V. Arvind, J. Kobler, and M. Mundhenk. Upper bounds for the complexity of sparse and tally descriptions. Mathematical Systems Theory, 29:63-94, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of attribution of the rst of these results) namely, 1. If (9Q) USATQ 2 P] then P = Few (and thus P = UP and P = FewP) 2. VV86] If (9Q) USATQ 2 P] then R = NP. Corollary 1.2 If SAT disjunctively reduces to a sparse set, then P = Few and R = NP. Furthermore, Arvind, K obler, and Mundhenk [AKM96] prove that if SAT disjunctively reduces to a sparse set, then PH = P . However, in light of Corollary 1.2, clearly the following can be claimed. Theorem 1.3 If SAT disjunctively reduces to a sparse set, then PH = P . 2 Background and Motivation The study of the consequences of NP having ....

V. Arvind, J. Kobler, and M. Mundhenk. Upper bounds for the complexity of sparse and tally descriptions. Mathematical Systems Theory, 29:63-94, 1996.

....of attribution of the first of these results) namely, 1. If (9Q) USATQ 2 P] then P = Few (and thus P = UP and P = FewP) 2. VV86] If (9Q) USATQ 2 P] then R = NP. Corollary 1.2 If SAT disjunctively reduces to a sparse set, then P = Few and R = NP. Furthermore, Arvind, Kobler, and Mundhenk [AKM96] prove that if SAT disjunctively reduces to a sparse set, then PH = P NP . However, in light of Corollary 1.2, clearly the following can be claimed. Theorem 1.3 If SAT disjunctively reduces to a sparse set, then PH = P R . 2 Background and Motivation The study of the consequences of NP ....

V. Arvind, J. Kobler, and M. Mundhenk. Upper bounds for the complexity of sparse and tally descriptions. Mathematical Systems Theory, 29:63--94, 1996.

....sized circuits is incompatible with being hard for NP under Turing reductions unless the polynomial hierarchy collapses to its second level. Besides, there have been many similar results for various other reducibilities to sparse sets; see, e.g. Lon82b, Ukk83, Yap83, Yes83, Kad89, OW91, RR92, AKM92, AH 93, AA95] Almost ten years after Mahaney s theorem, Watanabe and Ogiwara [OW91] were able to show that unless P = NP, sparseness even is incompatible with being hard for NP under bounded truth table reductions. For more information on this topic, the reader is referred to the survey by ....

Arvind, J. K obler, and M. Mundhenk, Upper bounds for the complexity of sparse and tally descriptions. To appear in MST. A preliminary version appeared in Proc. 3rd ISAAC (Springer, LNCS #650, 1992) 249--258.

....observe that similar collapse consequences downto P can be derived for other subclasses of P poly (see Corollary 3. 3) Some of these collapse consequences follow immediately from recent results investigating the complexity of sparse and tally descriptions for sets in P poly [BS92, Kob94, Gav95, AKM96] For the others we can exploit an interesting connection between the worst case complexity of a set L and the average case complexity of oracles used in the computation of an advice function for L. The following theorem shows that if an advice function h for some set L can be efficiently ....

....some oracle transducer M . Then h is computable in FP(T ) where T is the tally set T = fh0 n ; 0 i i j the ith query of M(0 n ) is in Dg: Since T many one reduces to D it follows from the assumption and from Theorem 2.3 that T 2 P, implying that h 2 FP. Now, using results from [BS92, AKM96, Kob94, Gav95] we can state similar collapse consequences as in Theorem 3.1 for several subclasses of P poly. We note that by using a different proof technique it has been shown in [BFNW93] that BPP = P follows from the assumption that every tally set in Sigma p 4 is contained in P. ....

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V. Arvind, J. K obler, and M. Mundhenk. Upper bounds for the complexity of sparse and tally descriptions. Mathematical Systems Theory, 29(1):63--94, 1996.

....problems, or they have low information content like sparse sets or sets reducible to (or equivalent to) sparse or tally sets via different kinds of reducibilities. In the past, a variety of language classes has been shown to be included in the low hierarchies (for an overview see for example [Sc86a, AH92, LS91, HOW92, AKM]) Allender and Hemachandra [AH92] and Long and Sheu [LS91] proved the optimality of the location of almost all these classes, at least in some relativized world. However, until now, the exact location of sets having polynomial size circuits remained open. In research from the early 1980 s to the ....

