7 research outputs found
Comparing disorder and adaptability in stochasticity
In the literature, there are various notions of stochasticity which measure
how well an algorithmically random set satisfies the law of large numbers. Such
notions can be categorized by disorder and adaptability: adaptive strategies
may use information observed about the set when deciding how to act, and
disorderly strategies may act out of order. In the disorderly setting, adaptive
strategies are more powerful than non-adaptive ones. In the adaptive setting,
Merkle et al. showed that disorderly strategies are more powerful than orderly
ones. This leaves open the question of how disorderly, non-adaptive strategies
compare to orderly, adaptive strategies, as well as how both relate to orderly,
non-adaptive strategies. In this paper, we show that orderly, adaptive
strategies and disorderly, non-adaptive strategies are both strictly more
powerful than orderly, non-adaptive strategies. Using the techniques developed
to prove this, we also make progress towards the former open question by
introducing a notion of orderly, ``weakly adaptable'' strategies which we prove
is incomparable with disorderly, non-adaptive strategies