Abstract
We consider Boolean exact threshold functions defined by linear equations, and in general degree d polynomials. We give upper and lower bounds on the maximum magnitude (absolute value) of the coefficients required to represent such functions. These bounds are very close and in the linear case in particular they are almost matching. The quantity is the same as the maximum magnitude of integer coefficients of linear equations required to express every possible intersection of a hyperplane in R n and the Boolean cube {0,1}n, or in the general case intersections of hypersurfaces of degree d in R n and the Boolean cube {0,1}n. In the process we construct new families of ill-conditioned matrices. We further stratify the problem (in the linear case) in terms of the dimension k of the affine subspace spanned by the solutions, and give upper and lower bounds in this case as well. Our bounds here in terms of k leave a substantial gap, a challenge for future work.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Agrawal, M., Arvind, V.: Geometric sets of low information content. Theoretical Computer Science 158(1-2), 193–219 (1996)
Alon, N., Vũ, V.H.: Anti-hadamard matrices, coin weighing, threshold gates, and indecomposable hypergraphs. Journal of Combinatorial Theory, Series A 79(1), 133–160 (1997)
Beigel, R.: Perceptrons, PP, and the polynomial hierarchy. Computational Complexity 4(4), 339–349 (1994)
Beigel, R., Tarui, J., Toda, S.: On probabilistic ACC circuits with an exact-threshold output gate. In: Ibaraki, T., Iwama, K., Yamashita, M., Inagaki, Y., Nishizeki, T. (eds.) ISAAC 1992. LNCS, vol. 650, pp. 420–429. Springer, Heidelberg (1992)
Faddeev, D.K., Sominskii, I.S.: Problems in Higher Algebra. W.H. Freeman, New York (1965)
Green, F.: A complex-number fourier technique for lower bounds on the mod-m degree. Computational Complexity 9(1), 16–38 (2000)
Hansen, K.A.: Computing symmetric boolean functions by circuits with few exact threshold gates. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 448–458. Springer, Heidelberg (2007)
Hansen, K.A.: Depth reduction for circuits with a single layer of modular counting gates. In: Frid, A., Morozov, A., Rybalchenko, A., Wagner, K.W. (eds.) Computer Science - Theory and Applications. LNCS, vol. 5675, pp. 117–128. Springer, Heidelberg (2009)
Hansen, K.A., Podolskii, V.V.: Exact threshold circuits. In: Proceedings of the 25th Annual IEEE Conference on Computational Complexity, pp. 270–279. IEEE Computer Society, Los Alamitos (2010)
Harkins, R.C., Hitchcock, J.M.: Dimension, halfspaces, and the density of hard sets. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 129–139. Springer, Heidelberg (2007)
Håstad, J.: On the size of weights for threshold gates. SIAM Journal on Discrete Mathematics 7(3), 484–492 (1994)
Hertz, J., Krogh, A., Palmer, R.G.: Introduction to the Theory of Neural Computation. Addison-Wesley Publishing Company, Reading (1991)
Muroga, S., Toda, I., Takasu, S.: Theory of majority decision elements. Journal of the Franklin Institute 271, 376–418 (1961)
Muroga, S.: Threshold Logic and its Applications. John Wiley & Sons, Inc., Chichester (1971)
Parberry, I.: Circuit Complexity and Neural Networks. MIT Press, Cambridge (1994)
Podolskii, V.V.: A uniform lower bound on weights of perceptrons. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds.) Computer Science – Theory and Applications. LNCS, vol. 5010, pp. 261–272. Springer, Heidelberg (2008)
Podolskii, V.V.: Perceptrons of large weight. Problems of Information Transmission 45(1), 46–53 (2009)
Smith, D.R.: Bounds on the number of threshold functions. IEEE Transactions on Electronic Computers EC 15(6), 368–369 (1966)
Yajima, S., Ibaraki, T.: A lower bound on the number of threshold functions. IEEE Transactions on Electronic Computers EC 14(6), 926–929 (1965)
Ziegler, G.M.: Lectures on 0/1-polytopes. In: Kalai, G., Ziegler, G.M. (eds.) Polytopes - Combinatorics and Computation, DMV Seminar, vol. 29, pp. 1–43. Birkhäuser, Basel (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Babai, L., Hansen, K.A., Podolskii, V.V., Sun, X. (2010). Weights of Exact Threshold Functions. In: Hliněný, P., Kučera, A. (eds) Mathematical Foundations of Computer Science 2010. MFCS 2010. Lecture Notes in Computer Science, vol 6281. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15155-2_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-15155-2_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15154-5
Online ISBN: 978-3-642-15155-2
eBook Packages: Computer ScienceComputer Science (R0)