Space

Abstract

. The complexity measure under consideration is SPACE ⇥ REVERSALS for Turing machines that are able to branch both existentially and universally. We show that, for any function h ( n ) between log log n and log n , ⇧ 1 SPACE ⇥ REVERSALS( h ( n )) is separated from ⌃ 1 SPACE ⇥ REVERSALS( h ( n )) as well as from co ⌃ 1 SPACE ⇥ REVERSALS( h ( n )), for middle , accept , and weak modes of this complexity measure. This also separates determinism from the higher levels of the alternating hierarchy. For “well-behaved” functions h ( n ) between log log n and log n , almost all of the above separations can be obtained by using unary witness languages. In addition, the construction of separating languages contributes to the research on minimal resource requirements for computational devices capable of recognizing nonregular languages. For any (arbitrarily slow growing) unbounded monotone recursive function f ( n ), a nonregular unary language is presented that can be accepted by a middle ⇧ 1 alternating Turing machine in s ( n ) space and i ( n ) input head reversals, with s ( n ) · i ( n ) 2 O (log log n · f ( n )). Thus, there is no exponential gap for the optimal lower bound on the product s ( n ) · i ( n ) between unary and general nonregular language acceptance—in sharp contrast with the one-way case.