## The complexity of decision versus search (1994)

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Venue: | SIAM Journal on Computing |

Citations: | 32 - 1 self |

### BibTeX

@ARTICLE{Bellare94thecomplexity,

author = {Mihir Bellare and Shafi Goldwasser},

title = {The complexity of decision versus search},

journal = {SIAM Journal on Computing},

year = {1994},

volume = {23},

pages = {97--119}

}

### Years of Citing Articles

### OpenURL

### Abstract

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 non-deterministic double-exponential 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: NP-completeness, self-reducibility, interactive proofs, program checking, sparse sets,

### Citations

479 |
The complexity of theoremâ€“ proving procedures
- Cook
- 1971
(Show Context)
Citation Context ...e that defines L, it is the case that search reduces to decision for ae (a consequence of the "self-reducibility" and NP-completeness of SAT as well as certain features of the proof of Cook'=-=s theorem [Co], thi-=-s fact is one of the most basic and well-known ones in the theory of computation). In particular, by our definition, search certainly reduces to decision for any NP-complete language. So the "har... |

405 | Nondeterministic Exponential Time has Two-Prover Interactive Protocols
- Babai, Fortnow, et al.
- 1991
(Show Context)
Citation Context ...a statistical ZK interactive proof possesses one with a prover who is a probabilistic, polynomial time machine with access to an NP oracle. In the case of multi-prover proofs, Babai, Fortnow and Lund =-=[BFL] show that-=- exponential time provers suffice for exponential time languages. We stress that all these works are concerned with upper bounding the complexity of provers in an "absolute" sense. The model... |

308 | Algebraic methods for interactive proof systems
- Lund, Fortnow, et al.
- 1992
(Show Context)
Citation Context ...r's result [Sh] implies that polynomial space provers suffice to prove PSPACE languages. The best upper bound on the complexity of a prover of a coNP language, due to Lund, Fortnow, Karloff and Nisan =-=[LFKN]-=-, is probabilistic, polynomial time with a #P oracle. Bellare and Petrank [BP] investigate the complexity of zero-knowledge (ZK) provers, and indicate that such provers can be reasonably efficient; sp... |

305 | Designing Programs that Check Their Work
- Blum, Kannan
- 1995
(Show Context)
Citation Context ...uired to hold with respect to any (computationally unbounded) prover. 4 1.3 Program Checking We briefly mention our results on program checking that are in the same vein as the above. Blum and Kannan =-=[BK]-=- introduced the notion of program checkers (see Section 5 for full definitions) . Negative results in this domain begin with Yao [Ya] who presented a language in deterministic space 2 n log log n that... |

240 |
A comparison of polynomial time reducibilities
- LADtqER, LYNCH, et al.
(Show Context)
Citation Context ... fact that search does not reduce to decision for L implies that L is not NP-complete. The existence of a non NP-complete language in NP \Gamma P can however be established assuming only P 6= NP (cf. =-=[La]-=-). 3 Deciders and their Properties Before extending the ideas of the previous section to interactive proofs and checking, we pause to develop some technical materiel. This materiel will be useful in p... |

193 | An Introduction to the Theory of Numbers - Niven, Zuckerman - 1991 |

131 | On the Power of Multiprover Interactive Protocols
- Fortnow, Rompel, et al.
- 1994
(Show Context)
Citation Context ...es are independent of the history). As we will see (cf. Lemma 4.3) it is the case that compIP ` frIP. Blum and Kannan also established that Check ` frIP. On the other hand, based on the techniques of =-=[FRS]-=-, one can show that frIP ` MIP. Function-restricted interactive proofs were introduced in order to relate program checking to interactive proofs. We introduce competitive interactive proofs to address... |

108 | On the theory of average case complexity
- Ben-David, Chor, et al.
- 1992
(Show Context)
Citation Context .... The usual reduction of search to decision has a strong sequential flavor, and Karp, Upfal and Wigderson [KUW] investigated the degree to which this is necessary. Ben-David, Chor, Goldreich and Luby =-=[BCGL] investiga-=-te the "decision versus search" question in the context of average-case complexity. Impagliazzo and Tardos [IT] consider the decision versus search question in the exponential case, and pres... |

107 |
The complexity of promise problems with applications to public-key cryptography
- Even, Selman, et al.
- 1984
(Show Context)
Citation Context ... with a "promise" that the input already belongs to some set, and we have only to "decide" whether or not it falls in a given subset of this set. Such problems have been considered=-= in many works; eg. [ESY]-=-. The formalization we use is different from (but equivalent to) the ones used in these works, and is as follows. The problem is specified by a pair of disjoint sets (A; B). Intuitively, the input is ... |

88 |
The Knowledge Complexity of Interactive Proof
- Goldwasser, Micali, et al.
- 1989
(Show Context)
Citation Context ... is also sufficient for almost all of the languages in IP that have been closely examined (specifically, the languages of graph isomorphism, graph non-isomorphism [GMW], and quadratic non-residuosity =-=[GMR]-=-). On the other hand all known interactive proofs for complete languages for coNP require the prover to do more than decide membership in the language. Similarly, all known interactive proofs for the ... |

