Algorithmic Randomness and Capacity of Closed Sets

We investigate the connection between measure, capacity and algorithmic randomness for the space of closed sets. For any computable measure m, a computable capacity T may be defined by letting T(Q) be the measure of the family of closed sets K which have nonempty intersection with Q. We prove an effective version of Choquet's capacity theorem by showing that every computable capacity may be obtained from a computable measure in this way. We establish conditions on the measure m that characterize when the capacity of an m-random closed set equals zero. This includes new results in classical probability theory as well as results for algorithmic randomness. For certain computable measures, we construct effectively closed sets with positive capacity and with Lebesgue measure zero. We show that for computable measures, a real q is upper semi-computable if and only if there is an effectively closed set with capacity q

Evolving Computability

We consider the degrees of non-computability (Weihrauch degrees) of finding winning strategies (or more generally, Nash equilibria) in infinite sequential games with certain winning sets (or more generally, outcome sets). In particular, we show that as the complexity of the winning sets increases in the difference hierarchy, the complexity of constructing winning strategies increases in the effective Borel hierarchy.Comment: An extended abstract of this work has appeared in the Proceedings of CiE 201

Rapid adaptation to invasive predators overwhelms natural gradients of intraspecific variation

Invasive predators can exert strong selection on native populations. If selection is strong enough, populations could lose the phenotypic variation caused by adaptation to heterogeneous environments. We compare frog tadpoles prior to and 14 years following invasion by crayfish. Prior to the invasion, populations differed in their intrinsic developmental rate, with tadpoles from cold areas reaching metamorphosis sooner than those from warm areas. Following the invasion, tadpoles from invaded populations develop faster than those from non-invaded populations. This ontogenetic shift overwhelmed the intraspecific variation between populations in a few generations, to the point where invaded populations develop at a similar rate regardless of climate. Rapid development can have costs, as fast-developing froglets have a smaller body size and poorer jumping performance, but compensatory growth counteracts some costs of development acceleration. Strong selection by invasive species can disrupt local adaptations by dampening intraspecific phenotypic variation, with complex consequences on lifetime fitness