M. Bellare and S. Goldwasser. The complexity of decision versus search. SIAM Journal on Computing, 23:97--119, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....If A is complete for any of the classes Sigma i , i 0, then A is coherent. In particular, all NP complete sets are coherent. These results first appeared in our two technical reports [6, 7] They were presented in preliminary form in [5] together with the results of Bellare and Goldwasser [8]. 2. History Trakhtenbrot [27] calls a set A autoreducible if A is Turing reducible to A via an oracle Turing machine that never queries the oracle about the input string. He considered time and space bounded versions of autoreducibility as well. Lynch, Meyer, and Fischer [23] considered ....

....answer to the question of Blum and Kannan [10] The other half is provided by Babai, Fortnow, and Lund [4] who show that there is a checkable set that is not in IP, unless EXP = PSPACE. We were the first to use sparse sets in the study of checkability [6, 7] Subsequently, Bellare and Goldwasser [8] used sparse sets in the direct construction of an uncheckable set B in NP, under the weaker assumption that NEE 6 BPEE. EE denotes the union, over all positive constants c, of DTIME(2 cn ) NEE and BPEE denote the analogous nondeterministic and bounded probabilistic time classes. Note that ....

M. Bellare and S. Goldwasser, The Complexity of Decision versus Search, MIT-LCS Technical Memorandum, TM-444, April, 1991.

....membership under the proof system. A remarkable property of NP complete languages is the fact that they are all self reducible, and hence their search versions are reducible by Cook reductions to their decision versions. Various refinements can be studied in this connection; see the papers [BF92, BG94, HNOS96, FFNR96] for several fine results in this direction. Being an (apparently) inherently sequential process with multiple queries, the task of reducing search to decision for NP languages offers a natural ground in which to compare the power of Cook reductions with that of the weaker ....

M. Bellare and S. Goldwasser. The complexity of decision versus search. SIAM Journal on Computing, 23(1):97--119, 1994.

....framework for proving strong lower bounds on the parameterized complexity of problems within FPT. To achieve a 2 o(k) p(n) lower bound, it su#ces to prove that a problem is MAX SNP hard and that the witness version nicely reduces to the decision version. As mentioned by Bellare and Goldwasser [6], it is well known that search is polynomial time Turing reducible to decision for every NP complete problem. To obtain Corollary 4.2, we require a more restrictive notion of reducibility between the witness and decision versions of parameterized problems. In particular, we require that the ....

Mihir Bellare and Shafi Goldwasser. The complexity of decision versus search. SIAM Journal on Computing, 23(1):97--119, February 1994.

....transformation from interactive proofs to zero knowledge proofs. The definition of a robust transformation does not require that the original verifier strategy be accessed only as a black box. The complexity of the prover in interactive proofs was previously studied by Bellare and Goldwasser [BG94] They showed that, under a complexity theoretic assumption, there is a problem in NP 3 which is harder to prove than it is to decide; that is, no interactive proof for has a prover which can be implemented in polynomial time given an oracle for deciding . 2 The main result Our proof of ....

....is, construct a problem with a private coin interactive proof (P; V ) such that has no public coin interactive proof where the prover can be implemented in polynomial time with oracle access to P . Presumably such a result would be under an intractability assumption. Bellare and Goldwasser [BG94] have given results of this nature for a different issue, namely separating the power needed to decide a language from the power needed to prove membership. 10 While we have considered the problems of converting interactive proofs to ones with public coins or perfect completeness, there are ....

Mihir Bellare and Shafi Goldwasser. The complexity of decision versus search. SIAM Journal on Computing, 23(1):97--119, February 1994.

....from more traditional hypotheses. Among these are the following. NP contains a P bi immune problem (Mayordomo [65] E #= NE and EE #= NEE (Lutz and Mayordomo [63] There is an NP search problem that does not reduce to the corresponding decision problem (Bellare and Goldwasser [12], Lutz and Mayordomo [63] Every problem that is # P m hard for NP has a dense exponential complexity core (Juedes and Lutz [41] There is a problem that is # P T complete (in fact, # P 2 T complete) but not # P m complete, for NP (Lutz and Mayordomo [63] For every k ....

