R. Beigel. NP-hard sets are p-superterse unless R = NP. Technical Report TR 4, Dept. of Computer Science, Johns Hopkins University, 1988.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....that if NP has a Turing hard p selective set, then NP P poly. Toda [Tod91] by exhibiting new combinatorial structural properties of p selective sets, complemented this result and showed that if NP has a tt hard p selective set, then NP = RP (this result also follows from the work of Beigel [Bei88] Ogihara [Ogi94] Beigel, Kummer, and Stephan [BKS94] and Agrawal and Arvind [AA94] showed that if NP has a bounded truth table hard p selective set, then NP = P (see also [HHO 93, TTW94] Thus the reducibility of NP sets to p selective sets offers precise characterizations of the ....

....under the weaker hypothesis that a Delta p 2 complete set is O(log n) membership comparable. Since the hypothesis that NP reduces to a p selective set by polynomial time truth table reductions implies that SAT is O(log n) membership comparable, we obtain another proof of the result of [Tod91, Bei88] We note, however, that our proof appears much more complicated and relies upon fairly heavy algebraic machinery unlike that of [Tod91, Bei88] Thus it appears that O(log n) membership comparability is a much weaker condition than polynomial time truth table reducibility to a p selective set. ....

[Article contains additional citation context not shown here]

R. Beigel. NP-hard sets are p-superterse unless R = NP. Technical Report TR 4, Dept. of Computer Science, Johns Hopkins University, 1988.

....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 class denoted by R. NP R P tt (SELECT) NP = R: 1) 2 In this paper, we show a deterministic upper bound on NP when considering bounded truth table reductions. Namely, we show NP R P btt ....

R. Beigel. NP-hard sets are P-superterse unless R = NP. Technical Report 88-04, Department of Computer Science, The John Hopkins University, 1988.

....all NP selective sets are P selective if and only if all pairs of disjoint NP sets are separable in P. Furthermore separability has been studied in the complexity theoretic settings of lower bounds for proof systems [Raz94,Raz95,KM98] complexity of Craig interpolants [SP98] P superterse sets [Bei88] and witness isomorphic reductions [FHT97] 1 A weak one way function is one that has some easy to compute extension but no easy to invert extension. The study of separability notions is very closely related to the study of partition classes in the context of the boolean hierarchy of ....

R. Beigel. NP-hard sets are P-superterse unless R=NP. Technical Report TR 88-04, Johns Hopkins University, Baltimore, 1988.

....to previous queries. For the language classes these two restrictions yield the same class, i.e. P NP[O(log n) P NP k [H89] For the function classes, we only have an inclusion, namely FP NP[O(log n) FP NP k and equality seems unlikely unless the Polynomial Hierarchy collapses [Be88, Se94, To91]. The Polynomial Hierarchy is defined as NP [ NP NP [ NP NP NP [ The Exponential Hierarchy is defined as E [ NE [ NE NP [ NE NP NP [ The Boolean Hierarchy is the closure of NP under the Boolean operations union, intersection, and complement. A subclass of the Boolean ....

Beigel, R.: NP-hard sets are P-superterse unless R = NP. Technical Report 88-04, Dept. of Computer Science, The John Hopkins University (1988).

....SUNY at Buffalo, Buffalo, NY 14260. Research partially supported by the National Science Foundationunder grant nos. CCR 9002292, INT 9123551, and CCR 9400229. 1 2 S. FENNER, S. HOMER, M. OGIHARA, A. SELMAN The class of partial functions with oracles in NP, namely, PF NP has been wellstudied [13, 1], as have been the corresponding class of partial functions that can be computed nonadaptively with oracles in NP, viz. PF NP tt [15] and the classes of partial functions that are obtained by limiting the number of queries to some value k 1, namely, PF NP[k] and PF NP[k] tt [2] A rich ....

