Résolution de contraintes du premier ordre dans la théorie des arbres finis ou infinis

National audienceWe present in this paper an algorithm, in the theory \Tt\, of (eventually infinite) trees, for solving constraints represented by full first order formulae, with equality as the only relation and with symbols of function taken in an infinite set \Ff. The algorithm consists of a set of 11 rewrite rules. It transforms a first order formula in a conjunction of ``solved'' formulae, equivalent in \Tt, which has not new free variables and which is such that, (1) the conjunction either is the constant logic \true or is reduced to \neg\true, or has at least one free variable and is equivalent neither to \true nor to \false, (2) each solved formula can be transformed immediately in a Boolean combination of basic formulae whose length does not exceed twice the length of the solved formula. The basic formulae are particular cases of existentially quantified conjunctions of equations. The correctness of the algorithm gives another proof of the completeness of \Tt demonstrated by Michael Maher. We end with benchmarks realized by an implementation, solving formulae with more than 160 nested alternated quantifiers

Complétude des extensions en arbres de théories

Nous présentons dans ce papier une méthode pour combiner une théorie T quelconque du premier ordre avec la théorie des arbres éventuellement infinis. Sémantiquement cette nouvelle théorie hybride n'est rien d'autre qu'une axiomatisation de l'extension en arbres des éléments des modèles de la théorie T. Tout d'abord, ayant une axiomatisation d'une théorie T, nous donnons l'axiomatisation de la théorie T de l'extension en arbre de T et présentons son modèle standard M. Nous introduisons ensuite une nouvelle classe de théories dite flexibles et montrons que si T est flexible alors T est complète. Les théories flexibles sont des théories ayant des propriétés élégantes qui nous permettent de manipuler aisément les formules. Enfin nous présentons un algorithme de décision de propositions dans T pour toute théorie T flexible. L'algorithme est donné sous forme d'un ensemble de six règles de réécriture qui pour toute proposition ' donnent soit vrai soit faux

Expressiveness of Full First-Order Constraints in the Algebra of Finite or Infinite Trees

International audienceWe are interested in the expressiveness of constraints represented by general first order formulae, with equality as unique relation symbol and function symbols taken from an infinite set F. The chosen domain is the set of trees whose nodes, in possibly infinite number, are labelled by elements of F. The operation linked to each element f of F is the mapping (a 1,..., a n ) map b, where b is the tree whose initial node is labelled f and whose sequence of daughters is a 1,..., a n . We first consider tree constraints involving long alternated sequences of quantifiers existforallexistforall.... We show how to express winning positions of two-person games with such constraints and apply our results to two examples. We then construct a family of strongly expressive tree constraints, inspired by a constructive proof of a complexity result by Pawel Mielniczuk. This family involves the huge number agr(k), obtained by top down evaluating a power tower of 2's, of height k. By a tree constraint of size proportional to k, it is then possible to define a tree having exactly agr(k) nodes or to express the multiplication table computed by a Prolog machine executing up to agr(k) instructions. By replacing the Prolog machine with a Turing machine we show the quasi-universality of tree constraints, that is to say, the ability to concisely describe trees which the most powerful machine will never have time to compute. We also rediscover the following result of Sergei Vorobyov: the complexity of an algorithm, deciding whether a tree constraint without free variables is true, cannot be bounded above by a function obtained from finite composition of simple functions including exponentiation. Finally, taking advantage of the fact that we have at our disposal an algorithm for solving such constraints in all their generalities, we produce a set of benchmarks for separating feasible examples from purely speculative ones. Among others we notice that it is possible to solve a constraint of 5000 symbols involving 160 alternating quantifiers

Solving First-Order Constraints in the Theory of the Evaluated Trees

International audienceWe describe in this paper a general algorithm for solving first-order constraints in the theory T of the evaluated trees which is a combination of the theory of finite or infinite trees and the theory of the rational numbers with addition, subtraction and a linear dense order relation. It transforms a first-order formula ϕ, which can possibly contain free variables, into a disjunction φ of solved formulas which is equivalent in T, without new free variables and such that φ is either $\mathit{true}$ or $\mathit{false}$ or a formula having at least one free variable and being equivalent neither to $\mathit{true}$ nor to $\mathit{false}$ in T

