More on PCP itself

We start by discussing variants of the PCP characterization of NP, and next turn to PCPs having expressing power beyond NP.

More on the PCP characterization of NP. Interestingly, the two com- plexity measures in the PCP-characterization of N P can be traded off, so that at the extremes we getN P=PCP(log, O(1)) andN P=PCP(0,poly), respectively.

Proposition 2.15 : There exist constantsα, β >0 such that for every inte- ger functionl(_·), so that 0_≤l(n)_≤αlog2n,

N P=_PCP(r(_·), q(_·)),

12_{Actually, it is not essential to use this fact, since one can easily convert any adap-}

tive system into a non-adaptive one while incurring an exponential blowup in the query complexity (which in our case is a constant).

wherer(n) =α·log2n−l(n)andq(n) =β·2l(n).

Proof Idea: Starting with Theorem 2.12, one tries all possibilities for the

l(n)-long prefix of the random tape of the verifier.

The above simple observation is but the tip of an iceberg. In the years which have passed since the establishment of Theorem 2.12 many far more interesting and technically involved facts regarding the PCP characterization of NP were discovered. Following is a brief summary of the various (still active) research directions.

• The length of PCPs: By definition, the number of possible different oracle queries in a PCP(log,log) system is polynomial (in the length of the input). Actually, in the PCP systems of Theorem 2.12 these queries refer only to a polynomially long prefix of the oracle, and so we may say that the length of these PCPs forN P is polynomial. It is known that the length of PCPs for_{N P}can be made nearly-linear [308].

• The number of queries in PCPs: Theorem 2.12 asserts that a constant number of queries suffice for PCPs with logarithmic randomness and soundness error 1/2 (for NP). It is currently known that this constant is at most 5 (whereas with 3 queries one may get arbitrary close to error 1/2) [208]. Allowing an arbitrary small constant error in the com- pleteness condition, 3 queries are sufficient [213] (and necessary, unless

P =_{N P}). The obvious trade-off between the number of queries and the soundness error gives rise to the robust notion of amortized query complexitydefined as the ratio of the number of queries and (minus) the logarithm (to based 2) of the soundness error. It is known that PCPs of logarithmic randomness and amortized query complexity 1 exist only for sets in_P [41]. On the other hand, PCPs of logarithmic randomness and amortized query complexity 2.5 +ǫexists for all_{N P}and anyǫ >0 (see [208] presenting a 5-query system of error 0.25 +ǫ). In case one allows arbitrary small constant error probability in the completeness condition, the amortized query complexity is practically 1 (since, 1 is again a lower bound, and 1 +ǫis an upper bound, for anyǫ >0) [328].

• The free-bit complexity: The motivation to this notion came from the PCP–MaxClique connection, but we find it intriguing for its own sake. Loosely speaking, here one distinguishes queries for which the verifier compares the answer with a value determined by previously obtained answers, from queries in which the verifier only records the answer for future usage [141]. The latter queries are called free (as the “acceptable answers” to them are not determined). Theamortized free-bit complexity is define analogously [55]. Interestingly, _{N P} has PCPs with logarithmic randomness and amortized free-bit complexity less than any positive constant (cf., H˚astad [212]).

2.4. PROBABILISTICALLY CHECKABLE PROOF SYSTEMS 63

• Adaptive versus non-adaptive: A PCP verifier is callednon-adaptive if its queries are determined solely based on its input and the outcome of its coin tosses. (A general verifier, called adaptive, may determine its queries also based on previously received oracle answers.) Recall that the PCP Characterization of NP (i.e., Theorem 2.12) is obtained using a non-adaptive verifier; however, it turns out that adaptive verifierare

more powerful than non-adaptive ones (in terms of quantitative results): Specifically, for everyǫ >0 and logarithmic randomness, (adaptive) 3- query PCPs with soundness error 0.5 +ǫ exist for_{N P} [208], whereas non-adaptive verifiers making 3 queries and having soundness error 5/8 exist only for_P [370].

• Non-binary queries: Our definition of PCP allows only binary queries. Certainly, non-binary queries can always be coded as binary ones, but the converse is not necessarily valid, in particular in adversarial settings. Note that the soundness condition constitutes an implicit adversarial setting, where a bad proof may be thought of as being selected by an adversary. Thus, when several binary queries are packed into one non- binary query, the adversary need not respect the packing (i.e., it may answer inconsistently on the same binary query depending on the other queries packed with it). For this reason, “parallel repetition” is highly non-trivial in the PCP (as well as the MIP) setting; see [316]. Still, using adequate “consistency tests” one may construct PCP systems for

N P using logarithmic randomness, a constant number of queries and soundness error exponential in the length of the answers (cf., [318] as well as [22]). (Currently, this is known only for sub-logarithmic answer lengths.) We comment that 2 non-binary queries are known to be less powerful (in terms of quantitative results) than an equivalent number of binary queries [346].

PCP with super-logarithmic randomness. The above text has focused on the important case where the verifier tosses logarithmically many coins, and hence the “effective proof length” is polynomial. Here we shortly mention that the above main results scale up as follows.

Proposition 2.16 (Proposition 2.11 – Generalized): For every integer func- tion r(_·), the class _PCP(r(_·),poly)is contained inNtime(2O(r(·)+log(·))_).

Theorem 2.17 (Theorem 2.12 – Generalized): Let t(_·) be an integer func- tion so that n < t(n)<2poly(n)_{, for all n’s. Then, the class Ntime(t(}

·)) is contained in the class_PCP(O(logt(_·)), O(1)).

We comment that_{N P}is unlikely to be in_PCP(o(log), o(log)) since_PCP(o(log), o(log)) =

P will follow (by iteratively applying the FGLSS-reduction [140] to Max- Clique, cf., [21]).

2.4.5

The Role of Randomness

No trade-off, between the number of bits examined and the confidence, is pos- sible if one requires the verifier to be deterministic. In particular,PCP(0, q(·)) contains only sets that are decidable by a deterministic algorithms of running time 2q(n)

·poly(n). It follows that_PCP(0,log) =_P. Furthermore, since it is unlikely that all NP-sets can be decided by (deterministic) algorithms of running time, say, 2n

·poly(n), it follows that_PCP(0, n) is unlikely to contain

N P.