Abstract

ABSTRACT Recent results on the internal, measure-theoretic structure of the exponential time complexity classes E and EXP are surveyed. The measure structure of these classes is seen to interact in informative ways with bi-immunity, complexity cores, polynomial-time reductions, completeness, circuit-size complexity, Kolmogorov complexity, natural proofs, pseudorandom generators, the density of hard languages, randomized complexity, and lowness. Possible implications for the structure of NP are also discussed. 1

Citations

1478	The probabilistic method - Spencer - 1992
1412	An Introduction to Kolmogorov Complexity and Its Applications - Li, Vitanyi - 1997
1117	Reducability Among Combinatorial Problems - Karp - 1972
409	The complexity of theorem proving procedures - Cook - 1971
250	Hardness vs. Randomness - Nisan, Wigderson - 1988
245	Measure Theory - Halmos - 1950
212	Some connections between nonuniform and uniform complexity classes - Karp, Lipton - 1980
166	On the computational complexity of algorithms - Hartmanis, Stearns - 1965
160	Structural Complexity - Balcazar, Diaz, et al. - 1988
158	Almost everywhere high nonuniform complexity - Lutz - 1992
152	Natural proofs - Razborov, Rudich - 1997
148	On isomorphisms and density of NP and other complete sets - Berman, Hartmanis - 1977
133	Zufälligkeit und Wahrscheinlichkeit - Schnorr - 1971
106	Some remarks on the theory of graphs - Erdös - 1947
94	Sparse complete sets for NP: Solution of a conjecture of berman and hartmanis - Mahaney - 1982
91	On the notion of a random sequence - Levin - 1973
77	A unified approach to the definition of random sequences - Schnorr - 1971
76	BPP and the polynomial hierarchy - Lautemann - 1983
72	Oracles and queries that are sufficient for exact learning - Bshouty, Cleve, et al. - 1996
69	Probabilistic methods in combinatorics - Erdos, Spencer - 1974
68	Measure and Category - Oxtoby
66	Equivalence of measures of complexity classes - Breutzmann, Lutz - 1999
56	New collapse consequences of NP having small circuits - Kobler, Watanabe - 1999
52	Cook versus Karp-Levin: separating completeness notions if NP is not small. Theoretical Computer Science 164:141--163 - Lutz, Mayordomo - 1996
51	The Graph Isomorphism Problem - Köbler, Schöning, et al. - 1993
51	Almost every set in exponential time is P-biimmune, Theoretical Computer Science 136 - Mayordomo - 1994
51	Unprovability of lower bounds on the circuit size in certain fragments of bounded arithmetic - Razborov - 1995
49	On the concept of a random sequence - Church - 1940
47	Bi-immune sets for complexity classes - Balcázar, Schöning - 1985
45	Measure on small complexity classes, with applications for BPP - Allender, Strauss - 1994
45	The complexity and distribution of hard problems - Juedes, Lutz - 1995
44	Provably difficult combinatorial games - Stockmeyer, Chandra - 1979
41	Comparison of polynomial-time reducibilities - Ladner, Lynch, et al. - 1975
41	Measure, stochasticity, and the density of hard languages - Lutz, Mayordomo - 1994
38	Contributions to the Study of Resource-Bounded Measure - Mayordomo - 1994
37	Structural Complexity II - Balcazar, Diaz, et al. - 1988
32	Resource-bounded measure - Lutz - 1998
30	The complexity of decision versus search - Bellare, Goldwasser - 1994
28	Pseudorandom generators, measure theory, and natural proofs - Regan, Sivakumar, et al. - 1995
26	Resource-bounded genericity - Ambos-Spies - 1996
25	Instance Complexity - Orponen, Ko, et al. - 1994
24	Klassifikation der Zufallsgesetze nach Komplexität und Ordnung. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete - Schnorr - 1970
23	Algorithms and randomness - Kolmogorov, Uspensky - 1987
21	Resource-bounded balanced genericity, stochasticity and weak randomness - Ambos-Spies, Mayordomo, et al. - 1996
21	Completeness, approximation and density - Ko, Moore - 1981
21	On reducibility to complex or sparse sets - Lynch - 1975
20	Diagonalizations over polynomial time computable sets - Ambos-Spies, Fleischhack, et al. - 1987
19	Notions of resource-bounded category and genericity - Fenner
18	The density and complexity of polynomial cores for intractable sets - Orponen, Schning - 1986
17	An excursion to the Kolmogorov random strings - Buhrman, Mayordomo - 1997