Minimal Unsatisfiability: Models, Algorithms and Applications (Invited Paper)

The task of modeling and reasoning about real-world problems often involves analyzing overconstrained representations, where not all constraints of a problem can be simultaneously satisfied. The need to analyze over-constrained (or unsatisfiable) problems occurs in many settings, including data and knowledge bases, artificial intelligence, applied formal methods, operations research and description logics. In most cases, the problem to solve is related with some form of minimal unsatisfiability, i.e. an irreducible set of constraints that explains unsatisfiability. This paper provides an overview of some of the computational problems related with minimal unsatisfiability in Boolean logic, including the identification of one minimal unsatisfiable sub-formula and the identification of all minimal unsatisfiable sub-formulas. In addition, the paper briefly overviews practical applications of minimal unsatisfiability. Finally, the paper highlights recent work on minimal unsatisfiability in other domains.

[1]  Karem A. Sakallah,et al.  Reveal: A Formal Verification Tool for Verilog Designs , 2008, LPAR.

[2]  Felip Manyà,et al.  MaxSAT, Hard and Soft Constraints , 2021, Handbook of Satisfiability.

[3]  James Bailey,et al.  Discovery of Minimal Unsatisfiable Subsets of Constraints Using Hitting Set Dualization , 2005, PADL.

[4]  Leslie E. Trotter,et al.  On the maximum feasible subsystem problem, IISs and IIS-hypergraphs , 2003, Math. Program..

[5]  Steven David Prestwich,et al.  CNF Encodings , 2021, Handbook of Satisfiability.

[6]  Sikun Li,et al.  Extracting Minimum Unsatisfiable Cores with a Greedy Genetic Algorithm , 2006, Australian Conference on Artificial Intelligence.

[7]  Hans Kleine Büning,et al.  Minimal Unsatisfiability and Autarkies , 2009, Handbook of Satisfiability.

[8]  Éric Grégoire,et al.  Does This Set of Clauses Overlap with at Least One MUS? , 2009, CADE.

[9]  Inês Lynce,et al.  A branch and bound algorithm for extracting smallest minimal unsatisfiable subformulas , 2008, Constraints.

[10]  Kenneth L. McMillan,et al.  Interpolation and SAT-Based Model Checking , 2003, CAV.

[11]  Nachum Dershowitz,et al.  A Scalable Algorithm for Minimal Unsatisfiable Core Extraction , 2006, SAT.

[12]  Christos H. Papadimitriou,et al.  The complexity of facets resolved , 1985, 26th Annual Symposium on Foundations of Computer Science (sfcs 1985).

[13]  Alberto Griggio,et al.  A Simple and Flexible Way of Computing Small Unsatisfiable Cores in SAT Modulo Theories , 2007, SAT.

[14]  Jennifer Ryan,et al.  Identifying Minimally Infeasible Subsystems of Inequalities , 1990, INFORMS J. Comput..

[15]  Vasco M. Manquinho,et al.  Algorithms for Weighted Boolean Optimization , 2009, SAT.

[16]  Ulrich Junker,et al.  QUICKXPLAIN: Preferred Explanations and Relaxations for Over-Constrained Problems , 2004, AAAI.

[17]  Joyce van Loon.  Irreducibly inconsistent systems of linear inequalities , 1981 .

[18]  Martin Gebser,et al.  Detecting inconsistencies in large biological networks with answer set programming , 2008, Theory and Practice of Logic Programming.

[19]  Maria Luisa Bonet,et al.  Solving (Weighted) Partial MaxSAT through Satisfiability Testing , 2009, SAT.

[20]  Felix Sheng-Ho Chang,et al.  Finding Minimal Unsatisfiable Cores of Declarative Specifications , 2008, FM.

[21]  Harvey J. Greenberg,et al.  Approaches to Diagnosing Infeasible Linear Programs , 1991, INFORMS J. Comput..

[22]  Hans van Maaren,et al.  Finding Guaranteed MUSes Fast , 2008, SAT.

[23]  Alain Hertz,et al.  Using heuristics to find minimal unsatisfiable subformulas in satisfiability problems , 2009, J. Comb. Optim..

[24]  Alain Hertz,et al.  Solution techniques for the Large Set Covering Problem , 2003, Discret. Appl. Math..

