Abstract
A general approximation technique to get lower bounds for the complexity of combinational circuits over an arbitrary algebras of operations is presented. The technique generalizes recent methods for monotone circuits and yields some new results. This report contains an exp(Ω(log2n)) lower bound for the complexity of realization of non-monotone Boolean functions by circuits over the basis (&,V,-) computing sufficiently many prime implicants, and of three-valued functions by circuits over some incomplete three-valued extensions of (&,V,-).
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M. Ajtai, Σ1-formulae on finite structures. Ann.of Pure and Appl.logic, 24 (1984), pp. 1–48.
N. Alon and R.B. Boppana, The monotone circuit complexity of Boolean functions, Combinatorica, 7, N.1 (1987), pp. 1–22.
A.E. Andreev, On one method of obtaining lower bounds of individual monotone function complexity, Doklady Akad. Nauk SSSR, 282 (1985), pp. 1033–1037.
A.E. Andreev, On one method of obtaining effective lower bounds of monotone complexity, Algebra i Logika, 26,N.1(1987),pp. 3–21.
M. Furst, J.B. Saxe and M. Sipser, Parity, circuits and the polynomial time hierarchy, Proc. 22nd FOCS (1981),pp.260–270.
A. Hajnal, W. Maass, P. Pudlak, M. Szegedy and G. Turan, Threshold circuits of bounded depth, Proc. 28th FOCS (1987), 99–110.
J. Hastad, Almost optimal lower bounds for small depth circuits, Proc. 18th STOC (1986), pp. 6–20.
S.P. Jukna, Lower bounds on the complexity of local circuits, Lect. Notes in Comput. Sci. 233 (Springer-Berlin, 1986), 440–448.
S.P. Jukna, Entropy of contact circuits and lower bounds on their complexity, Theoret. Comput.Sci., 57, N.1 (1988).
S.P. Jukna, On one entropic method of obtaining lower bounds on the complexity of Boolean functions, Doklady Akad. Nauk SSSR, 298, N.3 (1988), pp. 556–559.
E.A. Okol'nishnikova, On the influence of one type of restrictions to the complexity of combinational circuits, Discrete Analysis, 36 (Novosibirsk, 1981), pp. 46–58.
M.S. Paterson, Bonded-depth circuits over { ⊕, & }, Preprint, 1986.
A.A. Razbororov, Lower bounds for the monotone complexity of some Boolean functions, Doklady Akad. Nauk SSSR, 281 (1985), pp. 798–801.
A.A. Razborov, A lower bound on the monotone network complexity of the logical permanent, Mat. Zametki, 37, N.6 (1985)
A.A.Razborov, Lower bounds for the size of bounded-depth circuits over the basis { ⊕, & }, Preprint (Moscow, 1986).
R. Smolensky, Algebraic methods in the theory of lower bounds for Boolean circuit complexity, Proc. 19th STOC (1987), pp.77–87.
A. Yao, Lower bounds by probabilistic arguments, Proc. 24th FOCS (1983), pp. 420–428.
G.A. Tkachev, On the complexity of one sequence of functions of k-valued logic, Vestnik MGU, Ser.2, N.1 (1977), pp. 45–57.
A.B. Ugol'nikov, On the complexity of realization of Boolean functions by circuits over the basis with median and implication Vestnik MGU, Ser.1, N.4 (1987), pp. 76–78.
L.G. Valiant, Short monotone formulae for the majority function, Journal of Algorithms, v. 5 (1984), pp. 363–366.
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© 1988 Springer-Verlag Berlin Heidelberg
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Jukna, S.P. (1988). Two lower bounds for circuits over the basis (&, V, -). In: Chytil, M.P., Koubek, V., Janiga, L. (eds) Mathematical Foundations of Computer Science 1988. MFCS 1988. Lecture Notes in Computer Science, vol 324. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0017160
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DOI: https://doi.org/10.1007/BFb0017160
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