Results 1  10
of
18
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 complexity cla ..."
Abstract

Cited by 55 (5 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 32 (1 self)
 Add to MetaCart
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. ..."
Abstract

Cited by 25 (7 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 19 (14 self)
 Add to MetaCart
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...
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 ..."
Abstract

Cited by 18 (5 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
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 ecient pseudorandom generators exist and P = BPP. 2. If NP is easy on average then given an NP machine M we can easily on average nd accepting computations of M(x) when it accepts. 3. For any A in EXP, if NEXP ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
We show several results about derandomization including 1. If NP is easy on average then ecient pseudorandom generators exist and P = BPP. 2. If NP is easy on average then given an NP machine M we can easily on average nd accepting computations of M(x) when it accepts. 3. For any A in EXP, if NEXP A is in P A =poly then NEXP A = EXP A . 4. If A is p k complete then NEXP A = EXP = MA A . 1
Coding Complexity: The Computational Complexity of Succinct Descriptions
, 1996
"... For a given set of strings, the problem of obtaining a succinct description becomes an important subject of research, related to several areas of theoretical computer science. In structural complexity theory, researchers have developed a reasonable framework for studying the complexity of these prob ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
For a given set of strings, the problem of obtaining a succinct description becomes an important subject of research, related to several areas of theoretical computer science. In structural complexity theory, researchers have developed a reasonable framework for studying the complexity of these problems. In this paper, we survey how such investigation has proceeded, and explain the current status of our knowledge.