[25]  Manu Sridharan,et al.  Debugging overconstrained declarative models using unsatisfiable cores , 2003, 18th IEEE International Conference on Automated Software Engineering, 2003. Proceedings..

[26]  John W. Chinneck,et al.  Finding a Useful Subset of Constraints for Analysis in an Infeasible Linear Program , 1997, INFORMS J. Comput..

[27]  N. J.L.deSiqueira,et al.  Explanation-Based Generalisation of Failures , 1988, ECAI.

[28]  Linus Schrage,et al.  Generalized filtering algorithms for infeasibility analysis , 2008, Comput. Oper. Res..

[29]  P. M. Wognum,et al.  Diagnosing and Solving Over-Determined Constraint Satisfaction Problems , 1993, IJCAI.

[30]  Jie-Hong R. Jiang,et al.  To SAT or not to SAT: Ashenhurst decomposition in a large scale , 2008, ICCAD 2008.

[31]  Éric Grégoire,et al.  Local-search Extraction of MUSes , 2007, Constraints.

[32]  Mehrdad Tamiz,et al.  Detecting iis in infeasible linear programmes using techniques from goal programming , 1996, Comput. Oper. Res..

[33]  Joao Marques-Silva,et al.  Algorithms for Maximum Satisfiability using Unsatisfiable Cores , 2008, 2008 Design, Automation and Test in Europe.

[34]  Sharad Malik,et al.  Extracting small unsatis able cores from unsatis able boolean formulas , 2003 .

[35]  Niraj K. Jha,et al.  Efficient Design for Testability Solution Based on Unsatisfiability for Register-Transfer Level Circuits , 2007, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[36]  Igor L. Markov,et al.  AMUSE: a minimally-unsatisfiable subformula extractor , 2004, Proceedings. 41st Design Automation Conference, 2004..

[37]  Sharad Malik,et al.  On Solving the Partial MAX-SAT Problem , 2006, SAT.

[38]  Inês Lynce,et al.  On Computing Minimum Unsatisfiable Cores , 2004, SAT.

[39]  John W. Chinneck,et al.  Locating Minimal Infeasible Constraint Sets in Linear Programs , 1991, INFORMS J. Comput..

[40]  Éric Grégoire,et al.  Extracting MUSes , 2006, ECAI.

[41]  Jie-Hong Roland Jiang,et al.  Bi-decomposing large Boolean functions via interpolation and satisfiability solving , 2008, 2008 45th ACM/IEEE Design Automation Conference.

[42]  Lakhdar Sais,et al.  Extracting MUCs from Constraint Networks , 2006, ECAI.

[43]  Antonio Sassano,et al.  Errors Detection and Correction in Large Scale Data Collecting , 2001, IDA.

[44]  Renato Bruni,et al.  Approximating minimal unsatisfiable subformulae by means of adaptive core search , 2003, Discret. Appl. Math..

[45]  Karem A. Sakallah,et al.  Algorithms for Computing Minimal Unsatisfiable Subsets of Constraints , 2007, Journal of Automated Reasoning.

[46]  Inês Lynce,et al.  Conflict-Driven Clause Learning SAT Solvers , 2009, Handbook of Satisfiability.

[47]  Renato Bruni,et al.  On exact selection of minimally unsatisfiable subformulae , 2005, Annals of Mathematics and Artificial Intelligence.

[48]  Ernest Teniente,et al.  Providing Explanations for Database Schema Validation , 2008, DEXA.

[49]  Lakhdar Sais,et al.  Efficient Combination of Decision Procedures for MUS Computation , 2009, FroCoS.

[50]  Jinbo Huang,et al.  MUP: a minimal unsatisfiability prover , 2005, ASP-DAC.

[51]  Éric Grégoire,et al.  On Approaches to Explaining Infeasibility of Sets of Boolean Clauses , 2008, 2008 20th IEEE International Conference on Tools with Artificial Intelligence.

[52]  Bart Selman,et al.  Satisfiability Solvers , 2008, Handbook of Knowledge Representation.

[53]  Rafael Peñaloza,et al.  Axiom Pinpointing in General Tableaux , 2007, TABLEAUX.

[54]  John W. Chinneck,et al.  Analyzing Infeasible Mixed-Integer and Integer Linear Programs , 1999, INFORMS J. Comput..

[55]  Barry O'Sullivan,et al.  Reformulating Table Constraints using Functional Dependencies—An Application to Explanation Generation , 2008, Constraints.