This paper investigates the space hierarchies of the language classes for two-dimensional Turing machines (2-TM's), two-dimensional pushdown automata (2-PDA's) and two-dimensional counter automata (2-CA's) with small space. We show that (1) if L(n) is space constructible by a 2-TM, L(n) ≤ log n and L′(n) = o(L(n)), then strong 2-DSPACE(L(n)) – weak 2-ASPACE(L′(n)) ≠ ∅, (2) if L(n) is space constructible by a 2-PDA, L(n) ≤ log n and L′(n) = o(L(n)), then strong 2-DPDA(L(n)) – weak 2-ASPACE(L′(n)) ≠ ∅, and (3) if L(n) is space-constructible by a 2-CA, L(n) ≤ n and L′(n) = o(L(n)), then strong 2-DCA(L(n)) – weak 2-ACA(L′(n)) ≠ ∅, (4) where strong 2-DSPACE(L(n)) (strong 2-DPDA(L(n)), strong 2-DCA(L(n))) denotes the class of sets accepted by strongly L(n) space-bounded deterministic 2-TM's (2-PDA's, 2-CA's), and weak 2-ASPACE(L′(n)) (weak 2-ACA(L′(n))) denotes the class of sets accepted by weakly L′(n) space-bounded alternating 2-TM's (2-CA's). We also investigate the closure property of space-bounded alternating 2-PDA's and 2-CA's under complementation, and show that (1) if L(n) = o( log log n), then the class of sets accepted by L(n) space-bounded alternating 2-PDA's is not closed under complementation, and (2) if L(n) is space-constructible by a 2-CA, L(n) ≤ n and [Formula: see text], then the class of sets accepted by L′(n) space-bounded alternating 2-CA's is not closed under complementation.
On 1-inkdot alternating Turing machines with small space