Table Of ContentVladimir Lifschitz
Answer Set
Programming
Answer Set Programming
Vladimir Lifschitz
Answer Set Programming
123
VladimirLifschitz
DepartmentofComputerScience
UniversityofTexasatAustin
Austin,TX,USA
ISBN978-3-030-24657-0 ISBN978-3-030-24658-7 (eBook)
https://doi.org/10.1007/978-3-030-24658-7
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Tothecommunityofcomputerscientists,
mathematicians,andphilosopherswho
inventedanswersetprogrammingandare
workinghardtomakeitbetter
“Whatconstitutesthedignityofacraftisthat
itcreatesafellowship,thatitbindsmen
togetherandfashionsforthemacommon
language.”
–AntoinedeSaint-Exupéry
Preface
Answer set programming is a programming methodology rooted in research on
artificial intelligence and computational logic. It was created at the turn of the
century,anditisnowusedinmanyareasofscienceandtechnology.
ThisbookisabouttheartofprogrammingforCLINGO—oneofthemostefficient
and widely used answer set programming systems available today—and about the
mathematics of answer set programming. It is based on an undergraduate class
taught at the University of Texas at Austin. The book is self-contained; the only
prerequisiteissomefamiliaritywithprogramminganddiscretemathematics.
Chapter 1 describes the main ideas of answer set programming and its place
in the world of programming languages. Chapter 2 explains, in an informal way,
several constructs available in the input language of CLINGO. Chapter 3 shows
how they can be used to solve a number of computational problems. Chapters 4–
6putthediscussionofanswersetprogrammingonafirmmathematicalfoundation.
(ReadingChap.6isnotnecessaryforunderstandingtherestofthebook.)Chapter7
describes a few more programming constructs and gives examples of their use. In
Chap.8,answersetprogrammingisappliedtotheproblemofrepresentingactions
andgeneratingplans.
Many examples in this book are framed as exercises, and their solutions are
presentedintheappendix.Thisorganizationgivesthereaderanopportunitytolook
forasolutiononhisownandthencomparehisideaswiththesolutionproposedin
thebook.
This book, as well as everything else I have written, owes much to my wife,
Elena, who heroically accompanied me in long travels around the world in search
foranidealacademicenvironment.Eventually,IfounditintheComputerScience
DepartmentoftheUniversityofTexasatAustin.
In my younger years, I was fortunate to meet a few outstanding scientists and
to have an opportunity to take their classes and talk to them. I learned logic
fromNikolaiShanin(1919–2011),SergeiMaslov(1939–1982),andGrigoriMints
(1939–2014) and artificial intelligence from John McCarthy (1927–2011) and
Raymond Reiter (1939–2002). The most important influence on my professional
work, next to that of my teachers, came from Michael Gelfond, an old friend and
vii
viii Preface
oneofthefoundingfathersofanswersetprogramming.Hehasreadadraftofthis
bookandsuggestedmanywaystoimproveit.Usefulcommentshavebeenprovided
alsobyAmeliaHarrison,RolandKaminski,JoohyungLee,andAlfredZhongand
bystudentswhotookmyclassesonanswersetprogramming.
Finally, thanks to Susan Evans, senior editor, with the Springer Nature’s Com-
puter Science team, who gave me the idea of converting lecture notes into a
textbook.
Austin,TX,USA VladimirLifschitz
May2019
Contents
1 Introduction .................................................................. 1
1.1 DeclarativeProgramming............................................. 1
1.2 LogicProgramming ................................................... 2
1.3 AnswerSetSolvers.................................................... 4
1.4 BibliographicalandHistoricalRemarks.............................. 4
2 InputLanguageofCLINGO................................................. 7
2.1 Rules ................................................................... 8
2.2 DirectivesandComments............................................. 10
2.3 Arithmetic.............................................................. 13
2.4 Definitions ............................................................. 15
2.5 ChoiceRules........................................................... 19
2.6 VariablesinaChoiceRule ............................................ 22
2.7 Constraints............................................................. 23
2.8 AnonymousVariables ................................................. 26
2.9 BibliographicalandHistoricalRemarks.............................. 27
3 CombinatorialSearch....................................................... 29
3.1 SeatingArrangements................................................. 29
3.2 WhoOwnstheJackal? ................................................ 32
3.3 SchurNumbers ........................................................ 32
3.4 DigressiononGroundingandSolving................................ 35
3.5 SetPacking............................................................. 37
3.6 SearchinGraphs....................................................... 37
3.7 SearchinTwoDimensions............................................ 42
3.8 PartnerUnitsProblem................................................. 44
3.9 AdditionalProgrammingExercises................................... 45
3.10 BibliographicalandHistoricalRemarks.............................. 48
ix
x Contents
4 PropositionalProgramsandMinimalModels............................ 51
4.1 PropositionalFormulas................................................ 52
4.2 Equivalence ............................................................ 54
4.3 MinimalModels ....................................................... 56
4.4 StableModelsofPositivePropositionalPrograms................... 58
4.5 PropositionalImageofaCLINGOProgram .......................... 60
4.6 ValuesofaGroundTerm.............................................. 63
4.7 MoreonPropositionalImages........................................ 65
4.8 BibliographicalandHistoricalRemarks.............................. 68
5 ProgramswithNegation .................................................... 71
5.1 Examples............................................................... 71
5.2 StableModelsofaPropositionalProgram ........................... 73
5.3 StableModelsasFixpoints............................................ 76
5.4 SimplifyingPropositionalPrograms.................................. 78
5.5 CLINGOProgramswithNegation..................................... 81
5.6 TheLawofExcludedMiddle......................................... 83
5.7 CLINGOProgramswithChoice....................................... 86
5.8 TheoremonConstraints............................................... 88
5.9 BibliographicalandHistoricalRemarks.............................. 89
6 MathematicsofStableModels.............................................. 93
6.1 StrongEquivalence.................................................... 93
6.2 TheoremonStrongEquivalence...................................... 98
6.3 ProgramCompletion .................................................. 99
6.4 TheoremonCompletion............................................... 103
6.5 BibliographicalandHistoricalRemarks.............................. 106
7 MoreAbouttheLanguageofCLINGO..................................... 109
7.1 Counting ............................................................... 109
7.2 Summation............................................................. 112
7.3 MaximumandMinimum.............................................. 116
7.4 Optimization........................................................... 117
7.5 SymbolicFunctions ................................................... 122
7.6 ClassicalNegation..................................................... 124
7.7 BibliographicalandHistoricalRemarks.............................. 126
8 DynamicSystems............................................................. 129
8.1 Example:TheBlocksWorld .......................................... 129
8.2 TransitionDiagrams................................................... 131
8.3 Time.................................................................... 132
8.4 EffectsofActions...................................................... 134
8.5 NonexecutableActions................................................ 137
8.6 Prediction .............................................................. 138
8.7 Planning................................................................ 140
8.8 Concurrency ........................................................... 144
8.9 BibliographicalandHistoricalRemarks.............................. 145
Contents xi
9 Conclusion.................................................................... 147
Correctionto:PropositionalProgramsandMinimalModels............... C1
A AnswerstoExercises ........................................................ 149
Listings............................................................................ 179
Bibliography...................................................................... 181
Index............................................................................... 187
