3

Generalizing cl-reducibility on left-c.e. reals

The cl-reducibility is a Turing reducibility with use function bounded by $n+c$ where $c$ is a constant. For such reducibility, there is a Yu-Ding theorem saying that there is a pair of left-c.e. reals which are not both cl-reducible to any left-c.e. real, and a Barmpalias-Lewis theorem saying that there is a left-c.e. real which is not cl-reducible to any random left-c.e. real. Moreover, it is shown that both Yu-Ding theorem and Barmpalias-Lewis theorem characterize array computability for left-c.

Granularity of wagers in games and the (im)possibility of savings

In a casino where arbitrarily small bets are admissible, any betting strategy M can be modified into a savings strategy that not only is successful on each casino sequence where M is (thus accumulating unbounded wealth inside the casino) but also …