Implicit Hitting Set Algorithms for Reasoning Beyond NP

Lifting a recent proposal by Moreno-Centeno and Karp, we propose a general framework for so-called implicit hitting set algorithms for reasoning beyond NP. The framework is motivated by empirically successful specific instantiations of the approach—based on interactions between a Boolean satisfiability (SAT) solver and an integer programming (IP) solver—in the context of maximum satisfiability (MaxSAT). The framework opens up opportunities for developing implicit hitting set algorithms for various important reasoning problems in KR by implementing domain-specific reasoning modules with SAT and IP solvers. We detail instantiations of the framework for the minimum satisfiability problem— as a natural dual of MaxSAT—and, as a central KR problem, for propositional abduction, covering the second level of the polynomial hierarchy. We show empirically that an implementation of the instantiation for propositional abduction surpasses the efficiency of an approach based on encoding and solving propositional abduction instances as disjunctive logic programming under answer set semantics. We also study key properties of the general framework.

[1]  Bernd Finkbeiner,et al.  Lazy Synthesis , 2012, VMCAI.

[2]  Isil Dillig,et al.  Explain: A Tool for Performing Abductive Inference , 2013, CAV.

[3]  Fahiem Bacchus,et al.  Solving Weighted CSPs by Successive Relaxations , 2013, CP.

[4]  Nina Narodytska,et al.  Maximum Satisfiability Using Core-Guided MaxSAT Resolution , 2014, AAAI.

[5]  Mikolás Janota,et al.  Algorithms for computing backbones of propositional formulae , 2015, AI Commun..

[6]  Joao Marques-Silva,et al.  Iterative SAT Solving for Minimum Satisfiability , 2012, 2012 IEEE 24th International Conference on Tools with Artificial Intelligence.

[7]  Georg Gottlob,et al.  Propositional Circumscription and Extended Closed-World Reasoning are IIp2-Complete , 1993, Theor. Comput. Sci..

[8]  Eugene Santos,et al.  A Linear Constraint Satisfaction Approach to Cost-Based Abduction , 1994, Artif. Intell..

[9]  Fahiem Bacchus,et al.  Exploiting the Power of mip Solvers in maxsat , 2013, SAT.

[10]  Stefan Rümmele,et al.  Backdoors to Abduction , 2013, IJCAI.

[11]  Meir Kalech,et al.  Exploring the Duality in Conflict-Directed Model-Based Diagnosis , 2012, AAAI.

[12]  Helmut Veith,et al.  Counterexample-guided abstraction refinement for symbolic model checking , 2003, JACM.

[13]  Mikolás Janota,et al.  Solving QBF by Clause Selection , 2015, IJCAI.

[14]  Santosh S. Vempala,et al.  Algorithms for implicit hitting set problems , 2011, SODA '11.

[15]  Joao Marques-Silva,et al.  SAT-Based Horn Least Upper Bounds , 2015, SAT.

[16]  Georg Gottlob,et al.  On the complexity of propositional knowledge base revision, updates, and counterfactuals , 1992, Artif. Intell..

[17]  Vasco M. Manquinho,et al.  Incremental Cardinality Constraints for MaxSAT , 2014, CP.

[18]  Zhu Zhu,et al.  Optimizing with minimum satisfiability , 2012, Artif. Intell..

[19]  Mikolás Janota,et al.  Minimal Sets over Monotone Predicates in Boolean Formulae , 2013, CAV.

[20]  Alberto Griggio,et al.  A Modular Approach to MaxSAT Modulo Theories , 2013, SAT.

[21]  Radu Grigore,et al.  Counterexample Guided Abstraction Refinement Algorithm for Propositional Circumscription , 2010, JELIA.

[22]  Stefan Rümmele,et al.  The Parameterized Complexity of Abduction , 2012, AAAI.

[23]  Stefan Szeider,et al.  Compendium of Parameterized Problems at Higher Levels of the Polynomial Hierarchy , 2019, Electron. Colloquium Comput. Complex..

[24]  Joao Marques-Silva,et al.  Prime Compilation of Non-Clausal Formulae , 2015, IJCAI.

[25]  Xinming Ou,et al.  Theorem Proving Using Lazy Proof Explication , 2003, CAV.

[26]  Zhu Zhu,et al.  Minimum Satisfiability and Its Applications , 2011, IJCAI.

[27]  Youssef Hamadi,et al.  Efficiently solving quantified bit-vector formulas , 2010, Formal Methods in Computer Aided Design.

[28]  Harald Ruess,et al.  Lazy Theorem Proving for Bounded Model Checking over Infinite Domains , 2002, CADE.

[29]  Georg Gottlob,et al.  Complexity Results for Nonmonotonic Logics , 1992, J. Log. Comput..

[30]  Jonathan Stillman,et al.  It's Not My Default: The Complexity of Membership Problems in Restricted Propositional Default Logics , 1990, AAAI.

