The Log Space Oracle Hierarchy Collapses DRAFT Samuel R. Buss University of California, Berkeley Stephen A. Cook ∗ University of Toronto Patrick W. Dymond † University of California, San Diego Louise Hay University of Illinois July 20, 1987 The polynomial time hierarchy of Meyer and Stockmeyer has several equivalent characterizations — in particular it can be defined either in terms of polynomial time oracle Turing machines [Sto76], or in terms of polyno- mial time alternating Turing machines where the number of alternations is finitely bounded [CKS81]. One of the most important open questions in the area is whether or not this hierarchy collapses. (It collapses to P iff P = NP .) Similar hierarchies based on space bounded computations have been investigated by Ruzzo, Simon and Tompa [RST84]. They introduced a well-behaved model of space-bounded oracle Turing machine, and showed that the entire log space alternation hierarchy was contained in the second level of the log space oracle hierarchy. Subsequently Lange, Jenner and Kir- sig [LJK87] proved that the alternation hierarchy collapsed to its second level; this suggested the possibility that the log space oracle hierarchy might not coincide with the log space alternation hierarchy. In this paper we show that the log space oracle hierarchy also collapses, that it thus does coincide with the log space alternation hierarchy, and that the resulting complexity class L NL has other interesting characterizations in terms of circuits with oracle gates. A crucial aspect of these results is the exact definition of log space and nondeterministic log space oracle machines. We follow [RST84] which con- tains a discussion of the issues involved. In their model the oracle tape is * Research supported by Natural Sciences and Engineering Research Council † Research sponsored by the National Science Foundation under grant DCR-8604031 1