E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. SIAM Journal on Computing, 21(3):521--539, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....sparse sets. For example, the theorem below states that one truth table reduction to P selective sets yields the same class as one truth table equivalence to P selective sets. In contrast, the same result is known for sparse sets only under the (rather strong) additional assumption that P = NP [AHOW92] In the course of proving the theorem below, HHO94] in fact proves that R 2 tt (P selective) Gamma E T (P selective) is non empty. In contrast, it is known that proving the analogous result for sparse sets would immediately establish that P 6= NP [AHOW92] Adopting the standard notation ....

....additional assumption that P = NP [AHOW92] In the course of proving the theorem below, HHO94] in fact proves that R 2 tt (P selective) Gamma E T (P selective) is non empty. In contrast, it is known that proving the analogous result for sparse sets would immediately establish that P 6= NP [AHOW92] Adopting the standard notation in the literature, for any already defined reducibility r , let R r (P selective) denote fA j (9L 2 P selective) A r L]g, and let E r (P selective) denote fA j (9L 2 P selective) A r L and L r A]g. Theorem 3.1 ( HHO94] 1. P selective 1 T ....

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. SIAM Journal on Computing, 21(3):521--539, 1992.

.... simple relative to A 2. If a set A is reducible to a sparse set, then how easy is it to access all the relevant information contained in the set A when it is used as oracle Question 1 originates in the study of the notions of equivalence and reducibility to sparse sets (see for example [TB91, AHOW92, GW93] and concerns the complexity (relative to A) of small descriptions for sets A which are reducible to sparse sets. For the case of Turing reductions, Gavald a and Watanabe [GW93] established an NP co NP lower bound by constructing a set B that is Turing reducible to a sparse set but is ....

....to a sparse set but is not Turing reducible to a sparse set in NP(B) co NP(B) Hence, the class of sets Turing equivalent to some sparse set is a proper subclass of P=poly. The separation of equivalence and reduction classes for restricted truth table reducibilities is further investigated in [AHOW92] In this context, it is interesting to know, for various classes of sets that reduce to sparse sets, the complexity of the easiest sparse sets to which such sets reduce. In [AHH 93] this question is investigated for certain truth table reducibilities and upper bounds are established for the ....

[Article contains additional citation context not shown here]

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. SIAM Journal on Computing, 21(3):529-539, 1992.

....the work of this paper showed that there are relativized worlds in which there are sparse sets that are NP complete with respect to such bounds and yet (relativized) P 6= NP. With regard to Question 2: ffl In the context of recent comparisons between equivalence and reducibility to sparse sets [AHOW,GW91] it is interesting to know for various reduction classes to sparse sets how easy a sparse set can be relative to the set that is reduced to it. We show that any set A that disjunctively reduces (respectively, disjunctive bounded truth table reduces, 2 truth table reduces) to a sparse set in ....

....by Ladner, Lynch, and Selman [LLS75] see Table 1) We will use the following notation to describe downward closures of classes under various reductions; the interrelations among R p r (SPARSE) classes have been systematically studied by Book and Ko [BK88] and Ko [Ko89] Notation 2. 1 [BK88,AHOW] For any reducibility p r and any class of sets C, let R p r (C) fA fi fi (9B 2 C) A p r B]g. 3 Sets Reducing to Sparse Sets The study of sparse complete sets was started by the conjecture of L. Berman and J. Hartmanis [BH77] that there are no sparse NP complete sets; they were ....

[Article contains additional citation context not shown here]

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. SIAM Journal on Computing. To appear. Preliminary version appears as [AHOW91].

....For a class K of sets we denote the union of all sets in K by S K. Let h Delta; Deltai denote a standard pairing function. The reductions discussed in this paper are the polynomial bounded reductions defined by Ladner, Lynch, and Selman [LLS75] and by Adleman and Manders [AM77] Notation [AHOW] For any reducibility ff r and any class of sets C, let R ff r (C) fA j A ff r B for B 2 Cg, where ff 2 fp; np; co np; rp; co rpg and r 2 fm; c; d; b; Tg. Definition 2.1 The join of two sets A and B, denoted A Phi B is defined as A Phi B = f0x j x 2 Ag [ f1x j x 2 Bg Definition 2.2 A ....

