ACk is the class of problems solvable by an alternating Turing machine in space O(log n) and alternation depth O(logk n) [S. A. Cook, A taxonomy of problems with fast parallel algorithms, Inform. Contr. vol. 64]. We consider a game played by two persons: each player alternately moves a marker along an edge of a given digraph, and the first palyer who cannot move loses the game. It is shown that the problem to determine whether the first player can win the game on a digraph with n nodes exactly after logk n moves is complete for ACk nuder NC1 reducibility.

Let S(n) be a space constructible function such that S(n) log n. In this paper, we show that AuxSpTu (S(n),T(n)) NSPACE (S(n)log T(n)), where AuxSpTu (S(n),T(n)) is the class of languages accepted by nondeterministic auxiliary pushdown automata operating simultaneously in O(S(n)) space and O(T(n)) turns of the auxiliary tape head.

J. Bresnan and R. M. Kaplan introduced lexical-functional grammars (LFGs, for short) as a new formalism for human language syntax. It is important to show formal properties of this kind of grammars in order to characterize the formal complexity of human languages. In this paper, we will show that the emptiness problem for LFGs is undecidable.

CFGs (context-free grammars) with various types of memory are introduced and their generative capacities are investigated. For an automata-theoretic characterization, a new type of automaton called partitioning automaton is introduced and it is shown that the class of languages generated by CFGs with memory type X is equal to the class of languages accepted by partitioning automata of type X.