....queries to its oracle and that preserves the number of queries. For this reason, we do not distinguish classes NPMV F tt or NPMV F [k] tt . For k 1, SigmaMV k denotes NPMV Delta Delta Delta NPMV z k . Lemma 4.1. For every k 1, SigmaMV k = NPMV Sigma P k Gamma1 [1] and for every f 2 SigmaMV k , dom(f) 2 Sigma P k . Proof. The proof is by an induction on k. The statement trivially holds for k = 1. Let k = 2. We show that NPMV NPMV NPMV NP[1] Let f 2 NPMV NPMV via a machine M and a function g 2 NPMV. Let N be a machine witnessing g 2 NPMV. ....

[Article contains additional citation context not shown here]

R. Beigel. NP-hard sets are P-superterse unless R = NP. Technical Report 88-04, Department of Computer Science, The Johns Hopkins University, 1988.

....a set A 2a such that ffl (8m) F A m 2 FQ(k Sigma log m k 1 Upsilon ; A) ffl (8m) 8X) F A m = 2 FQ( k Gamma1) Xi log m k Pi ; X) Much less is known about how many queries to A are required to compute F A m . The following theorem is all that is known. Theorem 5. 8 [5] 3 Let A be any set. If there exists k such that F A k 2 FQ(k Gamma 1; A) then there exist m 0 and r 1 such that (8m m 0 ) F A m 2 FQ(m r ; A) It is an open problem to improve this result. 6 How hard is membership (A second look) The algorithm for F K 2 n Gamma1 2 FQ(n;K) ....

....to improve this result. 6 How hard is membership (A second look) The algorithm for F K 2 n Gamma1 2 FQ(n;K) has two features that we wish to examine: 2 In [4] this theorem was proven in a complexitytheory framework, but the proof is the same for our recursion theoretic framework. 3 In [5] this theorem was proven in a complexitytheory framework, but the proof is the same for our recursion theoretic framework. 5 1. The queries are made serially. 2. If incorrect answers are supplied then the computation must diverge. We show that if either of these luxuries are disallowed, then, ....

R. Beigel. NP-hard sets are p-superterse unless R=NP. Technical Report 4, The Johns Hopkins University, Dept. of Computer Science, 1988.

....of randomized reduction is used. However, USAT is not known to have OR . So, there wasn t an obvious way to amplify the Valiant Vazirani reduction from SAT to USAT. Nevertheless, the vv m reduction from SAT to USAT proved to be useful in many areas of research. For example, Richard Beigel [Bei88] used it to show that SAT is superterse unless RP = NP. Also, Toda [Tod89] used a similar reduction in his proof that PH P #P[1] This result, in turn, led to the Lund, Fortnow, Karloff and Nisan [LFKN90] result: PH IP. So, there should be little doubt in the reader s mind regarding the ....

R. Beigel. NP-hard sets are p-superterse unless R = NP. Technical Report 4, Department of Computer Science, The Johns Hopkins University, 1988.

....reducible to any nontrivial absolute and gap counting problem. 5.2 Approximable Sets A set A is approximable [4] i there is a constant j and an algorithm which computes for each input x 1 ; x j in polynomial time j bits y 1 ; y j such that A(x h ) y h for some h. Beigel [2, 3] analyzed the notion of approximable sets and showed that no NP hard set is approximable unless P = UP. This result can be transferred to the following theorem which is an extension of Conclusion 4.6. Theorem 5.3 If P 6= UP then no nontrivial absolute (gap, relative) counting problem is ....

....proofs are all direct combinations of Beigel s techniques with those to show the UPhardness of these sets. Thus we restrict ourselves to show the most involved case of relative counting sets. If a problem is NP complete or co NP complete then it is not approximable under the hypothesis P 6= UP [2, 3]. So one has only to adapt the main case in the proof of Theorem 4.5. So let L be any language in UP, note that UP SPP. The rst part of the Turing reduction 10 from L to Relative Counting(A) is exactly the same as in the proof of Theorem 4.5. In the second part it starts to di er at the de ....

Richard Beigel, NP-hard sets are P-superterse unless R = NP, Technical Report 88-04, John Hopkins University, Baltimore, 1988.

