Randomness conservation inequalities; information and independence in mathematical theories
Information and Control
Some consequences of the existence of pseudorandom generators
Journal of Computer and System Sciences
Algebraic methods for interactive proof systems
Journal of the ACM (JACM)
Journal of the ACM (JACM)
BPP has subexponential time simulations unless EXPTIME has publishable proofs
Computational Complexity
P = BPP if E requires exponential circuits: derandomizing the XOR lemma
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
A Pseudorandom Generator from any One-way Function
SIAM Journal on Computing
Resource-Bounded Kolmogorov Complexity Revisited
SIAM Journal on Computing
When Worlds Collide: Derandomization, Lower Bounds, and Kolmogorov Complexity
FST TCS '01 Proceedings of the 21st Conference on Foundations of Software Technology and Theoretical Computer Science
SIAM Journal on Computing
Decision versus search problems in super-polynomial time
SFCS '89 Proceedings of the 30th Annual Symposium on Foundations of Computer Science
Computational Complexity: A Modern Approach
Computational Complexity: A Modern Approach
The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory
Journal of Computer and System Sciences
The pervasive reach of resource-bounded Kolmogorov complexity in computational complexity theory
Journal of Computer and System Sciences
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It is a trivial observation that every decidable set has strings of length n with Kolmogorov complexity log n + O(1) if it has any strings of length n at all. Things become much more interesting when one asks whether a similar property holds when one considers resource-bounded Kolmogorov complexity. This is the question considered here: Can a feasible set A avoid accepting strings of low resource-bounded Kolmogorov complexity, while still accepting some (or many) strings of length n?. More specifically, this paper deals with two notions of resource-bounded Kolmogorov complexity: Kt and KNt. The measure Kt was defined by Levin more than three decades ago and has been studied extensively since then. The measure KNt is a nondeterministic analog of Kt. For all strings x, Kt(x) &KNt(x); the two measures are polynomially related if and only if NEXP ⊆ EXP/poly [5]. Many longstanding open questions in complexity theory boil down to the question of whether there are sets in P that avoid all strings of low Kt complexity. For example, the EXP vs ZPP question is equivalent to (one version of) the question of whether avoiding simple strings is difficult: (EXP = ZPP if and only if there exist ε 0 and a "dense" set in P having no strings x with Kt(x) = |x|ε [4]). Surprisingly, we are able to show unconditionally that avoiding simple strings (in the sense of KNt complexity) is difficult. Every dense set in NP ∩ co-NP contains infinitely many strings x such that KNt(x) ≤ |x|ε for some ε. The proof does not relativize. As an application, we are able to show that if E = NE, then accepting paths for nondeterministic exponential time machines can be found somewhat more quickly than the brute-force upper bound, if there are many accepting paths.