Klaus Ambos-Spies

Weakly Computable Real Numbers

A real number x is recursively approximable if it is a limit of a computable sequence of rational numbers. If, moreover, the sequence is increasing (decreasing or simply monotonic), then x is called left computable (right computable or semi-computable). x is called weakly computable if it is a difference of two left computable real numbers. We show that a real number is weakly computable if and only if there is a computable sequence (xs)s?N of rational numbers which converges to x weakly effectively, namely the sum of jumps of the sequence is bounded. It is also shown that the class of weakly computable real numbers extends properly the class of semi-computable real numbers and the class of recursively approximable real numbers extends properly the class of weakly computable real numbers.

Randomness, Relativizations, and Polynomial Reducibilities

We show that, for any set A which cannot be computed in polynomial time, the class of sets p-many-one incomparable with A has measure 1, whereas in case of p-Turing reducibility the class of sets incomparable with A has measure 1 if and only if A is not in the class BPP, the class of problems which can be probabilisticly solved with uniformly bounded error probability in polynomial time. Consequences for the reducibility relation between a randomly chosen pair of problems are discussed. Moreover, it is shown that any class, which in the relativized case collapses to P with probability one, is actually contained in BPP.

Diagonalizations over polynomial time computable sets

Abstract A formal notion of diagonalization is developed which allows to enforce properties that are related to the class of polynomial time computable sets (the class of polynomial time computable functions respectively), like, e.g., p-immunity. It is shown that there are sets—called p-generic— which have all properties enforceable by such diagonalizations. We study the behaviour and the complexity of p-generic sets. In particular, we show that the existence of p-generic sets in NP is oracle dependent, even if we assume P ≠ NP.

Sublattices of the polynomial time degrees

We show that any countable distributive lattice can be embedded in any interval of polynomial time degrees. Furthermore the embeddings can be chosen to preserve the least or the greatest element. This holds for both polynomial time bounded many-one and Turing reducibilities, as well as for all of the common intermediate reducibilities.

Minimal pairs for polynomial time reducibilities

Two recursive sets A and B form a minimal pair with respect to some polynomial time reducibility notion ≤pr if neither A nor B can be computed in polynomial time but every set which reduces to both A and B is polynomial time computable. We show that for every recursive set A∉P there is a recursive set B such that A and B form a minimal pair. Moreover, similar results for pairs without greatest predecessors are proved.

Resource-Bounded Balanced Genericity, Stochasticity and Weak Randomness

We introduce balanced t(n)-genericity which is a refinement of the genericity concept of Ambos-Spies, Fleischhack and Huwig [2] and which in addition controls the frequency with which a condition is met. We show that this concept coincides with the resource-bounded version of Churchs stochasticity [6]. By uniformly describing these concepts and weaker notions of stochasticity introduced by Wilber [19] and Ko [11] in terms of prediction functions, we clarify the relations among these resource-bounded stochasticity concepts. Moreover, we give descriptions of these concepts in the framework of Lutzs resource-bounded measure theory [13] based on martingales: We show that t(n)-stochasticity coincides with a weak notion of t(n)-randomness based on so-called simple martingales but that it is strictly weaker than t(n)-randomness in the sense of Lutz.

Resource-bounded genericity

Resource-bounded genericity concepts have been introduced by Ambos-Spies, Fleischhack and Huwig (1984, 1988), Lutz (1990), and Fenner (1991). Though it was known that these concepts are incompatible, the relations among these notions were not fully understood. We survey these notions and clarify the relations among them by specifying the types of diagonalizations captured by the individual concepts. Moreover, we introduce new, stronger resource-bounded genericity concepts corresponding to the fundamental diagonalization concepts in complexity theory. In particular we introduce general genericity, which generalizes the previous concepts and captures both standard finite extension arguments and slow diagonalizations. As we also point out, however, there is no strongest resource-bounded genericity concept. This is shown by giving a strict hierarchy of genericity notions corresponding to delayed diagonalizations. Finally we study some properties of the Baire category notions on E induced by the genericity concepts and we point out the relations between resource-bounded genericity and resource-bounded randomness.

A note on the complete problems for complexity classes

Abstract We show that every class of recursive sets which is closed under p-Turing equivalence possesses a ⩽ pm -complete set if and only if it possesses a ⩽ pT -complete set. One of the consequences hereof is that the classes Δ n p of the polynomial hierarchy possess ⩽ pm -complete problems.

Separating NP-Completeness Notions under Strong Hypotheses

Lutz (1993, “Proceedings of the Eight Annual Conference on Structure in Complexity Theory,” pp. 158?175) proposed the study of the structure of the class NP=NTIME(poly) under the hypothesis that NP does not have p-measure 0 (with respect to Lutzs resource bounded measure (1992, J. Comput. System Sci.44, 220?258)). Lutz and Mayordomo (1996, Theoret. Comput. Sci.164, 141?163) showed that, under this hypothesis, NP-m-completeness and NP-T-completeness differ, and they conjectured that additional NP-completeness notions can be separated. Here we prove this conjecture for the bounded-query reducibilities. In fact, we consider a new weaker hypothesis, namely the assumption that NP is not p-meager with respect to the resource bounded Baire category concept of Ambos-Spies et al. (1988, Lecture Notes in Computer Science, Vol. 329, pp. 1?16). We show that this category hypothesis is sufficient to get: (i)For k?2, NP-btt(k)-completeness is stronger than NP-btt(k+1)-completeness. (ii)For k?1, NP-bT(k)-completeness is stronger than NP-bT(k+1)-completeness. (iii)For every k?2, NP-bT(k?1)-completeness is not implied by NP-btt(k+1)-completeness and NP-btt(2k?2)-completeness is not implied by NP-bT(k)-completeness. (iv)NP-btt-completeness is stronger than NP-tt-completeness.

On the Structure of Polynomial Time Degrees

The main results of this paper are the following. 1) For both the polynomial time many-one and the polynomial time Turing degrees of recursive sets, every countable distributive lattice can be embedded in any interval of degrees. Furthermore, certain restraints — like preservation of the least or greatest element — can be imposed on the embeddings. 2) The upper semilattice of polynomial time many-one degrees is distributive, whereas that of the polynomial time Turing degrees is nondistributive. This gives the first (elementary) difference between the algebraic structures of p-many-one and p-Turing degrees, respectively.