In this paper we examine the natural interpretation of a ramified type hierarchy into Martin-Löf type theory with an infinite sequence of universes. It is shown that under this predicative interpretation some useful special cases of Russell’s reducibility axiom are valid, namely functional reducibility. This is sufficient to make the type hierarchy usable for development of constructive mathema...

How random is a real? Given two reals, which is more random? If we partition reals into equivalence classes of reals of the “same degrees of randomness”, what does the resulting structure look like? The goal of this paper is to look at questions like these, specifically by studying the properties of reducibilities that act as measures of relative randomness, as embodied in the concept of initia...

We discuss a reduction notion relating the random oracles in two cryptographic schemes A and B. Basically, the random oracle of scheme B reduces to the one of scheme A if any hash function instantiation of the random oracle (possibly still oracle based) which makes A secure also makes B secure. In a sense, instantiating the random oracle in scheme B is thus not more demanding than the one for s...