Logic programs with propositional connectives and aggregates

Answer set programming (ASP) is a logic programming paradigm that can be used to solve complex combinatorial search problems. Aggregates are an ASP construct that plays an important role in many applications. Defining a satisfactory semantics of aggregates turned out to be a difficult problem, and in this article we propose a new approach, based on an analogy between aggregates and propositional connectives. First we extend the definition of an answer set/stable model to cover arbitrary propositional theories; then we define aggregates on top of them both as primitive constructs and as abbreviations for formulas. Our definition of an aggregate combines expressiveness and simplicity, and it inherits many theorems about programs with nested expressions, such as theorems about strong equivalence and splitting.

[1]  Maurice Bruynooghe,et al.  Translation of Aggregate Programs to Normal Logic Programs , 2003, Answer Set Programming.

[2]  Ilkka Niemelä,et al.  DES: a Challenge Problem for Nonmonotonic Reasoning Systems , 2000, ArXiv.

[3]  Joohyung Lee,et al.  A generalization of the Lin-Zhao theorem , 2006, Annals of Mathematics and Artificial Intelligence.

[4]  Paolo Ferraris,et al.  Answer Sets for Propositional Theories , 2005, LPNMR.

[5]  Maurice Bruynooghe,et al.  Ultimate Well-Founded and Stable Semantics for Logic Programs with Aggregates , 2001, ICLP.

[6]  Pedro Cabalar,et al.  Propositional theories are strongly equivalent to logic programs , 2007, Theory Pract. Log. Program..

[7]  Martin D. F. Wong,et al.  Wire Routing and Satisfiability Planning , 2000, Computational Logic.

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

[9]  Hudson Turner,et al.  Strong equivalence made easy: nested expressions and weight constraints , 2003, Theory and Practice of Logic Programming.

[10]  C. R. Ramakrishnan,et al.  Fully Local and Efficient Evaluation of Alternating Fixed Points (Extended Abstract) , 1998, TACAS.

[11]  Jorge Lobo,et al.  Reasoning about Policies using Logic Programs , 2001, Answer Set Programming.

[12]  TurnerHudson.  Strong equivalence made easy: nested expressions and weight constraints , 2003 .

[13]  Vladimir Lifschitz,et al.  Answer Set Planning (Abstract) , 1999, LPNMR.

[14]  Esra Erdem,et al.  Reconstructing the Evolutionary History of Indo-European Languages Using Answer Set Programming , 2003, PADL.

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

[16]  Vladimir Lifschitz,et al.  Expressiveness of answer set languages , 2007 .

[17]  Wolfgang Faber,et al.  Declarative and Computational Properties of Logic Programs with Aggregates , 2005, IJCAI.

[18]  Ilkka Niemelä,et al.  Answer Set Programming and Bounded Model Checking , 2001, Answer Set Programming.

[19]  Paolo Ferraris.  A Logic Program Characterization of Causal Theories , 2007, IJCAI.

[20]  Anna Philippou,et al.  Tools and Algorithms for the Construction and Analysis of Systems , 2018, Lecture Notes in Computer Science.

[21]  Vladimir Lifschitz,et al.  Definitions in Answer Set Programming: (Extended Abstract) , 2003, ICLP.

[22]  Jürgen Dix,et al.  Non-Monotonic Extensions of Logic Programming , 1996, Lecture Notes in Computer Science.

[23]  Joohyung Lee,et al.  A New Perspective on Stable Models , 2007, IJCAI.

[24]  David Pearce,et al.  Strongly equivalent logic programs , 2001, ACM Trans. Comput. Log..

[25]  Joohyung Lee,et al.  On Reductive Semantics of Aggregates in Answer Set Programming , 2009, LPNMR.

[26]  Enrico Pontelli,et al.  Answer Sets for Logic Programs with Arbitrary Abstract Constraint Atoms , 2006, AAAI.

[27]  S. Shelah,et al.  Annals of Pure and Applied Logic , 1991 .

[28]  Wolfgang Faber,et al.  Recursive Aggregates in Disjunctive Logic Programs: Semantics and Complexity , 2004, JELIA.

[29]  Jorge Lobo,et al.  Formalizing (and Reasoning About) the Specifications of Workflows , 2000, CoopIS.

[30]  Dan Suciu,et al.  Journal of the ACM , 2006 .

[31]  Ilkka Niemelä,et al.  Developing a Declarative Rule Language for Applications in Product Configuration , 1999, PADL.

[32]  Ismail Hakki Toroslu,et al.  Specification of Workflow Process Using the Action Description Language C , 2001, Answer Set Programming.

[33]  R. Lathe.  Phd by thesis , 1988, Nature.

[34]  Vladimir Lifschitz,et al.  Splitting a Logic Program , 1994, ICLP.

[35]  Mauricio Osorio,et al.  Safe beliefs for propositional theories , 2005, Ann. Pure Appl. Log..

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

[37]  Chitta Baral,et al.  Declarative Specification and Solution of Combinatorial Auctions Using Logic Programming , 2001, LPNMR.

[38]  David Pearce,et al.  From Here to There: Stable Negation in Logic Programming , 1999 .

[39]  Hudson Turner,et al.  Causal Theories of Action and Change , 1997, AAAI/IAAI.

[40]  Vladimir Lifschitz,et al.  Mathematical Foundations of Answer Set Programming , 2005, We Will Show Them!.

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

[42]  Victor W. Marek,et al.  Autoepistemic logic , 1991, JACM.

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

[44]  Maurice Bruynooghe,et al.  Well-founded and stable semantics of logic programs with aggregates , 2007, Theory Pract. Log. Program..

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

[46]  J. Davenport.  Editor , 1960 .

[47]  Bernhard Nebel,et al.  Encoding Planning Problems in Nonmonotonic Logic Programs , 1997, ECP.

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

[49]  Danny De Schreye,et al.  Answer Set Planning , 1999 .

[50]  Victor W. Marek,et al.  Ultimate Approximations in Nonmonotonic Knowledge Representation Systems , 2002, KR.

[51]  Stefan Woltran,et al.  Encodings for Equilibrium Logic and Logic Programs with Nested Expressions , 2001, EPIA.

[52]  David Pearce,et al.  A New Logical Characterisation of Stable Models and Answer Sets , 1996, NMELP.

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

[54]  Michael Gelfond,et al.  Classical negation in logic programs and disjunctive databases , 1991, New Generation Computing.

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