Minimal non-two-colorable hypergraphs and minimal unsatisfiable formulas
Journal of Combinatorial Theory Series A
Intelligent backtracking on constraint satisfaction problems: experimental and theoretical results
Intelligent backtracking on constraint satisfaction problems: experimental and theoretical results
Handbook of combinatorics (vol. 1)
Subgraph counting identities and Ramsey numbers
Journal of Combinatorial Theory Series B
New methods for 3-SAT decision and worst-case analysis
Theoretical Computer Science
The SAT problem of signed CNF formulas
Labelled deduction
On subclasses of minimal unsatisfiable formulas
Discrete Applied Mathematics - Special issue on Boolean functions and related problems
Investigations on autark assignments
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
Parallel cooperative propositional theorem proving
Annals of Mathematics and Artificial Intelligence
Autarky Pruning in Propositional Model Elimination Reduces Failure Redundancy
Journal of Automated Reasoning
Polynomial-time recognition of minimal unsatisfiable formulas with fixed clause-variable difference
Theoretical Computer Science
A perspective on certain polynomial-time solvable classes of satisfiability
Discrete Applied Mathematics
An Upper Bound for Minimal Resolution Refutations
Proceedings of the 12th International Workshop on Computer Science Logic
An Application of Matroid Theory to the SAT Problem
COCO '00 Proceedings of the 15th Annual IEEE Conference on Computational Complexity
Generalized satisfiability problems: minimal elements and phase transitions
Theoretical Computer Science
Constraint Processing
On the structure of some classes of minimal unsatisfiable formulas
Discrete Applied Mathematics - The renesse issue on satisfiability
Lean clause-sets: generalizations of minimally unsatisfiable clause-sets
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)
Minimal unsatisfiable formulas with bounded clause-variable difference are fixed-parameter tractable
Journal of Computer and System Sciences
Matching Theory (North-Holland mathematics studies)
Matching Theory (North-Holland mathematics studies)
A Note on Unsatisfiable k-CNF Formulas with Few Occurrences per Variable
SIAM Journal on Discrete Mathematics
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)
Principles of Constraint Programming
Principles of Constraint Programming
On $k$-Term DNF with the Largest Number of Prime Implicants
SIAM Journal on Discrete Mathematics
Present and Future of Practical SAT Solving
Complexity of Constraints
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
Introduction to Mathematics of Satisfiability
Introduction to Mathematics of Satisfiability
SAT'07 Proceedings of the 10th international conference on Theory and applications of satisfiability testing
Mapping CSP into many-valued SAT
SAT'07 Proceedings of the 10th international conference on Theory and applications of satisfiability testing
SAT'07 Proceedings of the 10th international conference on Theory and applications of satisfiability testing
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
Resolution tunnels for improved SAT solver performance
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
TACAS'10 Proceedings of the 16th international conference on Tools and Algorithms for the Construction and Analysis of Systems
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
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We consider the problem of generalising boolean formulas in conjunctive normal form by allowing non-boolean variables, with the goal of maintaining combinatorial properties. Requiring that a literal involves only a single variable, the most general form of literals are the wellknown “signed literals”, corresponding to unary constraints in CSP. However we argue that only the restricted form of “negative monosigned literals” and the resulting generalised clause-sets, corresponding to “sets of no-goods” in the AI literature, maintain the essential properties of boolean conjunctive normal forms. In this first part of a mini-series of two articles, we build up a solid foundation for (generalised) clause-sets, including the notion of autarky systems, the interplay between autarkies and resolution, and basic notions of (DP-)reductions. As a basic combinatorial parameter of generalised clause-sets we introduce the (generalised) notion of deficiency, which in the boolean case is the difference between the number of clauses and the number of variables. Autarky theory plays a fundamental role here, and we concentrate especially on matching autarkies (based on matching theory). A natural task is to determine the structure of (matching) lean clause-sets, which do not admit non-trivial (matching) autarkies. A central result is the computation of the lean kernel (the largest lean subset) of a (generalised) clause-set in polynomial time for bounded maximal deficiency.