Hierarchy Theorems for Probabilistic Polynomial Time

Fortnow, Lance; Santhanam, Rahul

doi:10.1109/focs.2004.33

Cited by 43 publications

(56 citation statements)

References 22 publications

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“…A related line of research has shown strong time and space hierarchy theorems for randomised and other semantic classes which use advice [FS04,FST05,vMP07,KvM08]. All these results are not known to hold in the classical advice-free setting.…”

Section: Definition 3 ( [Bkm09]mentioning

confidence: 97%

Different Approaches to Proof Systems

Beyersdorff

Müller

2010

Lecture Notes in Computer Science

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Abstract. The classical approach to proof complexity perceives proof systems as deterministic, uniform, surjective, polynomial-time computable functions that map strings to (propositional) tautologies. This approach has been intensively studied since the late 70's and a lot of progress has been made. During the last years research was started investigating alternative notions of proof systems. There are interesting results stemming from dropping the uniformity requirement, allowing oracle access, using quantum computations, or employing probabilism. These lead to different notions of proof systems for which we survey recent results in this paper.

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Section: Definition 3 ( [Bkm09]mentioning

confidence: 97%

Different Approaches to Proof Systems

Beyersdorff

Müller

2010

Lecture Notes in Computer Science

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“…The advice requirement in Barak's result was subsequently improved by Fortnow and Santhanam [21] by making use of our construction of a PSPACEcomplete self-correctible and downward self-reducible problem. More recent improvements have reduced the amount of advice and extended the techniques to other classes [25,22,42].…”

Section: Conclusion and Subsequent Workmentioning

confidence: 99%

Pseudorandomness and Average-Case Complexity Via Uniform Reductions

Trevisan

Vadhan

2007

comput. complex.

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Abstract. Impagliazzo and Wigderson (1998) gave the first construction of pseudorandom generators from a uniform complexity assumption on EXP (namely EXP = BPP). Unlike results in the nonuniform setting, their result does not provide a continuous trade-off between worstcase hardness and pseudorandomness, nor does it explicitly establish an average-case hardness result. In this paper:• We obtain an optimal worst-case to average-case connection for EXP: if EXP ⊆ BPTIME(t(n)), then EXP has problems that cannot be solved on a fraction 1/2 + 1/t (n) of the inputs by BPTIME(t (n)) algorithms, for t = t Ω(1) .• We exhibit a PSPACE-complete self-correctible and downward self-reducible problem. This slightly simplifies and strengthens the proof of Impagliazzo and Wigderson, which used a #P-complete problem with these properties.• We argue that the results of Impagliazzo and Wigderson, and the ones in this paper, cannot be proved via "black-box" uniform reductions.

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“…Since instance checkers are the sole culprit of nonrelativization in Barak's argument, our proof shows that that component is not critical for obtaining a time hierarchy for randomized machines with two-sided error and one bit of advice. Apart from yielding stronger results and being more widely applicable, our approach also provides considerably simpler proofs for all the hierarchy theorems with one bit of advice that were known before [FS04,GST04,FST05]. We refer to Section 2 for a more detailed comparison of techniques.…”

Section: Theorem 2 For Any Reasonable Semantic Model Of Computation mentioning

confidence: 99%

“…This involves another level of padding to encode the a(n) < log n bits of the original advice for M in the padding length and using the one bit of new advice to indicate whether the padding length is valid. See [FS04,GST04] for the details. This way, we obtain a language which randomized machines with two-sided error can decide in polynomial time and one bit of advice but not in time n c and a log n bits of advice.…”

Section: Previous Workmentioning

confidence: 99%

“…In other words, he established the result for randomized machines with two-sided error whose descriptions can depend on the input length n in such a way that the size of the variable part is bounded by a(n). Subsequently, several authors tried to get as close as possible to the desired uniform result and managed to reduce the amount of advice to a single bit [FS04,GST04]. Barak's argument also applies to quantum machines with two-sided error but not to any of the other semantic models on our list.…”

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confidence: 99%

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A Generic Time Hierarchy for Semantic Models with One Bit of Advice

Melkebeek

Pervyshev

21st Annual IEEE Conference on Computational Complexity (CCC'06)

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Abstract. We show that for any reasonable semantic model of computation and for any positive integer a and rationals 1 ≤ c < d, there exists a language computable in time n d with a bits of advice but not in time n c with a bits of advice. A semantic model is one for which there exists a computable enumeration that contains all machines in the model but may also contain others. We call such a model reasonable if it has an efficient universal machine that can be complemented within the model in exponential time and if it is efficiently closed under deterministic transducers. Our result implies the first such hierarchy theorem for randomized machines with zero-sided error, quantum machines with one-or zero-sided error, unambiguous machines, symmetric alternation, Arthur-Merlin games of any signature, etc. Our argument yields considerably simpler proofs of known hierarchy theorems with one bit of advice for randomized and quantum machines with two-sided error. Our paradigm also allows us to derive stronger separation results in a unified way. For models that have an efficient universal machine that can be simulated deterministically in exponential time and that are efficiently closed under randomized reductions with two-sided error, we establish the following: For any constants a and c, there exists a language computable in polynomial time with one bit of advice but not in time n c with a log n bits of advice. In particular, we obtain such separation for randomized and quantum machines with two-sided error. For randomized machines with one-sided error, we get that for any constants a and c there exists a language computable in polynomial time with one bit of advice but not in time n c with a(log n) 1/c bits of advice.

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Hierarchy Theorems for Probabilistic Polynomial Time

Cited by 43 publications

References 22 publications

Different Approaches to Proof Systems

Different Approaches to Proof Systems

Pseudorandomness and Average-Case Complexity Via Uniform Reductions

A Generic Time Hierarchy for Semantic Models with One Bit of Advice

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