E.W. Allender and O. Watanabe, Kolmogorov complexity and degrees of tally sets, Information and Computation 86 (1990), pp. 160--178.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....that EXP is replaced by E and NEXP by NE throughout. Some simple algebra, together with the assumption that the class is contained in E=lin (rather than EXP=poly) allows us to achieve the collapse to E. 2 The results in this section relate to some of those obtained by Allender and Watanabe in [1] . There they considered a property Q which posits that for every honest function f : Sigma 0 there is a polynomial time computable weak inverse g such that for all x 2 f ( Sigma ) f(g(x) x. They prove that property Q is equivalent to the property of NE computations that, Every NE ....

....prove that property Q is equivalent to the property of NE computations that, Every NE predicate is E solvable . Meaning that, given an NE predicate R, there is an E computable function which, for any input x to R, computes a witness to R(x) if one exists. Allender and Watanabe (Proposition 2, [1] ) proved that E = E NP )Q) E = NE . It is not known if any of these arrows are reversible. Our results, when looked at in a similar light, can be seen as running parallel to theirs. Consider the property Q 0 stating that for every honest function f : Sigma 0 , the function g which ....

Allender E. & O. Watanabe. Kolmogorov Complexity and Degrees of Tally Sets. Information and Computation, 86 (1990) pp160--187.

.... Kolmogorov and Uspenskii [19] and Li and Vitanyi [21] In this paper we are primarily concerned with resource bounded Kolmogorov complexities, which have been investigated by Hartmanis [10] Sipser [39] Ko [17] Longpr e [22] Balc azar and Book [3] Huynh [13] Lutz [24] Allender and Watanabe [2], and many others. Martin Lof [29] showed that K(xjn) the conditional Kolmogorov complexity of infinite binary sequences x, exhibits a strong Shannon effect. Specifically, Martin Lof proved that if the series 1 X n=0 2 Gammaf (n) converges (e.g. if f(n) ff log n for some real ff 1) ....

E.W. Allender and O. Watanabe, Kolmogorov complexity and degrees of tally sets, Information and Computation 86 (1990), pp. 160--178.

....P T (SPARSE) classes was again shown much later when coding complexity was studied more explicitly, as described below. Isomorphism Degrees Book and Tang and their followers obtained many other classi cation results, just as the work of Book and Ko studied many other reduction classes. See, e.g. [AH92, AHOW91, AW90, TB88, TB91] for these results. We omit most of them here, but we will brie y mention what happens at the other end of the scale: the strongest degrees, de ned by polynomial time isomorphisms, applied to tally sets. Indeed, tally strings are the most natural examples of words of low Kolmogorov complexity ....

....Speci cally, consider the following purely complexity theoretic working hypothesis: Accepting computations for nondeterministic exponential time machines can be constructed deterministically in exponential time. This has been called sometimes hypothesis Q , and has been studied in depth in [AW90]. It is obviously stronger than the equality of deterministic and nondeterministic exponential time, but weaker than P = NP. More precisely, it is an intermediate step [IT89] of the Sewelson s conjecture; that is, E = NE implies Q, and Q implies E = E NP , and Sewelson conjectured that E = NE ....

[Article contains additional citation context not shown here]

Allender E, Watanabe O. Kolmogorov complexity and degrees of tally sets. Information and Computation 1990;86:160-178.

....T (SPARSE) classes was again shown much later when coding complexity was studied more explicitly, as described below. Isomorphism Degrees Book and Tang and their followers obtained many other classification results, just as the work of Book and Ko studied many other reduction classes. See, e.g. [AH92, AHOW91, AW90, TB88, TB91] for these results. We omit most of them here, but we will briefly mention what happens at the other end of the scale: the strongest degrees, defined by polynomial time isomorphisms, applied to tally sets. Indeed, tally strings are the most natural examples of words of low Kolmogorov complexity ....

....Specifically, consider the following purely complexity theoretic working hypothesis: Accepting computations for nondeterministic exponential time machines can be constructed deterministically in exponential time. This has been called sometimes hypothesis Q , and has been studied in depth in [AW90]. It is obviously stronger than the equality of deterministic and nondeterministic exponential time, but weaker than P = NP. More precisely, it is an intermediate step [IT89] of the Sewelson s conjecture; that is, E = NE implies Q, and Q implies E = E NP , and Sewelson conjectured that E = NE ....

[Article contains additional citation context not shown here]

Allender E, Watanabe O. Kolmogorov complexity and degrees of tally sets. Information and Computation 1990;86:160--178.