....of M relative to B. Our first result in this section is the equality R np m (SPARSE) R np T (SPARSE) Indeed, we show that every set in R np T (SPARSE) nondeterministically many one reduces to very sparse sets (in the sense that they contain at most one string of each length) Note that in [AHOW] it is shown, using the inclusion R p b (SPARSE) R p d (SPARSE) that R np c (SPARSE) R np T (SPARSE) and that R np m (SPARSE) R np d (SPARSE) Theorem 7.2 R np m (SPARSE) R np T (SPARSE) Proof Let A = L(M;S) 2 R np T (S) for a sparse set S and a nondeterministic Turing machine ....

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. SIAM Journal on Computing. To appear. Preliminary version appears as [AHOW91]. 21

....and NWO Grant SIR 13 603. y Department of Computer Science, Le Moyne College, Syracuse, NY 13214. Work done in part while at the University of Amsterdam. z College of Computer Science, Northeastern University, Boston, MA 02115. Supported in part by NSF Grant CCR 9211174. 1 [Ko89] AHOW92] and [AHH 92] Our main result refutes one of Ko s conjectures [Ko89] by showing that every sparse set is conjunctive truth table reducible to a tally set: SPARSE R ctt (TALLY) R ctt (SPARSE) R ctt (TALLY) The reduction uses polynomials over finite fields to encode any sparse set into ....

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. SIAM J. Comput., 21(3):521--539, 1992.

....P T (SPARSE) classes was again shown much later when coding complexity was studied more explicitly, as described below. Isomorphism Degrees Book and Tang and their followers obtained many other classi cation results, just as the work of Book and Ko studied many other reduction classes. See, e.g. [AH92, AHOW91, AW90, TB88, TB91] for these results. We omit most of them here, but we will brie y mention what happens at the other end of the scale: the strongest degrees, de ned by polynomial time isomorphisms, applied to tally sets. Indeed, tally strings are the most natural examples of words of low Kolmogorov complexity ....

Allender E, Hemachandra L, Ogiwara M, Watanabe O. Relating equivalence and reducibility to sparse sets. SIAM Journal of Computing 1992;21:521-539.

....T (SPARSE) classes was again shown much later when coding complexity was studied more explicitly, as described below. Isomorphism Degrees Book and Tang and their followers obtained many other classification results, just as the work of Book and Ko studied many other reduction classes. See, e.g. [AH92, AHOW91, AW90, TB88, TB91] for these results. We omit most of them here, but we will briefly mention what happens at the other end of the scale: the strongest degrees, defined by polynomial time isomorphisms, applied to tally sets. Indeed, tally strings are the most natural examples of words of low Kolmogorov complexity ....

Allender E, Hemachandra L, Ogiwara M, Watanabe O. Relating equivalence and reducibility to sparse sets. SIAM Journal of Computing 1992;21:521--539.

....then can be viewed as that of possessing small nonuniform complexity. This relation is interesting, because it provides fine classifications of P=poly: According to the power of the access, the reducibility closures of the sparse sets form proper hierarchies within P=poly (for such results, see [BK88, Ko89, AHOW92]) Thus, by products of the results on sparse hard set problems are nonuniform lower bounds of uniform complexity classes, from a totally different angle than usual circuit complexity lower bound arguments. 1.2 Progress on the Sparseness Conjecture for NP The sparseness conjecture by Berman and ....

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. SIAM Journal on Computing, 21(3):551--539, 1992.

....f(x;w) 2 S then T : T [ ff(x; w)g end end simulate M on input x using oracle T Figure 5: Oracle machine M 0 using oracle S Proposition 5.4 R p p pre (SPARSE) R p p post (SPARSE) R p p in (SPARSE) None of these inclusions is known to be proper. co SPARSE R p d (SPARSE) was shown in [AHOW92] follows also from the result above cited from [BLS93] Since SPARSE is additionally closed under pairing, we get from Proposition 4.7 and 4.5 Proposition 5.5 1. R p t in (SPARSE) R p c (co SPARSE) R p d (SPARSE) 2. R p p pre (SPARSE) R p c (R p d (co SPARSE) R p d (R p c ....

