Cancellation-Free Circuits in Unbounded and Bounded Depth
We study the notion of “cancellation-free” circuits. This is a restriction of linear Boolean circuits (XOR-circuits), but can be considered as being equivalent to previously studied models of computation. The notion was coined by Boyar and Peralta in a study of heuristics for a particular circuit minimization problem. They asked how large a gap there can be between the smallest cancellation-free circuit and the smallest linear circuit. We show that the difference can be a factor Ω(n/log2 n). This improves on a recent result by Sergeev and Gashkov who have studied a similar problem. Furthermore, our proof holds for circuits of constant depth. We also study the complexity of computing the Sierpinski matrix using cancellation-free circuits and give a tight Ω(nlogn) lower bound.
KeywordsBoolean Function Constant Depth Hadamard Matrice Strong Separation Nonzero Probability
Unable to display preview. Download preview PDF.
- 1.Aaronson, S.: Thread on cstheory.stackexchange.com, http://cstheory.stackexchange.com/questions/1794/circuit-lower-bounds-over-arbitrary-sets-of-gates
- 2.Boyar, J., Find, M.: Cancellation-free circuits in unbounded and bounded depth. arXiv preprint (1305.3041) (May 2013)Google Scholar
- 6.Find, M., Göös, M., Kaski, P., Korhonen, J.: Separating Or, Sum and XOR Circuits. arXiv preprint (1304.0513) (April 2013)Google Scholar
- 7.Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman (1979)Google Scholar
- 9.Hirsch, E., Melanich, O.: Personal communication (2012)Google Scholar
- 10.Jukna, S.: XOR versus OR circuits, http://www.thi.informatik.uni-frankfurt.de/~jukna/boolean/comment9.html
- 15.Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press (1997)Google Scholar
- 19.Nechiporuk, É.: Rectifier networks. Soviet Physics Doklady 8, 5 (1963)Google Scholar
- 20.Paar, C.: Some remarks on efficient inversion in finite fields. In: Whistler, B. (ed.) IEEE Internatiol Symposium on Information Theory. LNCS, vol. 5162, p. 58. Springer, Heidelberg (1995)Google Scholar
- 21.Pippenger, N.: On the evaluation of powers and related problems (preliminary version). In: FOCS, pp. 258–263. IEEE Computer Society (1976)Google Scholar
- 25.Sergeev, I.: On additive complexity of a sequence of matrices. arXiv preprint (1209.1645) (September 2012)Google Scholar