R. Chang and J. Kadin. On computing Boolean connectives of characteristic functions. Mathematical Systems Theory, 28(3):173-198, May/June 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that an NC circuit can be simulated by a balanced DLOGTIME uniform family of Boolean expressions made up of alternating layers of ANDs and ORs. Because of this fact, it suces to reduce the evaluation problem for these expressions to CTI. The core of the reduction is the simple construction from [18, 23] described in [5] page 45) for the purpose of simulating ANDs and ORs using graph isomorphism questions. We adapt this construction as follows. Consider four trees trees G 1 , G 2 , H 1 , and H 2 colored with two colors black and white (represented by black and white dots in the gures) G 1 G ....

R. Chang, J. Kadin, On computing boolean connectives of characteristic functions, Mathematical Systems Theory 28 (1995) 173-198.

....formulae. For any set A, P A[k] is the class of languages that are recognized by polynomial time Turing machines that access the oracle A at most k times on each input. The class P A[k] tt will allow only nonadaptive access to A. We note that P NP[1] P NP[2] if P NP[1] P NP[2] tt [CK95] so all our results could be stated assuming P NP[1] P NP[2] tt . If A is in P B[1] tt then there is a polynomial time function h(x) f ;g such that x is in A if and only if h(x) z; and z is in B or h(x) z; and z is not in B. The string z is the query made by ....

....If is Hard III then line 3 will accept. 2 Proof of Theorem 5.3(2) P NP = P NP[1] The proof follows from Lemmas 5.10 and 5.11. Lemma 5.10 (Chang Kadin) If P NP[1] P NP[2] then P NP[1] P NP tt . Lemma 5.11 If P NP[1] P NP[2] then P NP tt = P NP . Chang and Kadin [CK95] prove Lemma 5.10 by looking at computation trees. Their proof can not be used to generalize the result to k versus k 1 queries. We present a different proof using hard and easy strings. Chang [Cha97] uses the ideas of our proofs of Lemma 5.10 and 5.11 to extend Theorem 5.3(2) to show P NP[k] ....

R. Chang and J. Kadin. On computing boolean connectives of characteristic functions. Mathematical Systems Theory, 28:173--198, 1995. 9

....class C is closed under conjunctions (disjunctions) if for any finite set fa 1 ; a 2 ; am g of instances a i of some problem A in C, deciding whether each (at least one) a i for 1 i m is a positive instance of A lies in C. Closure under conjunctions and disjunctions was studied in [4], where it is noted that a large number of relevant complexity classes are closed under both conjunctions and disjunctions. In particular, all classes of the Polynomial Hierarchy as well as all relativizations of such classes are closed under conjunctions and under disjunctions. Theorem 4.6 If C ....

R. Chang and J. Kadin. On Computing Boolean Connectives of Characteristic Functions. To appear in Mathematical Systems Theory.

....formulae. For any set A, P A[k] is the class of languages that are recognized by polynomial time Turing machines that access the oracle A at most k times on each input. The class P A[k] tt will allow only nonadaptive access to A. We note that P NP[1] P NP[2] if P NP[1] P NP[2] tt [CK95] so all our results could be stated assuming P NP[1] P NP[2] tt . UP is the set of languages that are recognized by polynomial time nondeterministic Turing machines that have at most one accepting path on each input. We can generalize NP by defining the polynomial time hierarchy. We ....

....then line 2 will accept. If is Hard III then line 3 will accept. 2 Proof of Theorem 5.3(2) The proof follows from Lemmas 5.10 and 5.11. Lemma 5.10 (Chang Kadin) If P NP[1] P NP[2] then P NP[1] P NP tt . Lemma 5.11 If P NP[1] P NP[2] then P NP tt = P NP . Chang and Kadin [CK95] prove Lemma 5.10 by looking at computation trees. Their proof can not be used to generalize the result to k versus k 1 queries. We present a different proof using hard and easy strings. Chang [Cha97] uses the ideas of our proofs of Lemma 5.10 and 5.11 to extend Theorem 5.3(2) to show P NP[k] ....

R. Chang and J. Kadin. On computing boolean connectives of characteristic functions. Mathematical Systems Theory, 28:173--198, 1995.

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R. Chang and J. Kadin. On computing Boolean connectives of characteristic functions. Mathematical Systems Theory, 28(3):173-198, May/June 1995.

.... Hierarchy over 2 [CK96] This was further improved by Beigel, Chang and Ogihara to a class just above 2 [BCO93] Most recently Fortnow, Pavan and Sengupta pushed the collapse below the 2 level [FPS02] tt = PH S 2 : Separately, Chang and Kadin noted that P NP[O(log n) [CK95]. This was further improved by Buhrman and Fortnow to P [BF99] Since S [Cai01] we have the following situation: tt = PH ZPP This is almost a complete upward collapse of PH down to P except for the gap between P and ZPP . Closing this gap might be done with a ....

R. Chang and J. Kadin. On computing Boolean connectives of characteristic functions. Mathematical Systems Theory, 28(3):173-198, May/June 1995.

....Lemma 9, that we want to prove is stated in a very general form. To assist the reader in understanding this lemma, we first sketch a mind change proof on a concrete example. We give some details which help illustrate the statement of Lemma 9. The rest of the proof may be found in the literature [5, 12, 21, 24]. We will also use the following lemma in Section 4. Lemma 8. Let L be any language in . Then, L m L P C 4 L k where L P C and L NP C are respectively the m complete languages for P and DIFF k (NP ) Proof sketch: Let the language L NP m complete for NP . Fix a machine ....

