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14
The quantitative structure of exponential time
 Complexity theory retrospective II
, 1997
"... ABSTRACT Recent results on the internal, measuretheoretic 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 biimmunity, complexity cores, polynomialtime reductions, completeness, circuit ..."
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Cited by 90 (13 self)
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ABSTRACT Recent results on the internal, measuretheoretic 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 biimmunity, complexity cores, polynomialtime reductions, completeness, circuitsize 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
An oracle builder’s toolkit
, 2002
"... We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and ..."
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Cited by 47 (10 self)
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We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SPgenerics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SPgenerics, ULIN ∩ coULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ coNP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩coNP/1 ̸ ⊆ (NP∩coNP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.
ResourceBounded Measure and Randomness
 Complexity, Logic and Recursion Theory, Lecture Notes in Pure and Applied Mathematics
, 1997
"... We survey recent results on resourcebounded measure and randomness in structural complexity theory. In particular, we discuss applications of these concepts to the exponential time complexity classes E and E2 . Moreover, we treat timebounded genericity and stochasticity concepts which are weaker t ..."
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Cited by 40 (6 self)
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We survey recent results on resourcebounded measure and randomness in structural complexity theory. In particular, we discuss applications of these concepts to the exponential time complexity classes E and E2 . Moreover, we treat timebounded genericity and stochasticity concepts which are weaker than timebounded randomness but which suffice for many of the applications in complexity theory. 1 Introduction The first attempt for defining the concept of a random sequence goes back to von Mises [vM19] in 1919. He proposed that an infinite 01sequence S should be considered to be random if, in the limit, the number of the occurrences of the 0s and 1s in S is the same (i.e. the sequence S satisfies the law of large numbers) and if this stability property is inherited by every infinite subsequence of S obtained by an admissible selection rule. A fuzzyness in this concept, due to the lack of a formal definition of admissibility was later eliminated by Church [Ch40] in 1940, who proposed t...
InfinitelyOften Autoreducible Sets
 In Proceedings of the 14th Annual International Symposium on Algorithms and Computation
, 2003
"... A set A is autoreducible if one can compute, for all x, the value A(x) by querying A only at places y x. Furthermore, A is infinitelyoften autoreducible if, for infinitely many x, the value A(x) can be computed by querying A only at places y x. For all other x, the computation outputs a sp ..."
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Cited by 6 (0 self)
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A set A is autoreducible if one can compute, for all x, the value A(x) by querying A only at places y x. Furthermore, A is infinitelyoften autoreducible if, for infinitely many x, the value A(x) can be computed by querying A only at places y x. For all other x, the computation outputs a special symbol to signal that the reduction is undefined.
ResourceBounded Strong Dimension versus ResourceBounded Category
, 2005
"... Classically it is known that any set with packing dimension less than 1 is meager in the sense of Baire category. We establish a resourcebounded extension: if a class X has ∆strong dimension less than 1, then X is ∆meager. This has the applications of explaining some of Lutz’s simultaneous ∆meag ..."
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Cited by 5 (1 self)
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Classically it is known that any set with packing dimension less than 1 is meager in the sense of Baire category. We establish a resourcebounded extension: if a class X has ∆strong dimension less than 1, then X is ∆meager. This has the applications of explaining some of Lutz’s simultaneous ∆meager, ∆measure 0 results and providing a new proof of a Gu’s strong dimension result on infinitelyoften classes.
Collapsing PolynomialTime Degrees
"... For reducibilities r and r 0 such that r is weaker than r 0 , we say that the rdegree of A, i.e., the class of sets which are requivalent to A, collapses to the r 0 degree of A if both degrees coincide. We investigate for the polynomialtime bounded manyone, bounded truthtable, truthtable, and ..."
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Cited by 4 (0 self)
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For reducibilities r and r 0 such that r is weaker than r 0 , we say that the rdegree of A, i.e., the class of sets which are requivalent to A, collapses to the r 0 degree of A if both degrees coincide. We investigate for the polynomialtime bounded manyone, bounded truthtable, truthtable, and Turing reducibilities whether and under which conditions such collapses can occur. While we show that such collapses do not occur for sets which are hard for exponential time, we have been able to construct a recursive set such that its bounded truthtable degree collapses to its manyone degree. The question whether there is a set such that its Turing degree collapses to its manyone degree is still open; however, we show that such a set  if it exists  must be recursive.
