On the pre-AFL of [log n] space and related families of languages

Abstract

New insight into the 1965 conjecture of Lewis et al., that some Context-Free Languages (CFLs) require more than [log n] space for recognition on off-line Turing machines, is derived from an examination of the pre-AFL properties of [log n] space. General results about related AFLs are used to reduce this conjecture to the questions of whether the [log n] class is an AFL and of whether [log n] recognition is a decidable question for certain related AFLs. One of these is the AFL generated by the [log n] class itself, which AFL is shown to properly contain all CFLs (Theorem 1) and, also, to be generated via length-preserving homomorphisms from the [log n] class, using the result that the latter is a pre-AFL (Theorem 2). An example of a related family, which happens to be contained in this newly studied AFL, is the principal AFL, Q, of quasi-real-time languages. Also, the open question of whether [log n] space is the same deterministically or nondeterministically is related to the above questions.