# The Complexity Theory Companion

Language: English

Pages: 382

ISBN: 3540674195

Format: PDF / Kindle (mobi) / ePub

Here is an accessible, algorithmically oriented guide to some of the most interesting techniques of complexity theory. The book shows that simple algorithms are at the heart of complexity theory. The book is organized by technique rather than by topic. Each chapter focuses on one technique: what it is, and what results and applications it yields.

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2.7. Definition 2.8 is due to Beigel [Bei89] and Fact 2.9 is stated explicitly in Hemaspaandra and Zimand [HZ93], who study the structure and potential collapses of bounded-ambiguity classes such as UP

one weight function W E A such that the minimum-weight sets of F with respect to Ware ambiguous about inclusion of x. For each i, 1 ::; i ::; D - 1, there are R choices for Vi. So, there are at most RD-l weight functions W E Z such that the minimum-weight sets of F with respect to Ware ambiguous about inclusion of x. There are RD weight functions in Z, there are m choices for F, and there are D choices for x. Thus, the proportion of {W E Z I for some i, 1 ::; i ::; m, W is bad for F ii }} is at

(s(lxl)+p(lxl). As equation 4.1 holds for any polynomial r, we can select r so that this amount is less than So, transpose the table: We'll pick v first, then pick y. Then for more than ~ of v it holds that x E L if and only if the number of y such that (x#y)#v E B is odd. Hence we can switch the "BPP" part and the "parity" part. 0 This concludes the proof of Theorem 4.5. We now show below some immediate corollaries to Theorem 4.5. Since BPP ~ P /poly by Corollary 4.7 and pEBP = EBP by part 2 of

known to have the property that #P is closed under them if and only if #P is closed under all polynomial-time operations. Thus, for these operations, it remains possible that #P is closed under them, yet is not closed under proper subtraction. Again returning to the analogy with NP-completeness theory, these operations are in some sense analogous to potentially "intermediate" sets-sets, such as the set of all primes, that are in NP yet are neither known to be NPcomplete nor known to be in P.

[BOGKW88] yields a one-round "perfect zero-knowledge" protocol for each language in NEXP, and Feige and Lovasz [FL92] show that twoprover one-round interactive proof systems exist for all languages in NEXP. We noted in Sect. 6.5 that to construct a multiprover protocol for a EXP language a prover in EXP is sufficient. In other words, the oracle of a probabilistic oracle protocol for EXP languages can be in EXP. Based on this observation, Babai et al. [BFNW93] show that if EXP ~ P /poly then EXP