Parameters for minimal unsatisfiability: Smarandache primitive numbers and full clauses
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by
Oliver Kullmann, Xishun Zhao
2015
Abstract
We establish a new bridge between propositional logic and elementary number theory. A full clause in a conjunctive normal form (CNF) contains all variables, and we study them in minimally unsatisfiable clausesets (MU); such clauses are strong structural anchors, when combined with other restrictions. Counting the maximal number of full clauses for a given deficiency k, we obtain a close connection to the socalled "Smarandache primitive number" S2(k), the smallest n such that 2 k divides n!. The deficiency k ≥ 1 of an MU is the difference between the number of clauses and the number of variables. We also consider the subclass UHIT of MU given by unsatisfiable hitting clausesets, where every two clauses clash. While MU corresponds to irredundant (minimal) covers of the boolean hypercube {0, 1} n by subcubes, for UHIT the covers must indeed be partitions. We study the four fundamental quantities FCH, FCM, VDH, VDM : N → N, defined as the maximum number of full clauses in UHIT resp. MU, resp. the maximal minimal number of occurrences of a variable (the variable degree) in UHIT resp. MU, in dependency on the deficiency. We have the relations FCH(k) ≤ FCM(k) ≤ VDM(k) and FCH(k) ≤ VDH(k) ≤ VDM(k), together with VDM(k) ≤ nM(k) ≤ k + 1 + log 2 (k), for the "nonMersenne numbers" nM(k), enumerating the natural numbers except numbers of the form 2 n − 1. We show the lower bound S2(k) ≤ FCH(k); indeed we conjecture this to be exact. The proof rests on two methods: Applying an expansion process , fundamental since the days of Boole, and analysing certain recursions, combining an applicationspecific recursion with a recursion from the field of metaFibonacci sequences. The S2lower bound together with the nMupperbound yields a good handle on the four fundamental quantities, especially for those k with S2(k) = nM(k) (we show there are infinitely many such k), since then the four quantities must all be equal to S2(k) = nM(k). With the help of this we determine them for 1 ≤ k ≤ 13.
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