....Proposition 17. Let C be any class such that NP P = Sigma . C and L k be , and DIFF k (NP ) respectively. Using relativized versions of the mind change proof in [5] one can show that L P C 4 L k is k tt (for details consult Lemma 8 and the literature [12, 21, 24] ) Thus P if and only if Fix a polynomial time computable function h that performs that reduction. For each m, we will construct polynomial size advice allowing us to reduce L k to L k on strings of length m. Thus DIFF k (NP ) poly, so NP =poly, so PH ( Sigma . ....

R. Chang and J. Kadin. On computing Boolean connectives of characteristic functions. Technical Report TR 90-1118, Cornell Department of Computer Science, May 1990. To appear in Mathematical Systems Theory.

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R. Chang and J. Kadin. On computing Boolean connectives of characteristic functions. Mathematical Systems Theory, 28(3):173--198, May/June 1995.

....has infinitely many levels, much has been discovered about SATSAT, the D P P m complete set. For example, Kadin [Kad88, CK90a] showed that D P = co D P = PH collapses. So, SATSAT 62 co D P unless PH collapses. Similarly, one can show that SATSAT cannot have OR unless PH collapses [CK90b] These results show that SATSAT does not have the same robust properties of SAT. So, how does USAT, the vv m complete set for D P , compare with the P m complete set As it turns out, Chang and Kadin [CK90c] recently showed that USAT 62 co D P unless PH collapses, USAT does not ....

....SAT to USAT with probability 1=2 1=poly. Then, USAT would be complete for D P in a much stronger sense. In fact, such a theorem would answer the frequently posed question of whether USAT has OR 2 [CH86, GW86, CGH 89, GNW90] It is known that SATSAT does not have OR 2 unless PH collapses [CK90b] Corollary 4. If SAT rp m USAT with probability 1=2 1=p(n) for some polynomial bound p, then USAT does not have OR 2 unless PH collapses. Proof: We know that SAT P m USAT. By assumption, SAT rp m USAT with probability 1=2 1=poly. If USAT has OR 2 , then these two reductions can be ....

R. Chang and J. Kadin. On computing Boolean connectives of characteristic functions. Technical Report 90-1118, Department of Computer Science, Cornell University, May 1990.

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R. Chang and J. Kadin. On computing Boolean connectives of characteristic functions. Mathematical Systems Theory, 28(3):173--198, May/June 1995.

....Lemma 9, that we want to prove is stated in a very general form. To assist the reader in understanding this lemma, we rst sketch a mind change proof on a concrete example. We give some details which help illustrate the statement of Lemma 9. The rest of the proof may be found in the literature [5, 12, 21, 24]. We will also use the following lemma in Section 4. Lemma 8. Let L be any language in P NP k tt C . Then, L p m L P C 4 L k where L P C and L NP C are respectively the p m complete languages for P C and DIFF k (NP C ) Proof sketch: Let the language L NP C be p m ....

.... complete for P, P C , and DIFF k (NP C ) respectively. Using relativized versions of the mind change proof in [5] one can show that L P C 4 L k is p m complete for P NP k tt C and L P 4 L k is p m complete for P NP C k tt (for details consult Lemma 8 and the literature [12, 21, 24] ) Thus P NP C k tt = P NP k tt C if and only if L P C 4 L k p m L P 4 L k : Fix a polynomial time computable function h that performs that reduction. For each m, we will construct polynomial size advice allowing us to reduce L k to L k on strings of length m. Thus DIFF k ....

R. Chang and J. Kadin. On computing Boolean connectives of characteristic functions. Technical Report TR 90-1118, Cornell Department of Computer Science, May 1990. To appear in Mathematical Systems Theory.

No context found.

R. Chang and J. Kadin. On computing Boolean connectives of characteristic functions. Technical Report TR 90-1118, Cornell Department of Computer Science, May 1990. To appear in Mathematical Systems Theory.

....Lemma 9, that we want to prove is stated in a very general form. To assist the reader in understanding this lemma, we first sketch a mind change proof on a concrete example. We give some details which help illustrate the statement of Lemma 9. The rest of the proof may be found in the literature [5, 12, 21, 24]. We will also use the following lemma in Section 4. Lemma 8. Let L be any language in i P NP k tt j C . Then, L p m L P C 4 L k where L P C and L NP C are respectively the p m complete languages for P C and DIFF k (NP C ) Proof sketch: Let the language L NP C be p m ....

....m complete for P, P C , and DIFF k (NP C ) respectively. Using relativized versions of the mind change proof in [5] one can show that L P C 4 L k is p m complete for i P NP k tt j C and LP 4 L k is p m complete for P NP C k tt (for details consult Lemma 8 and the literature [12, 21, 24] ) Thus P NP C k tt = i P NP k tt j C if and only if L P C 4 L k p m LP 4 L k : Fix a polynomial time computable function h that performs that reduction. For each m, we will construct polynomial size advice allowing us to reduce L k to L k on strings of length m. Thus DIFF k (NP ....

R. Chang and J. Kadin. On computing Boolean connectives of characteristic functions. Technical Report TR 90-1118, Cornell Department of Computer Science, May 1990. To appear in Mathematical Systems Theory.

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R. Chang and J. Kadin, On computing Boolean connectives of characteristic functions. Math. Systems Theory 28, 173--198, 1995.

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