....n O(1) The following class relations are known: ffl For every smooth function f , For f(n) 1 2 log n, FP NP [f(n) Gamma1] ae FP NP [f(n) unless P = NP [Kre88, Theorem 4. 2] For f(n) 1 Gamma ffl) log n, ffl 2 (0; 1] FP NP [f(n) Gamma1] ae FP NP [f(n) unless P = NP [Bei88, Theorem 21] For f(n) 2 O(log n) FP NP [f(n) Gamma1] ae FP NP [f(n) unless Sigma p 3 = Pi p 3 [ABG91, Theorem 42] ffl FP NP [O(log n) ae FP NP unless P = NP [Kre88, Theorem 4.1] ffl FP NP [O(log n) ae FP NP jj unless R = NP [Sel91, Theorem 12] and FewP = P ....

....NP unless P = NP [Kre88, Theorem 4. 1] ffl FP NP [O(log n) ae FP NP jj unless R = NP [Sel91, Theorem 12] and FewP = P [Sel91, Corollary 4(i) ffl FP NP jj ae FP NP if and only if P NP [O(log n) ae P NP [Sel91, Theorem 1] Other separations hold under more exotic assumptions [Bei88, Bei91] Research to date has focused on all classes below FP NP [O(log n) and the class FP NP . Though many results have been imported directly from language based to function based classes, there have been some notable surprises, in particular the non equivalence of FP NP jj and FP NP ....

Beigel, R. NP-hard Sets are p-superterse unless R=NP. Technical Report 4, The Johns Hopkins University, Department of Computer Science, 1988.

....in extending Selman s [Sel82] result that NP does not positive truth table reduce to a P selective set unless NP = P. 7 This question has attracted many researchers, in particular after Ogiwara and Watanabe were able to generalize Mahaney s theorem to the bounded truth table case [OW91] In [Bei88, Tod91] it has been proved that if every 6 Note that besides A being disjunctive self reducible and many one reducible to a sparse set, the proof of Mahaney s theorem showing that A 2 P additionally requires that A is NP complete. 7 Actually, Selman proves that any disjunctive self reducible ....

R. Beigel, NP-hard sets are p-superterse unless R = NP, Tech. Rep. 88-04 (John Hopkins U., 1988). 13

....reducible to any nontrivial absolute and gap counting problem. 5.2 Approximable Sets A set A is approximable [4] iff there is a constant j and an algorithm which computes for each input x 1 ; x j in polynomial time j bits y 1 ; y j such that A(x h ) y h for some h. Beigel [2, 3] analyzed the notion of approximable sets and showed that no NP hard set is approximable unless P = UP. This result can be transferred to the following theorem which is an extension of Conclusion 7. Theorem 10 If P 6= UP then no nontrivial absolute (gap, relative) counting problem is ....

....proofs are all direct combinations of Beigel s techniques with those to show the UP hardness of these sets. Thus we restrict ourselves to show the most involved case of relative counting sets. If a problem is NP complete or co NP complete then it is not approximable under the hypothesis P 6= UP [2, 3]. So one has only to adapt the main case in the proof of Theorem 5. So let L be any language in UP, note that UP SPP. The first part of the Turing reduction from L to Relative Counting(A) is exactly the same as in the proof of Theorem 5. In the second part it starts to differ at the definition ....

R. Beigel. NP-hard sets are P-superterse unless R = NP, Technical Report 88-04, John Hopkins University, Baltimore, 1988.

....of hypotheses A and B are identical. This is not a coincidence, for we prove the following assertion: Theorem. Hypothesis A implies hypothesis B 1 . 1 This result has been obtained independently by T. Thierauf [Thi94] Also, this assertion can be obtained by strengthening results of Beigel [Bei87, Bei88]. Using a result of Jenner and Toran [JT93] and the above theorem, it follows that if every set in NP truth table reduces to a p selective set, then for all k 0, SAT DTIME[2 n=log k n ] We do not know whether Hypothesis B implies Hypothesis A. It is not known, for example, whether ....

