J. Hartmanis and L. Hemachandra. Robust machines accept easy sets. Theoretical Computer Science, (74):217--225, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and familiar notions of decision trees from computational complexity theory. While the results in this paper pertain to models for computation in higher types, the techniques used are closely related to work in Boolean decision tree complexity. Namely, a technique known as Blum s trick ( 2] [6], 21] which is used to show that Boolean functions with small nondeterministic and co nondeterministic complexity have smalldepth decision trees is generalized to show that in certain cases, sequential functionals can efficiently simulate continuous functionals. A lower bound on Boolean ....

....C sequential, a construction which is efficient, relative to B and C, is possible. This implies that, for a certain natural class of moduli, continuity and sequentiality do coincide. 4. An efficient simulation The result presented in this section is a generalization of Blum s trick , 2] [6], 21] which relates certificate size and decision tree complexity for Boolean functions, to the case of the type two functionals considered here. A similar proof is given in [8] but for a special case which is described in detail below. We begin with the following simple fact about ....

J. Hartmanis and L. Hemachandra. Robust machines accept easy sets. Theoretical Computer Science, (74):217--225, 1990.

....we only require that the probabilities be bounded away for that particular A. 8 Proposition 4.1 The classes NP coNP and BPP are irregular. Thus there exist oracles A and B such that NP 6= NP coNP) A] and BPP B 6= BPP[B] Proof. To show that NP coNP is irregular, we use the fact ( 3] [14], 26] 29] that (NP coNP) A] P NP PhiA for any oracle set A, where A Phi B = df f0x j x 2 Ag [ f1x j x 2 Bg. Thus it suffices to construct A so that NP , where SAT is any NP complete problem. Baker, Gill, and Solovay [1] construct an oracle A separating P and NP . This ....

Hartmanis, J., and Hemachandra, L. A. (1990), Robust machines accept easy sets, Theoret. Comput. Sci. 74, 217--225.

....and familiar notions of decision trees from computational complexity theory. While the results in this paper pertain to models for computation in higher types, the techniques used are closely related to work in Boolean decision tree complexity. Namely, a technique known as Blum s trick ( 2] [6], 21] which is used to show that Boolean functions with small nondeterministic and co nondeterministic complexity have smalldepth decision trees is generalized to show that in certain cases, sequential functionals can efficiently simulate continuous functionals. A lower bound on Boolean ....

....C sequential, a construction which is efficient, relative to B and C, is possible. This implies that, for a certain natural class of moduli, continuity and sequentiality do coincide. 4. An efficient simulation The result presented in this section is a generalization of Blum s trick , 2] [6], 21] which relates certificate size and decision tree complexity for Boolean functions, to the case of the type two functionals considered here. A similar proof is given in [8] but for a special case which is described in detail below. We begin with the following simple fact about ....

J. Hartmanis and L. Hemachandra. Robust machines accept easy sets. Theoretical Computer Science, (74):217--225, 1990.

....time Turing machines with access to an oracle (see, e.g. 16, page 294] and [11, Section 5. 3] for formal treatment) Bounds on the boolean decision trees complexity are useful tools in constructing oracles with desired relations between Turing complexity classes and in proving conditional results [2, 7, 8]. Conversely, all the facts proven for the corresponding Turing complexity classes that hold true under any oracle can be directly carried over decision trees. We mention three examples. 1. Arthur Merlin games are as powerful as a general interactive proof system [6] 2. The error in an ....

....1 , the maximum number of zeroes of f that differ from some one of f in disjoint blocks of variables. This is a simple extension of the bound r(f) Omega Gamma35 (f) from [12] Note that bound (4) together with relations nd(f) bs(f) bs( f) 5) and d(f) nd(f) nd( f) 6) proven in [12] and [2, 7, 14], respectively, implies the relation d(f) O(ip(f) 2 ip( f) 2 ) which is a qualitative generalization of (1) and (6) We suggest also a bound that is in a sense tighter. Namely, ip(f) sep(f) 2; 7) where sep(f) is a combinatorial characteristic of a boolean function that we call ....

