Strong positive reducibilities

The need of formalizing a satisfactory notion of relative computability of partial functions leads to enumeration reducibility, which can be viewed as computing with nondeterministic Turing machines using positive information. This paper is dedicated to certain reducibilities that are stronger than enumeration reducibility, with emphasis given to s-reducibility,which appears often in computability theory and applications. We review some of the most notable properties of s-reducibility, together with the main differences distinguishing the s-degrees from the e-degrees, both at the global and local level.