More on BPP and the Polynomial-time Hierarchy

doi:10.1016/0020-0190(96)00016-6

Information Processing Letters

Volume 57, Issue 5, 11 March 1996, Pages 237-241

https://doi.org/10.1016/0020-0190(96)00016-6 Get rights and content

Abstract

We simplify known proofs that BPP lies within the Polynomial-time Hierarchy. Furthermore, our simple proof shows that BPP is contained in a complexity class that presumably lies even lower in this Hierarchy than was previously known. Our technique also simplifies the proof that one-sided-error IP equals two-sided-error IP.

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To determine if two lists of numbers are the same set, we sort both lists and see if we get the same result. The sorted list is a canonical form for the equivalence relation of set equality. Other canonical forms arise in graph isomorphism algorithms. To determine if two graphs are cospectral (have the same eigenvalues), we compute their characteristic polynomials and see if they are equal; the characteristic polynomial is a complete invariant for cospectrality. Finally, an equivalence relation may be decidable in P without either a complete invariant or canonical form. Blass and Gurevich (1984) asked whether these conditions on equivalence relations—having an FP canonical form, having an FP complete invariant, and being in P—are distinct. They showed that this question requires non-relativizing techniques to resolve. We extend their results, and give new connections to probabilistic and quantum computation.
Proving SAT does not have small circuits with an application to the two queries problem
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We show that if SAT does not have small circuits, then there must exist a small number of satisfiable formulas such that every small circuit fails to compute satisfiability correctly on at least one of these formulas. We use this result to show that if $P^{NP [1]} = P^{NP [2]}$ , then the polynomial-time hierarchy collapses to $S_{2}^{p} \subseteq Σ_{2}^{p} \cap Π_{2}^{p}$ . Even showing that the hierarchy collapsed to $Σ_{2}^{p}$ remained open prior to this paper.
The consequences of eliminating NP solutions
2008, Computer Science Review
Given a function based on the computation of an $NP$ machine, can one in general eliminate some solutions? That is, can one in general decrease the ambiguity? This simple question remains, even after extensive study by many researchers over many years, mostly unanswered. However, complexity-theoretic consequences and enabling conditions are known. In this tutorial-style article we look at some of those, focusing on the most natural framings: reducing the number of solutions of $NP$ functions, refining the solutions of $NP$ functions, and subtracting from or otherwise shrinking #P functions. We will see how small advice strings are important here, but we also will see how increasing advice size to achieve robustness is central to the proof of a key ambiguity-reduction result for $NP$ functions.
Polylogarithmic-round interactive proofs for coNP collapse the exponential hierarchy
2007, Theoretical Computer Science
If every language in coNP has a constant-round interactive proof system, then the polynomial-time hierarchy collapses [R.B. Boppana, J. Håstad, S. Zachos, Does co-NP have short interactive proofs? Information Processing Letters 25 (2) (1987) 127–132]. On the other hand, the well-known LFKN protocol gives $O (n)$ -round interactive proof systems for all languages in coNP [C. Lund, L. Fortnow, H. Karloff, N. Nisan, Algebraic methods for interactive proof systems, Journal of the Association for Computing Machinery 39 (4) (1992) 859–868]. We consider the question of whether it is possible for coNP to have interactive proof systems with polylogarithmic-round complexity. We show that this is unlikely by proving that if a coNP-complete set has a polylogarithmic-round interactive proof system, then the exponential-time hierarchy collapses. We also consider exponential versions of the Karp–Lipton theorem and Yap’s theorem.
S<inf>2</inf><sup>p</sup> ⊆ ZPP<sup>NP</sup>
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Quantum Polynomial Hierarchies: Karp-Lipton, error reduction, and lower bounds
2024, arXiv

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¹: Supported by grant No. 92–00226 from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.

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More on BPP and the Polynomial-time Hierarchy

Abstract

J. Comput. System Sci.

Inform. Process. Lett.

Theoret. Comput. Sci.

Theoret. Comput. Sci.

The Probabilistic Method