Abstract
In computations realized by finite automata, a rich understanding has come from comparing the algebraic structure of the machines to the combinatorics of the languages being recognized. In this expository paper, we will first survey some basic ideas that have been useful in this model. In the second part, we sketch how this dual approach can be generalized to study some important class of boolean circuits, what results have been obtained, what questions are still open. The intuition gained in the simple model sometimes carry through, sometimes not, so that one has to be careful on what conjectures to make.
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References
Auinger, K., Steinberg, B.: Varieties of finite supersolvable groups with the M. Hall property. Math. Ann. 335, 853–877 (2006)
Barrington, D.A.: Bounded-width polynomial-size branching programs recognize exactly those languages in NC 1. Journal of Computer and System Sciences 38(1), 150–164 (1989)
Barrington, D.A.M., Straubing, H.: Superlinear lower bounds for bounded-width branching programs. Journal of Computer and System Sciences 50(3), 374–381 (1995)
Barrington, D.A.M., Thérien, D.: Finite monoids and the fine structure of NC 1. Journal of the ACM 35(4), 941–952 (1988)
Borodin, A., Dolev, D., Fich, F.E., Paul, W.: Bounds for width two branching programs. SIAM Journal on Computing 15(2), 549–560 (1986)
Brzozowski, J., Knast, R.: The dot-depth hierarchy of star-free languages is infinite. Journal of Computer and System Sciences 16(1), 37–55 (1978)
Eilenberg, S.: Automata, Languages and Machines, vol. B. Academic Press, New York (1976)
Furst, M., Saxe, J.B., Sipser, M.: Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory 17(1), 13–27 (1984)
Krohn, K., Rhodes, J.: The algebraic theory of machines I. Trans. Amer. Math. Soc. 116, 450–464 (1965)
Maurer, W., Rhodes, J.: A property of finite simple non-Abelian groups. Proc. Amer. Math. Soc 16, 552–554 (1965)
McKenzie, P., Péladeau, P., Thérien, D.: NC1: The automata theoretic viewpoint. Computational Complexity 1, 330–359 (1991)
Schützenberger, M.P.: On finite monoids having only trivial subgroups. Information and Computation 8(2), 190–194 (1965)
Sipser, M.: Borel sets and circuit complexity. In: Proceedings of the Fifteenth Annual ACM Symposium on the Theory of Computing, pp. 61–69 (1983)
Straubing, H.: Finite Automata, Formal Logic and Circuit Complexity. Birkhauser, Boston (1994)
Thérien, D.: Classification of finite monoids: The language approach. In: Theoretical Computer Science, vol. 14, pp. 195–208 (1981)
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Thérien, D. (2011). The Power of Diversity. In: Holzer, M., Kutrib, M., Pighizzini, G. (eds) Descriptional Complexity of Formal Systems. DCFS 2011. Lecture Notes in Computer Science, vol 6808. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22600-7_4
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DOI: https://doi.org/10.1007/978-3-642-22600-7_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22599-4
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