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9.3. PROBABILISTICALLY CHECKABLE PROOF SYSTEMS use none none integrating toinsert none in noneqr code builder sql .net upper-bounded b none for none y the logarithm of the (effective) length of the proofs employed (provided we allow non-uniform veri ers; see Exercise 9.16). On the role of randomness.

The PCP Theorem (i.e., N P PCP(log, O(1))) asserts that a meaningful probabilistic evaluation of proofs is possible based on a constant number of examined bits.

We note that, unless P = N P, such a phenomenon is impossible when requiring the veri er to be deterministic. Firstly, note that PCP(0, O(1)) = P holds (as a r special case of PCP(r, q) DTIME(22 q+r poly); see Exercise 9.17).

Secondly, as shown in Exercise 9.19, P = N P implies that N P is not contained in PCP(o(log), o(log)). Lastly, assuming that not all NP-sets have NP-proof systems that employ proofs of length (e.

g., (n) = n), it follows that if 2r (n) q(n) < (n) then PCP(r, q) does not contain N P (see Exercise 9.17 again).

. .NET Framework 2.0 9.3.2. The Power of Probabilistically Checkable Proofs The celebrated none none PCP Theorem asserts that N P = PCP(log, O(1)), and this result is indeed the focus of the current section. But before getting to it we make several simple observations regarding the PCP hierarchy. We rst note that PCP(poly, 0) equals coRP, whereas PCP(0, poly) equals N P.

It is easy to prove an upper bound on the non-deterministic time complexity of sets in the PCP hierarchy (see Exercise 9.17): Proposition 9.15 (upper bounds on the power of PCPs): For every polynomially bounded integer function r , it holds that PCP(r, poly) NTIME(2r poly).

In particular, PCP(log, poly) N P. The focus on PCP systems of logarithmic randomness complexity re ects an interest in PCP systems that utilize proof oracles of polynomial length (see discussion in Section 9.3.

1). We stress that such PCP systems (i.e.

, PCP(log, q)) are NP-proof systems with a (potentially amazing) extra property: The validity of the assertion can be probabilistically evaluated by examining a (small) portion (i.e., q(n) bits) of the proof.

Thus, for any xed polynomially bounded function q, a result of the form N P PCP(log, q) (9.6). is interesting (because it applies also to NP-sets having witnesses of length exceeding q). Needless to say, the smaller q the better. The PCP Theorem asserts the amazing fact by which q can be made a constant.

Theorem 9.16 (the PCP Theorem): N P PCP(log, O(1)). Thus, probabilistically checkable proofs in which the veri er tosses only logarithmically many coins and makes only a constant number of queries exist for every set in N P.

This constant is essentially three (see 9.3.4.

1). Before reviewing the proof of Theorem 9.16, we make a couple of comments.

Ef cient transformation of NP-witnesses to PCP oracles. The proof of Theorem 9.16 is constructive in the sense that it allows for ef ciently transforming any NP-witness (for an instance of a set in N P) into an oracle that makes the PCP veri er accept (with.

PROBABILISTIC PROOF SYSTEMS probability 1). That is, for every (NP-witness relation) R PC there exists a PCP veri er V as in Theorem 9.16 and a polynomial-time computable function such that for every (x, y) R the veri er V always accepts the input x when given oracle access to the proof (x, y) (i.

e., Pr[V (x,y) (x) = 1] = 1). Recalling that the latter oracles are themselves NPproofs, it follows that NP-proofs can be transformed into NP-proofs that offer a trade-off between the portion of the proof being read and the con dence it offers.

Speci cally, for every > 0, if one is willing to tolerate an error probability of then it suf ces to examine O(log(1/ )) bits of the (transformed) NP-proof. Indeed (as discussed in Section 9.3.

1), these bit locations need to be selected at random. The foregoing strengthening of Theorem 9.16 offers a wider range of applications than Theorem 9.

16 itself. Indeed, Theorem 9.16 itself suf ces for negative applications such as establishing the infeasibility of certain approximation problems (see Section 9.

3.3). But for positive applications (see 9.

3.4.2), typically some user (or a real entity) will be required to actually construct the PCP-oracle, and in such cases the strengthening of Theorem 9.

16 will be useful. A characterization of NP. Combining Theorem 9.

16 with Proposition 9.15 we obtain the following characterization of N P. Corollary 9.

17 (the PCP characterization of NP): N P = PCP(log, O(1)). Road map for the proof of the PCP Theorem. Theorem 9.

16 is a culmination of a sequence of remarkable works, each establishing meaningful and increasingly stronger versions of Eq. (9.6).

A presentation of the full proof of Theorem 9.16 is beyond the scope of the current work (and is, in our opinion, unsuitable for a basic course in Complexity Theory). Instead, we present an overview of the original proof (see 9.

3.2.2) as well as of an alternative proof (see 9.

3.2.3), which was found more than a decade later.

We will start, however, by presenting a weaker result that is used in both proofs of Theorem 9.16 and is also of independent interest. This weaker result (see 9.

3.2.1) asserts that every NPset has a PCP system with constant query complexity (albeit with polynomial randomness complexity); that is, N P PCP(poly, O(1)).

. Teaching note: none for none In our opinion, presenting in class any part of the proof of the PCP Theorem should be given low priority. In particular, presenting the connections between PCP and the complexity of approximation should be given a higher priority. As for relative priorities among the following three subsections, we strongly recommend giving 9.

3.2.1 the highest priority, because it offers a direct demonstration of the power of PCPs.

As for the two alternative proofs of the PCP Theorem itself, our recommendation depends on the intended goal. On the one hand, for the purpose of merely giving a taste of the ideas involved in the proof, we prefer an overview of the original proof (provided in 9.3.

2.2). On the other hand, for the purpose of actually providing a full proof, we de nitely prefer the new proof (which is only outlined in 9.

3.2.3).

. 9.3.2.

1. Provin g That N P PCP(poly, O(1)) The fact that every NP-set has a PCP system with constant query complexity (regardless of its randomness complexity) already testi es to the power of PCP systems. It asserts that probabilistic veri cation of proofs is possible by inspecting very few locations in a.

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