Minimal unsatisfiability describes the reduced kernel of unsatisfiable formulas. The investigation of this property is very helpful in understanding the reasons for unsatisfiability as well as the behaviour of SAT-solvers and proof calculi. Moreover, for propositional formulas and quantified Boolean formulas the computational complexity of various SAT-related problems are strongly related to the complexity of minimal unsatisfiable formulas. While “minimal unsatisfiability” studies the structure of problem instances without redundancies, the study of “autarkies” considers the redundancies themselves, in various guises related to partial assignments which satisfy some part of the problem instance while leaving the rest “untouched”. As it turns out, autarky theory creates many bridges to combinatorics, algebra and logic, and the second part of this chapter provides a solid foundation of the basic ideas and results of autarky theory: the basic algorithmic problems, the algebra involved, and relations to various combinatorial theories (e.g., matching theory, linear programming, graph theory, the theory of permanents). Also the general theory of autarkies as a kind of combinatorial “meta theory” is sketched (regarding its basic notions).
An Extension of Deficiency and Minimal Unsatisfiability of Quantified Boolean Formulas