....denotes the standard dictionary ordering. It is easy to see that L(r) is pselective, since a function that, given strings x and y, outputs the smaller string in fx;yg according to the dictionary ordering is a selector function for L(r) We will be referring to the following classes of functions [Sel94, Bei88]. Definition 2 Suppose m : N 7 N and f : S 7 S are functions. i) We say that f 2 PF NP [m(n) if there exist a polynomial time transducer T and a set A 2 NP such that for all strings x, T(x) computes f (x) by making at most m(jxj) queries to A. We say that f 2 PF NP [O(logn) if ....

R. Beigel. NP-hard sets are P-superterse unless R = NP. Technical Report 88-04, Department of Computer Science, The Johns Hopkins University, 1988.

....[Kre88] showed for any function f(n) 1 2 log n, that if FP SAT f(n) T = FP SAT (f(n) Gamma1) T , then P = NP. He asked whether the same statement holds for a larger function f . Krentel s proof directly applies to the case f(n) c log n for some constant c 1. Related to this, Beigel [Bei88] asked a question of whether FP SAT f(n) tt FP SAT (f(n) Gamma1) T . For the case f(n) c log n with c 1, Beigel [Bei88] showed for any p 1 tt hard set A for NP, that if FP A f(n) tt FP X (f(n) Gamma1) T for some X , then RP = NP and P = UP. Regarding general O(log n) case, Amir, ....

....whether the same statement holds for a larger function f . Krentel s proof directly applies to the case f(n) c log n for some constant c 1. Related to this, Beigel [Bei88] asked a question of whether FP SAT f(n) tt FP SAT (f(n) Gamma1) T . For the case f(n) c log n with c 1, Beigel [Bei88] showed for any p 1 tt hard set A for NP, that if FP A f(n) tt FP X (f(n) Gamma1) T for some X , then RP = NP and P = UP. Regarding general O(log n) case, Amir, Beigel, and Gasarch [ABG90] showed for any function f(n) O(log n) that if FP SAT f(n) tt FP X (f(n) Gamma1) T for ....

[Article contains additional citation context not shown here]

R. Beigel. NP-hard sets are P-superterse unless R=NP. Tech. Report 88-04, Department of Computer Science, The Johns Hopkins University, Baltimore, MD, August 1988.

....= c(set f (x) c(set f ) is a single valued total function. Given f 2 FewPF and a class of single valued functions G, define f 2 c G to mean that c(set f ) 2 G. The class of partial functions that are computable in polynomial time with oracles in NP, PF NP , has been wellstudied [Kre88, Bei88] as have been the corresponding class of partial functions that can be computed nonadaptively with oracles in NP [Sel94] PF NP tt , and the classes of partial functions that are obtained by limiting the number of queries to some value k 1, namely, PF NP[k] and PF NP[k] tt [Bei91] A rich ....

....time. Let L = ff j some output value of M 0 on input f is a satisfying assignmentg: Then, L 2 P and L is a solution of (SAT1;SAT) Thus, NP = R follows from the result of Valiant and Vazirani. 2 This proof is similar to earlier applications of the result of Valiant and Vazirani by Beigel [Bei88] and Toda [Tod91] Finally, we note that Jenner and Toran [JT95] proved that PF NP tt = PF NP (O(log n) implies that for all k 0, SAT 2 DTIME(2 n=log k n ) Their argument connects the hypothesis with a lowering of the amount of nondeterminism that is needed in a nondeterministic ....

R. Beigel. NP-hard sets are P-superterse unless R = NP. Technical Report 88-04, Department of Computer Science, The Johns Hopkins University, 1988.

....have subtitled this paper, On the complexity of inverting one way functions, but a meaningful explanation of this comment must wait until several technical definitions are given. For now, let us note that the inverse of any honest polynomial time computable function is itself a function, albeit a multivalued function, and that this inverse is solvable by an obvious nondeterministic algorithm. Thus, for this reason also, it makes sense to inquire directly about the complexity of multivalued functions that can be solved by nondeterministic algorithms. 1.1 Definitions of some complexity ....