J. Hartmanis and L. Hemachandra. Robust machines accept easy sets. Theoretical Computer Science, 74(2):217--226, 1990.

....in particular, the power to nd sets of inverses of honest polynomial time many one functions) Thus the above proof in fact proves Theorem 4. 6 below, whose oracle access mechanism is exactly that used in de ning the extended low two sets [5] a mechanism that appears in other applications also [14]. Of particular note is that the set L is as in Theorem 4.3 but unlike Theorem 4.7 queried only polynomially often. Theorem 4.6. If L 2 R p 2 tt (SPARSE) then there exists a sparse set S 0 such that: L p 5 tt S 0 , and S 0 2 P NP L . The above results are all conditioned upon ....

J. Hartmanis and L. Hemachandra, Robust machines accept easy sets, Theoretical Computer Science, 74 (1990), pp. 217-226.

....similar to an argument due to Riis [ which is itself similar to the proof that if a Boolean function and its negation both can be written in disjunctive normal form with terms of size d, then the function has a Boolean decision tree of height d 2 . This last result was implicit in [HH87] [HH90]. BI87] Tar] and appears explicitly in [IN88] We describe H j implicitly as a strategy for querying the purported matching ff. The strategy proceeds in at most k stages, and makes at most 2k queries in each stage. Let s represent the set of known edges of G at the beginning of stage s. Then ....

J. Hartmanis and L. A. Hemachandra, Robust machines accept easy sets, Theoret. Comput. Sci. 74 (1990), 217--225.

....by Schoning [26] We are interested in robustly unambiguous OTM s OTM s that behave unambiguously for every oracle. Hartmanis and Hemachandra showed that if a polynomial time bounded OTM is unambiguous for all oracles, then it accepts a language in P NP PhiA relative to all oracles A [14]. For a class L we denote by UP L the class of languages recognized by robustly unambiguous OTM s with an oracle in L, i.e. UP L = fL(M;A) j A 2 L; M is robustly unambiguous and polynomial time boundedg: Don t confuse UP L with a promise class that occurs only as an oracle. We show that ....

J. Hartmanis and L. A. Hemachandra. Robust machines accept easy sets. Theoretical Computer Science, 74(2):217--226, 1990.

....there are degrees in between. A similar term to that of independence are studied in the setting of standard Turing machines. Together with the concept of helping, the notion of robustness of oracle machines was introduced by Schoning [Sch85] and subsequently investigated in various ways, e.g. [Ko87, Sch88, HH90, AKS95, NRS95]. A robust oracle machine is a standard oracle Turing machine accepting the same set relative to every oracle. Because of Theorem 5.5, our study of independence is loosely connected with the notion of robust oracle machine, and in such sense independences can be viewed as some kind of partial ....

J. Hartmanis and L. A. Hemachandra. Robust machines accept easy sets. Theoretical Computer Science, 74(2):217--225, 1990.

....certificate. This approach will not work for time bounded computability. All is not lost, however. A converse of this implication may be obtained by extending results techniques which deal with 0 1 valued oracles, developed independently by a number of researchers (Blum and Impagliazzo 1987, Hartmanis and Hemachandra 1990, Tardos 1989) to the case of N valued oracles. We will now consider this approach. 4 Decision tree algorithms Continuity is essentially a non deterministic notion. That is, the value of a functional on some input depends on the existence of a certificate. However, there is no requirement that ....

J. Hartmanis and L. A. Hemachandra, Robust machines accept easy sets, Theoret Comput. Sci., 74 (1990), 217--225.

....the property that for no input does the machine have more than two accepting paths (see [Wat88] Let T A = f0 n j (9y) jyj = n and y 2 S A ]g 2 UP A 2 NP A . Now, using the standard type of oracle argument as to how hard it is for a Turing machine to maintain unambiguity (see, e.g. [HH90]) we can easily choose A to satisfy these conditions and yet al..so ensure that S A 6 P wi8 9r T A (via ensuring through a stage construction that each potential reducer either fails to appropriately reduce or reduces a string having one witness to one having two witnesses) Theorem 3.11 ....

