# Contributions to Group Theory by Appel K.I., Ratcliffe J.G., Schupp P.E. (eds.) By Appel K.I., Ratcliffe J.G., Schupp P.E. (eds.)

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Apply a k -SAT algorithm to Fi ’s; if at least one of them is satisﬁable, return “satisﬁable”; otherwise return “unsatisﬁable”. In the certiﬁcate settings, we take w = γn and we bound the length of certiﬁcates considering two cases: the case of a satisfying assignment of low weight (≤ w), and the case of application of Lemma 1. 1. If F is satisﬁed by an assignment of weight at most w then F has a certiﬁcate of length n lg + O(lg n). g. [Knu05]. (a) Consider the lexicographic order of all assignments (n-bit strings) of weight exactly w and consider the numbering of assignments in this list n by numbers from 0 to w −1.

Part 2b follows by i(k + 1) ≤ h(k + 1) + 2 and i(k + 1) = i(k) + 1, h(k + 1) = h(k). Now consider Part 3. Condition 3a implies condition 3b due to Δi(k) = 0 in case of k ∈ J by Part 2a. Condition 3b implies condition 3c, since Δh(k) = 2 implies Δi(k) = 0 (otherwise we had Δ nM(k) = 3), and so by Part 1c we have i(k) = i(k+1) = h(k+1), while the assumption says h(k + 1) = h(k) + 2. In turn condition 3c implies condition 3a, since by Part 2b we get Δi(k) = 0, and thus Δ nM(k) = Δh(k), while in case of Δh(k) ≤ 1 we would have i(k) − 1 ≥ nM(k − (i(k) − 1) + 1) contradicting the deﬁnition of i(k), due to nM(k − (i(k) − 1) + 1) = nM((k + 1) − i(k + 1) + 1) = h(k + 1) ≤ h(k) + 1 = i(k) − 1.

An EATCS Series. : On the complexity of k-SAT. : Which problems have strongly exponential complexity. : Generating All Combinations and Partitions. The Art of Computer Programming, fascicle 3, vol. 4, pp. 1–6. : A full derandomization of Sch¨ oning’s k-SAT algorithm. In: Proceedings of the 43rd Annual ACM Symposium on Theory of Computing, STOC 2011. : On the complexity of circuit satisfiability. In: Proceedings of the 42nd Annual ACM Symposium on Theory of Computing, STOC 2010, pp. 241–250. : An improved exponentialtime algorithm for k-SAT.