A classic result of Paul, Pippenger, Szemerédi and Trotter states that \({\textsf {DTIME}}(n) \subsetneq {\textsf {NTIME}}(n)\). The natural question then arises: could the inclusion \({\textsf {DTIME}}(t(n)) \subseteq {\textsf {NTIME}}(n)\) 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 \({\textsf {P}}~ \subsetneq {\textsf {NTIME}}(n)\). 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(n log n) steps on a nondeterministic multitape Turing machine, improving on the well-known O(n(log n)^c) bound for some constant but undetermined exponent c ≥ 1.

Paul、Pippenger、Szemerédi 和 Trotter 的一个经典结果指出\({\textsf {DTIME}}(n) \subsetneq {\textsf {NTIME}}(n)\)。那么自然的问题就出现了：包含\({\textsf {DTIME}}(t(n)) \subseteq {\textsf {NTIME}}(n)\) 是否适用于一些超线性时间可构造函数t ( n ) ？如果这样的函数t ( n) 确实存在，那么也存在有效的非确定性猜测策略来加速确定性计算。在这项工作中，我们通过证明有效的非确定性猜测策略的存在不会产生不太可能的后果，证明了非确定性猜测在加速确定性计算方面的有效性的局限性。特别是，我们表明，如果次多项式数量的不确定性猜测可用于通过多项式因子加速确定性计算，则\({\textsf {P}}~ \subsetneq {\textsf {NTIME}}(n)\ )。此外，即使以使每一步都不确定为代价实现对数加速，也会表明 SAT ∈ NTIME( n) 在适当的编码下。出于可能独立的兴趣，在这种编码下，我们还表明 SAT 可以在非确定性多磁带图灵机上以O ( n l o g n ) 步骤确定，改进了众所周知的O ( n ( l o g n ) ^c )绑定一些常数但未确定的指数c ≥ 1。