Genericity and Randomness over Feasible Probability Measures
 Theoretical Computer Science
"... This paper investigates the notion of resourcebounded genericity developed by AmbosSpies, Fleischhack, and Huwig. AmbosSpies, Neis, and Terwijn have recently shown that every language that is t(n)random over the uniform probability measure is t(n)generic. It is shown here that, in fact, every l ..."
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Cited by 4 (2 self)
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This paper investigates the notion of resourcebounded genericity developed by AmbosSpies, Fleischhack, and Huwig. AmbosSpies, Neis, and Terwijn have recently shown that every language that is t(n)random over the uniform probability measure is t(n)generic. It is shown here that, in fact, every language that is t(n)random over any strongly positive, t(n)computable probability measure is t(n)generic. Roughly speaking, this implies that, when genericity is used to prove a resourcebounded measure result, the result is not specific to the underlying probability measure. This research was supported in part by National Science Foundation Grant CCR9157382, with matching funds from Rockwell, Microware Systems Corporation, and Amoco Foundation. y Color LaserJet and Consumables Division, HewlettPackard Company, Boise, ID 83714, U.S.A. Email: amy lorentz@hpboiseom8.om.hp.com z Department of Computer Science, Iowa State University, Ames, IA 50011, U.S.A. Email: lutz@cs.iastate....
Baire Category and Nowhere Differentiability for Feasible Real Functions ⋆
"... Abstract. A notion of resourcebounded Baire category is developed for the class PC[0,1] of all polynomialtime computable realvalued functions on the unit interval. The meager subsets of PC[0,1] are characterized in terms of resourcebounded BanachMazur games. This characterization is used to pro ..."
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Cited by 3 (0 self)
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Abstract. A notion of resourcebounded Baire category is developed for the class PC[0,1] of all polynomialtime computable realvalued functions on the unit interval. The meager subsets of PC[0,1] are characterized in terms of resourcebounded BanachMazur games. This characterization is used to prove that, in the sense of Baire category, almost every function in PC[0,1] is nowhere differentiable. This is a complexitytheoretic extension of the analogous classical result that Banach proved for the class C[0, 1] in 1931. 1
Query Order and NPCompleteness
"... The effect of query order on NPcompleteness is investigated. A sequence ~ D = (D 1 ; : : : ; D k ) of decision problems is defined to be sequentially complete for NP if each D i 2 NP and every problem in NP can be decided in polynomial time with one query to each of D 1 ; : : : ; D k in this orde ..."
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Cited by 2 (1 self)
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The effect of query order on NPcompleteness is investigated. A sequence ~ D = (D 1 ; : : : ; D k ) of decision problems is defined to be sequentially complete for NP if each D i 2 NP and every problem in NP can be decided in polynomial time with one query to each of D 1 ; : : : ; D k in this order . It is shown that, if NP contains a language that is pgeneric in the sense of AmbosSpies, Fleischhack, and Huwig [3], then for every integer k 2, there is a sequence ~ D = (D 1 ; : : : ; D k ) such that ~ D is sequentially complete for NP, but no nontrivial permutation (D i 1 ; : : : ; D i k ) of ~ D is sequentially complete for NP. It follows that such a sequence ~ D exists if there is any strongly positive, pcomputable probability measure such that p (NP) 6= 0. 1 Introduction The success or efficiency of a computation sometimes dependsand sometimes does not dependupon the order in which it is allowed access to the various pieces of information that it may require. Altho...
AverageCase Complexity Theory and PolynomialTime Reductions
, 2001
"... This thesis studies averagecase complexity theory and polynomialtime reducibilities. The issues in averagecase complexity arise primarily from Cai and Selman's extension of Levin's denition of average polynomial time. We study polynomialtime reductions between distributional problems. Under stro ..."
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Cited by 2 (0 self)
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This thesis studies averagecase complexity theory and polynomialtime reducibilities. The issues in averagecase complexity arise primarily from Cai and Selman's extension of Levin's denition of average polynomial time. We study polynomialtime reductions between distributional problems. Under strong but reasonable hypotheses we separate ordinary NPcompleteness notions.