This paper investigates some fundamental properties of one-way alternating pushdown automata with sublinear space. We first show that one-way nondeterministic pushdown automata are incomparale with one-way alternating pushdown automata with only universal states, for spaces between log log n and log n, and also for spaces between log n and n/log n. We then show that there exists an infinite space hierarchy among one-way alternating pushdown automata with only universal states which have sublinear space.

This paper introduces an alternating rebound Turing machine and investigates some fundamental properties of it. Let DRTM (NRTM,ARTM) denote a deterministic (nondeterministic and alternating) rebound Turing machine, and URTM denote an ARTM with only universal states. We first investigate a relationship between the accepting powers of rebound machines and ordinary machines, and show, for example, that (1) there exists a language accepted by a deterministic rebound automaton, but not accepted by any o(log n) space-bounded alternating Turing machine, (2) alternating rebound automata are equivalent to two-way alternating counter automata, and (3) deterministic rebound counter automata are more powerful than two-way deterministic counter automata. We next investigate a relationship among the accepting powers of DRTM's, NRTM's, URTM's and ARTM's, and show that there exists a language accepted by alternating rebound automata, but not accepted by any o(logn) space-bounded NRTM (URTM). Then we show that there exists an infinite space hierarchy for DRTM's (NRTM's, URTM's) with spaces below log n. Furthermore, we investigate a relationship between the strong and weak modes of space complexity, and finally show that the classes of languages accepted by o(logn) space-bounded DRTM's (NRTM's, URTM's) are not closed under concatenation and Kleene .

This paper first investigates a relationship between inkdot-depth and inkdot-size of inkdot two-way alternating Turing machines and pushdown automata with sublogarithmic space, and shows that there exists a language accepted by a strongly loglog n space-bounded alternating pushdown automaton with inkdot-depth 1, but not accepted by any weakly o (log n) space-bounded and d (n) inkdot-size bounded alternating Turing machine, for any function d (n) such that limn [d (n)log n/n1/2] = 0. In this paper, we also show that there exists an infinite space hierarchy among two-way alternating pushdown automata with sublogarithmic space.