J. Hartmanis and L. Hemachandra. Robust machines accept easy sets. Theoretical Computer Science, 74(2):217--226, 1990.

....2.2 For each well defined reduction b a , let b a denote f(A; B) j A b a Bg. Proposition 2.3 p T RS T U T O T SN T . Using different terminology, robust underproductivity (though not U T ) has been introduced into the literature by Beigel ( Bei89] see also [HH90] and the following theorem will be of use in the present paper. Theorem 2.4 ( Bei89] see also [HH90] If NPTM N is robustly underproductive, then (8A) 9L 2 P SAT PhiA ) L rej (N A ) L L(N A ) Theorem 2.4 says that if a machine is robustly underproductive, then for every oracle ....

....p T RS T U T O T SN T . Using different terminology, robust underproductivity (though not U T ) has been introduced into the literature by Beigel ( Bei89] see also [HH90] and the following theorem will be of use in the present paper. Theorem 2. 4 ( Bei89] see also [HH90] If NPTM N is robustly underproductive, then (8A) 9L 2 P SAT PhiA ) L rej (N A ) L L(N A ) Theorem 2.4 says that if a machine is robustly underproductive, then for every oracle there is a relatively simple set that separates its acceptance set from its L rej set. In particular, if P ....

[Article contains additional citation context not shown here]

J. Hartmanis and L. Hemachandra. Robust machines accept easy sets. Theoretical Computer Science, 74(2):217--226, 1990.

....operator, Phi, and the symmetric difference operator, Delta, are similarly overloaded. Note, crucially, that whether a machine is categorical or not depends on its oracle. In fact, it is well known that machines that are categorical with respect to all oracles accept only easy languages [HH90] and create a polynomial hierarchy analog that is completely contained in a low level of the polynomial hierarchy (Allender and Hemaspaandra as cited in [HR92] Thus, when we speak of a UPOM, we will simply mean an NPOM that, with the oracle the machine has in the context being discussed, ....

J. Hartmanis and L. Hemachandra. Robust machines accept easy sets. Theoretical Computer Science, 74(2):217--226, 1990.

....Proof. If C is DT IME[2 log O(1) n ] second category, then it is also DSPACE[log O(1) n] second category. Now use Lemmas 3, 4, 5. 2 No complete p m degree that is noncollapsing is known. There are relativized worlds in which non callpsing complete p m degrees for various classes exist (see [HH90], Kur83] HS92] KMR89] In all these constructions, the non collapse is achieved by building an oracle relative to which there is a C p m complete set A having large gaps, i.e. such that there are infinitely many n with fx 2 A j n jxj 2 n g = Our next proposition, which is a ....

J. Hartmanis and L. Hemachandra. Robust machines accept easy sets. Theoretical Computer Science, 74(2):217--226, 1990.

....in particular, the power to find sets of inverses of honest polynomial time many one functions) Thus the above proof in fact proves Theorem 4. 6 below, whose oracle access mechanism is exactly that used in defining the extended low two sets [5] a mechanism that appears in other applications also [14]. Of particular note is that the set L is as in Theorem 4.3 but unlike Theorem 4.7 queried only polynomially often. Theorem 4.6. If L 2 R p 2 Gammatt (SPARSE) then there exists a sparse set S 0 such that: ffl L p 5 Gammatt S 0 , and ffl S 0 2 P NP PhiL . The above results are ....

J. Hartmanis and L. Hemachandra, Robust machines accept easy sets, Theoretical Computer Science, 74 (1990), pp. 217--226.

No context found.

J. Hartmanis and L. Hemachandra. Robust machines accept easy sets. Theoretical Computer Science, (74):217--225, 1990.

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