[31]  Raymond Reiter,et al.  A Theory of Diagnosis from First Principles , 1986, Artif. Intell..

[32]  Mikolás Janota,et al.  Solving QBF with Counterexample Guided Refinement , 2012, SAT.

[33]  Alexey Ignatiev,et al.  On Reducing Maximum Independent Set to Minimum Satisfiability , 2014, SAT.

[34]  Richard M. Karp,et al.  The Implicit Hitting Set Approach to Solve Combinatorial Optimization Problems with an Application to Multigenome Alignment , 2013, Oper. Res..

[35]  Adrian Kügel.  Natural Max-SAT Encoding of Min-SAT , 2012, LION.

[36]  Brian C. Williams,et al.  Diagnosing Multiple Faults , 1987, Artif. Intell..

[37]  Georg Gottlob,et al.  The Complexity of Logic-Based Abduction , 1993, STACS.

[38]  Zhu Zhu,et al.  A SAT-Based Approach to MinSAT , 2012, CCIA.

[39]  Mikolás Janota,et al.  Solving QBF with Counterexample Guided Refinement , 2012, SAT.

[40]  Stefan Szeider,et al.  Parameterized Complexity Results for Agenda Safety in Judgment Aggregation , 2015, AAMAS.

[41]  Ofer Strichman,et al.  SAT-based counterexample-guided abstraction refinement , 2004, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[42]  Joao Marques-Silva,et al.  Smallest MUS Extraction with Minimal Hitting Set Dualization , 2015, CP.

[43]  Paolo Mancarella,et al.  Abductive Logic Programming , 1992, LPNMR.

[44]  Zhu Zhu,et al.  Exact MinSAT Solving , 2010, SAT.

[45]  Josep Argelich,et al.  MinSAT versus MaxSAT for Optimization Problems , 2013, CP.

[46]  Hwee Tou Ng,et al.  Abductive Plan Recognition and Diagnosis: A Comprehensive Empirical Evaluation , 1992, KR.

[47]  Mikolás Janota,et al.  Abstraction-Based Algorithm for 2QBF , 2011, SAT.

[48]  Takeaki Uno,et al.  Enumerating Minimal Explanations by Minimal Hitting Set Computation , 2006, KSEM.

[49]  Miroslaw Truszczynski,et al.  Answer set programming at a glance , 2011, Commun. ACM.

[50]  Josep Argelich,et al.  A New Encoding from MinSAT into MaxSAT , 2012, CP.

[51]  David L. Dill,et al.  Checking Satisfiability of First-Order Formulas by Incremental Translation to SAT , 2002, CAV.

[52]  Georg Gottlob,et al.  On the computational cost of disjunctive logic programming: Propositional case , 1995, Annals of Mathematics and Artificial Intelligence.

[53]  Joao Marques-Silva,et al.  Core-Guided Binary Search Algorithms for Maximum Satisfiability , 2011, AAAI.

[54]  Joao Marques-Silva,et al.  Core-Guided MaxSAT with Soft Cardinality Constraints , 2014, International Conference on Principles and Practice of Constraint Programming.

[55]  Marius Thomas Lindauer,et al.  Potassco: The Potsdam Answer Set Solving Collection , 2011, AI Commun..

[56]  Stefan Woltran,et al.  The complexity of handling minimal solutions in logic-based abduction , 2015, J. Log. Comput..

[57]  Stefan Woltran,et al.  Complexity-sensitive decision procedures for abstract argumentation , 2012, Artif. Intell..

[58]  Joao Marques-Silva,et al.  Maximal Falsifiability - Definitions, Algorithms, and Applications , 2013, LPAR.

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

[60]  Stefan Szeider,et al.  Fixed-Parameter Tractable Reductions to SAT , 2014, SAT.

[61]  Ron Rymon.  An SE-tree-based prime implicant generation algorithm , 2005, Annals of Mathematics and Artificial Intelligence.

[62]  Joao Marques-Silva,et al.  Efficient MUS Enumeration of Horn Formulae with Applications to Axiom Pinpointing , 2015, SAT.

[63]  Fahiem Bacchus,et al.  Solving MAXSAT by Solving a Sequence of Simpler SAT Instances , 2011, CP.

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

[65]  Mikolás Janota,et al.  Efficient Model Based Diagnosis with Maximum Satisfiability , 2015, IJCAI.

[66]  Joao Marques-Silva,et al.  Fast, flexible MUS enumeration , 2015, Constraints.

[67]  Fahiem Bacchus,et al.  Postponing Optimization to Speed Up MAXSAT Solving , 2013, CP.

[68]  Carlos Ansótegui,et al.  MaxSAT Evaluation 2017 , 2017 .

[69]  Jonathan Stillman,et al.  The Complexity of Propositional Default Logics , 1992, AAAI.

[70]  Thomas Lukasiewicz,et al.  Default reasoning from conditional knowledge bases: Complexity and tractable cases , 2000, Artif. Intell..