....for lowtally sets remains open. To provide some context, let us mention the paper by Book and Ko [12] There, the classes of sets that can be reduced to sparse and tally sets under different notions of reducibilities are studied. On the other hand, Tang and Book [26] and Allender and Watanabe [3] studied sets that are not only reducible to arbitrary tally and sparse languages, but also inter reducible with them. With the same approach, we consider here reduction and equivalence classes to tally2 and lowtally sets, and study the relationships between them: although the truth table and the ....

E. Allender and O. Watanabe. Kolmogorov Complexity and Degrees of Tally Sets. Information and Computation, 86(2):160--178, 1990. 2 A direct proof exists although we prefer to mention previous results.

....is self reducible if and only if there exists a deterministic polynomialtime oracle machine M , such that the following holds: 1. A = L(M;A) 2. On each input of length n, M queries the oracle only about strings of length at most n Gamma 1. Hypothesis Q was introduced by Allender and Watanabe [2]. Q is a short name for the following statement, which is not known to hold: for each nondeterministic linear exponential time (2 o(n) machine accepting an input, it is possible to compute, deterministically, in linear exponential time, one accepting computation 1 . Note that trivially Q ....

E. Allender and O. Watanabe. Kolmogorov Complexity and Degrees of Tally Sets. Information and Computation, 86(2):160--178, 1990.

....main theorem below is that such a proof should not relativize. Specifically, our contribution here reduces the strength of the complexity theoretic hypothesis needed to obtain learnability, weakening the presence of an NP oracle to the so called hypothesis Q, introduced by Allender and Watanabe [AW90]. Hypothesis Q is short for the following statement, which is not known to hold: for each nondeterministic linear exponential time machine accepting an input, it is possible to compute deterministically, in linear exponential time, one accepting computation. Note that Q trivially implies that ....

....the disjunctive self reducibility of SATISFIABILITY (SAT) an NP complete problem. It turns out that Q has a number of beautiful and varied characterizations, in terms of various concepts like the bounded truth table degrees of tally sets and the invertibility of certain honest computations; see [AW90]. We give a new one here: this is our main theorem. Some preliminaries are needed for the proof, but for the time being we assume that the reader is familiar with the notation in [BBH95] Our proofs have some degree of missing details, in that specific names should be introduced for the constants ....

Eric Allender and Osamu Watanabe. Kolmogorov complexity and degrees of tally sets. Information and Computation, 86(2):160--178, June 1990.

....informally (formal definitions are given in Chapter 2) denote the notion solvable in deterministic time 2 O(n) The question Is every NE predicate E solvable is the natural exponential time analog of the so called witness finding question regarding NP and P. It was studied initially by [AW88] Even though the P = NP question is usually formulated as a question about the complexity of recognizing languages, it is equivalent to the question of witness finding (e.g. finding a satisfying assignment, instead of merely reporting that a satisfying assignment exists) However, for ....

....to time t(n) if 1. the set fx : 9y)R(x; y)g is infinite, and 2. for all f computable in time t(n) the set fx : R(x; f(x) g is finite. Call an NE predicate E immune if it is immune to Dtime(2 O(n) NE predicates and P printability are related by the following lemma by Allender and Watanabe [AW88] Lemma 2.2 Every infinite set in P has an infinite P printable subset ( No NEpredicate is E immune. Proof. Suppose every infinite set in P has an infinite P printable subset. Let R be an NE predicate defined by the NE machine M that accepts an infinite language. Let L = fx#y : R(x; y)g ....

[Article contains additional citation context not shown here]

E. Allender and O. Watanabe. Kolmogorov complexity and degrees of tally sets. In Proc. 3rd Structure in Complexity Theory Conference, pages 102-- 111, 1988.

....of reduction. Such a study was begun in [TB 88] Following their definitions, define E r (SPARSE) E r (TALLY ) to be the class of sets L such that there exists some sparse (tally) set S such that L# r S and S# r L. In addition to [TB 88] results concerning these classes may be found in [AW 90]. Our paper makes brief mention of the classes APT ( almost polynomial time , the class of sets that can be accepted by machines whose running time is polynomial outside of some sparse set) MP 79] and P close (the class of sets that are the symmetric di#erence of a set in P and a sparse set) ....

....sets. These results help clarify the structure underlying the low hierarchies. As the tables indicate, further progress on placing sets at the correct level of the low hierarchies may hinge on resolving the question of whether P poly = ET (SPARSE ) This question, mentioned both in [TB 88] and in [AW 90], is a special case of the more general question of whether it is the case for any standard reducibility r that P r (SPARSE) E r (SPARSE ) Recent results by the authors suggest that these questions may be quite di#cult, since, for example, one can show that P=NP =# Pm (SPARSE) Em (SPARSE ) ....

