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This paper first shows that (1) the class of languages recognized by o(log n) space-bounded two-way Monte Carlo Turing machines is not closed under concatenation, Kleene closure, and length-preserving homomorphism, and (2) the class of languages recognized by o(log log n) space-bounded two-way unbounded error probabilistic Turing machines is not closed under each of the above operations. We then show that the class of languages accepted by S(n) space-bounded two-way alternating Turing machines is not closed under each of the above operations for log log n ≤ S(n) ≤ o(log n).

CLOSURE PROPERTY OF PROBABILISTIC TURING MACHINES AND ALTERNATING TURING MACHINES WITH SUBLOGARITHMIC SPACES