Abstract
Nondeterministic circuits are a nondeterministic computation model in circuit complexity theory. In this paper, we prove a \(3(n-1)\) lower bound for the size of nondeterministic \(U_2\)-circuits computing the parity function. It is known that the minimum size of (deterministic) \(U_2\)-circuits computing the parity function exactly equals \(3(n-1)\). Thus, our result means that nondeterministic computation is useless to compute the parity function by \(U_2\)-circuits and cannot reduce the size from \(3(n-1)\). To the best of our knowledge, this is the first nontrivial lower bound for the size of nondeterministic circuits (including formulas, constant depth circuits, and so on) with unlimited nondeterminism for an explicit Boolean function. We also discuss an approach to proving lower bounds for the size of deterministic circuits via lower bounds for the size of nondeterministic restricted circuits.
This work was supported by JSPS KAKENHI Grant Number 15K11986.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arora, S., Barak, B.: Computational Complexity - A Modern Approach. Cambridge University Press (2009)
Blum, N.: A boolean function requiring 3n network size. Theor. Comput. Sci. 28, 337–345 (1984)
Demenkov, E., Kulikov, A.S.: An elementary proof of a 3n \(-\) o(n) lower bound on the circuit complexity of affine dispersers. In: Murlak, F., Sankowski, P. (eds.) MFCS 2011. LNCS, vol. 6907, pp. 256–265. Springer, Heidelberg (2011)
Iwama, K., Morizumi, H.: An explicit lower bound of 5n - o(n) for boolean circuits. In: Proc. of MFCS, pp. 353–364 (2002)
Klauck, H.: Lower bounds for computation with limited nondeterminism. In: Proc. of CCC, pp. 141–152 (1998)
Lachish, O., Raz, R.: Explicit lower bound of 4.5n - o(n) for boolean circuits. In: Proc. of STOC, pp. 399–408 (2001)
Schnorr, C.: Zwei lineare untere schranken für die komplexität boolescher funktionen. Computing 13(2), 155–171 (1974)
Tseitin, G.S.: On the complexity of derivation in propositional calculus. In: Slisenko, A.O. (ed.) Studies in Constructive Mathematics and Mathematical Logic, pp. 115–125 (1968)
Valiant, L.G.: Graph-theoretic properties in computational complexity. J. Comput. Syst. Sci. 13(3), 278–285 (1976)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Morizumi, H. (2015). Lower Bounds for the Size of Nondeterministic Circuits. In: Xu, D., Du, D., Du, D. (eds) Computing and Combinatorics. COCOON 2015. Lecture Notes in Computer Science(), vol 9198. Springer, Cham. https://doi.org/10.1007/978-3-319-21398-9_23
Download citation
DOI: https://doi.org/10.1007/978-3-319-21398-9_23
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-21397-2
Online ISBN: 978-3-319-21398-9
eBook Packages: Computer ScienceComputer Science (R0)