Self-Specifying Machines

Hemaspaandra, Lane A.; Hempel, Harald; Wechsung, Gerd

doi:10.1142/s0129054199000198

Int. J. Found. Comput. Sci.

1999

DOI: 10.1142/s0129054199000198

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Self-Specifying Machines

Lane A. Hemaspaandra

Harald Hempel

Gerd Wechsung

Abstract: We study the computational power of machines that specify their own acceptance types, and show that they accept exactly the languages that ≤ #P m -reduce to NP sets. A natural variant accepts exactly the languages that ≤ #P m -reduce to P sets. We show that these two classes coincide if and only if P #P[1] = P #P[1]:NP[O(1)] , where the latter class denotes the sets acceptable via at most one question to #P followed by at most a constant number of questions to NP.

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“…The work of this paper (especially Theorem 3.6), which first appeared in [21], should be compared with the independent work of Agrawal, Beigel, and Thierauf, which first appeared in [1]. In particular, let P BH j [1]:BH k [1] + denote the class of languages recognized by some polynomial-time machine making one query to a BH j oracle followed by one query to a BH k oracle and accepting if and only if the second query is answered "yes."…”

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Query Order

Hemaspaandra¹,

Hempel²,

Wechsung³

1998

SIAM J. Comput.

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We study the effect of query order on computational power, and show that P BH j [1]:BH k [1] -the languages computable via a polynomial-time machine given one query to the jth level of the boolean hierarchy followed by one query to the kth level of the boolean hierarchy-equals R p j+2k−1-tt (NP) if j is even and k is odd, and equals R p j+2k-tt (NP) otherwise. Thus, unless the polynomial hierarchy collapses, it holds that for each 1 ≤ j ≤ k:We extend our analysis to apply to more general query classes. IntroductionThis paper studies the importance of query order. Everyone knows that it makes more sense to first look up in your on-line datebook the date of the yearly computer science conference and then phone your travel agent to get tickets, as opposed to first phoning your travel agent (without knowing the date) and then consulting your on-line datebook to find the date. In real life, order matters.This paper seeks to determine-for the first time to the best of our knowledge-whether one's everyday-life intuition that order matters carries over to complexity theory.In particular, for classes C 1 and C 2 from the boolean hierarchy [10,11], we ask whether one question to C 1 followed by one question to C 2 is more powerful than one question to C 2 followed by one question to C 1 . That is, we seek the relative powers of the classes P C 1 [1]:C 2 [1] and P C 2 [1]:C 1 [1] . As is standard [26], for any constant m we say A is m-truth-table reducible to B (A ≤ p m-tt B) if there is a polynomial-time computable function that, on each input x, computes both

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Query Order

Hemaspaandra¹,

Hempel²,

Wechsung³

1998

SIAM J. Comput.

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“…This is a property that seems to be deeply dependent on the "const"-ness. For example, it is not known whether P #P[1] = P #P[2] , and indeed it is known that if this seemingly unlikely equality holds then two complexity classes associated with self-specifying machines are equal[18].…”

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A second step towards complexity-theoretic analogs of Rice's Theorem

Hemaspaandra

Rothe

2000

Theoretical Computer Science

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Rice's Theorem states that every nontrivial language property of the recursively enumerable sets is undecidable. Borchert and Stephan [4] initiated the search for complexity-theoretic analogs of Rice's Theorem. In particular, they proved that every nontrivial counting property of circuits is UP-hard, and that a number of closely related problems are SPP-hard.The present paper studies whether their UP-hardness result itself can be improved to SPP-hardness. We show that their UP-hardness result cannot be strengthened to SPP-hardness unless unlikely complexity class containments hold. Nonetheless, we prove that every P-constructibly bi-infinite counting property of circuits is SPP-hard. We also raise their general lower bound from unambiguous nondeterminism to constant-ambiguity nondeterminism. *

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Self-Specifying Machines

Cited by 2 publications

References 50 publications

Query Order

Query Order

A second step towards complexity-theoretic analogs of Rice's Theorem

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