Do most strong definitions of randomness exist?

The goal of our paper is to propose a way to obtain more refined definitions of randomness than the notions known so far (e.g. Martin-Löf randomness). We show that a ``perfect'' definition of randomness based on provability does not exist. We then weaken our requirements on the definition by replacing provability by consistency and obtain a formula that defines a set of random sequences that fulfills rather strong conditions.Nous proposons ici de raffiner les définitions classiques du caractère aléatoire des suites infinies (en particulier la très classique définition de Martin-Löf). Nous prouvons qu'il n'existe pas de définition parfaite fondée sur la notion de prouvabilité. En remplaçant la prouvabilité par la consistence, nous obtenons une définition des suites aléatoires très générale qui remplit des conditions raisonablement fortes

Do Most Strong Definitions of Randomness Exist?

The goal of our paper is to propose a way to obtain more refined definitions of randomness than the notions known so far (e.g. MartinL of randomness). We show that a "perfect" definition of randomness based on provability does not exist. We then weaken our requirements on the definition by replacing provability by consistency and obtain a formula that defines a set of random sequences that fulfills rather strong conditions. Keywords: Randomness, logics, Solovay model R'esum'e Nous proposons ici de raffiner les d'efinitions classiques du caract`ere al'eatoire des suites infinies (en particulier la tr`es classique d'efinition de Martin-Lof). Nous prouvons qu'il n'existe pas de d'efinition parfaite fond'ee sur la notion de prouvabilit'e. En remplacant la prouvabilit 'e par la consistence, nous obtenons une d'efinition des suites al'eatoires tr`es g'en'erale qui remplit des conditions raisonablement fortes. Mots-cl'es: Suites al'eatoire, logique, mod`ele de Solovay Do most..

Do most strong definitions of randomness exist?

(eng) The goal of our paper is to propose a way to obtain more refined definitions of randomness than the notions known so far (e.g. Martin-Löf randomness). We show that a ``perfect'' definition of randomness based on provability does not exist. We then weaken our requirements on the definition by replacing provability by consistency and obtain a formula that defines a set of random sequences that fulfills rather strong conditions.(fre) Nous proposons ici de raffiner les définitions classiques du caractère aléatoire des suites infinies (en particulier la très classique définition de Martin-Löf). Nous prouvons qu'il n'existe pas de définition parfaite fondée sur la notion de prouvabilité. En remplaçant la prouvabilité par la consistence, nous obtenons une définition des suites aléatoires très générale qui remplit des conditions raisonablement fortes

Plasticity and convergence in the evolution of short-necked plesiosaurs

Plesiosaurs were the longest-surviving group of secondarily marine tetrapods, comparable in diversity to today’s cetaceans. During their long evolutionary history, which spanned the Jurassic and the Cretaceous (201 to 66 Ma), plesiosaurs repeatedly evolved long- and short-necked body plans [1,2]. Despite this postcranial plasticity, short-necked plesiosaur clades have traditionally been regarded as being highly constrained to persistent and clearly distinct ecological niches: advanced members of Pliosauridae (ranging from the Middle Jurassic to the early Late Cretaceous) have been characterised as apex predators [2–5], whereas members of the distantly related clade Polycotylidae (middle–Late Cretaceous) were thought to have been fast-swimming piscivores [1,5–7]. We report a new, highly unusual pliosaurid from the Early Cretaceous of Russia that shows close convergence with the cranial structure of polycotylids: Luskhan itilensis gen. et sp. nov. Using novel cladistic and ecomorphological data, we show that pliosaurids iteratively evolved polycotylid-like cranial morphologies from the Early Jurassic until the Early Cretaceous. This underscores the ecological diversity of derived pliosaurids and reveals a more complex evolutionary history than their iconic representation as gigantic apex predators of Mesozoic marine ecosystems suggests. Collectively, these data demonstrate an even higher degree of morphological plasticity and convergence in the evolution of plesiosaurs than previously thought, and suggest the existence of an optimal ecomorphology for short-necked piscivorous plesiosaurs through time and across phylogeny