Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas
Journal of Combinatorial Theory Series A
The complexity of facets resolved
Journal of Computer and System Sciences - 26th IEEE Conference on Foundations of Computer Science, October 21-23, 1985
Sharp bounds for decompositions of graphs into complete r-partite subgraphs
Journal of Graph Theory
On subclasses of minimal unsatisfiable formulas
Discrete Applied Mathematics - Special issue on Boolean functions and related problems
An efficient algorithm for the minimal unsatisfiability problem for a subclass of CNF
Annals of Mathematics and Artificial Intelligence
An Application of Matroid Theory to the SAT Problem
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
Approximating minimal unsatisfiable subformulae by means of adaptive core search
Discrete Applied Mathematics - The renesse issue on satisfiability
Annals of Mathematics and Artificial Intelligence
Theory and Applications of Satisfiability Testing: 6th International Conference, Sat 2003, Santa Margherita Ligure, Italy, May 5-8 2003: Selected Revised Papers (Lecture Notes in Computer Science, 2919)
Extension and Equivalence Problems for Clause Minimal Formulae
Annals of Mathematics and Artificial Intelligence
Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable
Journal of Computer and System Sciences
Theory and Applications of Satisfiability Testing: 8th International Conference, SAT 2005, St Andrews, Scotland, June 19-23, 2005, Proceedings (Lecture Notes in Computer Science)
Theory and Applications of Satisfiability Testing: 7th International Conference, SAT 2004, Vancouver, BC, Canada, May 10-13, 2004, Revised Selected Papers (Lecture Notes in Computer Science)
On $k$-Term DNF with the Largest Number of Prime Implicants
SIAM Journal on Discrete Mathematics
Handbook of Satisfiability: Volume 185 Frontiers in Artificial Intelligence and Applications
Handbook of Satisfiability: Volume 185 Frontiers in Artificial Intelligence and Applications
Solving non-Boolean satisfiability problems with stochastic local search
IJCAI'01 Proceedings of the 17th international joint conference on Artificial intelligence - Volume 1
Redundancy in logic I: CNF propositional formulae
Artificial Intelligence
Mapping problems with finite-domain variables to problems with boolean variables
SAT'04 Proceedings of the 7th international conference on Theory and Applications of Satisfiability Testing
Polynomial time SAT decision, hypergraph transversals and the hermitian rank
SAT'04 Proceedings of the 7th international conference on Theory and Applications of Satisfiability Testing
On finding all minimally unsatisfiable subformulas
SAT'05 Proceedings of the 8th international conference on Theory and Applications of Satisfiability Testing
SAT'10 Proceedings of the 13th international conference on Theory and Applications of Satisfiability Testing
SAT'06 Proceedings of the 9th international conference on Theory and Applications of Satisfiability Testing
Constraint Satisfaction Problems in Clausal Form I: Autarkies and Deficiency
Fundamenta Informaticae
On variables with few occurrences in conjunctive normal forms
SAT'11 Proceedings of the 14th international conference on Theory and application of satisfiability testing
SAT'10 Proceedings of the 13th international conference on Theory and Applications of Satisfiability Testing
On davis-putnam reductions for minimally unsatisfiable clause-sets
SAT'12 Proceedings of the 15th international conference on Theory and Applications of Satisfiability Testing
On computing minimal equivalent subformulas
CP'12 Proceedings of the 18th international conference on Principles and Practice of Constraint Programming
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Concluding this mini-series of 2 articles on the foundations of generalised clause-sets, we study the combinatorial properties of non-boolean conjunctive normal forms (clause-sets), allowing arbitrary (but finite) sets of values for variables, while literals express that some variable shall not get some (given) value. First we study the properties of the direct translation (or “encoding”) of generalised clause-sets into boolean clause-sets. Many combinatorial properties are preserved, and as a result we can lift fixed-parameter tractability of satisfiability in the maximal deficiency from the boolean case to the general case. Then we turn to irredundant clause-sets, which generalise minimally unsatisfiable clause-sets, and we prove basic properties. The simplest irredundant clause-sets are hitting clause-sets, and we provide characterisations and generalisations. Unsatisfiable irredundant clause-sets are the minimally unsatisfiable clause-sets, and we provide basic tools. These tools allow us to characterise the minimally unsatisfiable clause-sets of minimal deficiency. Finally we provide a new translation of generalised boolean clause-sets into boolean clause-sets, the nested translation, which preserves the conflict structure. As an application, we can generalise results for boolean clause-sets regarding the hermitian rank/defect, especially the characterisation of unsatisfiable hitting clause-sets where between every two clauses we have exactly one conflict. We conclude with a list of open problems, and a discussion of the “generic translation scheme”.