Question: [40] Is it possible that #P problems have solutions of low timebounded Kolmogorov complexity relative to the input? Prove that if there is a polynomial-time

[40] Is it possible that #P problems have solutions of low timebounded Kolmogorov complexity relative to the input? Prove that if there is a polynomial-time Turing machine that on an input that is a Boolean formula

f, prints out a polynomial-sized list of numbers among which one number is the number of solutions of f (though we may not know which one), then P = P#P. This gives strong evidence that in general, the number of solutions of a Boolean formula has high timebounded Kolmogorov complexity relative to the formula.

Comments. Source: [J.-Y. Cai and L. Hemachandra, Inform. Process.

Lett., 38(1991), 215–219]. Hint: use A. Shamir’s polynomial interpolation technique in [Proc. 31st IEEE Found. Comput. Sci., 1990, pp. 11–15].

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