Complete First-Order Axiomatization of Finite or Infinite M-extended Trees

We present in this paper an axiomatization of the structure of finite or infinite $M$-extended trees. Having a structure $M=(D_M,F_M,R_M)$, we define the structure of finite or infinite $M$-extended trees $Ext_M=(D,F,R)$ whose domain $D$ consists of trees labelled by elements of $D_M\cup F$, where $F$ is a set of function symbols containing $F_M$ and another infinite set of function symbols disjoint from $F_M$. For each $n$-ary function symbol $f\in F$, the operation $f(a_1,..,a_n)$ is evaluated in $M$ and produces an element of $D_M$ if $f\in F_M$ and all the $a_i$ are elements of $D_M$, or is a tree whose root is labelled by $f$ and whose immediate children are $a_1,..,a_n$ otherwise. The set of relations $R$ contains $R_M$ and another relation which distinguishes the elements of $D_M$ from the others. Using a first-order axiomatization $T$ of $M$, we give a first-order axiomatization $\cal{T}$ of the structure $Ext_M$ and show that if $T$ is {\em flexible} then $\cal{T}$ is {\em complete}. The flexible theories are particular theories whose function and relation symbols have some elegant properties which enable us to handle formulae more easily

Expressiveness of Full First Order Constraints in the Algebra of Finite or Infinite Trees

International audienceWe are interested in the expressiveness of constraints represented by general first order formulae, with equality as unique relational symbol and functional symbols taken from an infinite set F. The chosen domain is the set of trees whose nodes, in possibly infinite number, are labeled by elements of F. The operation linked to each element f of F is the mapping , where b is the tree whose initial node is labeled f and whose sequence of daughters is . We first consider constraints involving long alternated sequences of quantifiers . We show how to express winning positions of two-person games with such constraints and apply our results to two examples. We then construct a family of strongly expressive constraints, inspired by a constructive proof of a complexity result by Pawel Mielniczuk. This family involves the huge number, obtained by evaluating top down a power tower of 2's, of height k. With elements of this family, of sizes at most proportional to k, we define a finite tree having nodes, and we express the result of a Prolog machine executing at most instructions. By replacing the Prolog machine by a Turing machine we rediscover the following result of Sergei Vorobyov: the complexity of an algorithm, deciding whether a constraint without free variables is true, cannot be bounded above by a function obtained by finite composition of elementary functions including exponentiation. Finally, taking advantage of the fact that we have at our disposal an algorithm for solving such constraints in all their generality, we produce a set of benchmarks for separating feasible examples from purely speculative ones. Among others we solve constraints involving alternated sequences of more than 160 quantifiers

Résolution de contraintes du premier ordre dans la théorie des arbres évalués

Nous présentons dans ce papier un algorithme général de résolution de contraintes du premier ordre dans la théorie T des arbres évalués. Cette théorie est une combinaison de la théorie des arbres finis ou infinis et de la théorie des rationnels munis de l'addition, de la soustraction et d'une relation d'ordre dense sans extrême. L'algorithme est donné sous forme d'un ensemble de 28 règles de réécriture et transforme toute formule du premier ordre ', qui peut éventuellement contenir des variables libres, en une disjonction D de formules résolues, équivalente à ' dans T, sans nouvelles variables libres, et telle que D est soit la formule vrai , soit la formule faux , soit une formule ayant au moins une variable libre et n'étant équivalente ni à vrai ni à faux dans T. En particulier, si ' est sans variables libres, D est soit la formule vrai soit la formule faux . Si D contient des variables libres, les solutions sur ces variables sont exprimées d'une façon explicite et D peut se transformer directement en une combinaison booléenne de conjonctions quantifiées de formules atomiques, qui n'acceptent pas d'élimination de quantificateurs. La correction de notre algorithme est une autre preuve de la complétude de la théorie T