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. SIAM Journal on Computing, 21(3):529--539, 1992.

.... we show first that distinguishing equivalence and reducibility to lowtally sets, under the many one reduction, would imply that P 6= NP, in the same way that it is not possible to separate equivalence and reducibility to sparse sets, for many one reductions, if P = NP (for more information see [2]) As a consequence of this result, we also obtain that separating Em(Lowtally) Sigma and E btt (Lowtally) becomes a difficult task too. Theorem 34 P = NP = Pm(Lowtally) Em(Lowtally) Sigma . Proof: The inclusion Em(Lowtally) Sigma Pm(Lowtally) is obvious. So, it is only ....

..... Proof: The inclusion Em(Lowtally) Sigma Pm(Lowtally) is obvious. So, it is only necessary to see the converse. let L and LT be sets such that L m LT via g, where LT is a lowtally set and L 6= Sigma . We define the set LT 0 , using the method from [22] and following the steps of [2], as follows: LT 0 = f0 hl;mi j 9y; jyj = l g(y) 0 m 2 LTg On the one hand, L m LT 0 via a function h defined in this way: h(y) 0 hjyj;jg(y)ji Indeed, h can be calculated in polynomial time, and for all y y 2 L ( h(y) 2 LT 0 When y 2 L, the word 0 hjyj;jg(y)ji is in LT 0 by ....

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating Equivalence and Reducibility to Sparse Sets. SIAM Journal on Computing, 21(3):521--539, 1992.

....sparse sets. For example, the theorem below states that one truth table reduction to P selective sets yields the same class as one truth table equivalence to P selective sets. In contrast, the same result is known for sparse sets only under the (rather strong) additional assumption that P = NP [AHOW92] In the course of proving the theorem below, HHO94] in fact proves that R p 2 tt (P selective) Gamma E p T (P selective) is non empty. In contrast, it is known that proving the analogous result for sparse sets would immediately establish that P 6= NP [AHOW92] Adopting the standard ....

....assumption that P = NP [AHOW92] In the course of proving the theorem below, HHO94] in fact proves that R p 2 tt (P selective) Gamma E p T (P selective) is non empty. In contrast, it is known that proving the analogous result for sparse sets would immediately establish that P 6= NP [AHOW92] Adopting the standard notation in the literature, for any already defined reducibility p r , let R p r (P selective) denote fA j (9L 2 P selective) A p r L]g, and let E p r (P selective) denote fA j (9L 2 P selective) A p r L and L p r A]g. Theorem 3.1 ( HHO94] 1. P selective ae ....

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. SIAM Journal on Computing, 21(3):521--539, 1992.

....is identical to the class of sets 1 truthtable reducible to P selective sets. Though our techniques bear no relation to the techniques used to study sparse sets, we note that reductions and equivalences to sparse sets have been satisfyingly studied in a long line of research (see, e.g. BK88,TB91,AHOW92,GW93,Ko89, AHH 93,Gav92,BLS93] the present paper constructs, for P selectivity, a theory roughly comparable in scope to the theory for sparse sets constructed in the just mentioned line of papers. 2 Equivalence and Reducibility to P Selective Sets For any class C and any reducibility ....

....theory for sparse sets constructed in the just mentioned line of papers. 2 Equivalence and Reducibility to P Selective Sets For any class C and any reducibility denoted t r , let R t r (C) and E t r (C) respectively denote fA j (9L 2 C) A t r L]g and fA j (9L 2 C) A t r L L t r A]g [AHOW92] In this section, we study the structure of the E(P sel) and R(P sel) classes stretching from P sel = E p m (P sel) R p m (P sel) up to R p T (P sel) which equals P=poly (see [Sel79,Sel82b, Ko83] the well studied class of sets having small circuits [KL80] Previous results along this ....

[Article contains additional citation context not shown here]

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. SIAM Journal on Computing, 21(3):521--539, 1992.

