Abstract
We extend the familiar program of understanding circuit complexity in terms of regular languages to visibly counter languages. Like the regular languages, the visibly counter languages are \(\mathrm {NC}^{1}\)- complete. We investigate what the visibly counter languages in certain constant depth circuit complexity classes are. We have initiated this in a previous work for \(\mathrm {AC}^{0}\). We present characterizations and decidability results for various logics and circuit classes. In particular, our approach yields a way to understand \(\mathrm {TC}^{0}\), where the regular approach fails.
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References
Alur, R., Madhusudan, P.: Visibly pushdown languages. In: Babai, L. (eds.) STOC, pp. 202–211. ACM (2004)
Bárány, V., Löding, C., Serre, O.: Regularity problems for visibly pushdown languages. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 420–431. Springer, Heidelberg (2006)
David, A., Barrington, M.: Bounded-Width Polynomial-Size Branching Programs Recognize Exactly Those Languages in NC\(^1\). J. Comput. Syst. Sci. 38(1), 150–164 (1989)
David, A., Thérien, D., Straubing, H., Compton, K.J., Barrington, M.: Regular Languages in NC\(^1\). J. Comput. Syst. Sci. 44(3), 478–499 (1992)
David, A., Thérien, D., Barrington, M.: Finite monoids and the fine structure of NC\(^{\text{1 }}\). J. ACM 35(4), 941–952 (1988)
Furst, M.L., Saxe, J.B., Sipser, M.: Parity, circuits, and the polynomial-time hierarchy. In: FOCS, pp. 260–270 (1981)
Håstad, J.: Almost optimal lower bounds for small depth circuits. In: STOC, pp. 6–20. ACM (1986)
Krebs, A., Lange, K., Ludwig, M.: Visibly Counter Languages and Constant Depth Circuits. In: Mayr, E.W., Ollinger, N. (eds.) 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015), Garching, Germany, vol. 30 of LIPIcs, pp. 594–607. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 4–7 March 2015
McKenzie, P., Thomas, M., Vollmer, H.: Extensional uniformity for boolean circuits. SIAM J. Comput. 39(7), 3186–3206 (2010)
Mehlhorn, K.: Pebbling mountain ranges and its application to DCFL-recognition. In: de Bakker, J., van Leeuwen, J. (eds.) Automata Languages and Programming. LNCS, pp. 422–435. Springer, Berlin Heidelberg (1980)
Smolensky, R.: Algebraic methods in the theory of lower bounds for boolean circuit complexity. In: STOC, pp. 77–82 (1987)
Straubing, H.: Finite Automata, Formal Logic, and Circuit Complexity. Birkhäuser, Boston (1994)
Vollmer, H.: Introduction to Circuit Complexity - A Uniform Approach. Texts in theoretical computer science. Springer, Heidelberg (1999)
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Hahn, M., Krebs, A., Lange, KJ., Ludwig, M. (2015). Visibly Counter Languages and the Structure of \(\mathrm {NC}^{1}\) . In: Italiano, G., Pighizzini, G., Sannella, D. (eds) Mathematical Foundations of Computer Science 2015. MFCS 2015. Lecture Notes in Computer Science(), vol 9235. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48054-0_32
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DOI: https://doi.org/10.1007/978-3-662-48054-0_32
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