....two function classes are equal, then P = NP. We will show the following concerning the two inner inclusions. We will show that that P NP = P NP tt if and only if PF NP = PF NP tt . i.e. the right two function classes in (2) are equal if and only if all three set classes in (1) are equal. Beigel, Hemachandra, and Wechsung [BHW89] showed that P NP (O(log n) PP. Thus, PF NP = PF NP tt implies P NP PP. We take this to be evidence that the classes PF NP and PF NP tt are not identical. We will show that if PF NP (O(log n) PF NP tt , then FewP = P and R = NP. Thus, ....

[Article contains additional citation context not shown here]

R. Beigel. NP-hard sets are P-superterse unless R = NP. Technical Report 88-04, Department of Computer Science, The Johns Hopkins University.

....function computes RM SAT r(n) using fewer than dlog(r(n) 1)e queries to X, then P = NP. In this paper, we need lower bounds on RM SAT r(n) for cases where r(n) 2 n O(1) We could obtain some lower bounds by translating the terseness results by Amir, Beigel and Gasarch [ABG90] and Beigel [Bei88] However, the lower bounds would not be optimal, because there would be some difficulty translating the terminology between the papers regarding the size of the input. 2 Thus, we prove the following lemma using the tree pruning techniques from Amir, Beigel and Gasarch. Theorem 7 Let r(n) 2 ....

....Previous versions of this paper [Cha94a, Cha95] stated that Corollary 6 can be used to obtain terseness results. While this is true, the statement of Corollary 9 in those versions is incorrect. In particular, the terseness results we could obtain would not supersede the theorems shown by Beigel [Bei88] Amir BeigelGasarch [ABG90] or Krentel [Kre88] The difficulty lies in the fact that in this paper we do not count queries based on the length of the input. Even though we can show that PF SAT[q(n) can solve RM SAT 2 q(n) Gamma1 but PF SAT[q(n) Gamma1] cannot (unless P = NP) the n here ....

R. Beigel. NP-hard sets are p-superterse unless R = NP. Technical Report 4, Department of Computer Science, The Johns Hopkins University, 1988.

....polynomial time honest functions. To wit, the inverse of every polynomial time honest function belongs to NPMV, and the inverse of every one one polynomial time honest function belongs to NPSV. The class of partial functions with oracles in NP, namely, PF NP has been well studied [Kre88, Bei88] as have been the corresponding class of partial functions that can be computed nonadaptively with oracles in NP, viz. PF NP tt [Sel92] and the classes of partial functions that are obtained by limiting the number of queries to some value k 1, namely, PF NP[k] and PF NP[k] tt [Bei91] A ....

R. Beigel. NP-hard sets are P-superterse unless R = NP. Technical Report 88-04, Department of Computer Science, The Johns Hopkins University, 1988.

....3.4 to generate l 0 k many different vectors, containing F A l 0 (q 0 1 ; q 0 l 0 ) From these vectors one can generate l 0 k many vectors containing F SAT l (q 1 ; q l ) FP NP jj = FP NP[log] follows from Lemma 2.9. 2 The following theorem is implicit in [Bei88, Tod91b] Theorem 3.5 (Beigel Toda) FP NP jj = FP NP[log] Unique SAT is in P Proof: We have to show that there is a polynomial time algorithm that tells formulae with exactly one satisfying assignment apart from ones that are unsatisfiable. Consider the function f(OE) that on input OE with ....

....In fact Toda s results hold for the more general k approximable sets. In this section we cite all results for k approximable sets. Since P selective sets are k approximable sets with k = 2, all these results also hold for P selective sets. Similar ideas were obtained independently by Beigel [Bei88] Theorem 4.1 (Beigel Toda) 1. P = UP if and only if UP p tt bAPP. 2. Unique SAT 2 P if and only if Unique SATQ p tt bAPP for some Q. 3. P = NP if and only iff Delta p 2 p tt bAPP 4. P = PSPACE if and only PSPACE p tt bAPP. 5. EXP 6 p tt bAPP The Turing reduction of bAPP ....

