## In Search of an Easy Witness: Exponential Time vs. Probabilistic Polynomial Time

### Cached

### Download Links

- [www.math.ias.edu]
- [www-cse.ucsd.edu]
- [publications.ias.edu]
- DBLP

### Other Repositories/Bibliography

Citations: | 55 - 6 self |

### BibTeX

@MISC{Impagliazzo_insearch,

author = {Russell Impagliazzo and Valentine Kabanets and Avi Wigderson},

title = {In Search of an Easy Witness: Exponential Time vs. Probabilistic Polynomial Time},

year = {}

}

### Years of Citing Articles

### OpenURL

### Abstract

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 hardness-randomness tradeoffs, we establish a number of results relating the complexity of exponential-time 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 promise-BPP) 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 double-exponential time class and BPE is the exponential-time analogue of BPP.

### Citations

2429 | Computational complexity
- Papadimitriou
- 1994
(Show Context)
Citation Context ...and open problems are given in Section 7. 2 Preliminaries 2.1 Complexity Classes We assume that the reader is familiar with the standard complexity classes such as P, NP, ZPP, RP, and BPP (see, e.g., =-=[Pap94]-=-). We will need the two exponential-time deterministic complexity classes E = DTIME(2 O(n) ) and EXP = DTIME(2 poly(n) ), and their nondeterministic analogues NE and NEXP. We dene SUBEXP = \ >0 DTIM... |

529 | Theory and applications of trapdoor functions - Yao - 1982 |

407 | Non-deterministic exponential time has two-prover interactive protocols
- Babai, Fortnow, et al.
- 1991
(Show Context)
Citation Context ...exity of NEXP. 11 4.1 NEXP versus MA Babai, Fortnow, and Lund [BFL91, Corollary 6.10], based on an observation by Nisan, improved a result of Albert Meyer [KL82] by showing the following. Theorem 22 (=-=[BFL91]-=-). EXP P=poly) EXP = MA. Here we will prove Theorem 23. NEXP P=poly, NEXP = MA. Buhrman and Homer [BH92] proved that EXP NP P=poly ) EXP NP = EXP 4 , but left open the question whether NEXP P=... |

317 | Arthur-merlin games: A randomized proof system, and a hierarchy of complexity classes - Babai - 1988 |

304 | Trading group theory for randomness - Babai - 1985 |

282 | Hardness vs. Randomness - Nisan, Wigderson - 1988 |

225 | How to go beyond the black-box simulation barrier - Barak - 2001 |

225 | On the (im)possibility of obfuscating programs - Barak, Goldreich, et al. - 2001 |

180 |
P = BPP if E requires exponential circuits: derandomizing the XOR lemma
- Impagliazzo, Wigderson
- 1997
(Show Context)
Citation Context ...results are known which are based on the assumption that EXP contains hard Boolean functions, i.e., those of \high" circuit complexity [NW94, BFNW93, ACR98, IW97, STV99]. For instance, it is shown in =-=[IW97]-=- that BPP = P if DTIME(2 O(n) ) contains a language that requires Boolean circuits of size 2 (n) . Results of this form, usually called hardnessrandomness tradeos, are proved by showing that the trut... |

174 | Natural proofs
- Razborov, Rudich
- 1997
(Show Context)
Citation Context ... for Circuit Approximation and Natural Properties In this section, we present two implications of our Theorem 23 for the problem of circuit approximation and natural properties of Razborov and Rudich =-=[RR97]-=-. In Section 5.1, we show that (for nondeterministic Turing machines with sublinear amount of advice) if the problem of circuit approximation can be solved eciently at all, then it can also be solved... |

128 | Pseudorandom generators without the XOR lemma - Sudan, Trevisan, et al. |

125 | Modern Cryptography, Probabilistic Proofs and Pseudorandomness
- Goldreich
- 1998
(Show Context)
Citation Context ...onclusion is immediate. 2.4 Pseudorandom Generators and Conditional Derandomization For more background on pseudorandom generators and derandomization, the reader is referred to the book by Goldreich =-=[Gol99]-=-, as well as the surveys by Miltersen [Mil01] and Kabanets [Kab02]. A generator is a function G : f0; 1g ! f0; 1g which maps f0; 1g l(n) to f0; 1g n , for some function l : N ! N; we are intereste... |

114 | BPP has subexponential time simulations unless EXPTIME has publishable proofs
- Babai, Fortnow, et al.
- 1993
(Show Context)
Citation Context ...losure results. Very few such results are known. For example, Impagliazzo and Naor [IN88] prove that P = NP ) DTIME(polylog(n)) = NTIME(polylog(n)) \ coNTIME(polylog(n)) = RTIME(polylog(n)); see also =-=[BFNW93]-=- and [HIS85]. We prove several downward closure results for probabilistic complexity classes. Along the way, we also obtain \gap" theorems for the complexity of BPE, ZPE, and MA. Note: Fortnow [For01]... |

