Effective guessing has unlikely consequences
Abstract
A classic result of Paul, Pippenger, Szemerédi and Trotter states that DTIME(n) is strictly contained in NTIME(n). The natural question then arises: could DTIME(t(n)) be contained in NTIME(n) for some superlinear timeconstructible function t(n)? If such a function t(n) does exist, then there also exist effective nondeterministic guessing strategies to speed up deterministic computations. In this work, we prove limitations on the effectiveness of nondeterministic guessing to speed up deterministic computations by showing that the existence of effective nondeterministic guessing strategies would have unlikely consequences. In particular, we show that if a subpolynomial amount of nondeterministic guessing could be used to speed up deterministic computation by a polynomial factor, then P is strictly contained in NTIME(n). Furthermore, even achieving a logarithmic speedup at the cost of making every step nondeterministic would show that SAT is in NTIME(n) under appropriate encodings. Of possibly independent interest, under such encodings we also show that SAT can be decided in O(n lg n) steps on a nondeterministic multitape Turing machine, improving on the wellknown O(n(lg n)^c) bound for some constant but undetermined exponent c which is at least 1.
 Publication:

arXiv eprints
 Pub Date:
 November 2021
 arXiv:
 arXiv:2111.02138
 Bibcode:
 2021arXiv211102138S
 Keywords:

 Computer Science  Computational Complexity
 EPrint:
 21 pages, submitted for journal review