....description relative to A In this subsection we discuss this well studied question and state applications (in the form of collapse results) of certain specific answers to this question. This study originates in the notions of equivalence and reducibility to sparse sets (see for example [TB91, AHOW92, GW93] It concerns the complexity (relative to A) of small descriptions for sets A which are reducible to sparse sets. Gavald a and Watanabe [GW93] obtained an important lower bound for the case of Turing reductions by constructing a set B that is Turing reducible to a sparse set but is not ....

....what was a long standing open question. Theorem 4. 12 [GW93] There is a set B that is in R p T (SPARSE) but is not Turing reducible to a sparse set in NP(B) co NP(B) The separation of equivalence and reduction classes for restricted truth table reducibilities is further investigated in [AHOW92] In a broader setting it is of interest to know for various classes of sets that reduce to sparse sets, the complexity of the easiest sparse sets to which such sets reduce. This question is first investigated in [AHH 93] where upper bounds for the relative complexity of sparse descriptions ....

[Article contains additional citation context not shown here]

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. SIAM Journal on Computing, 21(3):529-539, 1992.

.... simple relative to A 2. If a set A is reducible to a sparse set, then how easy is it to access all the relevant information contained in the set A when it is used as oracle Question 1 originates in the study of the notions of equivalence and reducibility to sparse sets (see for example [TB91, AHOW, GW91] and addresses the complexity (relative to A) of sparse descriptions for sets A which are reducible to sparse sets. For the case of Turing reductions, Gavald a and Watanabe [GW91] established an NP co NP lower bound by constructing a set B 2 R p T (SPARSE) in fact, B is even in R p c ....

....time, such that for all x, f computes a tuple f(x) hy 1 ; y 2 ; y 2k i such that 1. B (y 1 ) B (y 2 ) Delta Delta Delta B (y 2k ) and 2. A (x) W k i=1 [ B (y 2i Gamma1 ) B (y 2i ) We call f a bounded Hausdorff reduction (A p bhd B) if k is a constant. Notation [AHOW] For any reducibility ff r where ff 2 fp; np; co npg and r 2 fm; c; d; 1 tt; b; hd; bhdg and any class C of sets let R ff r (C) fA j A ff r B for some B 2 Cg. A class K of sets that includes ; and Sigma and is closed under union and intersection is said to be a set ring. The ....

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. SIAM Journal on Computing, to appear.

....) Prob[f(x; w) 2 B] 1; x 62 A ) Prob[f(x; w) 2 B] 1=2: Here, the string w is chosen uniformly at random from the set Sigma q(jxj) Let ff be any reducibility. Then the reduction class fA j 9B 2 C : A ff Bg of all sets that are ff reducible to some set in C is denoted by P C ff [BK88, AHOW92] Furthermore, let D be an oracle class and let C be a relativizable complexity class. Then the class of all sets (or functions) computable by a machine M of type C by asking on every computation path one round of parallel queries (at most k adaptive queries) to an oracle from D is denoted by C ....

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. In SIAM Journal on Computing, 21(3):521--539, 1992.

.... R p c (TALLY) BLS93] we conclude R p c (SPARSE) R p c (co SPARSE) Using Propositions 3.9 and 3.5 we get the inclusions Proposition 5.5 R p posS (SPARSE) R p negS (SPARSE) R p T (SPARSE) None of these inclusions is known to be proper. co SPARSE R p d (SPARSE) was shown in [AHOW92] follows also from the result above cited from [BLS93] Since SPARSE is additionally closed under pairing, we get from Proposition 4.7 and 4.5 Proposition 5.6 1. R p bS (SPARSE) R p c (co SPARSE) R p d (SPARSE) 2. R p posS (SPARSE) R p c (R p d (co SPARSE) R p d (R p c ....

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. SIAM Journal on Computing, 21(3):529--539, 1992.

.... P r (SPARSE) E r (SPARSE ) Recent results by the authors suggest that these questions may be quite di#cult, since, for example, one can show that P=NP =# Pm (SPARSE) Em (SPARSE ) and the question of whether Pm (TALLY ) Em (TALLY ) is equivalent to some basic open questions in complexity theory [AHOW 91]. At this point, however, nothing is known about the P poly = ET (SPARSE ) question. One other avenue for further work involves the notion of self reducibility. It can be shown that for some classes C, the self reducible elements of C have lowness properties not shared by other elements of C. ....