.... time [MP79] IC[log,poly] is the class of sets for which all input strings have low instance complexity [OKSW94] and the classes of P selective, P close, and P cheatable sets have been introduced in [Se79, Sc86b, Be91] respectively) EL P; Delta 2 : Tally sets [BB86] APT [LS91] IC[log,poly] [AKM], EL P; Sigma 2 : BPP [Sc86a] P selective and P cheatable sets [ABG90] EL P; Theta 3 : Sparse sets, P close sets, and sets that are 1 truth table reducible to some sparse set [LS91] sets that are bounded truth table, conjunctively, or disjunctively reducible to some sparse set [AKM] 3 ....

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V. Arvind, J. K obler, M. Mundhenk. Upper bounds on the complexity of sparse and tally descriptions. To appear in Mathematical Systems Theory.

....statements are equivalent: i A 2 Full P=log. ii A has a family of logarithmic advice words that can be obtained in polynomial time making queries to (A Phi SAT) in a nonadaptive way to A. Proof: The nontrivial direction is from i to ii , and it is a consequence of previous results from [4, 5] 2 . Actually, in [5] it was proven that IC[log, poly] is in the first level of the extended hierarchy. That is, NP(A) P(A Phi SAT) for all sets A in IC[log, poly] Moreover, the proof of this shows that for each language L in NP(A) there exists a deterministic algorithm that decides L in ....

....i A 2 Full P=log. ii A has a family of logarithmic advice words that can be obtained in polynomial time making queries to (A Phi SAT) in a nonadaptive way to A. Proof: The nontrivial direction is from i to ii , and it is a consequence of previous results from [4, 5] 2 . Actually, in [5] it was proven that IC[log, poly] is in the first level of the extended hierarchy. That is, NP(A) P(A Phi SAT) for all sets A in IC[log, poly] Moreover, the proof of this shows that for each language L in NP(A) there exists a deterministic algorithm that decides L in polynomial time querying ....

V. Arvind, J. Kobler, and M. Mundhenk. Upper Bounds on the Complexity of Sparse and Tally Descriptions. Mathematical Systems Theory, 29(1):63--94, 1996.

....have a single learning algorithm provided that this constant is given as input to it. First we show the fact that NP(A) P(A Phi SAT) That is, Full P log is in the first level of the extended low hierarchy. There are alternative proofs of this fact, e.g. via instance complexity (combine [4] and [5]) Let A be a set in Full P=log. That means 8n 9w n (jw n j c log n) 8x (jxj n) x 2 A ( hx; w n i 2 B) where B 2 P and c is a constant. Let M be the machine that decides the set B. Define the set C as follows: C = fhx; v; w; 0 k i j 9z jzj k and x v z and M(hz; vi) 6= M(hz; wi)g It is ....

V. Arvind, J. Kobler, and M. Mundhenk. Upper Bounds on the Complexity of Sparse and Tally Descriptions. Mathematical Systems Theory, 29(1):63--94, 1996.

....to have low instance complexity. But at least they cannot be NP complete. Theorem 6.2 [15] For no NP complete problem A and polynomial p holds ic p (x : A) O(log jxj) for almost every x, unless P = NP. Moreover, problems with low instance complexity are relatively close to problems in P (see [1]) Note that for every polynomial p, log C(x) log C p (x) log jxj. Therefore, problems with low instance complexity cover the polynomially time bounded cases of both Theorems 5.1 and 5.2. It is still open, whether ic p (x : A) log C p 0 (x) for polynomials p and p 0 ) implies that ....

V. Arvind, J. Kobler, and M. Mundhenk. Upper bounds for the complexity of sparse and tally descriptions. Mathematical Systems Theory, 29:63-94, 1996.