E. Allender and O. Watanabe, Kolmogorov complexity and the degrees of tally sets, Information and Control 86, 160--178.

....computation of M on input x (if such a string exists) In the case where the running time is 2 O(n) we call this an NE search problem. It is clear that if every NEXP search problem is solvable in deterministic exponential time, then NEXP = EXP; however the converse is not known to hold [AW90,IT89,BFL01]. 2.1 Levin s Kt Complexity There can be no doubt that the question of how to find accepting computations of a nondeterministic machine is of central importance to computer science. Levin observed that there is an easy to compute ordering on # # giving an essentiallyoptimal search strategy ....

Eric Allender and Osamu Watanabe. Kolmogorov complexity and degrees of tally sets. Information and Computation, 86:160--178, 1990.

.... Additionally, equivalence has been used by Balc azar and Book to characterize completely a natural subset of R p T (SPARSE) namely the sets with self producible circuits [4] The study of equivalence to sparse sets and the study of reducibility to sparse sets have each yielded a urry of results [9,31,10,16,18,20,3,1,21]. Nonetheless, many of the most basic questions have remained unanswered and, in some cases, unasked. In particular, the relationships between equivalence and reducibility to sparse sets have remained wholly 1 Though formal de nitions will be given in Section 2, it is useful to introduce some ....

....open questions in complexity theory. First, we present some de nitions. A function f is weakly invertible if there is a polynomial time computable function h such that f(h(x) x for all x 2 range(f ) Let E denote S k 0 DTIME[2 kn ] and let NE denote S k 0 NTIME[2 kn ] It is shown in [3] that the following are equivalent: 1. Every NE predicate is E solvable. 2. Every honest polynomial time computable function f : 0 is weakly invertible. 3. E p m (TALLY) S f g = E p 1 tt (TALLY) 4. E p m (TALLY) S f g = E p btt (TALLY) 6 Condition 1 above is the ....

[Article contains additional citation context not shown here]

E. Allender and O. Watanabe, Kolmogorov complexity and the degrees of tally sets, Information and Computation, 86 (1990), pp. 160-178.

....notion of immunity. Definition 5. An NE predicate R is immune with respect to time t(n) if (1) the set fx : 9y R(x; y)g is infinite, and (2) for all f computable in time t(n) the set fx : R(x; f(x) g is finite. The connections between NE predicates and Kolmogorov complexity were first drawn in [AW90]; the following theorem is a slight generalization of the results presented there. In short, it says that there are hard NE predicates if and only if there are sets L in P such that KL (n) grows quickly. Time Bounded Kolmogorov Complexity in Complexity Theory 5 Theorem 6. a) Every NE predicate ....

....have nontrivial growth rate is very closely related to the E=NE question. It is natural to ask if it is in fact equivalent to E=NE. Note that if every NE predicate is solvable in exponential time, then E=NE is a trivial consequence; does the converse hold This question was explicitly raised in [AW90] as a result of an investigation using Kolmogorov complexity as a tool for answering certain questions concerning classes of sets equivalent to tally sets 3 under varying notions of reducibility. 3 A set is a tally set if it is a subset of 0 3 . Eric Allender (See [Boo92] for a ....

[Article contains additional citation context not shown here]

E. Allender and O. Watanabe. Kolmogorov complexity and degrees of tally sets. Inform. and Computation 86:160--178, 1990.

....notion of immunity. Definition5. An NE predicate R is immune with respect to time t(n) if (1) the set fx : 9y R(x; y)g is infinite, and (2) for all f computable in time t(n) the set fx : R(x; f(x) g is finite. The connections between NE predicates and Kolmogorov complexity were first drawn in [AW90]; the following theorem is a slight generalization of the results presented there. In short, it says that there are hard NE predicates if and only if there are sets L in P such that KL (n) grows quickly. Time Bounded Kolmogorov Complexity in Complexity Theory 5 Theorem 6. a) Every NE predicate ....

....have nontrivial growth rate is very closely related to the E=NE question. It is natural to ask if it is in fact equivalent to E=NE. Note that if every NE predicate is solvable in exponential time, then E=NE is a trivial consequence; does the converse hold This question was explicitly raised in [AW90] as a result of an investigation using Kolmogorov complexity as a tool for answering certain questions concerning classes of sets equivalent to tally sets 3 under varying notions of reducibility. 3 A set is a tally set if it is a subset of 0 . Eric Allender (See [Boo92] for a ....