[Article contains additional citation context not shown here]

R. Beigel. NP-hard sets are P-superterse unless R = NP. Technical Report TR 884, Johns Hopkins University, 1988.

....happens in many problems, computing a cost can be basically as hard as computing a solution. For example, computing the cost of the best tour for the Traveling Salesperson Problem is complete in FP NP , and also there 4 This result was originally proved for ffl 1 2 . As observed by Beigel in [Bei88], the result is true for any ffl 1. 1. The Complexity of Obtaining Solutions for Problems in NP and NL 10 are cases like the function Max Assign considered in Section 2 in which the solution is encoded in the cost function. There are, however, many other optimization problems, like ....

....for the complexity of selecting a Boolean formula from a list. For a smooth function f , NPSV NP [f ] denotes the generalization of NPSV to functions computed by a nondeterministic single valued machine that can 5 Valiant and Vazirani [VV86] obtained the second and third consequences. Beigel [Bei88] observed that the weaker result UP = P follows from the equivalent hypothesis that Unique SAT and SAT are P separable. 6 Beigel [Bei88] and Toda [Tod91] contain other interesting applications of Theorem 5.7. 1. The Complexity of Obtaining Solutions for Problems in NP and NL 17 make f queries ....

[Article contains additional citation context not shown here]

R. Beigel. NP-hard sets are p-superterse unless R=NP. Technical Report TR4, Department of Computer Science, John Hopkins University, Baltimore, 1988.

....advice even with reference to A is impossible. 5 MOD classes, self reducibility and log advice We now consider self reducible languages in MOD k P log, and their relationship to MOD k P. We use standard definitions of structural complexity concepts as found in [1] and [2] Definition 5. 1 [4] A language L is in MOD k P if there exists a #P function f such that for all x 2 Sigma [x 2 L ( f(x) j 1 mod k] Only prime k will be considered in the remainder of this paper. Definition 5.2 A set A is self reducible if there exists a deterministic polynomial time oracle machine M such ....

Beigel, R. NP-hard sets are P-superterse unless R = NP. Technical Report 88-04, Department of Computer Science, The Johns Hopkins University.

....We prove that FL NP log coincides with any of the other two classes then L = P. The question whether the classes FP NP k and FP NP log are equal has attracted the attention of different researchers. It is known that the hypothesis FP NP k = FP NP log implies that FewP=P, NP=R and coNP=US, [4][39] 35] These three consequences follow in fact from a weaker hypothesis, namely from the existence of a polynomial time algorithm that decides correctly the satisfiability of a formula with at most one satisfying assignment. If the formula has more than one assignment the algorithm may ....

....is stated formally using the concept of promise problems (see [15] A promise problem is a pair of sets (Q; R) A set L is called a solution to the promise problem (Q; R) if 8x(x 2 Q ) x 2 L , x 2 R) 1SAT denotes the set of Boolean formulas with at most one satisfying assignment. Theorem 3. 3 [4, 39] If FP NP k FP NP log , then the promise problem (1SAT; SAT) has a solution in P. A polynomial time solution for the promise problem (1SAT; SAT) would imply the unexpected consequences expressed in the next theorem. The complexity classes FewP and R mentioned in the result are well known ....

R. Beigel, NP-hard sets are p-superterse unless R=NP. Tech. Report TR4, Dept. of Comp. Science, John Hopkins University, 1988.

....i 1, if 0 i 2 T , define w i to be the lexicographically largest witness for 0 i 2 T , and if 0 i 62 T , define w i = 0 n i . Define the infinite sequence w = w 1 Delta w 2 Delta w 3 Delta Delta Delta. 4 Theorem 1 and Proposition 1 also follow as consequences of a result of Beigel [Bei88] Consider the standard left cut L = L(w) Clearly L is p selective. We need to show that L 2 NP Gamma P and that search reduces to decision for L. For any string x, let k(x) be the smallest length k such that jxj n 1 n 2 Delta Delta Delta n k . Then, x 2 L if and only if there exist ....

R. Beigel. NP-hard sets are P-superterse unless R = NP. Technical Report 88-04, Department of Computer Science, The Johns Hopkins University, 1988.