M. Bellare and S. Goldwasser. The complexity of decision versus search. SIAM Journal on Computing, 23:97--119, 1994.

.... double exponential time disagree (and, by using still sparser sets in NP, similar separations for larger time classes can be obtained) The hypothesis EE 6= NEE in term implies the existence of an NP search problem which does not reduce to the corresponding decision problem (Bellare and Goldwasser [BeG94]) The Genericity Hypothesis suffices to distinguish many of the standard polynomial time reducibilities on NP: For n 3 generic A 2 NP, the separations in (4.3) with the possible exception of Pm (A) 6= P btt(1) A) are witnessed by sets in NP. So, by (the proof of) Theorem 4.12, p 1 , ....

M. Bellare and S. Goldwasser. The complexity of decision versus search. SIAM Journal on Computing 23 (1994) 97-119.

....for betting on all NP languages. Second, the hypothesis has a rapidly growing body of credible consequences. We summarize recently discovered such consequences [16, 7, 15] and prove two new consequences, namely the class separation E 6= NE and (building on recent work of Bellare and Goldwasser [1]) the existence of NP search problems that are not reducible to the corresponding decision problems. In section 4 we prove our Main Theorem. In section 5, we prove that, if NP is not small, then many truth table reducibilities are distinct in NP. Taken together, our results suggest that NP does ....

....0, then E 6= NE co GammaNE and EE 6= NEE co GammaNEE. Proof. This follows immediately from Theorem 3.6 and Lemma 3.9. 2 Corollary 3.11. If NP does not have p measure 0, then there is an NP search problem that does not reduce to the corresponding decision problem. Proof. Bellare and Goldwasser [1] have shown that, if EE 6= NEE, then there is an NP search problem that does not reduce to the corresponding decision problem. The present corollary follows immediately from this and Theorem 3.10. 2 4 Separating Completeness Notions in NP In this section we prove our main result, that the CvKL ....

M. Bellare and S. Goldwasser, The complexity of decision versus search, SIAM Journal on Computing 23 (1994), pp. 97--119.

.... every P n ff tt hard language for NP (ff 1) is exponentially dense [26] and every P m hard language for NP has an exponentially dense, exponentially hard complexity core [17] there is an NP search problem that is not efficiently reducible to the corresponding decision problem [4, 25]; there are problems that are P T complete, but not P m complete, for NP[25] and every P tt hard language for NP is p superterse[3, 32] Since fiNP NP, the hypothesis (fiNP j E 2 ) 6= 0 implies the hypothesis (NP j E 2 ) 6= 0. There does not appear to be any a priori reason for ....

M. Bellare and S. Goldwasser. The complexity of decision versus search. SIAM Journal on Computing, 23:97--119, 1994.

....) and NEE denotes the corresponding nondeterministic class) 1. E 6= NE and EE 6= NEE (Mayordomo [30] Lutz and Mayordomo [26] 2. There exist NP search problems which are not reducible to the corresponding decision problems (this follows from item 1 above and a result of Bellare and Goldwasser [1]) 3. Every P m complete language for NP contains a dense exponential complexity core (Juedes and Lutz [11] 4. For every real number ff 1, every P n ff Gammatt hard language for NP is dense (Lutz and Mayordomo [27] Our main result is that relative to a random oracle, NP does not ....

M. Bellare and S. Goldwasser. The complexity of decision versus search. To appear in SIAM Journal on Computing.

....E 2 = DTIME(2 polynomial ) the smallest deterministic time complexity class known to contain NP. For example, if p (NP) 6= 0, it is now known that NP contains P bi immune languages [25] there is an NP search problem that is not efficiently reducible to the corresponding decision problem [3, 23]; every P n ff Gammatt complete problem for NP (ff 1) is exponentially dense [22] every P m complete problem for NP has an exponentially dense exponential complexity core [6] and there are problems that are P T complete, but not P m complete, for NP [23] These conclusions, ....

M. Bellare and S. Goldwasser. The complexity of decision versus search. SIAM Journal on Computing, 23:97--119, 1994.

....the classes Sigma p i , Pi p i , or Delta p i , i 0, then A is coherent. In particular, all NP complete sets are coherent. These results first appeared in our two technical reports [6, 7] They were presented in preliminary form in [5] together with the results of Bellare and Goldwasser [8]. 2. History Trakhtenbrot [27] calls a set A autoreducible if A is Turing reducible to A via an oracle Turing machine that never queries the oracle about the input Incoherence 3 string. He considered time and space bounded versions of autoreducibility as well. Lynch, Meyer, and Fischer [23] ....

