Relativizations of the P =? NP and Other Problems: Developments in Structural Complexity Theory

SIAM Review ◽  
1994 ◽  
Vol 36 (2) ◽  
pp. 157-175 ◽  
Author(s):  
Ronald V. Book
Keyword(s):  
Complexity Theory ◽  
Structural Complexity

A proof of Beigel's cardinality conjecture

1992 ◽  
Vol 57 (2) ◽  
pp. 677-681 ◽  
Author(s):  
Martin Kummer
Keyword(s):  
Complexity Theory ◽  
Structural Complexity ◽  
Halting Problem ◽  
Parallel Queries

In 1986, Beigel [Be87] (see also [Od89, III.5.9]) proved the nonspeedup theorem: if A, B ⊆ ω, and as a function of 2n variables can be computed by an algorithm which makes at most n queries to B, then A is recursive (informally, 2n parallel queries to a nonrecursive oracle A cannot be answered by making n sequential (or “adaptive”) queries to an arbitrary oracle B). Here, 2n cannot be replaced by 2n − 1. In subsequent papers of Beigel, Gasarch, Gill, Hay, and Owings the theory of “bounded query classes” has been further developed (see, for example, [BGGOta], [BGH89], and [Ow89]). The topic has also been studied in the context of structural complexity theory (see, for example, [AG88], [Be90], and [JY90]).If A ⊆ ω and n ≥ 1, let . Beigel [Be87] stated the powerful “cardinality conjecture” (CC): if A, B ⊆ ω, and can be computed by an algorithm which makes at most n queries to B, then A is recursive. Owings [Ow89] verified CC for n = 1, and, for n 1, he proved that A is recursive in the halting problem. We prove that CC is true for all n.

scholarly journals Structural complexity theory: Recent surprises

1990 ◽  
pp. 1-12 ◽  
Author(s):  
Juris Hartmanis ◽  
Richard Chang ◽  
Desh Ranjan ◽  
Pankaj Rohatgi
Keyword(s):  
Complexity Theory ◽  
Structural Complexity

scholarly journals Arithmetization: A new method in structural complexity theory

1991 ◽  
Vol 1 (1) ◽  
pp. 41-66 ◽  
Author(s):  
L�szl� Babai ◽  
Lance Fortnow
Keyword(s):  
Complexity Theory ◽  
Structural Complexity ◽  
New Method

Guest Column

2021 ◽  
Vol 52 (3) ◽  
pp. 38-59
Author(s):  
Carlo Mereghetti ◽  
Beatrice Palano
Keyword(s):  
Quantum Computing ◽  
Complexity Theory ◽  
Quantum Computer ◽  
Structural Complexity ◽  
Research Area ◽  
Point Of View ◽  
Quantum Mechanical ◽  
Turing Machines ◽  
Computational Power ◽  
Quantum Turing Machine

Quantum computing is a prolific research area, halfway between physics and computer science [27, 29, 52]. Most likely, its origins may be dated back to 70's, when some works on quantum information began to appear (see, e.g., [34, 37]). In early 80's, R.P. Feynman suggested that the computational power of quantum mechanical processes might be beyond that of traditional computation models [25]. Almost at the same time, P. Benioff already proved that such processes are at least as powerful as Turing machines [9]. In 1985, D. Deutsch [22] proposed the notion of a quantum Turing machine as a physically realizable model for a quantum computer. From the point of view of structural complexity, E. Bernstein and U. Vazirani introduced in [20] the class BQP of problems solvable in polynomial time on quantum Turing machines, focusing attention on relations with the corresponding deterministic and probabilistic classes P and BPP, respectively. Several works in the literature explored classical issues in complexity theory from the quantum paradigm perspective (see, e.g., [7, 60, 61]).

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