....if there exists a polynomial time computable function f : Sigma Theta Sigma 7 Sigma such that (i) f(x; y) 2 fx; yg, and (ii) if f(x; y) y, then x 2 L y 2 L. The function f is called a p selector for L. We will be referring to the following classes of functions [Sel92, Bei88]. Definition 2 Suppose m : N 7 N and f : Sigma 7 Sigma are functions. i) We say that f 2 PF NP [m(n) if there exist a polynomial time transducer T and a set A 2 NP such that for all strings x, T (x) computes f(x) by making at most m(jxj) queries to A. We say that f 2 PF NP ....

R. Beigel. NP-hard sets are P-superterse unless R = NP. Technical Report 88-04, Department of Computer Science, The Johns Hopkins University, 1988.

....to obtain a stronger collapse of the polynomial hierarchy than the collapse known by Toda s theorem [Tod91] That is, recall that RP = NP implies 1 This result has been obtained independently by T. Thierauf [Thi94] Also, this assertion can be obtained by strengthening results of Beigel [Bei87, Bei88] a collapse of the polynomial hierarchy to ZPP NP because RP has polynomial size circuits [Adl78, KL80, KW94] In the following theorem we obtain a collapse of the polynomial hierarchy to P NP . A standard left cut is a special kind of a p selective set and is de ned in the next section. ....

....the standard dictionary ordering. It is easy to see that L(r) is p selective, since a function that, given strings x and y, outputs the smaller string in fx; yg according to the dictionary ordering is a selector function for L(r) We will be referring to the following classes of functions [Sel94, Bei88] De nition 2 Suppose m : N 7 N and f : Sigma 7 Sigma are functions. i) We say that f 2 PF NP [m(n) if there exist a polynomial time transducer T and a set A 2 NP such that for all strings x, T (x) computes f(x) by making at most m(jxj) queries to A. We say that f 2 PF NP ....

R. Beigel. NP-hard sets are P-superterse unless R = NP. Technical Report 88-04, Department of Computer Science, The Johns Hopkins University, 1988.

....queries to A. A set A is defined to be k query p superterse if k parallel queries to A allow us to compute more functions than only k Gamma 1 serial queries to B for every set B. The reason for defining p superterseness is that most proofs of p terseness are in fact proofs of p superterseness [2, 9, 7, 4, 5]; in addition k query psuperterseness is the strongest of our sixteen ways of stating that k queries to an oracle enable us to compute more functions in polynomial time than only k Gamma 1 queries. A special case is 2 query p terseness; it is known that a set A is 2 query p terse if and only if A ....

.... 1) query p superterse. In the special case k = 1, this failure is not so dramatic: a set A is 1 cheatable if and only if A is not 2 query p superterse. Efforts to extend a number of results about 2 query p superterseness to general results about k query p superterseness have been unsuccessful [2, 9, 7, 4]. However, some of those results can be extended to general results about k cheatability. We think that results about 2 query p superterseness are best understood as results about cheatability, not as results about p superterseness. A set A is p superterse (p terse) if A is k query p superterse ....

[Article contains additional citation context not shown here]

R. Beigel. NP-hard sets are p-superterse unless R = NP. TR 88-04, The Johns Hopkins University, Dept. of Computer Science, 1988.

.... dealing with bounded query computations relative to arbitrary oracles have been considered in [BGGO93, Bei88b] Bounded query classes for polynomial time computations relative to an oracle for the Boolean satisfiability problem have also been studied (see, for example, AG88, ABG90, Bei90, Bei91, Bei88a, BH91, CGH 88, KSW87, Wag88, WW85] 2 Notation and terminology We use lower case italic letters to refer to integers and functions, upper case italic letters to refer to sets of integers, upper case roman letters to refer to sets of sets of integers, and boldface letters to refer to ....

Richard Beigel. NP-hard sets are p-superterse unless R = NP. Technical Report 88-04, The Johns Hopkins University, Dept. of Computer Science, 1988.

