## The Complexity and Distribution of Hard Problems (1993)

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Venue: | SIAM JOURNAL ON COMPUTING |

Citations: | 48 - 18 self |

### BibTeX

@ARTICLE{Juedes93thecomplexity,

author = {David W. Juedes and Jack H. Lutz},

title = {The Complexity and Distribution of Hard Problems},

journal = {SIAM JOURNAL ON COMPUTING},

year = {1993},

volume = {24},

pages = {177--185}

}

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### Abstract

Measure-theoretic aspects of the P m -reducibility structure of the exponential time complexity classes E=DTIME(2 linear ) and E 2 = DTIME(2 polynomial ) are investigated. Particular attention is given to the complexity (measured by the size of complexity cores) and distribution (abundance in the sense of measure) of languages that are P m - hard for E and other complexity classes. Tight upper and lower bounds on the size of complexity cores of hard languages are derived. The upper bound says that the P m -hard languages for E are unusually simple, in the sense that they have smaller complexity cores than most languages in E. It follows that the P m -complete languages for E form a measure 0 subset of E (and similarly in E 2 ). This latter fact is seen to be a special case of a more general theorem, namely, that every P m -degree (e.g., the degree of all P m -complete languages for NP) has measure 0 in E and in E 2 .

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Citation Context ...1S 4P A language H is m-hard for a class C of languages if A P mH for all A 2C. A language P P C is m-complete for C if C 2Cand C is m-hard for C. IfC =NP, this is the usual notion of NP-completeness=-=[13]-=-. In this paper we are especially concerned with languages that are P P m-hard or m-complete for E or E2. 3 Resource-Bounded Measure Resource-bounded measure [17, 19] is a very general theory whose sp... |

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Citation Context ...reducible." (See section 2 for notation and terminology used in this introduction.) For example, in most usages, \NP-complete" means \ P m-complete for NP," the completeness notion introduced by Karp =-=[15]-=- and Levin [16]. This research was supported in part by National Science Foundation Grants CCR-8809238 and CCR9157382, with matching funds from Rockwell International and Microware Systems Corporation... |

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Citation Context ...) for the classes of languages decidable in 2linear time and 2polynomial time, respectively. Other complexity classes that we use here, suchasNP, PH, PSPACE, etc., have completely standard de nitions =-=[2, 3]-=-. P If A and B are languages, then a polynomial time, many-one reduction (brie y mreduction) of A to B is a function f 2 PF such thatA = f ;1(B) =fx j f(x) 2 Bg. A P P m-reduction of A is a function f... |

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Citation Context ...e in the sense of measure) of languages that are P m-hard for E (equivalently,E2) and other complexity classes, including NP. (By \measure" here, we mean resource-bounded measure as developed by Lutz =-=[17]-=- and described in section 3 of the present paper.) We giveatight lower bound and, perhaps surprisingly, atightupper bound on the sizes of complexity cores of hard languages. More generally, we analyze... |

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Citation Context ...e section 2 for notation and terminology used in this introduction.) For example, in most usages, \NP-complete" means \ P m-complete for NP," the completeness notion introduced by Karp [15] and Levin =-=[16]-=-. This research was supported in part by National Science Foundation Grants CCR-8809238 and CCR9157382, with matching funds from Rockwell International and Microware Systems Corporation, and in part b... |

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Citation Context ...ery such language has a non-sparse polynomial complexity core, though it is achieved at the cost of a stronger hypothesis. This hypothesis, originally proposed by Lutz, is discussed at some length in =-=[20, 22, 23]-=-. In section 5 we investigate the resource-bounded measure of the lower P m-spans, the upper P m-spans, and the P m-degrees of languages in E and E2. (The lower P m-span of A is the set of all languag... |

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Citation Context ...ery such language has a non-sparse polynomial complexity core, though it is achieved at the cost of a stronger hypothesis. This hypothesis, originally proposed by Lutz, is discussed at some length in =-=[20, 22, 23]-=-. In section 5 we investigate the resource-bounded measure of the lower P m-spans, the upper P m-spans, and the P m-degrees of languages in E and E2. (The lower P m-span of A is the set of all languag... |

