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In Search of an Easy Witness: Exponential Time vs. Probabilistic Polynomial Time
"... Restricting the search space f0; 1g to the set of truth tables of "easy" Boolean functions on log n variables, as well as using some known hardnessrandomness tradeoffs, we establish a number of results relating the complexity of exponentialtime and probabilistic polynomialtime comp ..."
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Cited by 58 (6 self)
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Restricting the search space f0; 1g to the set of truth tables of "easy" Boolean functions on log n variables, as well as using some known hardnessrandomness tradeoffs, we establish a number of results relating the complexity of exponentialtime and probabilistic polynomialtime complexity classes. In particular, we show that NEXP ae P=poly , NEXP = MA; this can be interpreted as saying that no derandomization of MA (and, hence, of promiseBPP) is possible unless NEXP contains a hard Boolean function. We also prove several downward closure results for ZPP, RP, BPP, and MA; e.g., we show EXP = BPP , EE = BPE, where EE is the doubleexponential time class and BPE is the exponentialtime analogue of BPP.
The complexity of decision versus search
 SIAM Journal on Computing
, 1994
"... A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and nondeterministic doubleexponential time are unequal) we construct a language in NP for which search does not red ..."
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Cited by 33 (1 self)
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A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and nondeterministic doubleexponential time are unequal) we construct a language in NP for which search does not reduce to decision. These ideas extend in a natural way to interactive proofs and program checking. Under similar assumptions we present languages in NP for which it is harder to prove membership interactively than it is to decide this membership, and languages in NP which are not checkable. Keywords: NPcompleteness, selfreducibility, interactive proofs, program checking, sparse sets,
ResourceBounded Kolmogorov Complexity Revisited
 In Proceedings of the 14th Symposium on Theoretical Aspects of Computer Science
, 2001
"... We take a fresh look at CD complexity, where CD (x) is the size of the smallest program that distinguishes x from all other strings in time t(jxj). We also look at CND complexity, a new nondeterministic variant of CD complexity, and timebounded Kolmogorov complexity, denoted by C complexity. ..."
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Cited by 26 (7 self)
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We take a fresh look at CD complexity, where CD (x) is the size of the smallest program that distinguishes x from all other strings in time t(jxj). We also look at CND complexity, a new nondeterministic variant of CD complexity, and timebounded Kolmogorov complexity, denoted by C complexity.
When Worlds Collide: Derandomization, Lower Bounds, and Kolmogorov Complexity
 OF REDUCTIONS,IN“PROC.29THACM SYMPOSIUM ON THEORY OF COMPUTING
, 1997
"... This paper has the following goals:  To survey some of the recent developments in the field of derandomization.  To introduce a new notion of timebounded Kolmogorov complexity (KT), and show that it provides a useful tool for understanding advances in derandomization, and for putting vario ..."
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Cited by 19 (5 self)
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This paper has the following goals:  To survey some of the recent developments in the field of derandomization.  To introduce a new notion of timebounded Kolmogorov complexity (KT), and show that it provides a useful tool for understanding advances in derandomization, and for putting various results in context.  To illustrate the usefulness of KT, by answering a question that has been posed in the literature, and  To pose some promising directions for future research.
Unambiguous Computation: Boolean Hierarchies and Sparse TuringComplete Sets
, 1994
"... This paper studies, for UP, two topics that have been intensely studied for NP: Boolean hierarchies and the consequences of the existence of sparse Turingcomplete sets. Unfortunately, as is often the case, the results for NP draw on special properties of NP that do not seem to carry over straightfor ..."
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Cited by 17 (13 self)
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This paper studies, for UP, two topics that have been intensely studied for NP: Boolean hierarchies and the consequences of the existence of sparse Turingcomplete sets. Unfortunately, as is often the case, the results for NP draw on special properties of NP that do not seem to carry over straightforwardly to UP. For example, it is known for NP (and more generally for any class containing \Sigma and ; and closed under union and intersection) that the symmetric difference hierarchy, the Boolean hierarchy, and the Boolean closure all are equal. We prove that closure under union is not needed for this claim: For any class K that contains \Sigma and ; and is closed under intersection (e.g., UP, US, and FewP), the symmetric difference hierarchy over K, the Boolean hierarchy over K, and the Boolean closure of K all are equal. On the other hand, we show that two hierarchiesthe Hausdorff hierarchy and the nested difference hierarchy which in the NP case are equal to the Boolean cl...
On the Cutting Edge of Relativization: The Resource Bounded Injury Method
, 1994
"... In this report we present a new method of diagonalization that is a refinement of the wellknown finite injury priority method discovered independently by Friedberg and Muchnik in 1957. In the resource bounded injury method , it is necessary in addition to proving that the number injuries for a give ..."
