Results 1  10
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15
Almost Everywhere High Nonuniform Complexity
, 1992
"... . We investigate the distribution of nonuniform complexities in uniform complexity classes. We prove that almost every problem decidable in exponential space has essentially maximum circuitsize and spacebounded Kolmogorov complexity almost everywhere. (The circuitsize lower bound actually exceeds ..."
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Cited by 177 (36 self)
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. We investigate the distribution of nonuniform complexities in uniform complexity classes. We prove that almost every problem decidable in exponential space has essentially maximum circuitsize and spacebounded Kolmogorov complexity almost everywhere. (The circuitsize lower bound actually exceeds, and thereby strengthens, the Shannon 2 n n lower bound for almost every problem, with no computability constraint.) In exponential time complexity classes, we prove that the strongest relativizable lower bounds hold almost everywhere for almost all problems. Finally, we show that infinite pseudorandom sequences have high nonuniform complexity almost everywhere. The results are unified by a new, more powerful formulation of the underlying measure theory, based on uniform systems of density functions, and by the introduction of a new nonuniform complexity measure, the selective Kolmogorov complexity. This research was supported in part by NSF Grants CCR8809238 and CCR9157382 and in ...
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.
Separability and Oneway Functions
, 2000
"... We settle all relativized questions of the relationships between the following ve propositions: P = NP P = UP P = NP \ coNP All disjoint pairs of NP sets are Pseparable. ..."
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Cited by 27 (13 self)
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We settle all relativized questions of the relationships between the following ve propositions: P = NP P = UP P = NP \ coNP All disjoint pairs of NP sets are Pseparable.
Applications of TimeBounded Kolmogorov Complexity in Complexity Theory
 Kolmogorov complexity and computational complexity
, 1992
"... This paper presents one method of using timebounded Kolmogorov complexity as a measure of the complexity of sets, and outlines anumber of applications of this approach to di#erent questions in complexity theory. Connections will be drawn among the following topics: NE predicates, ranking functi ..."
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Cited by 18 (4 self)
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This paper presents one method of using timebounded Kolmogorov complexity as a measure of the complexity of sets, and outlines anumber of applications of this approach to di#erent questions in complexity theory. Connections will be drawn among the following topics: NE predicates, ranking functions, pseudorandom generators, and hierarchy theorems in circuit complexity.
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
Automatic Forcing and Genericity: On the Diagonalization Strength of Finite Automata
 In Proceedings of the 4th International Conference on Discrete Mathematics and Theoretical Computer Science
, 2003
"... Algorithmic and resourcebounded Baire category and corresponding genericity concepts introduced in computability theory and computational complexity theory, respectively, have become elegant and powerful tools in these settings. Here we introduce some new genericity notions based on extension f ..."
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Cited by 2 (0 self)
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Algorithmic and resourcebounded Baire category and corresponding genericity concepts introduced in computability theory and computational complexity theory, respectively, have become elegant and powerful tools in these settings. Here we introduce some new genericity notions based on extension functions computable by nite automata which are tailored for capturing diagonalizations over regular sets and functions. We show that the generic sets obtained either by the partial regular extension functions of any given xed constant length or by all total regular extension of constant length are just the sets with saturated (also called disjunctive) characteristic sequences. Here a sequence is saturated if every string occurs in as a substring. We also show that these automatic generic sets are not regular but may be context free. Furthermore, we introduce stronger automatic genericity notions based on regular extension functions of nonconstant length and we show that the corresponding generic sets are biimmune for the classes of regular and context free languages.
Measure, Category and Learning Theory
"... Measure and category (or rather, their recursion theoretical counterparts) have been used in Theoretical Computer Science to make precise the intuitive notion "for most of the recursive sets." We use the Supported in part by NSF Grant CCR 9253582. y Supported in part by Latvian Council of Scie ..."
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Cited by 2 (1 self)
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Measure and category (or rather, their recursion theoretical counterparts) have been used in Theoretical Computer Science to make precise the intuitive notion "for most of the recursive sets." We use the Supported in part by NSF Grant CCR 9253582. y Supported in part by Latvian Council of Science Grant 93.599 and NSF Grant 9119540. z Supported in part by NSF Grant 9301339. x Supported in part by NSF Grants 9119540 and 9301339.  Supported by the Deutsche Forschungsgemeinschaft (DFG) Grant Me 672/41. notions of effective measure and category to discuss the relative sizes of inferrible sets, and their complements. We find that inferrible sets become large rather quickly in the standard hierarchies of learnability. On the other hand, the complements of the learnable sets are all large. 1 Introduction Determining the relative size of denumerable sets, and those with cardinality @ 1 , led mathematicians do develop the notions of measure and category [Oxt71]. Described in t...
On the Relative Sizes of Learnable Sets
"... Measure and category (or rather, their recursiontheoretical counterparts) have been used in theoretical computer science to make precise the intuitive notion "for most of the recursive sets." We use the notions of effective measure and category to discuss the relative sizes of inferrible sets, and ..."
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Cited by 1 (1 self)
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Measure and category (or rather, their recursiontheoretical counterparts) have been used in theoretical computer science to make precise the intuitive notion "for most of the recursive sets." We use the notions of effective measure and category to discuss the relative sizes of inferrible sets, and their complements. We find that inferable sets become large rather quickly in the standard hierarchies of learnability. On the other hand, the complements of the learnable sets are all large. 1 Introduction Determining the relative size of denumerable sets, and those with cardinality @ 1 , led mathematicians to develop the notions of measure and category [Oxt71]. We investigate an application of measure and category techniques to a branch of learning theory called inductive inference [AS83]. The models of learning used in this field have been inspired by features of human learning. The goal of this work is to determine the relative sizes of classes of inferable sets of functions. The idea ...
Effective Category and Measure in Abstract Complexity Theory (Extended Abstract)
, 1995
"... Complexity TheoryExtended Abstract #+ Cristian Calude # and Marius Zimand Abstract Strong variants of the Operator Speedup Theorem, Operator Gap Theorem and Compression Theorem are obtained using an e#ective version of Baire Category Theorem. It is also shown that all complexity classes of re ..."
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Cited by 1 (0 self)
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Complexity TheoryExtended Abstract #+ Cristian Calude # and Marius Zimand Abstract Strong variants of the Operator Speedup Theorem, Operator Gap Theorem and Compression Theorem are obtained using an e#ective version of Baire Category Theorem. It is also shown that all complexity classes of recursive predicates have e#ective measure zero in the space of recursive predicates and, on the other hand, the class of predicates with almost everywhere complexity above an arbitrary recursive threshold has recursive measure one in the class of recursive predicates. Keywords: Complexity measure, Operator Speedup Theorem, Operator Gap Theorem, Compression Theorem, e#ective Baire classification, e#ective measure. 1 Introduction The abstract complexity theory initiated by Blum [2] (see also Bridges [5], Calude [8], Hartmanis and Hopcroft [17], Machtey and Young [23], Seiferas [34]) has revealed fundamental properties of complexity measures. The striking importance of this theory relies in i...