Effective Guessing Has Unlikely Consequences

TL;DR

The paper investigates whether nondeterministic guessing can meaningfully speed up deterministic computations, introducing time-witness classes TIWI and the NTIGU framework to formalize time-plus-witness tradeoffs. It proves two conditional results: a strong effective guessing hypothesis would force P to be strictly contained in NTIME$(n)$ (and related consequences), while a weaker hypothesis would imply SAT belongs to NTIME$(n)$ and yield tighter upper bounds such as SAT ∈ NTIGU$(n \, log\, n, O(n/\log n))$. The authors also establish a concrete SAT upper bound by showing SAT ∈ TIWI$(n\, log\, n, 4n/\log n)$, leveraging precise variable-bound and sorting-time analyses. Collectively, the results suggest that even modest effective guessing would have striking, unlikely implications, and they propose an ineffective guessing conjecture to delimit the potential power of nondeterminism. The work contributes a structured, conditional landscape around speedups via nondeterminism, with concrete implications for SAT and classical complexity separations, and highlights key open questions about the power of nondeterministic time.

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 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 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 well-known O(n(lg n)^c) bound for some constant but undetermined exponent c which is at least 1.

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Theorems & Definitions (47)

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