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Citation Context ..." any element ofE.Very early, Berman [6] established limits to this interpretation by P proving that no m-complete language is P-immune, even though E contains P-immune languages. (In fact, Mayordomo =-=[25]-=- has recently shown that almost every language in E is P-bi-immune.) In section 6 below weprove avery strong limitation on the complexity P P of m-hard languages for E. We provethat every m-hard langu... |

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Citation Context ...) for the classes of languages decidable in 2linear time and 2polynomial time, respectively. Other complexity classes that we use here, suchasNP, PH, PSPACE, etc., have completely standard de nitions =-=[2, 3]-=-. P If A and B are languages, then a polynomial time, many-one reduction (brie y mreduction) of A to B is a function f 2 PF such thatA = f ;1(B) =fx j f(x) 2 Bg. A P P m-reduction of A is a function f... |

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Citation Context ...ery such language has a non-sparse polynomial complexity core, though it is achieved at the cost of a stronger hypothesis. This hypothesis, originally proposed by Lutz, is discussed at some length in =-=[20, 22, 23]-=-. In section 5 we investigate the resource-bounded measure of the lower P m-spans, the upper P m-spans, and the P m-degrees of languages in E and E2. (The lower P m-span of A is the set of all languag... |

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Citation Context ...e prove that P m-hard problems are rare, in the sense that they form a p-measure 0 set. We also prove thatevery P m-degree has measure 0 in exponential time. Complexity cores, rst introduced by Lynch =-=[24]-=- have been studied extensively [8, 9, 10, 11, 12, 14, 27, 28, 29, etc.]. Intuitively, a complexity core of a language A is a xed set K of inputs such that every machine whose decisions are consistent ... |

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Citation Context ...ly on all but nitely many elements of K. The meaning of \e ciently" is a parameter of the de nition that varies according to the context. (See section 4 for a precise de nition.) Orponen and Schoning =-=[28]-=- have established two lower bounds on the sizes of complexity cores of hard languages. First, every P m-hard language for E has a dense P-complexity core. Second, if P 6= NP, then every P m-hard langu... |

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Citation Context ...at is latter fact holds with P m replaced by P T would imply that E 6 BPP. P Languages that are m-hard for E are typically considered to be \at least as complex as" any element ofE.Very early, Berman =-=[6]-=- established limits to this interpretation by P proving that no m-complete language is P-immune, even though E contains P-immune languages. (In fact, Mayordomo [25] has recently shown that almost ever... |

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(Show Context)
Citation Context ...every element ofEis P m-reducible to A. A language A is weakly P m-hard for E if every element of some nonnegligible, i.e., non-measure 0, set of languages in E is reducible to A. Very recently, Lutz =-=[21]-=- has proven that \weakly P m-hard" is more general than \ P m-hard.") Speci cally, we prove that every language that P is weakly m-hard for E or E2 has a dense exponential complexity core. It follows ... |

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9 |
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(Show Context)
Citation Context ... Zi. It follows by Lemma 3.3 that p(P ;1 m (A)) = (P;1 m (A) E) = 0. This completes the proof of 1. The proof of 2 is identical. One need only note that, if A 2 E2, then d 2 p2. 2 Remark. Ambos-Spies =-=[1]-=- has shown that Pm(A) has Lebesgue measure 0 whenever A 62 P. Lemma 5.2 obtains a stronger conclusion (resource-bounded measure 0) from a stronger hypothesis on A. It is now straightforward to derive ... |

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2 |
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Citation Context ... to fewer questions \f(x) 2 B?" If A is incompressible by mreductions, then very little such compression can occur. Our rst observation, an obvious generalization of a result of Balcazar and Schoning =-=[4]-=- (see Corollary 4.2 below), relates incompressibility to complexity cores. Lemma 4.1. If t : N ! N is time constructible, then every language that is incompressible by -reductions has f0� 1g as a DTIM... |

2 | A classi cation of complexity core lattices - Orponen - 1986 |