108 | Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses
- Klivans, Melkebeek
- 1999
(Show Context)
Citation Context ...ment (1) above are such that, for every d 2 N and innitely many n 2 N, f n has circuit complexity greater than n d , then, for every > 0, MA io-[NTIME(2 n )=a(n )]. Klivans and Van Melkebeek =-=[KM99]-=- show that a quick SIZE SAT (n)-pseudorandom generator G : f0; 1g n ! f0; 1g n allows one to simulate every language in AM in nondeterministic time 2 n k , for some k 2 N. Thus, if the truth tables... |

73 | Randomness vs. time: De-randomization under a uniform assumption
- Impagliazzo, Wigderson
- 1998
(Show Context)
Citation Context ...lish the following Theorem 38. EXP = BPP, EE = BPE. 5 Usually, by an explicit Boolean function, one means a function in NP. 16 Our proof will rely on the following result by Impagliazzo and Wigderson =-=[IW98]-=- on the derandomization of BPP under a uniform hardness assumption. Theorem 39 ([IW98]). Suppose that EXP 6= BPP. Then, for every binary language L 2 BPP and every > 0, there is a deterministic 2 n ... |

64 | Derandomizing arthur-merlin games using hitting sets
- Miltersen, Vinodchandran
- 1999
(Show Context)
Citation Context .... Thus, if the truth tables of Boolean functions of superpolynomial SAT-oracle circuit complexity can be generated nondeterministically in time polynomial in their length, then AM NSUBEXP (see also =-=[MV99]-=- for derandomization of AM under weaker assumptions). More precisely, we have the following. Theorem 13 (following [KM99]). 1. Suppose there is a poly(2 n )-time algorithm which, given an advice strin... |

51 |
Turing machines that take advice. L’Enseignement Mathématique
- Karp, Lipton
- 1982
(Show Context)
Citation Context ...heorems relating uniform and nonuniform complexity of NEXP. 11 4.1 NEXP versus MA Babai, Fortnow, and Lund [BFL91, Corollary 6.10], based on an observation by Nisan, improved a result of Albert Meyer =-=[KL82]-=- by showing the following. Theorem 22 ([BFL91]). EXP P=poly) EXP = MA. Here we will prove Theorem 23. NEXP P=poly, NEXP = MA. Buhrman and Homer [BH92] proved that EXP NP P=poly ) EXP NP = EXP 4 ... |

50 |
On sparse sets
- Hartmanis
- 1983
(Show Context)
Citation Context ...s. Very few such results are known. For example, Impagliazzo and Naor [IN88] prove that P = NP ) DTIME(polylog(n)) = NTIME(polylog(n)) \ coNTIME(polylog(n)) = RTIME(polylog(n)); see also [BFNW93] and =-=[HIS85]-=-. We prove several downward closure results for probabilistic complexity classes. Along the way, we also obtain \gap" theorems for the complexity of BPE, ZPE, and MA. Note: Fortnow [For01] gives much ... |

43 | Nonrelativizing separations
- Buhrman, Fortnow, et al.
- 1998
(Show Context)
Citation Context ...) contradict Corollary 8. Corollary 25. If NEXP P=poly, then NEXP = MA. Proof. If NEXP P=poly, then NEXP = EXP by Theorem 24, and EXP = MA by Theorem 22. Remark 26. Buhrman, Fortnow, and Thierauf =-=[BFT98]-=- show that MA-E 6 P=poly. Combined with a simple padding argument, their result yields the following implication: MA = NP) NEXP 6 P=poly. Our Corollary 25 is a signicant strengthening of this impli... |

40 | Easiness assumptions and hardness tests: Trading time for zero error
- Kabanets
- 2000
(Show Context)
Citation Context ...similar downward closures for ZPP, RP, and MA. 1 Main Techniques One of the main ideas that we use to derive our results can be informally described as the \easy witness" method, invented by Kabanets =-=[Kab01]-=-. It consists in searching for a desired object (e.g., a witness in a NEXP search problem) among those objects that have concise descriptions (e.g., truth tables of Boolean functions of low circuit co... |

36 |
A new general derandomization method
- Andreev, Clementi, et al.
- 1998
(Show Context)
Citation Context ... : ; r t , and tries f x (r i ) for each i. (If G is a pseudo-random generator, thesnal output is the majority of the bits f x (r i ). Other constructions such as the hitting-set derandomization from =-=[ACR98]-=- are more complicated, but have the same general form.) In particular, 1. We never use the acceptance probability guarantees for the algorithm on other inputs. Thus, we can derandomize algorithms even... |

