Effective Guessing Has Unlikely Consequences

Abstract

AbstractA classic result of Paul, Pippenger, Szemerédi and Trotter states that ${\textsf {DTIME}}(n) \subsetneq {\textsf {NTIME}}(n)$ DTIME ( n ) ⫋ NTIME ( n ) . The natural question then arises: could the inclusion ${\textsf {DTIME}}(t(n)) \subseteq {\textsf {NTIME}}(n)$ DTIME ( t ( n ) ) ⊆ NTIME ( n ) hold for some superlinear time-constructible 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 ${\textsf {P}}~ \subsetneq {\textsf {NTIME}}(n)$ P ⫋ NTIME ( n ) . Furthermore, even achieving a logarithmic speedup at the cost of making every step nondeterministic would show that SAT ∈NTIME(n) under appropriate encodings. Of possibly independent interest, under such encodings we also show that SAT can be decided in O(nlogn) steps on a nondeterministic multitape Turing machine, improving on the well-known O(n(logn)c) bound for some constant but undetermined exponent c ≥ 1.