A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Mathematical Systems Theory, 13:55--65, 1979.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....complexity, but they become computationally tractable when a small amount of extra advice (in the sense of Karp and Lipton [33] is available for each word length. For example, only one advice bit per word length is needed to decide a tally set. A less trivial example are P selective sets [53], which can be decided by an NP machine [30] if n 1 advice bits are available for the words of length n, and by a P machine [34] if n bits are available. Much research has focused on the question of whether sets with low information content can be helpful in deciding natural computationally ....

....of SAC is closed under complementation [16] and this closure property holds for the logspace uniform SAC as well. For de nitions of polynomialtime truth table reductions and Turing reduction see [37] for the logspace counterparts see [36] The notion of P selectivity is due to Selman [53], while the notion of membership comparability is due to Ogihara [46] Both notions can readily be generalized to logarithmic space. De nition 13 ( 53] A selector for a language A is a binary function g such that for all x; y 2 1. g(x; y) 2 fx; yg, 2. if x 2 A or y 2 A, then g(x; y) 2 A. ....

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A. L. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Math. Systems Theory, 13:55-65, 1979. 16

....two could be demonstrated. So far, as many papers state, these hopes have been in vain. On the way, many notions from recursion theory have been transferred to the world of complexity theory, and the flurry of notions that was originated by Post s program has not escaped this vogue. A.L. Selman [Sel79] introduced both the resource bounded version of semirecursiveness, called p selectivity, and the notion of polynomial time positive 1 reducibility (in [Sel82] which translated Jockush s [Joc66] positive reducibility. In his paper [Sel82] Selman showed that the combination of positive truth ....

....there exists a polynomial time bounded oracle machine M , such that A = L(M;B) Selman (after Jockusch) extended this notion to: A language A is polynomial time positive Turing reducible to a language B pos B) iff A T B via a positive oracle machine M . Selman introduced p selective sets in [Sel79]. A set A is called p selective if there exists a polynomial time computable function f : Sigma Theta Sigma 7 Sigma , called a p selector , such that for any x; y 2 Sigma both f(x; y) 2 fx; yg and A (f(x; y) maxfA (x) A (y)g 3 Main Result This section is dedicated to ....

A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Math. Systems Theory, 13:55-- 65, 1979.

.... (and the associated semi recursive sets those sets having recursive semi membership algorithms) were introduced in the 1960s by Jockush [Joc79] Deterministic polynomialtime semi membership algorithms (and the associated P selective or semi feasible sets) were introduced in the 1970s by Selman [Sel79] Nondeterministic polynomial time semimembership algorithms, both partial and total (and the associated NPSV p selective and NP selective sets) were introduced in the 1990s by, respectively, Hemaspaandra, Naik, Ogihara, and Selman [HNOS94] and Hemaspaandra, Hoene, Ogiwara, Selman, Thierauf, ....

....Theorem 3.16] 3 Reductions to P Selective Sets, and Two Puzzling Results In the sense of Definition 2.2, consider the PF single Gammavalued; total selective sets. This class is commonly referred to as the P selective sets (and we will also use this shorthand) and was first studied by Selman [Sel79] In Section 4 of this paper, we will survey a line of research asking what consequences follow if, e.g. NP has a bounded truth table hard P selective set. It is now known (see Theorem 4.6) that this hypothesis implies P = NP. However, arriving at that result took a long time. On the other hand, ....

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A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Mathematical Systems Theory, 13:55--65, 1979.

....questions are studied in Section 3. It seems that, for the questions we are interested in, the structure of the sparse set is more important than its computational complexity. Therefore, in a part of this section we study structurally simple rather than computationally simple sparse sets. Selman [Sel79] introduced p selective sets as a resource bounded analog of semi recursive sets introduced by Jockusch [Joc68] For any tally set, a polynomial time Turing equivalent p selective set can be found. Therefore p selective sets can be computationally very complex, Nonetheless, p selective sets are ....

....a sparse set that destroys completeness which is also polynomial time computable does not exist. To show this we take a little more general view on the complexity of the set S. Instead of taking S polynomial time computable, we let S be p selective. p selective sets were introduced by Selman [Sel79] as a resource bounded analog of semi recursive sets, introduced by Jockusch [Joc68] Definition 7 A set A is called p selective iff there exists a polynomial time computable function f : Sigma Theta Sigma 7 Sigma , called a p selector, such that for any x; y 2 Sigma 9 1. f(x; y) ....

A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Math. Systems Theory, 13:55--65, 1979.

