u and w are indeed of the same order of magnitude as the roughness obstacle size and decrease with increasing density. Instead of varying progressively with the packing density, they appear to be constrained by the presence of the canopy for the densest investigated configurations consistently with the change of the flow regime reported in the literature

Spatial modulations of kinetic energy in the roughness sublayer

High-Reynolds-number experiments are conducted in the roughness sublayer of a turbulent boundary-layer developing over a cubical canopy. Stereoscopic particle image velocimetry is performed in a wall-parallel plane to evidence a high degree of spatial modulation of the small-scale turbulence around the footprint of large-scale motions, despite the suppression of the inner layer by the high roughness elements. Both Fourier and wavelets analyses show that the near-wall cycle observed in smooth-wall-bounded flows is severely disrupted by the canopy, whose wake in the roughness sublayer generates a new range of scales, closer to that of the outer-layer large-scale motions. This restricts significantly scale separation, hence a diagnostic method is developed to divide carefully and rationally the fluctuating velocity fields into large- and small-scale components. Our analysis across all turbulent kinetic energy terms sheds light on the spatial imprint of the modulation mechanism, revealing a very different signature on each velocity component. The roughness sublayer shows a preferential arrangement of the modulated scales similar to what is observed in the outer-layer of smooth-wall-bounded flows – small-scale turbulence is enhanced near the front of high momentum regions and damped at the front of low-momentum regions. More importantly, accessing spanwise correlations reveals that modulation intensifies the most along the flanks of the large-scale motions

Effects of plan area densities of cubical roughness elements on turbulent boundary layers

During the past few years, large-scale motions (LSM) in turbulent boundary layers over smooth-walls have received renewed attention from the research community. Common features of the LSMs found in wall-bounded flows are that they consist in elongated low- and high-speed regions, the length of which scale with the boundary-layer depth (?) and can reach several times ?, they populate the log- and outer layer, they are animated by a meandering motion in the horizontal plane (Hutchins & Marusic, JFM, 2007) and interact with near-wall turbulence through an amplitude-modulation mechanism (Mathis et al, JFM, 2009). At the same time, attention has been devoted to the structure of boundary layer flows developing over rough walls, at laboratory scales or in the framework of atmospheric flows over urban or vegetation canopies, demonstrating similarities between flows over smooth and rough wall. In particular, the presence of streaky patterns of low- and high-speed regions, of ejection and sweep motions associated to the hairpin model and the organization of hairpin vortices in packets have been evidenced (Jimenez, Ann Rev Fluid Mech, 2004; Finnigan et al, JFM, 2009; Inagaki & Kanda, BLM, 2010; Takimoto et al, BLM, 2011). The results obtained in flows over rough-walls suggest that, in spite of the strong disturbance of the flow at the wall, LSMs exist and interact with the canopy flow in a similar manner as in smooth wall boundary layers. However, in configurations representative of flows over urban canopies, it is likely that the arrangement, the shape and plan density of the roughness elements have a strong influence on the flow dynamics in the lower part of the boundary layer. Building upon these recent results, the aim of the present work is to analyze the statistical and spectral characteristics of turbulent boundary layer developing over cubical roughness arrays of different plan densities, at high Reynolds number, which has never been fully properly documented. This study, conducted in an atmospheric wind tunnel, is based on single hot-wire measurements performed along a vertical profile across the boundary layer, in a flow configuration representative of the atmospheric boundary layer (ABL) developing over an urban canopy. Three different rough walls of plan area density (i.e the ratio between the area of the surface occupied by the roughness elements and that of the total surface) of 6,25%, 25% and 44% are studied. Each roughness configuration is composed of a staggered array of cubes of height of h = 50 mm distributed on the wind tunnel floor over a total length of 22 m and a width of 2 m. The experiments have been performed with two free-stream velocities Ue = 5.8 and 9.1 m/s corresponding to Reynolds numbers h+ = h.u-/? ~ 1300, 2000 and Re? = ?.u-/? ~ 27000, 42000, respectively (where u- is the friction velocity). One-point statistics up to the fourth order are computed using hot-wire data acquired at 10000 kHz over a period corresponding to about 27000 turnover times, ?/Ue, to ensure proper statistical convergence of both large- and small-scale contents. It should me emphasized that the large-scales convergence criterion requires one hour and twenty minutes of acquisition per measured point. Besides the computation of mean velocity, standard deviation, skewness and kurtosis, this unprecedented database is also used to investigate the evolution of the spectral density content of the streamwise velocity component as a function of the wall-distance, the roughness element distribution and the Reynolds number. The ultimate goal of this ongoing study is to identify the mechanisms of interaction between the ABL and the canopy flow in order to contribute to the development of wall and/or canopy model

Parallel Discourse Annotations on a Corpus of Short Texts

We present the first corpus of texts annotated with two alternative approaches to discourse structure, Rhetorical Structure Theory (Mann and Thompson, 1988) and Segmented Discourse Representation Theory (Asher and Lascarides, 2003). 112 short argumentative texts have been analyzed according to these two theories. Furthermore, in previous work, the same texts have already been annotated for their argumentation structure, according to the scheme of Peldszus and Stede (2013). This corpus therefore enables studies of correlations between the two accounts of discourse structure, and between discourse and argumentation. We converted the three annotation formats to a common dependency tree format that enables to compare the structures, and we describe some initial findings

