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  Accepted Manuscript Vicious circle principle, aggregates, and formation of sets in ASP based languagesMichael Gelfond, Yuanlin ZhangPII: S0004-3702(18)30569-1DOI: ARTINT 3136To appear in:  Artificial Intelligence Received date: 8 October 2018Revised date: 30 March 2019Accepted date: 15 April 2019Please cite this article in press as: M. Gelfond, Y. Zhang, Vicious circle principle, aggregates, and formation of sets in ASP basedlanguages,  Artif. Intell.  (2019), is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providingthis early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it ispublished in its final form. Please note that during the production process errors may be discovered which could affect the content, and alllegal disclaimers that apply to the journal pertain.  Vicious Circle Principle, Aggregates, and Formation of Sets in ASP Based Languages Michael Gelfond and Yuanlin Zhang Texas Tech University, Lubbock, Texas 79414, USA { michael.gelfond, y.zhang } Abstract The paper introduces an extension of the srcinal Answer Set Prolog (ASP) byseveral set constructs including aggregates, defined as functions on sets. The newlanguage, called A   log  allows creating sets based on the Vicious Circle Principleby Poincar´e and Russell which eliminates a number of problems found in existingextensions of ASP by aggregates. We argue that, despite the fact that A   log  isnot as expressive as other extensions of ASP by aggregates, clarity of its syntaxandsemantics, additionofseveralnewset-basedconstructs, andsimplicityandtheeaseofusemakeitaviablecompetitortotheselanguages. Wealsostudyanumberof important properties of the language and show how ideas used in its design canbe utilized to generalize and simplify the definition of another important extensionof ASP by aggregates. Keywords:  Aggregates; Answer Set Programming; Logic Programming;Knowledge Representation; Language Design 2010 MSC:  68N17, 68T27, 68T30 1. Introduction The development of answer set semantics for logic programs [1, 2] led to thecreation of a powerful knowledge representation language, Answer Set Prolog(ASP) [3], capable of representing recursive definitions, defaults, effects of ac-tions and other important phenomena of natural language. A program of ASP 5 consists of rules understood as constraints on so called  answer sets  – possiblecollections of beliefs of a rational agent associated with the program. The rule head   ←  body  requires the agent who believes the body of the rule to also be-lieve the rule’s  head  . In forming its beliefs the agent is supposed to satisfy the Preprint submitted to Artificial Intelligence April 24, 2019  rules, avoid contradictions, and adhere to Rationality Principle:  believe nothing 10  you are not forced to believe (by the rules of the program) . This intuition is cap-tured by the original definition of answer sets [3]. On the theoretical side, thiswork helped some people to better understand formation of rational beliefs andother forms of non-monotonic reasoning. In addition, the design of algorithms forcomputing answer sets and their efficient implementations in systems called  ASP 15 solvers  for instance, [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]) allowed the languageto become a powerful tool for building non-trivial knowledge intensive applica-tions such as [15, 16, 17, 18] among others. There are a number of extensionsof ASP which also contributed to its success. Some, such as CR-prolog [19] andP-log [20], which enhanced ASP by abductive and probabilistic reasoning respec- 20 tively, preserved the epistemic character of the language. Others, such as [21] and[22] abandoned the srcinal idea in favor of expanding the syntax of the languageto include arbitrary formulae and establishing closer relationship with traditionalsuper-intuitionistic and classical logics. Extensions of the srcinal language byother new constructs, such as choice rules, weight constraints, and minimization 25 statement [23, 4], weak constraints [24], etc. were motivated by the desire touse ASP to solve optimization problems and by other practical needs of ASP. Inthis paper we are especially interested in the large collection of works aimed at ex-panding ASP with  aggregates  - functions defined on sets of objects of the domain.Here is a typical example of the use of aggregates for knowledge representation. 30 Example 1 (Classes That Need Teaching Assistants).  Suppose that we have acomplete list of students enrolled in a class  c  that is represented by the followingcollection of atoms: enrolled(c,mike).enrolled(c,john). 35 ... Suppose also that we would like to define a new relation  need ta ( C  )  that holdsiff the class  C   needs a teaching assistant. In this particular school  need ta ( C  ) is true iff the number of students enrolled in the class is greater than 20. Thedefinition can be given by the following rules in the language of logic programs 40 with aggregates: need_ta(C) :- card{X : enrolled(C,X)} > 20.-need_ta(C) :- not need_ta(C). where  card   stands for the cardinality function. Let us call the resulting program TA .   45 2  The program is simple, has a clear intuitive meaning, and can be run on mostof the existing ASP solvers. However, the situation is more complex than that.Unfortunately, currently there is no single recognized language of logic programswith aggregates. Instead there is a comparatively large collection of such lan-guages with different syntax and, even more importantly, different semantics (see 50 [25, 4, 26, 27, 28, 29] among others).To illustrate the difficulty, consider the programs in the following example. Example 2.  Program  P 0  consists of a rule: p(1) :- card{X: p(X)} != 1. Program  P 1  consists of rules: 55 p(1) :- p(0).p(0) :- p(1).p(1) :- card{X: p(X)} != 1. Program  P 2  consists of rules: p(1) :- card{X: p(X)} >= 0. 60 Even for these seemingly simple programs, there are different opinions about theirmeaning. To the best of our knowledge, all ASP based semantics, including that of [27, 26, 30], view  P 0  as a bad specification. It is inconsistent, i.e., has no answersets. Opinions differ, however, about the meaning of the other two programs.[27] views  P 1  as a reasonable specification having one answer set  {  p ( 0 ) ,  p ( 1 ) } . 65 According to [26, 30],  P 1  is inconsistent. 1 According to most semantics,  P 2  hasone answer set  {  p ( 1 ) } . However, the same semantics view seemingly equivalentprogram  P 3 p(1) :- card{X: p(X)} = Y, Y >= 0. as inconsistent.   70 Inourjudgmentthisandothersimilar“clashesofintuition”causeaseriousimped-iment to the use of aggregates in ASP (as well as in some other KR languages). It 1 In the rest of the paper we often refer to languages from [27] and [26] as F  log  and S   log respectively. 3

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