What is answer set programming to propositional satisfiability

Propositional satisfiability (or satisfiability) and answer set programming are two closely related subareas of Artificial Intelligence that are used to model and solve difficult combinatorial search problems. Satisfiability solvers and answer set solvers are the software systems that find satisfying interpretations and answer sets for given propositional formulas and logic programs, respectively. These systems are closely related in their common design patterns. In satisfiability, a propositional formula is used to encode problem specifications in a way that its satisfying interpretations correspond to the solutions of the problem. To find solutions to a problem it is then sufficient to use a satisfiability solver on a corresponding formula. Niemelä, Marek, and Truszczyński coined answer set programming paradigm in 1999: in this paradigm a logic program encodes problem specifications in a way that the answer sets of a logic program represent the solutions of the problem. As a result, to find solutions to a problem it is sufficient to use an answer set solver on a corresponding program. These parallels that we just draw between paradigms naturally bring up a question: what is a fundamental difference between the two? This paper takes a close look at this question.

[1]  Cesare Tinelli,et al.  Solving SAT and SAT Modulo Theories: From an abstract Davis--Putnam--Logemann--Loveland procedure to DPLL(T) , 2006, JACM.

[2]  Yuliya Lierler,et al.  One More Decidable Class of Finitely Ground Programs , 2009, ICLP.

[3]  Vasco M. Manquinho,et al.  Pseudo-Boolean and Cardinality Constraints , 2021, Handbook of Satisfiability.

[4]  Toby Walsh,et al.  Handbook of Constraint Programming (Foundations of Artificial Intelligence) , 2006 .

[5]  Tomi Janhunen,et al.  Representing Normal Programs with Clauses , 2004, ECAI.

[6]  Armin Biere,et al.  A survey of recent advances in SAT-based formal verification , 2005, International Journal on Software Tools for Technology Transfer.

[7]  Bart Selman,et al.  Planning as Satisfiability , 1992, ECAI.

[8]  Michael Codish,et al.  Compiling finite domain constraints to SAT with BEE* , 2012, Theory and Practice of Logic Programming.

[9]  Stephan Schulz.  A Comparison of Different Techniques for Grounding Near-Propositional CNF Formulae , 2002, FLAIRS Conference.

[10]  Johan Wittocx,et al.  The IDP system: A model expansion system for an extension of classical logic , 2008 .

[11]  G. S. Tseitin.  On the Complexity of Derivation in Propositional Calculus , 1983 .

[12]  Geoff Sutcliffe.  The 6th IJCAR automated theorem proving system competition - CASC-J6 , 2013, AI Commun..

[13]  Marc Denecker,et al.  Extending Classical Logic with Inductive Definitions , 2000, Computational Logic.

[14]  Martin Gebser,et al.  GrinGo : A New Grounder for Answer Set Programming , 2007, LPNMR.

[15]  I. Niemelä,et al.  Extending the Smodels system with cardinality and weight constraints , 2001 .

[16]  Sharad Malik,et al.  Chaff: engineering an efficient SAT solver , 2001, Proceedings of the 38th Design Automation Conference (IEEE Cat. No.01CH37232).

[17]  Francesco Scarcello,et al.  Disjunctive Stable Models: Unfounded Sets, Fixpoint Semantics, and Computation , 1997, Inf. Comput..

[18]  Martin Gebser,et al.  Conflict-Driven Answer Set Solving , 2007, IJCAI.

[19]  Armin Biere,et al.  Bounded model checking , 2003, Adv. Comput..

[20]  Miroslaw Truszczynski,et al.  Predicate-calculus-based logics for modeling and solving search problems , 2006, TOCL.

[21]  Yuliya Lierler,et al.  Abstract answer set solvers with backjumping and learning , 2010, Theory and Practice of Logic Programming.

[22]  David Pearce,et al.  Correct Reasoning: essays on logic-based AI in honor of Vladimir Lifschitz , 2012 .

[23]  Mario Alviano,et al.  Advances in WASP , 2015, LPNMR.

[24]  V. S. Costa,et al.  Theory and Practice of Logic Programming , 2010 .

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

[26]  Fangzhen Lin,et al.  ASSAT: computing answer sets of a logic program by SAT solvers , 2002, Artif. Intell..

[27]  J. Lloyd.  Foundations of Logic Programming , 1984, Symbolic Computation.

[28]  Timo Soininen,et al.  Extending and implementing the stable model semantics , 2000, Artif. Intell..

[29]  David G. Mitchell,et al.  A Framework for Representing and Solving NP Search Problems , 2005, AAAI.

[30]  Sharad Malik,et al.  Efficient conflict driven learning in a Boolean satisfiability solver , 2001, IEEE/ACM International Conference on Computer Aided Design. ICCAD 2001. IEEE/ACM Digest of Technical Papers (Cat. No.01CH37281).

[31]  Martin Gebser,et al.  Repair and Prediction (under Inconsistency) in Large Biological Networks with Answer Set Programming , 2010, KR.

[32]  Tomi Janhunen,et al.  Some (in)translatability results for normal logic programs and propositional theories , 2006, J. Appl. Non Class. Logics.

[33]  Roman Barták,et al.  Constraint Programming: In Pursuit of the Holy Grail , 1999 .

[34]  Miroslaw Truszczynski,et al.  Connecting First-Order ASP and the Logic FO(ID) through Reducts , 2012, Correct Reasoning.

[35]  Toby Walsh,et al.  Handbook of Satisfiability: Volume 185 Frontiers in Artificial Intelligence and Applications , 2009 .

[36]  Vladimir Lifschitz,et al.  Answer set programming and plan generation , 2002, Artif. Intell..