[Article contains additional citation context not shown here]

E. Allender and O. Watanabe. Kolmogorov complexity and degrees of tally sets. Inform. and Computation 86:160--178, 1990.

....are not always clearly equivalent. For example, the E=NE question is the natural exponential time analog of the P=NP question, using the language recognition framework. The related witness finding question: Is every NE predicate solvable in exponential time was initially studied in [AW90]. In [IT89] it was shown that there is an oracle relative to which E=NE but not all NE predicates are solvable in exponential time. Thus at least in some relativized worlds, assuming that all NE predicates are solvable in exponential time is strictly stronger than merely assuming E=NE. We will ....

....Proposition 7 For any oracle A, the following are equivalent: a) Every NE A predicate is E A solvable. b) For every set L in P A , K A L (n) O(log n) c) K A S(A) n) O(log n) Proof: The implication (a) b) is proved (in the unrelativized case) as Theorem 6 in [Al92] see also [AW90, Theorem 4]) Since S(A) is clearly in P A , it follows that (b) implies (c) For the remaining implication, suppose K A S(A) n) c log n for some constant c. To solve the NE A predicate defined by the NE A machine M A i , we can use the following algorithm: On input x, let m = hi; xi. Since K A ....

[Article contains additional citation context not shown here]

E. Allender and O. Watanabe. Kolmogorov complexity and degrees of tally sets. Information and Computation 86 (1990) 160--178.

....T (SPARSE) classes was again shown much later when coding complexity was studied more explicitly, as described below. Isomorphism Degrees Book and Tang and their followers obtained many other classification results, just as the work of Book and Ko studied many other reduction classes. See, e.g. [AH92, AHOW91, AW90, TB88, TB91] for these results. We omit most of them here, but we will briefly mention what happens at the other end of the scale: the strongest degrees, defined by polynomial time isomorphisms, applied to tally sets. Indeed, tally strings are the most natural examples of words of low Kolmogorov complexity ....

....Specifically, consider the following purely complexity theoretic working hypothesis: Accepting computations for nondeterministic exponential time machines can be constructed deterministically in exponential time. This has been called sometimes hypothesis Q , and has been studied in depth in [AW90]. It is obviously stronger than the equality of deterministic and nondeterministic exponential time, but weaker than P = NP. More precisely, it is an intermediate step [IT89] of the Sewelson s conjecture; that is, E = NE implies Q, and Q implies E = E NP , and Sewelson conjectured that E = NE ....

[Article contains additional citation context not shown here]

Allender E, Watanabe O. Kolmogorov complexity and degrees of tally sets. Information and Computation 1990;86:160--178.

....p r (SPARSE) as the class of sets L such that, for some sparse set S, L p r S, and (2) R p r (TALLY) as the class of sets L such that, for some tally set T , L p r T . The study of equivalence to sparse sets and the study of reducibility to sparse sets have each yielded a flurry of results [9,31,10,16,18,20,3,1,21]. Nonetheless, many of the most basic questions have remained unanswered and, in some cases, unasked. In particular, the relationships between equivalence and reducibility to sparse sets have remained wholly unknown. The first results along this line are those of the present paper and the ....

....open questions in complexity theory. First, we present some definitions. A function f is weakly invertible if there is a polynomial time computable function h such that f(h(x) x for all x 2 range(f ) Let E denote S k0 DTIME[2 kn ] and let NE denote S k0 NTIME[2 kn ] It is shown in [3] that the following are equivalent: 1. Every NE predicate is E solvable. 2. Every honest polynomial time computable function f : Sigma 0 is weakly invertible. 3. E p m (TALLY) S f Sigma g = E p 1 Gammatt (TALLY) 4. E p m (TALLY) S f Sigma g = E p btt (TALLY) Condition 1 ....

[Article contains additional citation context not shown here]

E. Allender and O. Watanabe, Kolmogorov complexity and the degrees of tally sets, Information and Computation, 86 (1990), pp. 160--178.

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E.W. Allender and O. Watanabe, Kolmogorov complexity and degrees of tally sets, Information and Computation 86 (1990), pp. 160--178.

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E. W. Allender and O. Watanabe, Kolmogorov complexity and degrees of tally sets, Information and Computation 86 (1990), pp. 160--178.

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E. W. Allender and O. Watanabe. Kolmogorov complexity and degrees of tally sets. Information and Computation 86:160--178, 1990.

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