....1 x 2 , y 1 y 2 , i) i# (f(x 1 , y 1 ) f(x 2 , y 2 ) 1) f(x i , y i ) 1. Selection is equivalent to elimination where you are forced to eliminate one of 01, 10 . The complexity of enumeration, elimination, and selection has been studied in the context of both polynomial time [1, 2, 10, 16, 17, 25, 30, 48, 50, 49, 51] and computability theory [8, 7, 22, 26, 32] Let i k. Clearly D(ENUM(2 , f iD(f) Alice and Bob can transmit iD(f) bits to compute b 1 b 2 b i = f x i , y 1 y 2 y i ) and output the set of strings b 1 b 2 as candidates. We state (for the first time) the following ....

A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Mathematical Systems Theory, 13:55--65, 1979.

....A(x)A(y) Thus 1 cheatable sets are also a subset of 2membership comparable sets. We show that if P = NP, then every 1 cheatable set is 1 tt reducible to a p selective set. Thus showing 1 cheatable sets form a much larger class than p selective sets would be dicult. 1. 1 Previous Work Selman [Sel79] de ned p selective sets as a polynomial time analogue of semirecursive sets [Joc68] Beigel [Bei87a] while studying bounded query classes de ned k membership comparable sets using the terminology of approximable and superterse. A set is k membership comparable if and only if it is ....

....[Ogi95] showed that if A is f(n) truth table reducible to a p selective set, then A is ( 1 ) log f(n) membership comparable for any 0. From now, we write k mc sets for k membership comparable sets. Membership comparable sets p selective sets and are studied extensively in the literature [Sel79, Sel82, Ko83, Bei87a, Bei87b, Bei88, ABG90, Tod91, BvHT96, BKS95, Ogi95, AA96, HNOS96]. Ko [Ko83] showed that p selective sets have polynomial size circuits. This result was improved by Amir, Beigel, and Gasarch [ABG90] who showed poly mc sets have polynomial size circuits. Membership comparable sets and p selective sets have played an important role in understanding the structure ....

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A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Mathematical Systems Theory, 13:55-65, 1979.

....to R is the mapping that takes a given string x, on condition x 2 L, to some (all) y such that R(x; y) holds. The question of whether fR can be computed in polynomial time by a deterministic machine with L as oracle was studied implicitly by Borodin and Demers [BD76] and explicitly by Selman [Sel79] and others in recent years [BBFG91, NOS93] The quasilinear time analogue of this idea is: Definition 5 Given a language L, say that search reduces to decision for L in quasilinear time if there exists a quasilinear witness predicate R for L and a quasilinear time bounded oracle TM M such that ....

....above equivalence extends our results in the last section to the notion of self 1 helping. In particular, Corollary 9 can now be reformulated as: Corollary 15 Let A 2 NP. If A is a self 1 helper in quasilinear time, then A belongs to quasi polynomial time. We also note that a lemma of Selman [Sel79] carries over for quasilinear time reductions. Lemma 16 If L 1 and L 2 are such that L 1 j ql m L 2 and search reduces to decision in quasilinear time for L 1 , then search reduces to decision in quasilinear time for L 2 . Proof. Let g 1 and g 2 be QL functions such that L 1 ql m L 2 via g ....

A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Math. Sys. Thy., 13:55--65, 1979.

....computable function that, given k(n) instances of A of length at most n, excludes one of the 2 k(n) possibilities for the memberships of the given strings in A. The study of membership comparability has two main motivations: 1) Membership comparability generalizes the notion of p selectivity [Sel79] and is closely related to truth table reducibilities to p selective sets; and (2) Membership comparability is closely related to various notions in the study of query complexity of the characteristic function of the k fold product of a set with itself (see, for example, BKS94] A set A is ....

.... to truth table reducibilities to p selective sets; and (2) Membership comparability is closely related to various notions in the study of query complexity of the characteristic function of the k fold product of a set with itself (see, for example, BKS94] A set A is said to be p selective [Sel79] if there is a polynomial time computable function f such that for all x; y, f(x; y) 2 fx; yg and (x 2 A y 2 A) f(x; y) 2 A. Informally, the selector function tells which of its two input strings is more likely to be a string in A. At first glance, p selectivity appears to be a notion of ....

A. Selman. P-selective sets, tally languages, and behavior of polynomial time reducibilities on NP. Math. Systems Theory, 13:55--65, 1979.

....sparse sets. For example, Watanabe [Wat90] showed R P tt (SELECT) 6= R P tt (SPARSE) see [HHO 93] for more separations) Hence, it is interesting to investigate the consequences of NP being reducible to P selective sets with respect to some more restrictive type of reducibility. Selman [Sel79] showed that if every NP set is many one reducible to some Pselective set then P = NP. Assuming that NP sets are (unbounded) truth table reducible to P selective sets, Toda [Tod91] and Beigel [Bei88] showed that NP problems can be solved efficiently by randomized Las Vegas type algorithms, a ....

