Mathematics

Download A characteristic set method for solving boolean equations by Chai F., Gao X.-S., Yuan C. PDF

By Chai F., Gao X.-S., Yuan C.

This paper offers a attribute set technique for fixing Boolean equations, that's extra effective and has larger houses than the overall attribute set procedure. particularly, the authors provide a disjoint and monic 0 decomposition set of rules for the 0 set of a Boolean equation approach and an specific formulation for the variety of options of a Boolean equation method. The authors additionally end up attribute set may be computed with a polynomial variety of multiplications of Boolean polynomials by way of the variety of variables. As experiments, the proposed approach is used to resolve equations from cryptanalysis of a category of flow ciphers in accordance with nonlinear clear out turbines. broad experiments exhibit that the strategy is sort of potent.

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Definition 2. Let μ be a nonadditive measure on X. The Choquet integral + of a nonnegative function f : X → R with respect to μ is defined by ∞ (C) f dμ := X μ({f > t})dt, 0 where the right hand side of the above equation is the usual Lebesgue integral. Remark 1. For any nonadditive measure μ on X and any function f : X → + ∞ ∞ R , the two Lebesgue integrals 0 μ({f > t})dt and 0 μ({f ≥ t})dt are equal, since μ({f ≥ t}) ≥ μ({f > t}) ≥ μ(f ≥ t + ε}) for every ε > 0 and 0 ≤ t < ∞. This fact will be used implicitly in this paper.

1] Borel-Cantelli Lemma Let {An , n ≥ 1} be a sequence of events in F and (V, v) be a pair of upper and lower probability generated by P. ∞ (1) If ∞ ∞ V(An ) < ∞, then V Ai n=1 = 0. n=1 i=n (2) If further v(·) is lower continuous and {An , n ≥ 1} are pairwise independent with respect to v(·), that is for any n ∈ N, v ∞ i=n v(Aci ). Also if ∞ i=n ∞ Aci = V(An ) = ∞, then n=1 ∞ ∞ V Ai = 1. n=1 i=n Lemma 2. Given upper expectation E[·], let (Xi )∞ i=1 be a sequence of independent random variables such that sup E[|Xi |1+α ] < ∞ for some constant i≥1 α > 0.

33, 109–116 (1971) 22. : Theory of fuzzy integrals and its applications. Ph. D. Thesis. Tokyo Institute of Technology (1974) 23. : Pseudo-additive measures and integrals. J. Math. Anal. Appl. 122, 197–222 (1987) 24. : On the continuity of the concave integral. Fuzzy Sets and Systems 160, 1318–1326 (2009) 25. : Generalized Measure Theory. Springer, Boston (2009) 26. : Generalized Lebesgue integral. Int. J.

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