The atmospheric boundary layer over urban-like terrain: influence of the plan density on roughness sublayer dynamics

We investigate the effect of the packing density of cubical roughness elements on the characteristics of both the roughness sublayer and the overlying turbulent boundary layer, in the context of atmospheric flow over urban areas. This is based on detailed wind-tunnel hot-wire measurements of the streamwise velocity component with three wall-roughness configurations and two freestream flow speeds. The packing densities are chosen so as to obtain the three near-wall flow regimes observed in urban canopy flows, namely isolated-wake, wake-interference and skimming-flow regimes. Investigation of the wall-normal profiles of the one-point statistics up to third order demonstrates the impossibility of finding a unique set of parameters enabling the collapse of all configurations, except for the mean streamwise velocity component. However, spectral analysis of the streamwise velocity component provides insightful information. Using the temporal frequency corresponding to the peak in the pre-multiplied energy spectrum as an indicator of the most energetic flow structures at each wall-normal location, it is shown that three main regions exist, in which different scaling applies. Finally, scale decomposition reveals that the flow in the roughness sublayer results from a large-scale intrinsic component of the boundary layer combined with canopy-induced dynamics. Their relative importance plays a key role in the energy distribution and influences the near-canopy flow regime and its dynamics, therefore suggesting complex interactions between the near-wall scales and those from the overlying boundary layer

Analyse des structures du dialogue et de l'argumentation

Le présent manuscrit présente de nouvelles techniques d'extraction des structures : du dialogue de groupe, d'une part; de textes argumentatifs, d'autre part. Déceler la structure de longs textes et de conversations est une étape cruciale afin de reconstruire leur signification sous-jacente. La difficulté de cette tâche est largement reconnue, sachant que le discours est une description de haut niveau du langage, et que le dialogue de groupe inclut de nombreux phénomènes linguistiques complexes. Historiquement, la représentation du discours a fortement évolué, partant de relations locales, formant des collections non-structurées, vers des arbres, puis des graphes contraints. Nos travaux utilisent ce dernier paradigme, via la Théorie de Représentation du Discours Segmenté. Notre recherche se base sur un corpus annoté de discussions en ligne en anglais, issues du jeu de société Les Colons de Catane. De par la nature stratégique des conversations, et la liberté que permet le format électronique des discussions, ces dialogues contiennent des Unités Discursives Complexes, des fils de discussion intriqués, parmi d'autres propriétés que la littérature actuelle sur l'analyse du discours ignore en général. Nous discutons de deux investigations liées à notre corpus. La première étend la définition de la contrainte de la frontière droite, une formalisation de certains principes de cohérence de la structure du discours, pour l'adapter au dialogue de groupe. La seconde fait la démonstration d'un processus d'extraction de données permettant à un joueur artificiel des Colons d'obtenir un avantage stratégique en déduisant les possessions de ses adversaires à partir de leurs négociations. Nous proposons de nouvelles méthodes d'analyse du dialogue, utilisant conjointement apprentissage automatisé, algorithmes de graphes et optimisation linéaire afin de produire des structures riches et expressives, avec une précision supérieure comparée aux efforts existants. Nous décrivons notre méthode d'analyse du discours par contraintes, d'abord sur des arbres en employant la construction d'un arbre couvrant maximal, puis sur des graphes orientés acycliques en utilisant la programmation linéaire par entiers avec une collection de contraintes originales. Nous appliquons enfin ces méthodes sur les structures de l'argumentation, avec un corpus de textes en anglais et en allemand, parallèlement annotés avec deux structures du discours et une argumentative. Nous comparons les trois couches d'annotation et expérimentons sur l'analyse de l'argumentation, obtenant de meilleurs résultats, relativement à des travaux similaires.This work presents novel techniques for parsing the structures of multi-party dialogue and argumentative texts. Finding the structure of extended texts and conversations is a critical step towards the extraction of their underlying meaning. The task is notoriously hard, as discourse is a high-level description of language, and multi-party dialogue involves many complex linguistic phenomena. Historically, representation of discourse moved from local relationships, forming unstructured collections, towards trees, then constrained graphs. Our work uses the latter framework, through Segmented Discourse Representation Theory. We base our research on a annotated corpus of English chats from the board game The Settlers of Catan. Per the strategic nature of the conversation and the freedom of online chat, these dialogues exhibit complex discourse units, interwoven threads, among other features which are mostly overlooked by the current parsing literature. We discuss two corpus-related experiments. The first expands the definition of the Right Frontier Constraint, a formalization of discourse coherence principles, to adapt it to multi-party dialogue. The second demonstrates a data extraction process giving a strategic advantage to an artificial player of Settlers by inferring its opponents' assets from chat negotiations. We propose new methods to parse dialogue, using jointly machine learning, graph algorithms and linear optimization, to produce rich and expressive structures with greater accuracy than previous attempts. We describe our method of constrained discourse parsing, first on trees using the Maximum Spanning Tree algorithm, then on directed acyclic graphs using Integer Linear Programming with a number of original constraints. We finally apply these methods to argumentative structures, on a corpus of English and German texts, jointly annotated in two discourse representation frameworks and one argumentative. We compare the three annotation layers, and experiment on argumentative parsing, achieving better performance than similar works