[37]  Martin Gebser,et al.  Conflict-driven answer set solving: From theory to practice , 2012, Artif. Intell..

[38]  Victor W. Marek,et al.  Stable models and an alternative logic programming paradigm , 1998, The Logic Programming Paradigm.

[39]  Wolfgang Faber,et al.  The DLV system for knowledge representation and reasoning , 2002, TOCL.

[40]  F. RICCA,et al.  Team-building with answer set programming in the Gioia-Tauro seaport , 2011, Theory and Practice of Logic Programming.

[41]  Johannes Klaus Fichte,et al.  Clause-Learning Algorithms with Many Restarts and Bounded-Width Resolution , 2011, J. Artif. Intell. Res..

[42]  Keith L. Clark,et al.  Negation as Failure , 1987, Logic and Data Bases.

[43]  Mario Alviano,et al.  WASP: A Native ASP Solver Based on Constraint Learning , 2013, LPNMR.

[44]  Martin Gebser,et al.  aspcud: A Linux Package Configuration Tool Based on Answer Set Programming , 2011, LoCoCo.

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

[46]  Martin Gebser,et al.  Abstract gringo , 2015, Theory Pract. Log. Program..

[47]  Martin Gebser,et al.  What's a Head Without a Body? , 2006, ECAI.

[48]  Vladimir Lifschitz,et al.  Weight constraints as nested expressions , 2003, Theory and Practice of Logic Programming.

[49]  Jussi Rintanen,et al.  Planning as satisfiability: Heuristics , 2012, Artif. Intell..

[50]  John S. Schlipf,et al.  The Expressive Powers of the Logic Programming Semantics , 1995, J. Comput. Syst. Sci..

[51]  Miroslaw Truszczynski,et al.  A Tarskian Informal Semantics for Answer Set Programming , 2012, ICLP.

[52]  V. Lifschitz,et al.  Foundations of Logic Programming , 1997 .

[53]  Vladimir Lifschitz,et al.  Formalizing Common Sense: Papers by John McCarthy , 1998 .

[54]  Adnan Darwiche,et al.  On the power of clause-learning SAT solvers as resolution engines , 2011, Artif. Intell..

[55]  Joohyung Lee,et al.  A Model-Theoretic Counterpart of Loop Formulas , 2005, IJCAI.

[56]  Armin Biere,et al.  Effective Preprocessing in SAT Through Variable and Clause Elimination , 2005, SAT.

[57]  J. P. Marques,et al.  GRASP : A Search Algorithm for Propositional Satisfiability , 1999 .

[58]  Vladimir Lifschitz,et al.  Nested expressions in logic programs , 1999, Annals of Mathematics and Artificial Intelligence.

[59]  Yuliya Lierler,et al.  Logic Programs vs. First-Order Formulas in Textual Inference , 2013, IWCS.

[60]  Dov M. Gabbay,et al.  What Is Negation as Failure? , 2012, Logic Programs, Norms and Action.

[61]  Giovambattista Ianni,et al.  Computable Functions in ASP: Theory and Implementation , 2008, ICLP.

[62]  Martin Gebser,et al.  The BioASP Library: ASP Solutions for Systems Biology , 2010, 2010 22nd IEEE International Conference on Tools with Artificial Intelligence.

[63]  Miroslaw Truszczynski,et al.  Transition systems for model generators—A unifying approach , 2011, Theory and Practice of Logic Programming.

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

[65]  Martin Gebser,et al.  Writing Declarative Specifications for Clauses , 2016, JELIA.

[66]  Martin Gebser,et al.  On the Implementation of Weight Constraint Rules in Conflict-Driven ASP Solvers , 2009, ICLP.

[67]  François Fages,et al.  A new fixpoint semantics for general logic programs compared with the well-founded and the stable model semantics , 1990, New Generation Computing.

[68]  Donald W. Loveland,et al.  A machine program for theorem-proving , 2011, CACM.

[69]  Yuliya Lierler,et al.  Answer Set Programming Based on Propositional Satisfiability , 2006, Journal of Automated Reasoning.

[70]  John McCarthy,et al.  Mathematical logic in artificial intelligence , 1989 .

[71]  Johan Wittocx,et al.  GidL: A grounder for FO+ , 2008, Non-Monotonic Reasoning.

[72]  Yuliya Lierler,et al.  On elementary loops of logic programs , 2010, Theory and Practice of Logic Programming.

[73]  Alexander A. Razborov,et al.  Why are there so many loop formulas? , 2006, TOCL.

[74]  Enrico Giunchiglia,et al.  On the Relation Between Answer Set and SAT Procedures (or, Between cmodels and smodels) , 2005, ICLP.

[75]  Marco Calautti,et al.  Checking Termination of Bottom-Up Evaluation of Logic Programs with Function Symbols , 2015, Theory Pract. Log. Program..

[76]  Andrei Voronkov,et al.  Planning with Effectively Propositional Logic , 2013, Programming Logics.

[77]  Georg Gottlob,et al.  Complexity Results for Disjunctive Logic Programming and Application to Nonmonotonic Logics , 1993, ILPS.

[78]  Michael Gelfond,et al.  An A Prolog decision support system for the Space Shuttle , 2001, Answer Set Programming.

[79]  John S. Schlipf,et al.  Answer Set Programming with Clause Learning , 2004, LPNMR.

[80]  Matthew W. Moskewicz,et al.  Cha : Engineering an e cient SAT solver , 2001, DAC 2001.

[81]  Ilkka Niemelä,et al.  Logic programs with stable model semantics as a constraint programming paradigm , 1999, Annals of Mathematics and Artificial Intelligence.

[82]  Tommi Syrjänen.  Omega-Restricted Logic Programs , 2001, LPNMR.