Observations on complete sets between linear time and polynomial time

AbstractThere is a single set that is complete for a variety of nondeterministic time complexity classes with respect to related versions of m-reducibility. This observation immediately leads to transfer results for determinism versus nondeterminism solutions. Also, an upward transfer of collapses of certain oracle hierarchies, built analogously to the polynomial-time or the linear-time hierarchies, can be shown by means of uniformly constructed sets that are complete for related levels of all these hierarchies. A similar result holds for difference hierarchies over nondeterministic complexity classes. Finally, we give an oracle set relative to which the nondeterministic classes coincide with the deterministic ones, for several sets of time bounds, and we prove that the strictness of the tape-number hierarchy for deterministic linear-time Turing machines does not relativize

Computational Capabilities of Analog and Evolving Neural Networks over Infinite Input Streams

International audienceAnalog and evolving recurrent neural networks are super-Turing powerful. Here, we consider analog and evolving neural nets over infinite input streams. We then characterize the topological complexity of their ω-languages as a function of the specific analog or evolving weights that they employ. As a consequence, two infinite hierarchies of classes of analog and evolving neural networks based on the complexity of their underlying weights can be derived. These results constitute an optimal refinement of the super-Turing expressive power of analog and evolving neural networks. They show that analog and evolving neural nets represent natural models for oracle-based infinite computation

Real Hypercomputation and Continuity

By the sometimes so-called 'Main Theorem' of Recursive Analysis, every computable real function is necessarily continuous. We wonder whether and which kinds of HYPERcomputation allow for the effective evaluation of also discontinuous f:R->R. More precisely the present work considers the following three super-Turing notions of real function computability: * relativized computation; specifically given oracle access to the Halting Problem 0' or its jump 0''; * encoding real input x and/or output y=f(x) in weaker ways also related to the Arithmetic Hierarchy; * non-deterministic computation. It turns out that any f:R->R computable in the first or second sense is still necessarily continuous whereas the third type of hypercomputation does provide the required power to evaluate for instance the discontinuous sign function.Comment: previous version (extended abstract) has appeared in pp.562-571 of "Proc. 1st Conference on Computability in Europe" (CiE'05), Springer LNCS vol.352

On W[1]-Hardness as Evidence for Intractability

The central conjecture of parameterized complexity states that FPT !=W[1], and is generally regarded as the parameterized counterpart to P !=NP. We revisit the issue of the plausibility of FPT !=W[1], focusing on two aspects: the difficulty of proving the conjecture (assuming it holds), and how the relation between the two classes might differ from the one between P and NP. Regarding the first aspect, we give new evidence that separating FPT from W[1] would be considerably harder than doing the same for P and NP. Our main result regarding the relation between FPT and W[1] states that the closure of W[1] under relativization with FPT-oracles is precisely the class W[P], implying that either FPT is not low for W[1], or the W-Hierarchy collapses. This theorem also has consequences for the A-Hierarchy (a parameterized version of the Polynomial Hierarchy), namely that unless W[P] is a subset of some level A[t], there are structural differences between the A-Hierarchy and the Polynomial Hierarchy. We also prove that under the unlikely assumption that W[P] collapses to W[1] in a specific way, the collapse of any two consecutive levels of the A-Hierarchy implies the collapse of the entire hierarchy to a finite level; this extends a result of Chen, Flum, and Grohe (2005). Finally, we give weak (oracle-based) evidence that the inclusion W[t]subseteqA[t] is strict for t>1, and that the W-Hierarchy is proper. The latter result answers a question of Downey and Fellows (1993)

On the structure of intractable sets

There are two parts to this dissertation. The first part is motivated by nothing less than a reexamination of what it means for a set to be NP-complete. Are there sets in NP that in a mathematically meaningful sense should be considered to be complete for NP, but that are not NP-complete in the usual sense that every set in NP is ≤q[subscript]spmP-reducible to it? We define a noneffective binary relation that makes precise the notion that the complexity of A is polynomially related to the complexity of B, This relation yields new completeness and hardness notions for complexity classes, and we show that there are sets that are hard for NP that are not NP-hard in the usual sense. We also show that there are sets that must be considered to be complete for E that are not even ≤q[subscript]spTP-complete for E;In a certain way, hardness and completeness with respect to the relation we define is related to the notion of almost everywhere (a.e.) complexity, and so we initiate this study by first investigating this notion. We state and prove a deterministic time hierarchy theorem for a.e. complexity that is as tight as the Hartmanis-Stearns hierarchy theorem for infinitely often complexity. This result is a significant improvement over all previously known hierarchy theorems for a.e. complex sets. We derive similar, very tight, hierarchy theorems for sets that cannot be a.e. complex for syntactic reasons, but for which, intuitively, a.e. complex notions should exit. Similar results are applied to the study of P-printable sets and sets of low generalized Kolmogorov complexity;The second part of this study deals with relativization. Does the fact that DTIME(O (n)) ≠ NTIME(n) help in leading us to a proof that P ≠ NP? Does one imply the other? We seek evidence that this is a hard . We construct an oracle that answers this question in the affirmative, and we construct an oracle that answers this question in the negative. We conclude that the result that DTIME(O (n)) ≠ NTIME(n) does not imply P ≠ NP by recursive theoretic techniques;Finally, we study the relationships between P, NP, and the unambiguous and random time classes UP, and RP. Questions concerning these relationships are motivated by complexity issues to public-key cryptosystems. We prove that there exists a recursive oracle A such that P[superscript]A ≠ UP[superscript]A≠ NP[superscript]A, and such that the first inequality is strong, i.e., there exists a P[superscript]A-immune set in UP[superscript]A. Further, we constructed a recursive oracle B such that UP[superscript]B contains an RP[superscript]B-immune set. As a corollary we obtain P[superscript]B ≠ RB[superscript]B≠ NP[superscript]B and both inequalities are strong. By use of the techniques employed in the proof that P[superscript]A≠ UP[superscript]A≠ NP[superscript]A, we are also able to solve an open problem raised by Book, Long and Selman

A SURVEY OF LIMITED NONDETERMINISM IN COMPUTATIONAL COMPLEXITY THEORY

Nondeterminism is typically used as an inherent part of the computational models used incomputational complexity. However, much work has been done looking at nondeterminism asa separate resource added to deterministic machines. This survey examines several differentapproaches to limiting the amount of nondeterminism, including Kintala and Fischer\u27s βhierarchy, and Cai and Chen\u27s guess-and-check model