## Cook versus Karp-Levin: Separating Completeness Notions If NP Is Not Small (1992)

Venue: | Theoretical Computer Science |

Citations: | 58 - 15 self |

### BibTeX

@INPROCEEDINGS{Lutz92cookversus,

author = {Jack H. Lutz and Elvira Mayordomo},

title = {Cook versus Karp-Levin: Separating Completeness Notions If NP Is Not Small},

booktitle = {Theoretical Computer Science},

year = {1992},

pages = {415--426},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

Under the hypothesis that NP does not have p-measure 0 (roughly, that NP contains more than a negligible subset of exponential time), it is show n that there is a language that is P T -complete ("Cook complete "), but not P m -complete ("Karp-Levin complete"), for NP. This conclusion, widely believed to be true, is not known to follow from P 6= NP or other traditional complexity-theoretic hypotheses. Evidence is presented that "NP does not have p-measure 0" is a reasonable hypothesis with many credible consequences. Additional such consequences proven here include the separation of many truthtable reducibilities in NP (e.g., k queries versus k+1 queries), the class separation E 6= NE, and the existence of NP search problems that are not reducible to the corresponding decision problems. This research was supported in part by National Science Foundation Grant CCR9157382, with matching funds from Rockwell International. 1 Introduction The NP-completeness of decision problems has...

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Citation Context ...nown formulations. These are the polynomial-time Turing completeness ( P T - completeness) introduced by Cook [5] and the polynomial-time many-one completeness ( P m -completeness) introduced by Karp =-=[8] and Levin [11]. The-=-se two completeness notions, sometimes called "Cook completeness" and "Karp-Levin completeness," have been widely conjectured, but not proven, to be distinct. The main purpose of t... |

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Citation Context ...ational. 1 Introduction The NP-completeness of decision problems has two principal, well-known formulations. These are the polynomial-time Turing completeness ( P T - completeness) introduced by Cook =-=[5] and the polyno-=-mial-time many-one completeness ( P m -completeness) introduced by Karp [8] and Levin [11]. These two completeness notions, sometimes called "Cook completeness" and "Karp-Levin complete... |

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Citation Context ...em of this paper, Theorem 4.1 below, says that the CvKL Conjecture follows from the hypothesis that "NP does not have pmeasure 0". This hypothesis, whose formulation involves resource-bounde=-=d measure [14, 13] (a comple-=-xity-theoretic generalization of Lebesgue measure) , is explained in detail in section 3 below. Very roughly speaking, the hypothesis says that "NP is not small," in the sense that NP contai... |

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Citation Context ...where b = [[s n 2 A]]. Condition (3.3) ensures that the betting is fair. The strategy succeeds on A if its capital is unbounded as the betting progresses. Martingales were used extensively by Schnorr =-=[20, 21, 22, 23]-=- in his investigation of random and pseudorandom sequences. Recently, martingales have been shown to characterize p-measure 0 sets: Theorem 3.5 ([14, 13]). A set X of languages has p-measure 0 if and ... |

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Citation Context ...t notsP m -complete, for NP. The CvKL conjecture immediately implies that P 6= NP, so it may be very difficult to prove. We mention five items of evidence that the conjecture is reasonable. 1. Selman =-=[24]-=- proved that the widely-believed hypothesis E 6= NE implies that the reducibilitiessP T andsP m are distinct in NP[co\GammaNP. That is, if DTIME(2 linear ) 6= NTIME(2 linear ), then there exist A,B 2 ... |

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Citation Context ...e of a surprisingly efficient algorithm 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 corre... |

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Citation Context ...ons. These are the polynomial-time Turing completeness ( P T - completeness) introduced by Cook [5] and the polynomial-time many-one completeness ( P m -completeness) introduced by Karp [8] and Levin =-=[11]. These two complete-=-ness notions, sometimes called "Cook completeness" and "Karp-Levin completeness," have been widely conjectured, but not proven, to be distinct. The main purpose of this paper is to... |

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Citation Context ... that everysP m - complete language for NP issP T -complete for NP. Conversely, all known, naturalsP T -complete languages for NP are alsosP m -complete. Nevertheless, it is widely conjectured (e.g., =-=[10, 29, 12, 6]) that Coo-=-k completeness is more general than Karp-Levin completeness: CvKL Conjecture. ("Cook versus Karp-Levin"). There exists a language that issP T -complete, but notsP m -complete, for NP. The Cv... |

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Citation Context ...e of a surprisingly efficient algorithm 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 corre... |

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Citation Context ... P-immune, it follows that A " T 62 P, whence oe(A) 62 EE. Thus oe(A) 2 NEE \Gamma EE, so EE 6= NEE. Note also that A " T is a tally language in NP \Gamma P. The existence of such a language=-= is known [3] to be equ-=-ivalent to E 6= NE. The proof of 2 is similar. 2 Theorem 3.10. 1. If NP does not have p-measure 0, then E 6= NE and EE 6= NEE. 2. If NP " co\GammaNP does not have p-measure 0, then E 6= NE "... |

41 | Measure, stochasticity, and the density of hard languages
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Citation Context ...ard language for NP is nonsparse (i.e., is not polynomially sparse). More recently, it has been proven that the �� p (NP) 6= 0 hypothesis yields a stronger conclusion: Theorem 3.8 (Lutz and Mayord=-=omo [15]-=-). If NP does not have p-measure 0, then for every real number ff ! 1 (e.g., ff = 0:99), everysP n ff \Gammatt -hard language for NP is dense. We conclude this section by noting some new consequences ... |

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Citation Context ...em of this paper, Theorem 4.1 below, says that the CvKL Conjecture follows from the hypothesis that "NP does not have pmeasure 0". This hypothesis, whose formulation involves resource-bounde=-=d measure [14, 13] (a comple-=-xity-theoretic generalization of Lebesgue measure) , is explained in detail in section 3 below. Very roughly speaking, the hypothesis says that "NP is not small," in the sense that NP contai... |

33 | The Complexity of Decision versus Search
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Citation Context ...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 ... |

33 |
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Citation Context ...is E 6= NE " co\GammaNE, Selman proved that the reducibilitiessP T andsP m are distinct in NP. 2. Ko and Moore [9] constructed a language that issP T -complete, but notsP m -complete, for E. Wata=-=nabe [26, 27]-=- refined this by separating a spectrum of completeness notions in E. 3. Watanabe and Tang [28] exhibited reasonable complexity-theoretic hypotheses implying the existence of languages that aresP T -co... |

25 |
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Citation Context ...with SAT performs exponentially badly (either by running for more than 2 jxj ffl steps or by failing to decide) on all but finitely many inputs x 2 K. (The weaker hypothesis P 6= NP was already known =-=[19] to -=-imply the weaker conclusion that everysP m -complete language for NP has a nonsparse polynomial complexity core). The third consequence of �� p (NP) 6= 0 to be mentioned here concerns the density ... |

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15 |
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