....sets as can be accepted in polynomial time with k serial queries to an NP set. In fact, the same is true for any class in place of NP that is closed under polynomial time positive bounded truth table reductions. This contrasts with the expected result for function computations with an NP oracle [6]. In addition we show that the Boolean hierarchy and the bounded query hierarchies (of languages) either stand or collapse together. Finally we show that if the Boolean hierarchy collapses to any level but the zeroth (deterministic polynomial time) then for all k there are functions computable in ....

....set and parallel queries to an NP set. Especially in light of Theorem 14.ii, we might wonder whether the converse is true. Namely, is PF NP (2 k Gamma1) tt PF NP k T Since this would imply that NP complete sets are not p superterse, we expect that the answer is no. In fact, we have shown [6] that all NP hard sets (under truth table reductions) are p superterse unless P = UP and R = NP. For the same reason, we do not expect that the second containment in part (ii) can be tightened. Can the first containment in (ii) be tightened 7. Discussion For decision problems, we have shown that ....

[Article contains additional citation context not shown here]

R. Beigel. NP-hard sets are p-superterse unless R = NP. TR 88-04, The Johns Hopkins University, Dept. of Computer Science, 1988.

....to a set A [6, 11] The corresponding complexity classes are called bounded query classes. While the results obtained in this framework do not depend on special properties of the oracle set, they often yield corollaries about computations with a bounded number of queries to an NP oracle [14, 12]; in addition they yield answers to questions ostensibly unrelated to counting oracle queries [13] A natural function to look at in this context is the following: Definition 1.1. If A f0; 1g and k 2 N then F A k : Sigma ) k f0; 1g k is defined by F A k (x 1 ; x k ) ....

....n O(k) s and depth d O(1) that computes F A =n k(n) without making any queries. By applying the lower bound on parity ( 21, 38, 77] we easily obtain Cai s result. 1 Krentel actually showed this just for f(n) 1 2 log n, but his proof can be modified to f(n) 1 Gamma ffl) log n. See [12]. ffl Cai and Hemachandra [27] proved that if P 6= P #P then, for all ffl 1, #SAT is not n ffl enumerable, i.e. there is no polynomial algorithm that, with formula as input, produces n ffl numbers one of which is #SAT( We extend their definition of enumerability to ....

[Article contains additional citation context not shown here]

R. Beigel. NP-hard sets are p-superterse unless R=NP. Technical Report 4, The Johns Hopkins University, Dept. of Computer Science, 1988.

No context found.

R. Beigel. NP-hard Sets are P-Superterse unless R = NP. Technical report 88-04, Dept. of Computer Science, The Johns Hopkins University, 1988.

....Komplexitat und Deduktionssysteme, Universitat Karlsruhe, D 76128 Karlsruhe, Germany, Email: fstephan ira.uka.de) Supported by the Deutsche Forschungsgemeinschaft (DFG) grant Me 672 4 1. The notion p superterse was introduced by Beigel [6] and was studied in several recent papers, e.g. in [2] [8], 10] Originally, it was defined via bounded query classes : A set A is p superterse iff for every k 1 and all oracles X, F A k cannot be computed by any polynomial time oracle Turing machine (OTM) with fewer than k queries to X. Intuitively, there is no way to save a query, regardless of ....

....notion NP hard we have to specify a polynomial time reduction. Previously, it was not even known whether an m complete set for NP must be p superterse if P 6= NP. The following results were the best known. Fact 4. 1 (1) If SAT is p tt reducible to an approximable set then R = NP and P = UP [8]. Independently Toda [33] obtained the special case for p selective sets. 2) If there is an approximable set that is p k tt hard for the Boolean hierarchy then P = NP [2, Corollary 39] 3) If SAT is p 1 tt reducible to a p selective set then P = NP [13, Corollary 15] 4) If SAT is ....

[Article contains additional citation context not shown here]

R. Beigel. NP-hard sets are p-superterse unless R = NP. TR 4, Johns Hopkins Univ., Dept. of C.S., 1988.

No context found.

R. Beigel. NP-hard sets are p-superterse unless R = NP. Technical Report 4, Department of Computer Science, The Johns Hopkins University, 1988.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC