R. Beigel and J. Feigenbaum. On being incoherent without being very hard. Computational Complexity, 2(1):1--17, October 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....1. A = L(M;A) 2. for all x: x = 2 Q(M; x; A) Where Q(M; x; A) is the set of queries M generates on input x with A as an oracle. We say that a set A is r autoreducible if the machine M in fact is a r reduction from A to A. Note that the randomized version of autoreducibility, studied in [Yao90, BF92], is called coherence. Translating the notion of mitoticity to a polynomial time setting is less obvious. In [AS84] two definitions are given. 1. A recursive set A is polynomial time m(T) mitotic (m(T) mitotic for short) if there exists a set B 2 P such that: A j B j B. 2. A ....

....better understood. Surprisingly, the parallel with recursion theory disappears with respect to complete sets. Of course the same remarks about m complete sets that were made for mitoticity are true for autoreducibility but the T complete sets seem to behave differently. Theorem 4. 6 ( [BF92, BvHT93]) Every T complete set for NP is autoreducible. In fact in [BF92] it is shown that all T degrees that contain a self reducible set are completely autoreducible hence: Theorem 4.7 ( BF92] All T complete sets for all levels of the Polynomial Hierarchy and PSPACE are autoreducible. ....

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R. Beigel and J. Feigenbaum. On being incoherent without being very hard. Computational Complexity, 2(1):1--17, October 1992.

....of 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 ....

R. Beigel and J. Feigenbaum. On being incoherent without being very hard. Computational Complexity, 2:1--17, 1992.

....if x 62 A, then a string z such that R(x; z) does not exist . Hence such a string can not be output of the computation. 2 It is known that all sets in NP that are complete under P T reductions have SRTD [Sel92] Together with proposition 7, this gives the following corollary. Corollary 8 [BF92] If A is P T complete for NP then A is auto reducible This corollary is also interesting seen in the light that such a fact is plainly not true for r.e. sets, since Ladner [Lad73] has demonstrated T complete r.e. sets that are not mitotic and hence not auto reducible (for recursive ....

R. Beigel and J. Feigenbaum. On being incoherent without being very hard. Computational Complexity, 2(1):1--17, October 1992.

....sets and complete sets for some classes in the exponential time hierarchy. These autoreductions use game characterizations of classes creating a contest between a player trying to show a string x is in a set A and a player trying to show that x is not in A. Earlier Beigel and Feigenbaum [BF92] used a di erent technique to show that all of the Turing complete sets for PSPACE are autoreducible. 6 3.2 Intersecting Finite Automata Karakostas, Lipton and Viglas [KLV00] give an interesting approach to the L 6= NP problem by looking at the complexity of determining whether a collection of ....

R. Beigel and J. Feigenbaum. On being incoherent without being very hard. Computational Complexity, 2(1):1-17, 1992.

....both the unbounded and space bounded models. Ladner [10] showed that there exist Turing complete computably enumerable sets that are not autoreducible. Ambos Spies [1] rst transferred the notion of autoreducibility to the polynomial time setting. More recently, Yao [19] and Beigel and Feigenbaum [5] have studied a probabilistic variant of autoreducibility known as coherence. In this paper, we ask for what complexity classes do all the complete sets have the autoreducibility property. In particular we show: All Turing complete sets for EXP k are autoreducible for any constant k, ....

....technique of Theorem 3.3 also applies to EXP. In particular, we obtain: Theorem 3.6 There is a 6 P 3 tt complete set for EXP that is not 6 P btt autoreducible. 4 Autoreducibility Results For small complexity classes, all complete sets turn out to be autoreducible. Beigel and Feigenbaum [5] established this property of all levels of the polynomial time hierarchy as well as of PSPACE, the largest class for which it was known to hold before our work. In this section, we will prove it for the levels of the exponential time hierarchy. As to nonadaptive reductions, the question was ....

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R. Beigel and J. Feigenbaum. On being incoherent without being very hard. Computational Complexity, 2(1):1-17, 1992.

....concepts in complexity theory that are intimately related to checking. Two such concepts are coherence and random self reducibility. These concepts have been considered extensively in the literature. Definitions of these concepts and their relation to program checking can be found for example in [5, 14]. The rest of this paper is organized as follows: A more formal description of the program checking model is given in section 2. In section 3 we illustrate the concept with the prototypical example of the graph isomorphism problem. In section 4 we derive structural theorems which allow us to ....

R. Beigel and J. Feigenbaum. On Being Incoherent Without Being Very Hard. Computational Complexity 2 (1992), 1--17.

....and resource bounded models. Ladner [Lad73] showed that there existed Turing complete recursively enumerable sets that are not autoreducible. Ambos Spies [AS84] first transferred the notion of autoreducibility to the polynomial time settings. More recently, Yao [Yao90] and Beigel and Feigenbaum [BF92] have studied a probabilistic variant of autoreducibility known as coherence. In this paper, we ask for what complexity classes do all the complete sets have the autoreducibility property. In particular we show: ffl All Turing complete sets for EXPSPACE are autoreducible. ffl There exists a ....

....and will cause M(x) to accept. If x 62 L then B 0 will play according to a winning strategy for B and will cause M(x) to reject. 2 Similar though simpler proofs yield the following corollary: Corollary 3.4 All Turing complete sets for PSPACE and EXP are autoreducible. Beigel and Feigenbaum [BF92] had previously shown that Turing complete sets for PSPACE as well as all the levels of the polynomial time hierarchy are autoreducible. We can get more stronger autoreducibilities of complete sets if we allow nonuniformity, i.e. a polynomial amount of advice (see [KL82] that depends only on ....

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R. Beigel and J. Feigenbaum. On being incoherent without being very hard. Computational Complexity, 2:1--17, 1992.

....for short) if there exists a polynomial time oracle Turing machine M such that: 1. A = L(M;A) 2. for all x: x = 2 Q(M; x; A) Where Q(M; x; A) is the set of queries M generates on input x with A as an oracle. There also exists a randomized version of auto reducibility called coherence [Yao90, BF92]. Translating mitoticity [AS84] to the polynomial time setting gives a less clean transition. Definition 40 1. A recursive set A is polynomial time m(T) mitotic (m(T) mitotic for short) if there exists a set B 2 P such that: A j p m(T ) A T B j p m(T ) A T B. 2. A recursive set A is ....

....better understood. Surprisingly, the parallel with recursion theory disappears with respect to complete sets. Of course the same remarks about p m complete sets that were made for mitoticity are true for autoreducibility but the p T complete sets seem to behave differently. Theorem 44 ( [BF92, BT96]) Every p T complete set for NP is autoreducible. In fact in [BF92] it is shown that all p T degrees that contain a selfreducible set are completely autoreducible hence: Theorem 45 ( BF92] All p T complete sets for all levels of the Polynomial Hierarchy and PSPACE are ....

[Article contains additional citation context not shown here]

R. Beigel and J. Feigenbaum. On being incoherent without being very hard. Computational Complexity, 2(1):1--17, October 1992.

....theory and space bounded models. Ladner [Lad73] showed that there exist Turing complete computably enumerable sets that are not autoreducible. Ambos Spies [AS84] rst transferred the notion of autoreducibility to the polynomial time setting. More recently, Yao [Yao90] and Beigel and Feigenbaum [BF92] have studied a probabilistic variant of autoreducibility known as icoherence.j In this paper, we ask for what complexity classes do all the complete sets have the autoreducibility property. In particular we show: ffl All Turing complete sets for Delta EXP k are autoreducible for any constant ....

....technique of Theorem 3.3 also applies to EXP. In particular, we obtain: Theorem 3.6 There is a 6 P 3 Gammatt complete set for EXP that is not 6 P btt autoreducible. 4 Autoreducibility Results For small complexity classes, all complete sets turn out to be autoreducible. Beigel and Feigenbaum [BF92] established this property of all levels of the polynomial time hierarchy as well as of PSPACE, the largest class for which it was known to hold before our work. In this section, we will prove it for the Delta levels of the exponential time hierarchy. As to nonadaptive reductions, the question ....

[Article contains additional citation context not shown here]

R. Beigel and J. Feigenbaum. On being incoherent without being very hard. Computational Complexity, 2(1):117, 1992.

....notion of randomized self reductions, Yao [15] introduced coherence. A function is coherent if it can be computed by a probabilistic polynomial time Turing machine which can query the function on any point but the input. Every random selfreducible set is also coherent with polynomial advice [3]. It is natural to consider self reductions in which the queries can be made adaptively, that is, in several rounds, or nonadaptively (in parallel. A few results are known about the relationship between the various flavors of self reductions. Feigenbaum, Fortnow, Lund and Spielman [8] have shown ....

.... again the distribution would be skewed. To transform an arbitrary random self reduction into a coherent reduction, one may have to encode the value of the function on the (few) inputs on which the input is one of the queries. This is the main idea behind the proposition below. Proposition 1 [3] Every nonadaptively random selfreducible function is nonadaptively coherent with polynomial advice. For this reason, we compare random self reducibility with nonuniform coherence. 2.2 Multilinear extensions The low degree polynomial trick, or arithmetization is a standard way to construct ....

R. Beigel and J. Feigenbaum. On being incoherent without being very hard. Computational Complexity, 2:1--17, 1992.

....by NSF grant CCR 92 53582. z University of Amsterdam, Plantage Muidergracht 24, 1024 TV, Amsterdam. E mail: leen fwi.uva.nl. Partially supported by HC M grant nr. ERB4050PL93 0516. first transferred the notion of autoreducibility to the polynomial time settings. More recently, Yao and others [Yao90, BF92] have studied a probabilistic variant of autoreducibility known as coherence. In this paper, we ask for what complexity classes do all the complete sets have the autoreducibility property. In particular we show: ffl All Turing complete sets for EXPSPACE are autoreducible. ffl There exists a ....

R. Beigel and J. Feigenbaum. On being incoherent without being very hard. Computational Complexity, 2:1--17, 1992.

....theory and spacebounded models. Ladner [10] showed that there exist Turing complete computably enumerable sets that are not autoreducible. Ambos Spies [1] first transferred the notion of autoreducibility to the polynomial time setting. More recently, Yao [18] and Beigel and Feigenbaum [5] have studied a probabilistic variant of autoreducibility known as coherence. In this paper, we ask for what complexity classes do all the complete sets have the autoreducibility property. In particular we show: ffl All Turing complete sets for Delta EXP k are autoreducible for any constant ....

....technique of Theorem 3.3 also applies to EXP. In particular, we obtain: Theorem 3.6 There is a 6 P 3 Gammatt complete set for EXP that is not 6 P btt autoreducible. 4 Autoreducibility Results For small complexity classes, all complete sets turn out to be autoreducible. Beigel and Feigenbaum [5] established this property of all levels of the polynomial time hierarchy as well as of PSPACE, the largest class for which it was known to hold before our work. In this section, we will prove it for the Delta levels of the exponential time hierarchy. As to nonadaptive reductions, the question ....

[Article contains additional citation context not shown here]

R. Beigel and J. Feigenbaum. On being incoherent without being very hard. Computational Complexity, 2(1):1--17, 1992.

....that: 1. A = L(M;A) 2. for all x: x = 2 Q(M; x; A) Where Q(M; x; A) is the set of queries M generates on input x with A as an oracle. We say that a set A is r autoreducible if the machine M in fact is a r reduction from A to A. Note that the randomized version of autoreducibility, studied in [Yao90, BF92], is called coherence. Translating the notion of mitoticity to a polynomial time setting is less obvious. In [AS84] two definitions are given. Definition 4.2 1. A recursive set A is polynomial time m(T) mitotic (m(T) mitotic for short) if there exists a set B 2 P such that: A j P m(T ) A T ....

....better understood. Surprisingly, the parallel with recursion theory disappears with respect to complete sets. Of course the same remarks about P m complete sets that were made for mitoticity are true for autoreducibility but the P T complete sets seem to behave differently. Theorem 4. 6 ( [BF92, BvHT93]) Every P T complete set for NP is autoreducible. In fact in [BF92] it is shown that all P T degrees that contain a self reducible set are completely autoreducible hence: Theorem 4.7 ( BF92] All P T complete sets for all levels of the Polynomial Hierarchy and PSPACE are ....

[Article contains additional citation context not shown here]

R. Beigel and J. Feigenbaum. On being incoherent without being very hard. Computational Complexity, 2(1):1--17, October 1992.

....both the unbounded and space bounded models. Ladner [10] showed that there exist Turing complete computably enumerable sets that are not autoreducible. Ambos Spies [1] rst transferred the notion of autoreducibility to the polynomialtime setting. More recently, Yao [19] and Beigel and Feigenbaum [5] have studied a probabilistic variant of autoreducibility known as icoherence.j In this paper, we ask for what complexity classes do all the complete sets have the autoreducibility property. In particular we show: ffl All Turing complete sets for Delta EXP k are autoreducible for any constant ....

....technique of Theorem 3.3 also applies to EXP. In particular, we obtain: Theorem 3.6 There is a 6 P 3 Gammatt complete set for EXP that is not 6 P btt autoreducible. 4 Autoreducibility Results For small complexity classes, all complete sets turn out to be autoreducible. Beigel and Feigenbaum [5] established this property of all levels of the polynomial time hierarchy as well as of PSPACE, the largest class for which it was known to hold before our work. In this section, we will prove it for the Delta levels of the exponential time hierarchy. As to nonadaptive reductions, the question ....

[Article contains additional citation context not shown here]

R. Beigel and J. Feigenbaum. On being incoherent without being very hard. Computational Complexity, 2(1):117, 1992.

....as an oracle without querying the input. Buhrman et al. 1995) used the property of deterministic coherence (called autoreducibility) as a tool to separate complexity classes. It is known that all nonadaptively randomself reducible functions are nonadaptively coherent with polynomial sized advice (Beigel Feigenbaum 1992). Despite notable progress in our understanding of these topics (see Beigel Feigenbaum 1992, Feigenbaum Fortnow 1993, Feigenbaum et al. 1994, Buhrman et al. 1995) many complexity theoretic questions about random selfreducibility and coherence remain open. This paper examines two of them. We ....

