Turing machines with few accepting computations and low sets for PP

Köbler, Johannes; Schöning, Uwe; Toda, Seinosuke; Torán, Jacobo

doi:10.1016/0022-0000(92)90022-b

Cited by 53 publications

(27 citation statements)

References 16 publications

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“…Clearly, Corollary 3.5 represents an improvement on the trivial inclusion FewP ⊆ NP. However, how does it compare with the nontrivial result of Köbler et al [30] and Fenner, Fortnow, and Kurtz [20] that FewP ⊆ Few ⊆ SPP ⊆ ⊕P ∩ C = P? Informally stated, Few [15] is what a P machine can do given one call to a #P function that obeys the promise that its value is always at most polynomial.…”

Section: Discussionmentioning

confidence: 98%

“…3 Our proof technique builds (e.g., by adding a rate-of-growth argument) on that used by Cai and Hemachandra [15] to prove FewP ⊆ ⊕P, where ⊕P [23,32] is the class of languages L such that for some nondeterministic polynomial-time Turing machine N , on each x it holds that x ∈ L ⇐⇒ #acc N (x) ≡ 1 (mod 2). We note that Köbler, Schöning, Toda, and Torán [30] interestingly built on that technique in their proof that FewP ⊆ C = P, where C = P [39] is the class of languages L such that there is a polynomial-time function f and 3 Though this result is stated in a relatively general format, we mention in passing that even the restriction employed can be relaxed to the case of nonempty sets of positive integers for which, for some uniform constant, given any integer in the set finding another larger but at most multiplicatively-constantly-larger integer in the set is a polynomial-time task. One can even slightly relax the growth rate, but one has to be very careful to avoid a "bootstrapping" growth-explosion effect via clocking growth rates always with respect to the input.…”

Section: Resultsmentioning

confidence: 99%

“…(Note: UP ⊆ UP ≤2 ⊆ UP ≤3 ⊆ · · · ⊆ UP O(1) ⊆ FewP ⊆ NP, where UP O(1) = k≥1 UP ≤k .) These classes are also connected to the existence of one-way functions and have been extensively studied in a wide variety of contexts, such as class containments [20,30], complete sets [28], reducibilities [27], boolean hierarchy equivalences [29], complexity-theoretic analogs of Rice's Theorem [12], and upward separations [33].…”

Section: Introductionmentioning

confidence: 99%

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Restrictive Acceptance Suffices for Equivalence Problems

Borchert¹,

Hemaspaandra²,

Rothe³

2000

LMS J. Comput. Math.

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One way of suggesting that an NP problem may not be NP-complete is to show that it is in the class UP. We suggest an analogous new approach-weaker in strength of evidence but more broadly applicable-to suggesting that concrete NP problems are not NP-complete. In particular we introduce the class EP, the subclass of NP consisting of those languages accepted by NP machines that when they accept always have a number of accepting paths that is a power of two. Since if any NP-complete set is in EP then all NP sets are in EP, it follows-with whatever degree of strength one believes that EP differs from NP-that membership in EP can be viewed as evidence that a problem is not NP-complete.We show that the negation equivalence problem for OBDDs (ordered binary decision diagrams [22,13]) and the interchange equivalence problem for 2-dags are in EP. We also show that for boolean negation [25] the equivalence problem is in EP NP , thus tightening the existing NP NP upper bound. We show that FewP [2], bounded ambiguity polynomial time, is contained in EP, a result that is not known to follow from the previous SPP upper bound. For the three problems and classes just mentioned with regard to EP, no proof of membership/containment in UP is known, and for the problem just mentioned with regard to EP NP , no proof of membership in UP NP is known. Thus, EP is indeed a tool that gives evidence against NP-completeness in natural cases where UP cannot currently be applied.

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Section: Discussionmentioning

confidence: 98%

Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Restrictive Acceptance Suffices for Equivalence Problems

Borchert¹,

Hemaspaandra²,

Rothe³

2000

LMS J. Comput. Math.

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“…Additionally, SPP has nice closure properties [10]: P SPP = SPP SPP = SPP. Another class that is low for PP [14] is BPP (the class of languages with polynomial-time randomized algorithms with error probability bounded by, say, 1/3.) Subsequently, the complexity class AWPP was introduced 1 in [9].…”

Section: Spp and Other Counting Complexity Classesmentioning

confidence: 99%

On the Complexity of Computing Units in a Number Field

Arvind

Kurur

2004

Lecture Notes in Computer Science

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Given an algebraic number field K, such that [K : Q] is constant, we show that the problem of computing the units group O * K is in the complexity class SPP. As a consequence, we show that principal ideal testing for an ideal in OK is in SPP. Furthermore, assuming the GRH, the class number of K, and a presentation for the class group of K can also be computed in SPP. A corollary of our result is that solving PELL S EQUATION, recently shown by Hallgren [12] to have a quantum polynomial-time algorithm, is also in SPP.

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“…Proof: The first implication follows along the lines of the previous theorem. The conséquence Few = P follows from [37] using the containment Few Ç P FewP [26], and USAT G co-NP follows from [20].…”

Section: [13]: a Promise Problem Is A Pair Of Sets (Q R) A Set L Ismentioning

confidence: 99%