....answer to the question of Blum and Kannan [10] The other half is provided by Babai, Fortnow, and Lund [4] who show that there is a checkable set that is not in IP, unless EXP = PSPACE. We were the first to use sparse sets in the study of checkability [6, 7] Subsequently, Bellare and Goldwasser [8] used sparse sets in the direct construction of an uncheckable set B in NP, under the weaker assumption that NEE 6 BPEE. EE denotes the union, over all positive constants c, of DTIME(2 2 cn ) NEE and BPEE denote the analogous nondeterministic and bounded probabilistic time classes. Note ....

M. Bellare and S. Goldwasser, The Complexity of Decision versus Search, MIT-LCS Technical Memorandum, TM-444, April, 1991. 16 Beigel & Feigenbaum

....of interesting complexity theoretic conclusions have been derived, which are not known to follow from P 6= NP. Two prominent examples of such results are: there are Turing complete sets for NP that are not many one complete [15] there are NP problems for which search does not reduce to decision [15, 7]. Recently, Lutz [14] has shown that the hypothesis p (NP) 6= 0 (in fact, the possibly weaker hypothesis p ( Delta P k ) 6= 0, k 2) implies that BP Delta Delta P k = Delta P k (in other words, BP Delta Delta P k can be derandomized) This has an improved lowness consequence: it ....

M. Bellare and S. Goldwasser. The complexity of decision versus search. SIAM Journal on Computing, 23:97--119, 1994.

.... double exponential time disagree (and, by using still sparser sets in , similar separations for larger time classes can be obtained) The hypothesis = in term implies the existence of an search problem which does not reduce to the corresponding decision problem (Bellare and Goldwasser [BeG94]) The Genericity Hypothesis suffices to distinguish many of the standard polynomial time reducibilities on : For generic , the separations in (4.3) with the possible exception of ( are witnessed by sets in . So, by (the proof of) Theorem 4.12, for 2, and are mutually different on ....

M. Bellare and S. Goldwasser. The complexity of decision versus search. 23 (1994) 97-119.

....of search reduces to decision for L to interactive proof systems. We call a prover natural if it is an oracle for L, and we define a natural proof system to be one in which all honest provers are natural. Natural proof systems are similar to competitive proof systems defined in [BBFG91, BG] If a language L has a natural interactive proof system how much interaction is required We show under assumption NE 6 BPE that there is a language L 2 NP Gamma BPP such that L has a natural proof system but L does not have any 1 round natural proof system even if we permit a polynomial ....

....reduction that computes f relative to a set L 2 NP A . Recall that g A (x) is a set of queries Q = fq 1 ; q 2 ; q k g (suitably encoded as a string) and that f(x) e A (x; L(q 1 ) L(q 2 ) L(q k ) Define code(e A ) fhw; i; bi j the i th bit of e A (w) is bg; and define code(g A ) fhw; i; j; bi j the i th bit of the j th element of g A (w) is bg; where w is a string in f0; 1g , i 1, j 1, and b 2 f0; 1g. Since g A and e A are in PF A , code(g A ) 2 P A and code(e A ) 2 P A . Thus, the following language L 0 = code(g ....

[Article contains additional citation context not shown here]

M. Bellare and S. Goldwasser. The complexity of decision versus search. SIAM J. Comput. In press.

....(Note that the prover in Construction 1 has this property. This interpretation generalizes the notion of self reducibility of NP sets. By self reducibility of an NP set we mean that the search problem of finding an NP witness is polynomial time reducible to deciding membership in the set. See [12]. 3. A prover is considered relatively efficient if it can be implemented by a probabilistic machine which runs in time which is polynomial in the deterministic complexity of the set. This interpretation relates the difficulty of convincing a lazy verifier to the complexity of finding the truth ....

M. Bellare and S. Goldwasser. The Complexity of Decision versus Search. SIAM Journal on Computing, Vol. 23, pages 97--119, 1994.

....12.5 (Lutz and Mayordomo [LM] If NP does not have pmeasure 0, then E 6= NE and EE 6= NEE. Corollary 12.6 (Lutz and Mayordomo [LM] If NP does not have pmeasure 0, then there is an NP search problem that does not reduce to the corresponding decision problem. Proof. Bellare and Goldwasser [BG94] have shown that, if EE 6= NEE, then there is an NP search problem that does not reduce to the corresponding decision problem. The present corollary follows immediately from this and Theorem 12.5. 2 We now consider complexity cores of languages that are P m hard for NP. The following result is ....

