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Some remarks on the alternating hierarchy and closure under complement for sublogarithmic space. (English) Zbl 0684.68062

From the author’s introduction: “It is shown that, for any unbounded fully space constructible function S(n), if \(S(n)=o(\log n)\), then \[ \Pi_ 2-SPACE(S(n))-NSPACE(o(\log n))\neq \emptyset. \] This means that if \(L(n)=o(\log n)\) and L(n)\(\geq S(n)\) for some unbounded fully space constructible S(n), then \(\Pi_ 2\)-SPACE(L(n)) properly contains NSPACE(L(n)), hence the L(n) space-bounded alternating hierarchy does not collapse to the \(\Sigma_ 1\) level. Taking these facts into consideration, a natural question arises: whether for such functions L(n) the inequality \[ \Pi_ 2-SPACE(L(n))\neq NSPACE(L(n)) \] entails that NSPACE(L(n)) is not closed under complement or that \[ DSPACE(L(n))\neq NSPACE(L(n)). \] We discuss this problem and, although we do not give the complete answer, prove some weaker result, viz. that strongly L(n) space- bounded nondeterministic Turing machines are not closed under the so- called independent (from the starting configuration) complement. More precisely, there is a strongly L(n) space-bounded nondeterministic Turing machine M (even with a single-letter input alphabet) such that there is no strongly L(n) space-bounded nondeterministic Turing machine \(M'\) with the property that M started on any input x in any L(\(| x|)\) space-bounded configuration c accepts if and only if \(M'\) started on x in c does not accept. Furthermore, since strongly L(n) space-bounded deterministic Turing machines with a single-letter input alphabet are closed under independent complement, it is shown that there exists an L(n) space-bounded nondeterministic Turing machine (even with a single- letter input alphabet) which cannot be simulated in a similar strong way by any L(n) space-bounded deterministic Turing machine.”

MSC:

68Q25 Analysis of algorithms and problem complexity
68Q05 Models of computation (Turing machines, etc.) (MSC2010)
03D15 Complexity of computation (including implicit computational complexity)
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