Conjunctive Normal Forms with Non-boolean Variables, Autarkies and Hypergraph Colouring Conjunctive Normal Forms with Non-boolean Variables, Autarkies and Hypergraph Colouring

We investigate in depth the properties of a natural generalisation, called “sets of no-goods” in the AI literature, of boolean formulas in conjunctive normal forms such that non-boolean variables can be used. After building up a solid foundation, we generalise the notion of deficiency, and we obtain polynomial time satisfiability decision for generalised clause-sets with fixed maximal deficiency (generalising the boolean case, where the deficiency is the difference between the number of clauses and the number of variables). The main tool here is autarky theory (where autarkies are generalisations of satisfying assignments), and we concentrate especially on matching autarkies (based on matching theory) and special cases of linear autarkies (based on linear programming and linear algebra). As an application we study the problem of hypergraph colouring, considering early results of P.D. Seymour on the number of edges in minimally non-2-colourable hypergraphs, which are reinterpreted and generalised in the light of autarky theory.