M. Bellare and S. Goldwasser. The complexity of decision versus search. SIAM Journal on Computing, 23:97--119, 1994.

....of search reduces to decision for L to interactive proof systems. We call a prover natural if it is an oracle for L, and we define a natural proof system to be one in which all honest provers are natural. Natural proof systems are similar to competitive proof systems defined in [BBFG91, BG94] If a language L has a natural interactive proof system how much interaction is required We show under the assumption NE 6 BPE that there is a language L 2 NP Gamma BPP such that L has a natural proof system but L does not have any 1 round natural proof system even if we permit a polynomial ....

....Tradeoffs in multiprover interactive proof systems Now we turn to interactive proof systems and show that our techniques are applicable to this area. We assume readers are familiar with interactive proof systems [GMR89, Bab85] and with multiprover interactive proof systems [BOGKW88] In [BBFG91, BG94] the authors consider competitive proof systems. An interactive proof system for a language L is competitive if the honest prover is a probabilistic polynomial time oracle Turing machine that accesses the language L. We consider a variation of this model. Define a prover to be natural if it is an ....

M. Bellare and S. Goldwasser. The complexity of decision versus search. SIAM J. Comput., 23(1):97--119, 1994.

....(Note that the prover in Construction 1 has this property. This interpretation generalizes the notion of self reducibility of NP sets. By self reducibility of an NP set we mean that the search problem of finding an NP witness is polynomial time reducible to deciding membership in the set. See [10]. 3. A prover is considered relatively efficient if it can be implemented by a probabilistic machine which runs in time which is polynomial in the deterministic complexity of the set. This interpretation relates the difficulty of convincing a lazy verifier to the complexity of finding the truth ....

M. Bellare and S. Goldwasser. The Complexity of Decision versus Search. SIAM Journal on Computing, Vol. 23, pages 97--119, 1994.

....generalizes the notion of self reducibility of NP languages. By self reducibility of an NP language we mean that the search problem of finding an NP witness is Cook reducible to deciding membership in the language. Thus, every NP complete language has a relatively efficient proof system. See [16]. 3rd interpretation: A prover is considered relatively efficient if it can be implemented by a probabilistic machine that runs in time that is polynomial in the deterministic complexity of the language. This interpretation relates the difficulty of convincing a lazy verifier to the complexity ....

....Hierarchy. Open Problem 3: Regarding subsection 2.1.3, it will be interesting to understand the limitations of relatively efficient provers according to the 2nd and 3rd interpretations. A better understanding of the self reducibility on NP languages is also long due. A specific challenge (cf. [16]) provide an NP proof system for Quadratic Non Residucity (QNR) using a probabilistic polynomial time prover with access to QNR language. Open Problem 4: Try to provide firm grounds for the heuristics of making proof systems noninteractive by use of random public functions (cf. 34, 63] I ....

M. Bellare and S. Goldwasser. The Complexity of Decision versus Search. SIAM J. on Computing, Vol. 23, No. 1, pages 97--119, 1994.

....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 under the assumption that Sigma 2 6= Pi 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 proofs [BGKW] and coherence [Ya] Here we discuss ....

M. Bellare and S. Goldwasser. The Complexity of Decision versus Search. MITLCS Technical Memo. TM-444, April 1991.

....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] 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 proofs [BGKW] and coherence [Ya] Here we discuss ....

M. Bellare and S. Goldwasser. The Complexity of Decision versus Search. MITLCS Technical Memo. TM-444, April 1991.

....ae such that L ae 2 NP Gamma P but still search does not reduce to decision for ae. With even stronger assumptions one can show there is an A 2 NP Gamma P such that search does not reduce to decision for any of the many possible NP relation ae for which A = L ae . See for example reference [1]. ....

M. Bellare and S. Goldwasser. The complexity of decision versus search. SIAM J. on Computing, Vol. 23, No. 1, February 1994.

....Supported in part by NSF grant No. CCR 8657527, DARPA grant No. DAAL03 86 K 0171 and grant No. 89 00312 from the United States Israel Binational Science Foundation (BSF) This paper is a combination of the results by Beigel and Feigenbaum[BF] and Bellare and Goldwasser [BG]. Some proofs have been omitted because of space limitations. 1 Introduction The work on interactive proofs brought back to light an old question: how powerful does a prover need to be to convince a verifier of membership in a language L Clearly, the prover needs at least the power to decide ....

M. Bellare and S. Goldwasser. The Complexity of Decision versus Search, MIT-LCS Technical Memo. TM-444 (April 1991).

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M. Bellare and S. Goldwasser. The complexity of decision versus search. SIAM Journal on Computing, 23:97--119, 1994.

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M. Bellare and S. Goldwasser. The complexity of decision versus search. SIAM Journal on Computing, 23:97-119, 1994.

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