....that P = NP if there is a sparse Turing hard set for NP. Indeed, this question is open even for stronger reducibilities. In this paper we consider the question of existence of sparse hard sets for NP w. r. t. disjunctive truth table reductions. We briefly recall some known results: it is shown in [AKM96] that if there is a sparse hard set for NP under disjunctive reductions then PH collapses to Delta p 2 . More recently, it is shown in [CNS96] that if there are sparse hard sets for NP under the disjunctive reducibility then RP = NP. The proof technique in [CNS96] is based on powerful algebraic ....

V. Arvind, J. K obler and M. Mundhenk. Upper bounds for the complexity of sparse and tally descriptions. In Mathematical Systems Theory, 29:63--94, 1996.

.... R p d (HIGH) R p bhd (R p c (HIGH) The third holds since the existence of a sparse hard set for NP (or PSPACE) with respect to the composed Hausdorff and conjunctive reducibility implies the collapse of the polynomial time hierarchy to Delta p 2 (respectively, PSPACE = Delta p 2 ) [3]. The fourth part follows from the result that NP is not contained in R p bhd (R co rp m (SPARSE) unless NP = RP (cf. 2] It is easy to see from Theorem 4.5, that the above corollary also holds for honest reductions to EA sets for the considered reducibilities. Acknowledgments. We thank ....

V. Arvind, J. Kobler, and M. Mundhenk. Upper bounds on the complexity of sparse and tally descriptions. Mathematical Systems Theory, to appear.

....under monotonous reductions exist for NP. Theorem 6.1 1. If SAT p t in or p p pre reduces to a sparse set, then the Polynomial Time Hierarchy collapses to P NP . 2. If SAT p t in reduces to a tally set, then P = NP. The first part of the theorem follows from Theorem 5. 2 and results in [AKM95b] the second part follows from Proposition 5.6 and results in [AHH 93] For ETIME we can prove that p pre hard sparse sets do not exist. This extends results of Watanabe [Wat87] who proved that ETIME cannot have sparse hard sets under conjunctive reducibility. Remember that R p c (SPARSE) is ....

V. Arvind, J. Kobler, and M. Mundhenk. Upper bounds on the complexity of sparse and tally descriptions. Mathematical Systems Theory, 1995. to appear.

....sparse sets, i.e. sets which only contain a polynomially bounded number of strings up to each length. This study has its roots in a conjecture by L. Berman and J. Hartmanis [BH77] that there are no sparse NP complete sets under many one Parts of this work have been presented at FST TCS 1992 [AKM92] reductions. Mahaney settled the conjecture by proving that if any NP complete set many one reduces to a sparse set then P = NP [Mah82] Related work has been done in [Ber78, For79, Yap83, Yes83] see Section 4 for a detailed discussion. From a different perspective, the possible existence ....

....fy 1 ; y j Gamma1 g (x) rejects. Since M is monotonous, also M B (x) rejects, a contradiction. The next proposition should be compared with the characterization of the composition of the Hausdorff and conjunctive reducibilities in terms of the non monotonic Hausdorff reducibility given in [AKM] Proposition 3.4 For every set B, R p hd (R p c (B) R p h (B) Proof If A 2 R p hd (R p c (B) via a Hausdorff reduction function f and a conjunctive reduction function g, then x 2 A can be easily decided knowing the maximum initial subsequence s of (y 1 1 ; y 1 k 1 ; ....

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V. Arvind, J. Kobler, and M. Mundhenk. Upper bounds on the complexity of sparse and tally descriptions. Mathematical Systems Theory, to appear.

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V. Arvind, J. Kobler, and M. Mundhenk. Upper bounds on the complexity of sparse and tally descriptions. Mathematical Systems Theory, to appear.

..... Figure 6: Monotonic reductions to sparse sets 2. If SAT p bS reduces to a tally set, then P = NP. The first part of the theorem follows from Theorem 5. 2 and results in [AKM96b] the second part from Proposition 5.7 and results in [AHH 93] For ETIME we can prove that hard sparse sets do not exist w.r.t. positive search reducibility. This extends results of Watanabe [Wat87] who proved that ETIME cannot have sparse hard sets under conjunctive reducibility. Note that ....

V. Arvind, J. Kobler, and M. Mundhenk. Upper bounds for the complexity of sparse and tally descriptions. Mathematical Systems Theory, 29:63--94, 1996.

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