....coherence (called autoreducibility) as a tool to separate complexity classes. It is known that all nonadaptively randomself reducible functions are nonadaptively coherent with polynomial sized advice (Beigel Feigenbaum 1992) Despite notable progress in our understanding of these topics (see Beigel Feigenbaum 1992, Feigenbaum Fortnow 1993, Feigenbaum et al. 1994, Buhrman et al. 1995) many complexity theoretic questions about random selfreducibility and coherence remain open. This paper examines two of them. We first address the power of adaptiveness and advice in coherence. Feigenbaum et al. 1994) ....

[Article contains additional citation context not shown here]

R. Beigel and J. Feigenbaum, On being incoherent without being very hard. Comput complexity 2 (1992), 1--17.

....querying the input. Buhrman, Fortnow, and Torenvliet [BFT95] used the property of deterministic coherence (called autoreducibility) as a tool to separate complexity classes. It is known that all nonadaptively random self reducible functions are nonadaptively coherent with polynomial sized advice [BF92]. Despite notable progress in our understanding of these topics [BF92, FF93, FFLS94, BFT95] many complexity theoretic questions about random self reducibility and coherence remain open. This paper examines two of them. We first address the power of adaptiveness and advice in coherence. Feigenbaum ....

....property of deterministic coherence (called autoreducibility) as a tool to separate complexity classes. It is known that all nonadaptively random self reducible functions are nonadaptively coherent with polynomial sized advice [BF92] Despite notable progress in our understanding of these topics [BF92, FF93, FFLS94, BFT95], many complexity theoretic questions about random self reducibility and coherence remain open. This paper examines two of them. We first address the power of adaptiveness and advice in coherence. Feigenbaum et al. FFLS94] showed that there is a random selfreducible function f that is not ....

[Article contains additional citation context not shown here]

R. Beigel and J. Feigenbaum. On being incoherent without being very hard. Computational Complexity, 2:1--17, 1992.

....input. Buhrman, Fortnow and Torenvliet [BFT95] have recently used the property of deterministic coherence (called autoreducibility) as a tool to separate complexity classes. It is known that all nonadaptively random self reducible functions are nonadaptively coherent with polynomial sized advice [BF92]. Despite notable progress in our understanding of these topics [BF92, FF93, FFLS94, BFT95] many complexity theoretic questions about random self reducibility and coherence remain open. This paper examines two of them. We rst address the power of adaptiveness and advice in coherence. Feigenbaum ....

....property of deterministic coherence (called autoreducibility) as a tool to separate complexity classes. It is known that all nonadaptively random self reducible functions are nonadaptively coherent with polynomial sized advice [BF92] Despite notable progress in our understanding of these topics [BF92, FF93, FFLS94, BFT95], many complexity theoretic questions about random self reducibility and coherence remain open. This paper examines two of them. We rst address the power of adaptiveness and advice in coherence. Feigenbaum et al. FFLS94] show that there is a random self reducible function f that is not ....

[Article contains additional citation context not shown here]

R. Beigel and J. Feigenbaum. On being incoherent without being very hard. Computational Complexity, 2:117, 1992.

....as an oracle without querying the input. Buhrman et al. 1995) used the property of deterministic coherence (called autoreducibility) as a tool to separate complexity classes. It is known that all nonadaptively randomself reducible functions are nonadaptively coherent with polynomial sized advice (Beigel Feigenbaum 1992). Despite notable progress in our understanding of these topics (see Beigel Feigenbaum 1992, Feigenbaum Fortnow 1993, Feigenbaum et al. 1994, Buhrman et al. 1995) many complexity theoretic questions about random selfreducibility and coherence remain open. This paper examines two of them. We ....

....coherence (called autoreducibility) as a tool to separate complexity classes. It is known that all nonadaptively randomself reducible functions are nonadaptively coherent with polynomial sized advice (Beigel Feigenbaum 1992) Despite notable progress in our understanding of these topics (see Beigel Feigenbaum 1992, Feigenbaum Fortnow 1993, Feigenbaum et al. 1994, Buhrman et al. 1995) many complexity theoretic questions about random selfreducibility and coherence remain open. This paper examines two of them. We first address the power of adaptiveness and advice in coherence. Feigenbaum et al. 1994) ....

[Article contains additional citation context not shown here]

R. Beigel and J. Feigenbaum, On being incoherent without being very hard. Computational complexity 2 (1992), 1--17.

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R. Beigel and J. Feigenbaum. On being incoherent without being very hard. Computational Complexity, 2(1):1--17, October 1992.

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R. Beigel and J. Feigenbaum, "On being incoherent without being very hard," Computational Complexity, Vol. 2, pp. 1--17, 1992. Response to questions of [13, 40], including a proof that all NP-complete languages are coherent.

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