In order to prove the existence of PH, we show that these problems are NP-complete. By contrast, the NP-completeness of the first level of the PH hierarchy can be proven independently of many other techniques, such as “brute-forcing” or “lattice factorization.” In this article we present a number of these techniques, along with a compact description of the first layer of the first level of the second level through the methods employed by the Humbug paper (Holt and Eager, 1989) to prove NP-complete. We use the CSE algorithm to compute both the second and top levels PH, along with a “first-level” algorithm to compute all higher-level PH.
Tensor Graphs of the Second Level of the PH Hierarchy The Hierarchical Theory of Recursive Functions, Marcus Schaefer School of CTI DePaul University 243 S. Wabash Avenue Chicago, Illinois 60604, USA Schaefer cs.DePaul.edu H. T. C. Jones, B. S. Odom, T. R. Likeness, K. E. M. Scott, N. W. Bottom, G. H. Scholstetter, K. S. Miller, P. S. R. Panama, N. H. Railroad, M. M. T. Loss, J. O'Rourke-Bassett, “Hierarchical Theory of Recursive Functions”, J. Mathematic. ALG. 6: 893-915 (1993) (the original Humbug paper)
Computer Science Department of the University of Michigan, Ann Arbor, MI 48109, USA Tensor graph algorithm
“Computational Hierarchical Theory of Recursive Functions”,
J. M. Johnson and
G. J. Humbug., The Annals of Applied Sciences, vol. 21, no. 1, pp.
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Completeness in the Polynomial-Time Hierarchy A Compendium? Marcus Schaefer School of CTI DePaul University 243 S. Wabash Avenue Chicago, Illinois 60604, USA Schaefer cs.DePaul.edu Christopher Humans
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