Constrained Distance Based Clustering for Satellite Image Time-Series

International audienceThe advent of high-resolution instruments for time-series sampling poses added complexity for the formal definition of thematic classes in the remote sensing domain-required by supervised methods-while unsupervised methods ignore expert knowledge and intuition. Constrained clustering is becoming an increasingly popular approach in data mining because it offers a solution to these problems, however, its application in remote sensing is relatively unknown. This article addresses this divide by adapting publicly available constrained clustering implementations to use the dynamic time warping (DTW) dissimilarity measure, which is sometimes used for time-series analysis. A comparative study is presented, in which their performance is evaluated (using both DTW and Euclidean distances). It is found that adding constraints to the clustering problem results in an increase in accuracy when compared to unconstrained clustering. The output of such algorithms are homogeneous in spatially defined regions. Declarative approaches and k-Means based algorithms are simple to apply, requiring little or no choice of parameter values. Spectral methods, however, require careful tuning, which is unrealistic in a semi-supervised setting, although they offer the highest accuracy. These conclusions were drawn from two applications: crop clustering using 11 multi-spectral Landsat images non-uniformly sampled over a period of eight months in 2007; and tree-cut detection using 10 NDVI Sentinel-2 images non-uniformly sampled between 2016 and 2018

Safety and efficacy of fluoxetine on functional outcome after acute stroke (AFFINITY): a randomised, double-blind, placebo-controlled trial

Background Trials of fluoxetine for recovery after stroke report conflicting results. The Assessment oF FluoxetINe In sTroke recoverY (AFFINITY) trial aimed to show if daily oral fluoxetine for 6 months after stroke improves functional outcome in an ethnically diverse population. Methods AFFINITY was a randomised, parallel-group, double-blind, placebo-controlled trial done in 43 hospital stroke units in Australia (n=29), New Zealand (four), and Vietnam (ten). Eligible patients were adults (aged ≥18 years) with a clinical diagnosis of acute stroke in the previous 2–15 days, brain imaging consistent with ischaemic or haemorrhagic stroke, and a persisting neurological deficit that produced a modified Rankin Scale (mRS) score of 1 or more. Patients were randomly assigned 1:1 via a web-based system using a minimisation algorithm to once daily, oral fluoxetine 20 mg capsules or matching placebo for 6 months. Patients, carers, investigators, and outcome assessors were masked to the treatment allocation. The primary outcome was functional status, measured by the mRS, at 6 months. The primary analysis was an ordinal logistic regression of the mRS at 6 months, adjusted for minimisation variables. Primary and safety analyses were done according to the patient's treatment allocation. The trial is registered with the Australian New Zealand Clinical Trials Registry, ACTRN12611000774921. Findings Between Jan 11, 2013, and June 30, 2019, 1280 patients were recruited in Australia (n=532), New Zealand (n=42), and Vietnam (n=706), of whom 642 were randomly assigned to fluoxetine and 638 were randomly assigned to placebo. Mean duration of trial treatment was 167 days (SD 48·1). At 6 months, mRS data were available in 624 (97%) patients in the fluoxetine group and 632 (99%) in the placebo group. The distribution of mRS categories was similar in the fluoxetine and placebo groups (adjusted common odds ratio 0·94, 95% CI 0·76–1·15; p=0·53). Compared with patients in the placebo group, patients in the fluoxetine group had more falls (20 [3%] vs seven [1%]; p=0·018), bone fractures (19 [3%] vs six [1%]; p=0·014), and epileptic seizures (ten [2%] vs two [<1%]; p=0·038) at 6 months. Interpretation Oral fluoxetine 20 mg daily for 6 months after acute stroke did not improve functional outcome and increased the risk of falls, bone fractures, and epileptic seizures. These results do not support the use of fluoxetine to improve functional outcome after stroke