Non-uniform automata over groups

Abstract

A new model, non-uniform deterministic finite automata (NUDFA's) over general finite monoids, has recently been developed as a strong link between the theory of finite automata and low-level parallel complexity. Achievements of this model include the proof that width 5 branching programs recognize exactly the languages in non-uniform NC 1 , NUDFA characterizations of several important subclasses of NC 1 , and a new proof of the old result that the dot-dephth hierarchy is infinite, using M. Sipser's (1983, in “Proceedings, 15th ACM Symposium on the Theory of Computing,” Association for Computing Machinery, New York, pp. 61–69) work on constant depth circuits. Here we extend this theory to NUDFA's over solvable groups (NUDFA's over non-solvable groups have the maximum possible computing power). We characterize the power of NUDFA's over nilpotent groups and prove some optimal lower bounds for NUDFA's over certain groups which are solvable but not nilpotent. Most of these results appeared in preliminary form in ( D. A. Barrington and D. Thérien, 1987 , in “Automata, Languages, and Programming: 14th International Colloquium,” Springer-Verlag, Berlin, pp. 163–173).