Non-deterministic computation and the Jayne-Rogers Theorem

Non-deterministic type-2 machines (NDTMs) were suggested by Ziegler [34, 35, 33] as a model for hypercomputation in computable analysis. As demonstrated by Brattka, de Brecht and Pauly [2, 6], the strength of various kinds of type-2 non-deter-minism neatly classifies various important classes of noncomputable functions; and a characterization of such classes as those functions computable by certain NDTMs opens up new, simple ways to prove closure properties for them. A NDTM with advice space Z is a Turing machine with an input tape, an oracle tape, some work tapes and a write-once output tape. The input is an infinite sequence written on the input tape, the oracle tape is initialized with a guess, an infinite sequence from the set Z. The machine either halts eventually, which is seen as a rejection of the guess, or continues to write an infinite sequence on the output tape. For any valid input there must be an acceptable guess. Thus, a NDTM naturally computes a multivalued function f :⊆ {0,1}N ⇒ {0,1}N. The notion of non-deter-ministic computability is then lifted to arbitrary represented spaces: Some f : X⇒ Y is nondeterministally computable with advice space Z, iff there is an NDTM such that any p∈ {0,1}N denoting an element of X is accepted, and every successful computation produces a name for some y ∈ f (x). The power of NDTMs severely depends on the advice space. The spaces {0,1}N and N yield incomparable computational power, N×{0,1}N is more powerful than both, and NN again significantly more powerful than N×{0,1}N. The crucial property for us is that the additional computational power of N×{0,1}N over N only applies to multivalued functions—any single-valued f : X→Y (with Y computably admissible) non-deter-ministically computable with advice space N×{0,1}N already is non-deter-ministically computable with advice space N. We will apply the theory of non-deter-ministic computations to descriptive set theory. A subset of a metric space is called ∆2, if it is both the countable union of closed sets and the countable intersection of open sets. A function is called ∆2-measurable, iff the preimage of any open set is a ∆ 0 2-set. A function will be called A -piecewise continuous, iff there is a countable cover of its domain by closed sets, such that the restriction to any such closed set is continuous.