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 [Ba86], NUDFA characterizations of several important subclasses of NC 1 [BT87], and a new proof [BT87] of the old result [BK78] that the dot-depth hierarchy is infinite, using Sipser's work [Si83] on constant depth circuits.
Here we outline these earlier results and extend this theory to NUDFA's over solvable groups (NUDFA's over non-solvable groups have the maximum possible computing power [Ba86]). 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.
Supported by NSF grant MCS-8304769 and US Air Force grant AFOSR-82-0326.
Supported by NSERC grant A4546 and FCAR grant 86-EQ-2933.
Address for 1986-87: Laboratoire Informatique Théorique et Programmation, Université P. et M. Curie, 2, Place Jussieu, 75221 Paris, France.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
9. References
M. Ajtai, ‘ε 11 formulae on finite structures', Annals of Pure and Applied Logic 24 (1983), 1–48.
D. A. Barrington, ‘Width 3 Permutation Branching Programs', Technical Memorandum TM-291, M.I.T. Laboratory for Computer Science, Dec. 1985.
D. A. Barrington, ‘Bounded-width polynomial-size branching programs recognize exactly those languages in NC 1', Proc. 18th ACM STOC, 1986, 1–5.
D. A. Barrington, ‘Bounded-width branching programs', Ph.D. thesis, Dept. of Mathematics, M.I.T., May 1986. Available as Technical Report TR-361, M.I.T. Laboratory for Computer Science.
A. Borodin, D. Dolev, F. E. Fich, and W. Paul, ‘Bounds for width two branching programs', Proc. 15th ACM STOC, 1983, 87–93.
J. A. Brzozowski and R. Knast, ‘The dot-depth hierarchy of star-free languages is infinite', J. Comp. Sys. Sci. 16 (1978) 37–55.
D. A. Barrington and D. Thérien, ‘Finite monoids and the fine structure of NC 1', Proc. 19th ACM STOC, 1987, to appear.
R. S. Cohen and J. A. Brzozowski, ‘Dot-depth of star-free events', J. Comp. Sys. Sci. 5 (1971) 1–16.
A. K. Chandra, S. Fortune, and R. Lipton, ‘Unbounded fan-in circuits and associative functions', Proc. 15th ACM STOC, 1983, 52–60.
A. K. Chandra, L. Stockmeyer, and U. Vishkin, ‘Constant-depth reducibility', SIAM J. Computing 13 (May 1984) 423–439.
S. A. Cook, ‘The taxonomy of problems with fast parallel algorithms', Information and Control 64 (Jan. 1985), 2–22.
S. Eilenberg, Automata, Languages, and Machines, Vol. B (Academic Press, New York, 1976).
R. Fagin, M. M. Klawe, N. J. Pippenger, and L. Stockmeyer, ‘Bounded depth, polynomial-size circuits for symmetric functions', Theoretical Computer Science 36 (1985) 239–250.
M. Furst, J. B. Saxe, and M. Sipser, ‘Parity, circuits, and the polynomial-time hierarchy', Proc. 22nd IEEE FOCS, 1981, 260–270.
D. S. Johnson, ‘The NP-completeness column: An ongoing guide', Journal of Algorithms 7:2 (June 1986), 289–305.
G. Lallement, Semigroups and Combinatorial Applications, (New York: John Wiley & Sons, 1979).
N. Pippenger, ‘On simultaneous resource bounds (preliminary version), Proc. 20th IEEE FOCS, 1979, 307–311.
J. E. Pin, Varieties of Formal Languages (New York: Plenum Press, 1986).
A. A. Razborov, ‘Lower bounds for the size of circuits of bounded depth with basis {&, ⊕}, preprint (in Russian). To appear in Mathematical Notes of the Academy of Sciences of the USSR.
J. E. Savage, The Complexity of Computing, (New York, J. Wiley and Sons, 1976).
M. P. Schützenberger, ‘On finite monoids having only trivial subgroups', Information and Control 8 (1965) 190–194.
M. Sipser, ‘Borel sets and circuit complexity', Proc. 15th ACM STOC, 1983, 61–69.
R. Smolensky, ‘Algebraic methods in the theory of lower bounds for Boolean circuit complexity', Proc. 19th ACM STOC, 1987, to appear.
P. M. Spira, ‘On time-hardware complexity tradeoffs for Boolean functions', Proc. 4th Hawaii Symposium on System Sciences (North Hollywood, Calif., Western Periodicals Co., 1971) 525–527.
H. Straubing, ‘Finite semigroup varieties of the form V * D', J. of Pure and Applied Algebra 36 (1985) 53–94.
H. Straubing, ‘Semigroups and languages of dot-depth 2', Proc. 13th ICALP, Lecture Notes in Computer Science, No. 226, (New York, Springer-Verlag, 1986) 416–423.
H. Straubing, ‘Finite monoids and Boolean circuit complexity', draft.
D. Thérien, ‘Classification of finite monoids: the language approach', Theoretical Computer Science 14 (1981) 195–208.
D. Thérien, ‘Subword counting and nilpotent groups', in L. J. Cummnings, ed., Combinatorics on Words: Progress and Perspectives, (Academic Press, New York, 1983), 297–305.
H. J. Zassenhaus, The Theory of Groups, 2nd ed. (New York, Chelsea Publ. Co., 1958).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1987 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Barrington, D.A., Thérien, D. (1987). Non-uniform automata over groups. In: Ottmann, T. (eds) Automata, Languages and Programming. ICALP 1987. Lecture Notes in Computer Science, vol 267. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-18088-5_13
Download citation
DOI: https://doi.org/10.1007/3-540-18088-5_13
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-18088-3
Online ISBN: 978-3-540-47747-1
eBook Packages: Springer Book Archive