Abstract
Paulet al. [12] proved that deterministic Turing machines can be speeded up by a factor of log*t (n) using four alternations; that is, DTIME(t(n) log*t(n)) $$ \subseteq $$ σ4(t(n)). Balcazaret al. [1] noted that two alternations are sufficient to achieve this speed-up of deterministic Turing machines; that is, DTIME(t(n) log*t(n)) $$ \subseteq $$ Σ2(t (n)). We supply a proof of this speed-up and using that show that for each time-constructible functiont(n), DTIME(t(n)) ⊂ Σ2(t(n)); that is, two alternations are strictly more powerful than deterministic time. An important corollary is that at least one (or both) of the immediate generalizations of the result DTIME(n) ⊂ NTIME(n) [12] must be true: NTIME(n) ≠ co-NTIME(n) or, for each time-constructible functiont(n), DTIME(t (n)) ⊂ NTIME(t (n)).
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