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Cited by 8 (0 self)
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In this report we present a new method of diagonalization that is a refinement of the wellknown finite injury priority method discovered independently by Friedberg and Muchnik in 1957. In the resource bounded injury method , it is necessary in addition to proving that the number injuries for a given requirement is finite to carefully count these injuries and prove that this number does not exceed a bound given by the index of the requirement. The method is used to construct an oracle relative to which the polynomial time hierarchy collapses to an extent that the second level of this hierarchy (P NP A ) captures nondeterministic exponential time. This oracle is an answer to an open problem posed by Heller in 1984 that has thus far resisted existing methods and that has recently regained interest by work of Fu et. al. and by work of Homer and Mocas. Moreover, our oracle provides a constructive counterexample to Sewelson's conjecture that does not make use of information theoretical l...
The Complexity of Generating and Checking Proofs of Membership
, 1996
"... We consider the following questions: 1. Can one compute satisfying assignments for satisfiable Boolean formulas in polynomial time with parallel queries to NP? 2. Is the unique optimal clique problem (UOCLIQUE) complete for P NP[O(log n)] ? 3. Is the unique satisfiability problem (USAT) NP ha ..."
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Cited by 6 (3 self)
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We consider the following questions: 1. Can one compute satisfying assignments for satisfiable Boolean formulas in polynomial time with parallel queries to NP? 2. Is the unique optimal clique problem (UOCLIQUE) complete for P NP[O(log n)] ? 3. Is the unique satisfiability problem (USAT) NP hard? We define a framework that enables us to study the complexity of generating and checking proofs of membership. We connect the above three questions to the complexity of generating and checking proofs of membership for sets in NP and P NP[O(log n)] . We show that an affirmative answer to any of the three questions implies the existence of coNP checkable proofs for P NP[O(log n)] that can be generated in FP NP k . Furthermore, we construct an oracle relative to which there do not exist coNP checkable proofs for NP that are generated in FP NP k . It follows that relative to this oracle all of the above questions are answered negatively. 1
Two Oracles that Force a Big Crunch
, 1999
"... The central theme of this paper is the construction of an oracle A such that NEXP A = P NP A . The construction of this oracle answers a long standing open question rst posed by Heller, and unsuccessfully attacked many times since. For the rst construction of the oracle, we present a new ty ..."
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Cited by 5 (1 self)
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The central theme of this paper is the construction of an oracle A such that NEXP A = P NP A . The construction of this oracle answers a long standing open question rst posed by Heller, and unsuccessfully attacked many times since. For the rst construction of the oracle, we present a new type of injury argument that we call \resource bounded injury." In the special case of the construction of this oracle, a tree method can be used to transform unbounded search into exponentially bounded, hence recursive, search. This transformation of the construction can be interleaved with another construction so that relative to the new combined oracle also P = UP = NP\coNP. This leads to the curious situation where LOW(NP) = P, but LOW(P NP ) = NEXP, and the complete p m degree for P NP collapses to a single pisomorphism type. 1 Introduction In 1978, Seiferas, Fischer and Meyer [SFM78] showed a very strong separation theorem for nondeterministic time: For time constru...
Some Results on Derandomization
 Theory of Computing Systems
"... We show several results about derandomization including 1. If NP is easy on average then e#cient pseudorandom generators exist and P = BPP. ..."
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Cited by 2 (2 self)
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We show several results about derandomization including 1. If NP is easy on average then e#cient pseudorandom generators exist and P = BPP.
Avoiding Simplicity is Complex
"... Abstract. 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 resourcebounded Kolmogorov c ..."
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Cited by 1 (1 self)
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Abstract. 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 resourcebounded Kolmogorov complexity. This is the question considered here: Can a feasible set A avoid accepting strings of low resourcebounded Kolmogorov complexity, while still accepting some (or many) strings of length n? More specifically, this paper deals with two notions of resourcebounded 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 instance, EXP = ZPP if and only if there is a “dense ” set in P (i.e., a set that contains at least 2 n /n k strings of each length n, forsomek) that contains no strings x of length n with Kt(x) ≤ n ɛ,forsomeɛ>0 [4]. That is, the EXP vs ZPP question is equivalent to (one version of) the question of whether avoiding simple strings is difficult. Surprisingly, we are able to show unconditionally that avoiding simple strings (in the sense of KNt complexity) is difficult. Every dense set in NP ∩ coNP contains infinitely many strings x such that KNt(x) ≤x  ɛ for some ɛ. 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 bruteforce upper bound, if there are many accepting paths. 1