Osamu Watanabe. Personal communication, 1994. 21  Home/Search   Document Not in Database   Summary   Related Articles   Check

....polynomial hierarchy. Open Question 2.4 Suppose Hypothesis U is relaxed to allow h to be 2 valued (or even polynomially valued) rather than single valued. Does this relaxed hypothesis still collapse the polynomial hierarchy to NP This question is currently open (and interesting ) Watanabe [Wat94] has taken a first step towards a resolution. He shows that one may relax the hypothesis to polynomially valued functions if one adds a certain constraint on the behavior of h. Below, we quickly summarize some of the other main results of Hemaspaandra et al. HNOS94] Definition 2.5 1. KL80] ....

O. Watanabe, May 1994. Personal Communication.

....thresholds is a good assignment that eliminates a large portion of the functions. The probabilistic estimation of two threshold functions employs the approximate uniform generation of Jerrum et al. JVV86] Finally, we show an application of our results to structural complexity theory. Watanabe [W94] has observed that our results about learning with equivalence queries and an NP oracle imply the following complexity theoretic result: if every NP set has polynomial size circuits Kushilevitz and Mansour [KM91] have shown that depth O(logn) decision trees are exactly learnable with high ....

....[AHK93] have shown that monotone readonce formulas are exactly learnable from membership queries alone. The proof of Theorem 22(1) rules out the possibility for monotone read twice DNF (since the CNF size might be exponentially large) 7 Applications to Structural Complexity Theory Watanabe [W94] has observed that a consequence of Theorem 7(b) in Section 3 is an improvement of the following result of Karp and Lipton [KL80] Theorem 23 [KL80] If every NP set has polynomial size circuits then the polynomial hierarchy collapses to Sigma 2 . It is clear that ZPP is contained in ....

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Osamu Watanabe. Personal communication, 1994. 23

....7.5 demonstrate Turing machines that correctly compute in nitely many accepting computations of M in 2 n time, we have a contradiction to Hypothesis H 0 . Thus L is not p m complete for NP. Next we note some extensions. These extensions are independenty observed by Regan and Watanabe [71]. In Hypothesis H we can replace the UP machine by an NP machine under a stronger intractability assumption. Consider the following hypothesis: There is a NP machine M that accepts 0 such that 1. no probabilistic polynomial time bounded Turing machine correctly outputs in nitely many ....

K. Regan and O. Watanabe. Personal communication.

....can say that if the size of the order is o the use of memory is o S for the whole graph with S nodes. Chapter 5 Partial matching 5. 1 Definition Partial matching is a different matching technique that can be used instead of treematching in the matching step of mapping, in this case DAG mapping[Wat98] The power of partial matching is that it is only looking one stage back from the root. It tries to find new partial matches out of partial matches that are already found at the inputs of the root cell. It is possible to use partial information to calculate the partial cost and store the best ....

Y. Watanabe, March 1998. Personal Communication.

....and Rogers [FR94] obtained an oracle relative to which NP #= co NP and Hypothesis H # fails. We know of no oracle relative to which P #= NP and every # P T complete set is # P m complete. 4. 4 Extensions The extensions in this section are independently observed by Regan and Watanabe [RW01]. In Hypothesis H we can replace the UP machine by an NP machine under a stronger intractability assumption. Consider the following hypothesis: There is a NP machine M that accepts 0 # such that 1. no probabilistic polynomial time bounded Turing machine correctly outputs infinitely many accepting ....

K. Regan and O. Watanabe. Personal communication. 17

....that primality and irredundancy of a cube in a cover, is sufficient for the corresponding gate to be robustly gate delay fault testable. This would imply that each prime in the cover contains an relatively essential vertex that is distance 1 from the OFF set. However, Touati and Watanabe [32] have given a two level circuit that is prime and irredundant which contains a prime whose only relatively essential vertex is, in fact, distance 2 from the OFF set. This same function is given as Figure 2 4 in [4] Thus, as was the case for robust path delay fault testability modified covering ....

H. Touati and Y. Watanabe. personal communication, 1989.

....on the number of membership queries required (regardless of the computational power) Finally, we show that any class of functions that is learnable from membership query with unlimited computational power is learnable from membership query with an NP oracle in expected polynomial time. Watanabe [W94] has observed that our results about learning with equivalence queries and an NP oracle imply the following complexity theoretic result: if every NP set has polynomial size circuits then the polynomial hierarchy collapses to ZPP NP . This improves the previous result of Karp and Lipton [KL80] ....

....we assert that any membership query c must be such that c 2 A. Otherwise if c = 2 A then m(c Phi a) 1 (since c Phi a 6j 0) for all a 2 A. So now the adversary may say YES for all a 2 A asked by the learner except for the last one.2 8 Applications to Structural Complexity Theory Watanabe [W94] has observed that a consequence of Theorem 7(b) in Section 3 is an improvement of the following result of Karp and Lipton [KL80] Theorem 23 [KL80] If every NP set has polynomial size circuits then the polynomial hierarchy collapses to Sigma p 2 . It is clear that ZPP NP is contained in ....

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Osamu Watanabe. Personal communication, 1994. 21

....technique, but computationally expensive. In addition, it is difficult to decide automatically which nodes to choose to optimize. Finally, an attempt to duplicate the results presented in that work indicated that the area savings achieved could be achieved by well known combinational techniques [Wat93]. In [MSBSV91, Mal90] a method was given that applies existing techniques for combinational optimization to optimize signals that span over several time periods. Termed retiming and resynthesis, the basic idea is to push all registers to the boundary of the logic (peripherally retime) optimize ....

Yosinori Watanabe, June 1993. Personal communication.

....f(r) Since A f in P(D) and since by assumption, P(D) AP FP , it follows from Corollary 4.3 that the result follows from the assumption that P(D) AP FP . By using results from [BCG 96, KW95, Lip91, FF93, GKR 95, KT96] it is easy to derive the following corollary. Corollary 4.5 1. Wat96] If Delta p 2 AP FP then every self reducible set in P poly is in ZPP . 2. If NP AP FP(NP) then every self reducible set in P poly is in ZPP. 3. For K 2 fP(PP) MP;ModP ; PSPACEg, K is not contained in AP FP unless K = P. Proof. 1. Since, as shown in [BCG 96, KW95] every ....

O. Watanabe, 1996. Personal communication.

....thresholds is a good assignment that eliminates a large portion of the functions. The probabilistic estimation of two threshold functions employs the approximate uniform generation of Jerrum et al. JVV86] Finally, we show an application of our results to structural complexity theory. Watanabe [W94] has observed that our results about learning with equivalence queries and an NP oracle imply the following complexity theoretic result: if every NP set has polynomial size circuits 2 Kushilevitz and Mansour [KM91] have shown that depth O(logn) decision trees are exactly learnable with high ....

....[AHK93] have shown that monotone readonce formulas are exactly learnable from membership queries alone. The proof of Theorem 22(1) rules out the possibility for monotone read twice DNF (since the CNF size might be exponentially large) 7 Applications to Structural Complexity Theory Watanabe [W94] has observed that a consequence of Theorem 7(b) in Section 3 is an improvement of the following result of Karp and Lipton [KL80] Theorem 23 [KL80] If every NP set has polynomial size circuits then the polynomial hierarchy collapses to Sigma p 2 . It is clear that ZPP NP is contained in ....

[Article contains additional citation context not shown here]

Osamu Watanabe. Personal communication, 1994.

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Osamu Watanabe. Personal communication, 1994. 21

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