....the many one reducibility. For any class C of languages, let R P T (C) R P tt (C) R P b(n) tt (C) and R P btt (C) respectively denote the class of sets that are P T , P tt , P b(n) tt , and P btt reducible to some set in C. P selective sets were introduced by Selman [Sel79] as the polynomial time analog of semi recursive sets [Joc68] A set A is P selective, if there exists a polynomial time computable function f , called a P selector for A, such that for all x; y 2 Sigma , 1. f(x; y) 2 fx; yg, and 2. if x 2 A or y 2 A, then f(x; y) 2 A. Intuitively, f selects ....

A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Mathematical Systems Theory, 13:55--65, 1979.

...., i) # SELECT(f 2 ) i# (f(x 1 , y 1 ) 1 # f(x 2 , y 2 ) 1) # f(x i , y i ) 1. Selection is equivalent to elimination where you are forced to eliminate one of 01, 10 . The complexity of enumeration, elimination, and selection has been studied in the context of both polynomial time [1, 2, 10, 16, 17, 25, 30, 48, 50, 49, 51] and computability theory [8, 7, 22, 26, 32] Let i # k. Clearly D(ENUM(2 k i , f k ) # iD(f) Alice and Bob can transmit iD(f) bits to compute b 1 b 2 b i = f i (x 1 x 2 x i , y 1 y 2 y i ) and output the set of strings b 1 b 2 b i 0, 1 k i as candidates. We ....

A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Mathematical Systems Theory, 13:55--65, 1979.

....languages for languages which have a recursive selector. The class of semirecursive languages, denoted by REC[sel] in the following, plays a key role in the solution of Post s Problem [19] 1 Electronic Colloquium on Computational Complexity, Report No. 77 (2000) ISSN1433 8092 Selman [22] introduced P selective languages. Such languages have a polynomial time computable selector. The class of P selective languages will be denoted by P[sel] see De nitions 9 and 14 for the reasons for this notation. Selman proved that the satis ability problem is not P selective, unless P = NP. ....

....see De nitions 9 and 14 for the reasons for this notation. Selman proved that the satis ability problem is not P selective, unless P = NP. This result has been considerably strengthened by extending it to reduction closures of the P selective languages, see the following Facts 2 and 3. Fact 1 ([22]) If some NP hard language is P selective, then NP = P. Fact 2 ( 6, 20, 1] If some NP hard language is sublinear truth table reducible to a P selective language, then NP = P. Fact 3 ( 22, 15] If some NP hard language is Turing reducible to a P selec tive language, then NP P=poly. These ....

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A. L. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Math. Syst. Theory, 13:55-65, 1979.

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A. Selman. P-selective sets, tally languages, and the behavior of polynomial-time reducibilities on NP. Mathematical Systems Theory, 13:55--65, 1979.

....HIS85] Every P printable set is sparse and belongs to P. A set A is P printable immune if no infinite subset of A is P printable. A set L is p selective if there is a polynomial time bounded function f such that for every x, y # # , f(x, y) # x, y , and x, y #= # # f(x, y) L [Sel79]. A function f PF is almost always one way [FPS01] if no polynomial time Turing machine inverts f correctly on more than a finite subset of range(f) Proposition 5.1 1. A, B) is symmetric if and only if (B, A) is symmetric. 2. If (A, B) is P separable, then (A, B) is symmetric. Proof The ....

....f(x) A. So g will output f(x) and M will reject x. Therefore, A L(M) B. # Now we give evidence for the existence of nonsymmetric disjoint NP pairs. Theorem 5.4 If E coNP such that (A, A) is not symmetric. Proof If E coNE, then there is a tally set T P. From Selman [Sel79, Theorem 5], the existence of such a tally set implies that there is a p selective set A P. Clearly, A, A) is not P separable. Hence, by Proposition 5.3, A, A) is nonsymmetric. As a corollary, if E coNP such that (A, A)## m (A, A) yet clearly (A, A)# T (A, A) We will show that the ....

A. Selman. P-selective sets, tally languages, and the behavior of polynomial-time reducibilities on NP. Mathematical Systems Theory, 13:55--65, 1979.

....hypotheses. Our separations are obtained easily from existing techniques to separate reductions between NP sets [PS01] A set L is p selective if there is a polynomial time bounded function g such that for every x, y # # , g(x, y) # x, y , and x, y L #= # # g(x, y) L [Sel79] The function g is called the selector function for L. Given a finite alphabet, let # denote the set of all strings of infinite length of order type #. For r , the standard left cut of r [Sel79, Sel82] is the set L(r) x r , where is the ordinary dictionary ordering of ....

....such that for every x, y # # , g(x, y) # x, y , and x, y L #= # # g(x, y) L [Sel79] The function g is called the selector function for L. Given a finite alphabet, let # denote the set of all strings of infinite length of order type #. For r , the standard left cut of r [Sel79, Sel82] is the set L(r) x r , where is the ordinary dictionary ordering of strings with 0 less than 1. It is obvious that every standard left cut is p selective with selector g(x, y) min (x, y) For any A NP, there is a polynomial p( and a polynomial time predicate R such ....

A. Selman. P-selective sets, tally languages, and the behavior of polynomial-time reducibilities on NP. Mathematical Systems Theory, 13:55--65, 1979.

....that P i does not recognize B. There is an oracle relative to which BPP#NP is not contained in L 2 . There is an oracle relative to which coAM#NP is not contained in L 2 . The next result concerns P selective and P cheatable sets. P selective sets were defined by Selman in [Se 79] and have been studied since then in [Se 81, KS 85] and numerous other papers. It was shown in [KS 85] that every P selective set in NP is in L 2 , and in [ABG 89] it was shown that all P selective sets are in EL 2 . P cheatable sets have been studied by several authors in the last few years (see ....

A. Selman, P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP, Math. Systems Theory 13, 55--65.

....The genericity hypothesis asserts that NP contains a p generic language. Lutz s measure hypothesis implies the genericity hypothesis, but the converse is not known to be true. The genericity hypothesis also implies the the consequences in Theorems 2.3 and 2.4 [6] 2. 6 P Selective Sets Selman [77] introduced p selective sets. De nition 2.3. 77] A language L is p selective if there exists a polynomialtime computable function f : such that for all x and y, f(x; y) 2 fx; yg and if either x 2 L or y 2 L, then f(x; y) 2 L. We call f a selector for L. Given a nite ....

....a p generic language. Lutz s measure hypothesis implies the genericity hypothesis, but the converse is not known to be true. The genericity hypothesis also implies the the consequences in Theorems 2.3 and 2.4 [6] 2.6 P Selective Sets Selman [77] introduced p selective sets. De nition 2.3. [77]) A language L is p selective if there exists a polynomialtime computable function f : such that for all x and y, f(x; y) 2 fx; yg and if either x 2 L or y 2 L, then f(x; y) 2 L. We call f a selector for L. Given a nite alphabet , let denote the set of all strings of ....

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A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Mathematical Systems Theory, 13:55-65, 1979.

....P r complete for NP if S # NP and every set in NP is # P r reducible to S. Recall that a set L is p selective if there exists a polynomial time computable function f : S # S # # S # such that for all x and y, f (x,y) # x,y and f (x,y) belongs to L, if either x # L or y # L [Sel79]. The function f is called a selector for L. Given a finite alphabet, let S w denote the set of all strings of infinite length of order type w. For r # S # #S w , the standard left cut of r [Sel79, Sel82] is the set L(r) x x r , where is the ordinary dictionary ordering of ....

....for all x and y, f (x,y) # x,y and f (x,y) belongs to L, if either x # L or y # L [Sel79] The function f is called a selector for L. Given a finite alphabet, let S w denote the set of all strings of infinite length of order type w. For r # S # #S w , the standard left cut of r [Sel79, Sel82] is the set L(r) x x r , where is the ordinary dictionary ordering of strings with 0 less than 1. It is obvious that every standard left cut is p selective with selector f (x,y) min(x,y) Given a p selective set L such that the function f defined by f (x,y) min(x,y) is a selector ....

A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Mathematical Systems Theory, 13:55--65, 1979.

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A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Mathematical Systems Theory, 13:55--65, 1979.

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A. L. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Mathematical Systems Theory, 13:55--65, 1979.

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A.L. Selman: P-Selective Sets, Tally Languages, and the Behavior of Polynomial Time Reducibilities on NP. Mathematical Systems Theory 13 (1979), 55--65.

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A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Mathematical Systems Theory, 13:55-65, 1979.

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A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Mathematical Systems Theory, 13(1):55--65, 1979.

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A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Mathematical Systems Theory, 13:55--65, 1979.

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A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Mathematical Systems Theory, 13:55-65, 1979.

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A. Selman. P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP. Mathematical Systems Theory, 13:55--65, 1979. 11

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