83 |
Relative complexity of checking and evaluation
- Valiant
- 1976
(Show Context)
Citation Context ...NP-languages. So, in that sense, the conclusions are weaker than ours. Let us now describe some of this work. In what follows we let ae denote a polynomial time computable binary predicate. 5 Valiant =-=[Va] appe-=-ars to have been the first to indicate that there are NP-relations ae for which search is unlikely to reduce to decision; specifically, assuming P 6= NP " coNP, he presents a particular NP-relati... |

38 |
personal communication
- Impagliazzo, Naor, et al.
- 1998
(Show Context)
Citation Context ...e of studying the complexity of provers in a different way; namely, in terms of the "comparative" complexity of proving versus deciding. Recent Work. Independently of this work, Impagliazzo =-=and Sudan [IS]-=- show that if NE 6= coNE then there is a language in NP for which search does not reduce to decision. Here the conclusion 6 is the same as in Theorem 1.1, but the assumption is different (and not know... |

33 |
Decision trees downward closure
- Impagliazzo, Naor
- 1988
(Show Context)
Citation Context ...for which L ae 2 NP \Gamma P but search still does not reduce to decision for ae. Hartmanis and Hemachandra [HH] present results similar to Valiant's but assuming P 6= UP " coUP. Impagliazzo and =-=Naor [IN] indi-=-cate that, at least in relativized worlds, the assumption P 6= NP " coNP is not necessary for the conclusion of Valiant's result. More precisely, they present a relativized world in which P = NP ... |

26 |
Coherent functions and program checkers
- Yao
- 1990
(Show Context)
Citation Context ...m checking that are in the same vein as the above. Blum and Kannan [BK] introduced the notion of program checkers (see Section 5 for full definitions) . Negative results in this domain begin with Yao =-=[Ya]-=- who presented a language in deterministic space 2 n log log n that does not have a checker. Beigel and Feigenbaum [BF] and Krawczyk [Kr] improved this to deterministic space n log n . The question of... |

25 |
Proofs that Yield Nothing but their Validity
- Goldreich, Micali, et al.
- 1991
(Show Context)
Citation Context ...embership in the language. Such power is also sufficient for almost all of the languages in IP that have been closely examined (specifically, the languages of graph isomorphism, graph non-isomorphism =-=[GMW]-=-, and quadratic non-residuosity [GMR]). On the other hand all known interactive proofs for complete languages for coNP require the prover to do more than decide membership in the language. Similarly, ... |

24 |
The knowledge complexity ofinteractive proofs
- Goldwasser, Micali, et al.
- 1989
(Show Context)
Citation Context ...wer is also su cient for almost all of the languages in IP that have been closely examined (speci cally, the languages of graph isomorphism, graph non-isomorphism [GMW], and quadratic non-residuosity =-=[GMR]-=-). On the other hand all known interactive proofs for complete languages for coNP require the prover to do more than decide membership in the language. Similarly, all known interactive proofs for the ... |

21 |
The complexity of parallel search
- Karp, Upfal, et al.
- 1988
(Show Context)
Citation Context ... search problem for ae to its corresponding decision. Decision versus Search in other settings. The usual reduction of search to decision has a strong sequential flavor, and Karp, Upfal and Wigderson =-=[KUW] investiga-=-ted the degree to which this is necessary. Ben-David, Chor, Goldreich and Luby [BCGL] investigate the "decision versus search" question in the context of average-case complexity. Impagliazzo... |

20 |
Decision versus search problems in super-polynomial time
- Impagliazzo, Tardos
- 1989
(Show Context)
Citation Context ...d the degree to which this is necessary. Ben-David, Chor, Goldreich and Luby [BCGL] investigate the "decision versus search" question in the context of average-case complexity. Impagliazzo a=-=nd Tardos [IT]-=- consider the decision versus search question in the exponential case, and present an oracle relative to which E = NE but there is an exponential time binary predicate whose search problem is not solv... |

18 | Making Zero-Knowledge Provers Efficient
- Bellare, Petrank
- 1992
(Show Context)
Citation Context ...nguages. The best upper bound on the complexity of a prover of a coNP language, due to Lund, Fortnow, Karloff and Nisan [LFKN], is probabilistic, polynomial time with a #P oracle. Bellare and Petrank =-=[BP]-=- investigate the complexity of zero-knowledge (ZK) provers, and indicate that such provers can be reasonably efficient; specifically, they show that any language possessing a statistical ZK interactiv... |

18 | Some comments on functional self-reducibility and the NP hierarchy - Borodin, Demers - 1976 |