33 |
Decision trees and downward closures
- Impagliazzo, Naor
- 1988
(Show Context)
Citation Context ...g that a collapse of higher complexity classes implies a collapse of lower complexity classes are known as downward closure results. Very few such results are known. For example, Impagliazzo and Naor =-=[IN88]-=- prove that P = NP ) DTIME(polylog(n)) = NTIME(polylog(n)) \ coNTIME(polylog(n)) = RTIME(polylog(n)); see also [BFNW93] and [HIS85]. We prove several downward closure results for probabilistic complex... |

27 | Another proof that BPP⊆PH (and more
- Goldreich, Zuckerman
- 1997
(Show Context)
Citation Context ...ed in [Yao82, NW94], a quick SIZE(n)-pseudorandom generator G : f0; 1g n ! f0; 1g n allows one to simulate every BPP algorithm in deterministic time 2 n k , for some k 2 N. Goldreich and Zuckerman =-=[GZ97]-=- show that a quick SIZE(n)-pseudorandom generator G : f0; 1g n ! f0; 1g n allows one to decide every MA language in nondeterministic time 2 n k , for some k 2 N. Thus, if we can \eciently" generat... |

21 | Superpolynomial circuits, almost sparse oracles and the exponential hierarchy
- Buhrman, Homer
- 1992
(Show Context)
Citation Context ...n by Nisan, improved a result of Albert Meyer [KL82] by showing the following. Theorem 22 ([BFL91]). EXP P=poly) EXP = MA. Here we will prove Theorem 23. NEXP P=poly, NEXP = MA. Buhrman and Homer =-=[BH92]-=- proved that EXP NP P=poly ) EXP NP = EXP 4 , but left open the question whether NEXP P=poly) NEXP = EXP. Resolving this question is the main step in our proof of Theorem 23. Theorem 24. If NEXP ... |

19 |
Decision versus search problems in super-polynomial time
- Impagliazzo, Tardos
- 1989
(Show Context)
Citation Context ...e". Thus, apparently, the assumption NEXP = EXP does not suce to conclude that every NEXP search problem is solvable in deterministic time 2 poly(n) . The following theorem of Impagliazzo and Tardos =-=[IT89]-=- gives some evidence to this eect. Theorem 28 ([IT89]). There is an oracle relative to which NEXP = EXP, and yet there is a NEXP search problem that cannot be solved deterministically in less than do... |

15 | Comparing notions of full derandomization
- Fortnow
(Show Context)
Citation Context ...BFNW93] and [HIS85]. We prove several downward closure results for probabilistic complexity classes. Along the way, we also obtain \gap" theorems for the complexity of BPE, ZPE, and MA. Note: Fortnow =-=[For01]-=- gives much simpler proofs of the downward closures presented in this section. However, our techniques also allow us to establish the gap theorems that do not seem to follow from [For01]. 6.1 Case of ... |

13 |
Derandomizing complexity classes
- Miltersen
- 2001
(Show Context)
Citation Context ...rators and Conditional Derandomization For more background on pseudorandom generators and derandomization, the reader is referred to the book by Goldreich [Gol99], as well as the surveys by Miltersen =-=[Mil01]-=- and Kabanets [Kab02]. A generator is a function G : f0; 1g ! f0; 1g which maps f0; 1g l(n) to f0; 1g n , for some function l : N ! N; we are interested only in the generators with l(n) < n. For a... |

12 | Efficiently approximable real-valued functions
- Kabanets, Rackoff, et al.
- 2000
(Show Context)
Citation Context ...ce, MA can be derandomized), then NEXP requires superpolynomial circuit size. Thus, hard Boolean functions are also required for derandomizing promise-RP, promise-BPP, and the class APP introduced in =-=[KRC00]-=-. We would like to point out which of our theorems relativize, and which do not. It follows from the results in [BFT98] that the collapse of NEXP to MA when NEXP P=poly (Corollary 25) does not relat... |

12 | Derandomizing Arthur-Merlin games under uniform assumptions
- Lu
(Show Context)
Citation Context ...er assumption, we show that AM = NP. The same conclusion is known to hold under certain nonuniform hardness assumptions [KM99, MV99], and the assumption that NP is hard in a certain \uniform" setting =-=[Lu01]-=-. Theorem 19. If NE \ coNE 6 io-DTIME(2 2 n ) for some > 0, then AM = NP. Proof. Consider all pairs (R + ; R ) of polynomial-time decidable relations dened on f0; 1g n f0; 1g 2 n such that f ... |

11 |
Derandomization: A brief overview. Bulletin of the EATCS
- Kabanets
- 2002
(Show Context)
Citation Context ...l Derandomization For more background on pseudorandom generators and derandomization, the reader is referred to the book by Goldreich [Gol99], as well as the surveys by Miltersen [Mil01] and Kabanets =-=[Kab02]-=-. A generator is a function G : f0; 1g ! f0; 1g which maps f0; 1g l(n) to f0; 1g n , for some function l : N ! N; we are interested only in the generators with l(n) < n. For any oracle A, we say t... |