多项式时间置换下资源有界随机性概念的封闭性
Closure of resource-bounded randomness notions under polynomial time permutations
摘要：
一个无限位序列称为递归随机的，如果没有可计算的策略投注沿着序列有无限的资本。众所周知，递归随机性的性质在可计算置换下是封闭的。我们调查类似的语句定义的投注策略，可计算的资源范围内的随机性概念。假设S是一个多项式时间可计算的置换的字符串集合在一元字母表（标识为N）。如果S的逆不是多项式有界的，则很容易构造多项式时间随机比特序列Z，使得ZoS不是多项式时间随机的。因此，我们应该只考虑满足逆是多项式有界的额外条件的置换S。现在，封闭依赖于复杂性理论中的附加假设。我们的第一个结果表明，如果BPP包含一个超多项式确定性时间类，那么多项式时间随机性不被某些置换S保持，使得事实上S及其逆都在P中。我们的第二个结果表明，多项式空间随机性被具有多项式有界逆的多项式时间置换保持，所以如果P = PSPACE，那么多项式时间随机性被保持。
英文摘要：
An infinite bit sequence is called recursively random if no computable strategy betting along the sequence has unbounded capital. It is well-known that the property of recursive randomness is closed under computable permutations. We investigate analogous statements for randomness notions defined by betting strategies that are computable within resource bounds. Suppose that S is a polynomial time computable permutation of the set of strings over the unary alphabet (identified with N). If the inverse of S is not polynomially bounded, it is easy to build a polynomial time random bit sequence Z such that Z o S is not polynomial time random. So one should only consider permutations S satisfying the extra condition that the inverse is polynomially bounded. Now the closure depends on additional assumptions in complexity theory. Our first result shows that if BPP contains a superpolynomial deterministic time class then polynomial time randomness is not preserved by some permutation S such that in fact both S and its inverse are in P. Our second result shows that polynomial space randomness is preserved by polynomial time permutations with polynomially boun\-ded inverse, so if P=PSPACE then polynomial time randomness is preserved.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
