The equivalence of theories that characterize ALogTime

A number of theories have been developed to characterize ALogTime (or uniform NC ¹, or just NC ¹), the class of languages accepted by alternating logtime Turing machines, in the same way that Buss’s theory characterizes polytime functions. Among these, ALV′ (by Clote) is particularly interesting because it is developed based on Barrington’s theorem that the word problem for the permutation group S ₅ is complete for ALogTime. On the other hand, ALV (by Clote), T ⁰ NC ⁰ (by Clote and Takeuti) as well as Arai’s theory and its two-sorted version VNC ¹ (by Cook and Morioka) are based on the circuit characterization of ALogTime. While the last three theories have been known to be equivalent, their relationship to ALV′ has been an open problem. Here we show that ALV′ is indeed equivalent to the other theories.