This paper introduces a 1-inkdot two-way alternating pushdown automaton which is a two-way alternating pushdown automaton (2apda) with the additional power of marking at most 1 tape-cell on the input (with an inkdot) once. We first investigate a relationship between the accepting powers of sublogarithmically space-bounded 2apda's with and without 1 inkdot, and show, for example, that sublogarithmically space-bounded 2apda's with 1 inkdot are more powerful than those which have no inkdots. We next investigate an alternation hierarchy for sublogarithmically space-bounded 1-inkdot 2apda's, and show that the alternation hierarchy on the first level for 1-inkdot 2apda's holds, and we also show that 1-inkdot two-way nondeterministic pushdown automata using sublogarithmic space are incomparable with 1-inkdot two-way alternating pushdown automata with only universal states using the same space.

This paper introduces a new class of machines called multihead marker finite automata, and investigates how the number of markers affects its accepting power. Let HM{0}(i, j)(NHM{0}(i, j))denote the class of languages over a one-letter alphabet accepted by two-way deterministic (nondeterminstic) i-head finite automata with j markers. We show that HM{0} (i, j) HM{0}(i, j1) and NHM{0}(i, j) NHM{0}(i, j+1) for each i2, j0.

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 investigates some fundamental properties of 2-way alternating multi-counter automata (2amca's) with only existential (universal) states which have sublinear space and 1 inkdot. It is shown that for any function s(n) log n such that log s(n)=o(log n), s(n) space-bounded 1-inkdot 2amca's with only existential states are incomparable with the ones with only universal states, and the ones with only existential (universal) states are not closed under complementation.