Non-uniform automata over groups
Reads0
Chats0
TLDR
The power of NUDFA's over nilpotent groups is characterized and some optimal lower bounds for NUD FA's over certain groups which are solvable but not nilpotsent are proved.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 NC1, NUDFA characterizations of several important subclasses of NC1, 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. Therien, 1987 , in “Automata, Languages, and Programming: 14th International Colloquium,” Springer-Verlag, Berlin, pp. 163–173).read more
Citations
More filters
Proceedings ArticleDOI
The polynomial method in circuit complexity
TL;DR: The basic techniques for using polynomials in complexity theory are examined, emphasizing intuition at the expense of formality and closure properties, upper bounds, and lower bounds obtained.
Journal ArticleDOI
Nonuniform ACC Circuit Lower Bounds
TL;DR: The high-level strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms entail these lower bounds, while the second step requires a strengthening of the author’s prior work.
Proceedings ArticleDOI
Non-uniform ACC Circuit Lower Bounds
TL;DR: The high-level strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms can be applied to obtain the above lower bounds.
Journal ArticleDOI
Regular languages in NC 1
TL;DR: Several characterizations are given of the regular languages in the circuit complexity class AC 0, thus answering a question of Chandra, Fortune, and Lipton, and to determine effectively whether a given regular language is in AC 0 and to solve in part an open problem originally posed by McNaughton.
Journal ArticleDOI
Representing Boolean functions as polynomials modulo composite numbers
TL;DR: In this article, the authors defined the smallest degree of any polynomial function over the ring of integers modulom, such that for all 0-1 assignments, the degree is 0 if the number of input ones is a multiple ofn and is one otherwise.
References
More filters
Proceedings ArticleDOI
Algebraic methods in the theory of lower bounds for Boolean circuit complexity
TL;DR: It is proved that depth k circuits with gates NOT, OR and MODp where p is a prime require Exp(&Ogr;(n1/2k)) gates to calculate MODr functions for any r ≠ pm.
Journal ArticleDOI
Parity, circuits and the polynomial time hierarchy
TL;DR: A super-polynomial lower bound is given for the size of circuits of fixed depth computing the parity function and connections are given to the theory of programmable logic arrays and to the relativization of the polynomial-time hierarchy.
Proceedings ArticleDOI
Bounded-width polynomial-size branching programs recognize exactly those languages in NC1
TL;DR: The method of proof is extended to investigate the complexity of the word problem for a fixed permutation group and show that polynomial size circuits of width 4 also recognize exactly nonuniform NC 1.
Journal ArticleDOI
The NP-completeness column: An ongoing guide
TL;DR: This is the fourteenth edition of a quarterly column that provides continuing coverage of new developments in the theory of NP-completeness, and readers who have results they would like mentioned (NP-hardness, PSPACE- hardness, polynomialtime-solvability, etc.), or open problems they wouldlike publicized, should send them to David S. Johnson.
Journal ArticleDOI
On finite monoids having only trivial subgroups
TL;DR: An alternative definition is given for a family of subsets of a free monoid that has been considered by Trahtenbrot and by McNaughton.