3 research outputs found
Revising Type-2 Computation and Degrees of Discontinuity
By the sometimes so-called MAIN THEOREM of Recursive Analysis, every
computable real function is necessarily continuous. Weihrauch and Zheng
(TCS'2000), Brattka (MLQ'2005), and Ziegler (ToCS'2006) have considered
different relaxed notions of computability to cover also discontinuous
functions. The present work compares and unifies these approaches. This is
based on the concept of the JUMP of a representation: both a TTE-counterpart to
the well known recursion-theoretic jump on Kleene's Arithmetical Hierarchy of
hypercomputation: and a formalization of revising computation in the sense of
Shoenfield.
We also consider Markov and Banach/Mazur oracle-computation of discontinuous
fu nctions and characterize the computational power of Type-2 nondeterminism to
coincide with the first level of the Analytical Hierarchy.Comment: to appear in Proc. CCA'0
Nondeterminism, Fairness and a Fundamental Analogy
In this note we propose a model for unbounded nondeterministic computation which provides a very natural basis for the structural analogy between recursive function theory and computational complexity theory: P : NP ¸ = REC : RE At the same time this model presents an alternative version of the halting problem which has been known for a decade to be highly intractable. 1 Introduction Structural complexity theory is often presented as the theory in which the results obtained for classes of languages recognized by Turing machines are transferred to a resource bounded setting. Notions like reduction, simplicity, immunity, the arithmetical hierarchy, relativizations etc. were all first defined in recursive function theory and later (relativizations of) these notions were introduced in complexity theory. All of this work was inspired by the frustration originating from the difficulty of the fundamental problem in computational complexity theory which has become known as the P ? = NP pr..