A game theoretical approach to the algebraic counterpart of the Wagner hierarchy

La hiérarchie de Wagner constitue à ce jour la plus fine classification des langages ω-réguliers. Par ailleurs, l'approche algébrique de la théorie de langages formels montre que ces ensembles ω-réguliers correspondent précisément aux langages reconnaissables par des ω-semigroupes finis pointés. Ce travail s'inscrit dans ce contexte en fournissant une description complète de la contrepartie algébrique de la hiérarchie de Wagner, et ce par le biais de la théorie descriptive des jeux de Wadge. Plus précisément, nous montrons d'abord que le degré de Wagner d'un langage ω-régulier est effectivement un invariant syntaxique. Nous définissons ensuite une relation de réduction entre ω-semigroupes pointés par le biais d'un jeu infini de type Wadge. La collection de ces structures algébriques ordonnée par cette relation apparaît alors comme étant isomorphe à la hiérarchie de Wagner, soit un quasi bon ordre décidable de largeur 2 et de hauteur ω. Nous exposons par la suite une procédure de décidabilité de cette hiérarchie algébrique : on décrit une représentation graphique des ω-semigroupes finis pointés, puis un algorithme sur ces structures graphiques qui calcule le degré de Wagner de n'importe quel élément. Ainsi le degré de Wagner de tout langage ω-régulier peut être calculé de manière effective directement sur son image syntaxique. Nous montrons ensuite comment construire directement et inductivement une structure de n''importe quel degré. Nous terminons par une description détaillée des invariants algébriques qui caractérisent tous les degrés de cette hiérarchie. Abstract The Wagner hierarchy is known so far to be the most refined topological classification of ω-rational languages. Also, the algebraic study of formal languages shows that these ω-rational sets correspond precisely to the languages recognizable by finite pointed ω-semigroups. Within this framework, we provide a construction of the algebraic counterpart of the Wagner hierarchy. We adopt a hierarchical game approach, by translating the Wadge theory from the ω-rational language to the ω-semigroup context. More precisely, we first show that the Wagner degree is indeed a syntactic invariant. We then define a reduction relation on finite pointed ω-semigroups by means of a Wadge-like infinite two-player game. The collection of these algebraic structures ordered by this reduction is then proven to be isomorphic to the Wagner hierarchy, namely a well-founded and decidable partial ordering of width 2 and height $\omega^\omega$. We also describe a decidability procedure of this hierarchy: we introduce a graph representation of finite pointed ω-semigroups allowing to compute their precise Wagner degrees. The Wagner degree of every ω-rational language can therefore be computed directly on its syntactic image. We then show how to build a finite pointed ω-semigroup of any given Wagner degree. We finally describe the algebraic invariants characterizing every Wagner degree of this hierarchy.

Neuro-mechanical and metabolic adjustments to the repeated anaerobic sprint test in professional football players.

PURPOSE: This study aimed to determine the neuro-mechanical and metabolic adjustments in the lower limbs induced by the running anaerobic sprint test (the so-called RAST). METHODS: Eight professional football players performed 6 × 35 m sprints interspersed with 10 s of active recovery on artificial turf with their football shoes. Sprinting mechanics (plantar pressure insoles), root mean square activity of the vastus lateralis (VL), rectus femoris (RF), and biceps femoris (BF) muscles (surface electromyography, EMG) and VL muscle oxygenation (near-infrared spectroscopy) were monitored continuously. RESULTS: Sprint time, contact time and total stride duration increased from the first to the last repetition (+17.4, +20.0 and +16.6 %; all P < 0.05), while flight time and stride length remained constant. Stride frequency (-13.9 %; P < 0.001) and vertical stiffness decreased (-27.2 %; P < 0.001) across trials. Root mean square EMG activities of RF and BF (-18.7 and -18.1 %; P < 0.01 and 0.001, respectively), but not VL (-1.2 %; P > 0.05), decreased over sprint repetitions and were correlated with the increase in running time (r = -0.82 and -0.90; both P < 0.05). Together with a better maintenance of RF and BF muscles activation levels over sprint repetitions, players with a better repeated-sprint performance (lower cumulated times) also displayed faster muscle de- (during sprints) and re-oxygenation (during recovery) rates (r = -0.74 and -0.84; P < 0.05 and 0.01, respectively). CONCLUSION: The repeated anaerobic sprint test leads to substantial alterations in stride mechanics and leg-spring behaviour. Our results also strengthen the link between repeated-sprint ability and the change in neuromuscular activation as well as in muscle de- and re-oxygenation rates.