17 |
Natural self-reducible sets
- Selman
- 1988
(Show Context)
Citation Context ...for ae. The assumption P 6= NP suffices to indicate that the "usual" method of self-reduction (where one constructs a witness bit-by-bit, given an oracle for the language) may not always wor=-=k: Selman [Se]-=- shows that under this assumption there is a NP-relation ae for which L ae 2 NP \Gamma P but, given a pair of strings (x; u) and an oracle for L ae , it is impossible to decide in polynomial time whet... |

16 | Limitations of the upward separation technique
- Allender
- 1991
(Show Context)
Citation Context ... L 0 2 NEE \Gamma EE. Define L = fy:0 g(jyj)\Gammajyj : y 2 L 0 g, where g(k) = 2 2 k . We claim that L is uniformly log-sparse and L 2 NP \Gamma P. y [HSI] claimed the converse as well, but Allender =-=[Al]-=- points out that their proof is flawed and the theorem cannot be proved using techniques that relativize. 10 Define A to be the algorithm which on input x 2 f0; 1g n behaves as follows. If n is not in... |

13 | Languages that are easier than their proofs
- Beigel, Bellare, et al.
- 1991
(Show Context)
Citation Context ...d an uncheckable set in \Sigma P 2 under the assumption that \Sigma E 2 6= \Pi E 2 . Publication Notes. These results appeared in a preliminary form in [BG]. Later, merged with [BF], they appeared in =-=[BBFG]-=-. 1.6 Relations to other Notions We focus in this paper on (competitive) interactive proofs and checking. Related notions are function-restricted interactive proofs [BK], multi-prover interactive proo... |

10 | Lecture notes on the complexity of some problems in number the6 In particular, the idea of multi-prover interactive proofs of Benor, Goldwasser, Kilian, and Wigderson [15] (which has become better known as probabilistic checkable proofs) has led to a rice - Angluin - 1982 |

6 |
Multiprover Interactive Proof Systems: How to Remove Intractability Assumptions
- Ben-Or, Goldwasser, et al.
(Show Context)
Citation Context ....6 Relations to other Notions We focus in this paper on (competitive) interactive proofs and checking. Related notions are function-restricted interactive proofs [BK], multi-prover interactive proofs =-=[BGKW]-=- and coherence [Ya]. Here we discuss how these notions relate to ours and also how our results impinge on them. First, let us list the (complexity classes corresponding to) the notions in this area. T... |

4 |
Achieving zero-knowledge robustly
- Kilian
- 1990
(Show Context)
Citation Context ...ter conclusion may be established with stronger assumptions such as the existence of "ideal" secure circuit evaluation or the existence of "oblivious transfer." This follows from t=-=he result of Kilian [Ki]-=- (and we refer the reader to that paper for details on what exactly are these assumptions). All statistical ZK languages known to possess competitive interactive proofs are also known to possess stati... |

4 |
Fast Checkers for Cryptography
- Kompella, Adleman
- 1990
(Show Context)
Citation Context ... ; QNR s ) in the sense of Definition 6.2. Theorem 6.3 Let s be an integers1. Then the promise problem (QR s ; QNR s ) possesses a competitive interactive proof. In related work, Kompella and Adleman =-=[KA]-=- present checkers for this same special case of quadratic residuosity when the modulus is the product of a constant number of primes (i.e. they present checkers for the promise problem (QR s ; QNR s )... |

4 |
Making ZeroKnowledge Provers E cient
- Bellare, Petrank
- 1992
(Show Context)
Citation Context ...languages. The best upper bound on the complexity of a prover of a coNP language, due to Lund, Fortnow, Karlo and Nisan [LFKN], is probabilistic, polynomial time with a #P oracle. Bellare and Petrank =-=[BP]-=- investigate the complexity of zero-knowledge (ZK) provers, and indicate that such provers can be reasonably e cient; speci cally, they show that any language possessing a statistical ZK interactive p... |

1 | Complexity Classes without Machines - Hartmanis, Hemachandra - 1988 |

1 |
Private communication via Beigel and Feigenbaum
- Spielman
- 1991
(Show Context)
Citation Context ...on as in the above-mentioned result of Borodin and Demers [BD], but under an assumption which is different from that of [BD] (but, again, not known to be either weaker or stronger). Finally, Spielman =-=[Sp]-=- has constructed an uncheckable set in \Sigma P 2 under the assumption that \Sigma E 2 6= \Pi E 2 . Publication Notes. These results appeared in a preliminary form in [BG]. Later, merged with [BF], th... |

1 |
The Complexity ofParallel Search
- Karp, Upfal, et al.
- 1988
(Show Context)
Citation Context ...e the search problem for to its corresponding decision. Decision versus Search in other settings. The usual reduction of search to decision has a strong sequential avor, and Karp, Upfal and Wigderson =-=[KUW] inve-=-stigated the degree to which this is necessary. Ben-David, Chor, Goldreich and Luby [BCGL] investigate the \decision versus search" question in the context of average-case complexity. Impagliazzo... |