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Re lating Equivalence and Reducibility to Sparse Sets, To appear in Proc. 6th Structure in Complexity Theory Conference.

No context found.

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. In Proceedings of the 6th Structure in Complexity Theory Conference, pages 220--229. IEEE Computer Society Press, June/July 1991.

....the work of this paper showed that there are relativized worlds in which there are sparse sets that are NP complete with respect to such bounds and yet (relativized) P 6= NP. With regard to Question 2: ffl In the context of recent comparisons between equivalence and reducibility to sparse sets [AHOW,GW91] it is interesting to know, for various classes of sets that reduce to sparse sets, the complexity of the easiest sparse sets to which such sets reduce. We show that any set A that disjunctively reduces (respectively, disjunctive bounded truth table reduces, 2 truth table reduces) to a ....

....reduces) to a sparse set that is in P NP A (respectively, P NP A [log] P NP A [log] 3 Thus, for such sets, reducing to some sparse set implies reducing to some relatively simple sparse set. The nearest previous result is one of Allender, Hemachandra, Ogiwara, and Watanabe [AHOW] If P = NP and set A 2 truth table reduces to a sparse set, then A truth table reduces to some sparse set that itself truth table reduces to A. However, A does not two truth table reduce to the particular sparse set constructed in [AHOW] Via census functions, graph coloring, and the Erdos Rado ....

[Article contains additional citation context not shown here]

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. SIAM Journal on Computing. To appear. Preliminary version appears as [AHOW91].

....is identical to the class of sets 1 truthtable reducible to P selective sets. Though our techniques bear no relation to the techniques used to study sparse sets, we note that reductions and equivalences to sparse sets have been satisfyingly studied in a long line of research (see, e.g. BK88,TB91,AHOW92,GW,Ko89, AHH ,Gav92b,BLS93] Section 2 of the present paper constructs, for P selectivity, a theory roughly comparable in scope to the theory for sparse sets constructed in the just mentioned line of papers. The power of nondeterministic computation is one unifying theme of complexity ....

....result that applies to these and other selectivity classes. 2 Equivalence and Reducibility to P Selective Sets For any class C and any reducibility denoted t r , let R t r (C) and E t r (C) respectively denote fA fi fi (9L 2 C) A t r L]g and fA fi fi (9L 2 C) A t r L L t r A]g [AHOW92] In this section, we study the structure of the E(P sel) and R(P sel) classes stretching from P sel = E p m (P sel) R p m (P sel) up to R p T (P sel) which equals P=poly (see [Sel79,Sel82b,Ko83] the well studied class of sets having small circuits [KL80] Previous results along this line ....

[Article contains additional citation context not shown here]

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. SIAM Journal on Computing, 21(3):521-- 539, 1992.

....T (SPARSE) classes was again shown much later when coding complexity was studied more explicitly, as described below. Isomorphism Degrees Book and Tang and their followers obtained many other classification results, just as the work of Book and Ko studied many other reduction classes. See, e.g. [AH92, AHOW91, AW90, TB88, TB91] for these results. We omit most of them here, but we will briefly mention what happens at the other end of the scale: the strongest degrees, defined by polynomial time isomorphisms, applied to tally sets. Indeed, tally strings are the most natural examples of words of low Kolmogorov complexity ....

Allender E, Hemachandra L, Ogiwara M, Watanabe O. Relating equivalence and reducibility to sparse sets. SIAM Journal of Computing 1992;21:521--539.

No context found.

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. In Proceedings of the 6th Structure in Complexity Theory Conference, pages 220--229. IEEE Computer Society Press, June/July 1991.

No context found.

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe. Relating equivalence and reducibility to sparse sets. Proceeding 6th Structure in Complexity Theory Conference, 1991.

No context found.

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe, Relating equivalence and reducibility to sparse sets. In Proceeding 6th Structure in Complexity Theory Conference, 1991, 220-237.

No context found.

E. Allender, L. Hemachandra, M. Ogiwara, and O. Watanabe, Relating equivalence and reducibility to sparse sets. SIAM Journal on Computing 21(3) (1992), 521-539.

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