Apakah ada versi berkelanjutan dari teorema pengulangan paralel


13

Raz Paralel pretition teorema adalah hasil yang penting dalam PCP, inapproximation, dll Teorema ini fomalized sebagai berikut.

G=(S,T,A,B,π,V)S,T,A,BπS×TV:S×T×A×B{0,1}

v(G)=maxhAHA,hBHBs,tπ(s,t)V(s,t,hA(s),hB(t))
And n-fold game Gn=(Sn,Tn,An,Bn,πn,Vn). The theorem says if v(G)1ϵ, then v(Gn)(1ϵc)Ω(nlogmax{|A|,|B|}).

My quesion is what happen if the sets are infinite, in a continuous space. Say if S,T,A,B are subsets of a space, say Rn, or more abstract spaces. All the rest are same. Raz's theorem only gives a trivial upper bound 1 since the sizes of answer sets are infinite. Obviously n-fold value is upper bounded by single copy. Does exponential decrease also happen in continuous case? Would it be more interesting to restrict HA,HB to be collections of continuous functions or C functions or measureable functions?

Jawaban:


8

Does exponential decrease also happen in continuous case?

No. Feige and Verbitsky [FV02] showed that for every n, there is a game G (with finite sets of questions and answers) such that v(G)≤3/4 and v(Gn)≥1/8. Because your formulation generalizes games with finite sets of questions and answers of any size, parallel repetition (of any finitely many times) cannot decrease the value of a game from 3/4 to 1/8.

[FV02] Uriel Feige and Oleg Verbitsky. Error reduction by parallel repetition—A negative result. Combinatorica, 22(4):461–478, Oct. 2002. doi:10.1007/s00493-002-0001-0.

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