AbstractWe show that the total space in resolution, as well as in any other reasonable proof system, is equal (up to a polynomial and $${(\log n)^{O(1)}}$$(logn)O(1) factors) to the minimum refutation depth. In particular, all these variants of total space are equivalent in this sense. The same conclusion holds for variable space as long as we penalize for excessively (that is, super-exponential) long proofs, which makes the question about equivalence of variable space and depth about the same as the question of (non)-existence of “supercritical” tradeoffs between the variable space and the proof length. We provide a partial negative answer to this question: for all $${s(n) \leq n^{1/2}}$$s(n)≤n1/2 there exist CNF contradictions $${\tau_n}$$τn that possess refutations with variable space s(n) but such that every refutation of $${\tau_n}$$τn with variable space $${o(s^2)}$$o(s2) must have double exponential length $${2^{2^{\Omega(s)}}}$$22Ω(s). We also include a much weaker tradeoff result between variable space and depth in the opposite range $${s(n) \ll \log n}$$s(n)≪logn and show that no supercritical tradeoff is possible in this range.

中文翻译：

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关于分辨率的空间和深度

摘要我们表明分辨率的总空间以及任何其他合理证明系统中的总空间是相等的（直到多项式和 $${(\log n)^{O(1)}}$$(logn)O( 1) 因素) 到最小反驳深度。特别是，总空间的所有这些变体在这个意义上是等价的。只要我们惩罚过长的（即超指数的）长证明，同样的结论就适用于变量空间，这使得关于变量空间和深度的等价性的问题与“不存在”的问题大致相同超临界”变量空间和证明长度之间的权衡。我们对这个问题给出了部分否定的答案：对于所有 $${s(n) \leq n^{1/2}}$$s(n)≤n1/2 都存在 CNF 矛盾 $${\tau_n}$$τn ，它们具有可变空间 s( n) 但是这样每个对 $${\tau_n}$$τn 的反驳都必须有双指数长度 $${2^{2^{ \Omega(s)}}}$$22Ω(s)。我们还在相反的范围 $${s(n) \ll \log n}$$s(n)≪logn 中包含了可变空间和深度之间更弱的权衡结果，并表明在这个范围内不可能有超临界权衡。