Parameters for minimal unsatisfiability: Smarandache primitive numbers and full clauses

We establish a new bridge between propositional logic and elementary number theory. A full clause contains all variables, and we study them in minimally unsatisfiable clause-sets (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 so-called “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 clause-sets (every two clauses clash). 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+ log2(k), using the “non-Mersenne numbers” nM(k) as established in [21]. We show the lower bound S2(k) ≤ FCH(k); indeed we conjecture this to be exact. The proof rests on two methods: Applying subsumption resolution and its inverse, and analysing certain recursions, combining an application-specific recursion with a recursion from the field of metaFibonacci sequences. The S2-lower bound together with the nM-upperbound yields a good handle on the four quantities, which we determine for 1 ≤ k ≤ 13.