Origins of Answer-Set Programming - Some Background And Two Personal Accounts

Origins of Answ er -Set Pr o grammi ng – Som e Background And T wo P er sonal Accounts V ictor W . Mare k Departmen t of C omputer Science University of Kentucky Lexington, KY 40506-06 33, USA Ilkka Niemel ¨ a Departmen t of Information and Comp uter Science Aalto Univ ersity Finland Mirosław T ruszczy ´ nski Departmen t of C omputer Science University of Kentucky USA Abstract: W e discuss the ev olutio n of a spects of nonm onoton ic reasoning to wards the co mputation al pa radigm of an swer-set program ming (A SP). W e giv e a general overview of the roots of ASP and follo w up with the person al perspecti ve on research developments that helped verbalize the main pr inciples of ASP and d ifferentiated it from the classical logic progra mming. 1 Introd uction — Answer -Set Programming No w Merely ten years since the term was ﬁrst u sed and its meaning f ormally elaborated , answer-set prog rammin g h as re ached the status of a household name, at least in th e logic programm ing and knowledge representation c ommun ities. In this pap er , we present ou r person al perspective on inﬂuences and id eas — most of which can be traced back to research in kno wled ge representation, especially nonmon otonic reason- ing, logic program ming with negation, constraint s atisfaction and satisﬁability testing — that led to the two papers Marek and T ruszczy ´ nski ( 1999); Nieme l ¨ a (19 99) mark- ing the beginning of answer-set prog rammin g as a computational paradigm. Answer-set pr ogramming ( ASP , fo r short) is a par adigm f or declarative program - ming aimed at solving sear ch pro blems and their o ptimization v ar iants. Spea king in- formally , in ASP a search problem is modeled as a theo ry in some langua ge of lo gic. This repr esentation is designed s o that once appended with an encoding of a particular instance o f the problem , it re sults in a th eory who se models , under th e semantics of the formalism, correspond to solutions to the problem for this instance. The paradigm was ﬁrst f ormulated in these terms by Marek an d T ruszczy ´ nski (1 999) an d Niemel ¨ a (1999). 2 V . W . Marek, I. Niemel ¨ a and M. Truszczy ´ nski The ASP paradigm is most widely used with th e formalism of logic pr ogram- ming without func tion sy mbols, with programs interpreted by the stable-mod el se- mantics intro duced by Gelfond and Lifschitz (1988). Sometimes the syntax of pro- grams is extended with the str ong negation operato r and disjun ctions of literals are allowed in the heads of pr ogram rules. The semantics f or such p rogra ms was also deﬁned by Gelfond and Lifschitz (1 991). They pr oposed to use the term answer sets for sets of literals, by wh ich program s in the e xtended syntax were to be interpreted . T en ye ars after the answer-set semantics was in trodu ced, answer sets lent th eir name to the budding paradigm. Howe ver , there is mo re to answer-set p rogram ming than logic pro gramm ing with the stable-model and answer-set semantics. Answer-set pro- grammin g languages ro oted dir ectly in ﬁrst-order logic, extend ing it in some sim- ple intuitive ways to model deﬁnition s, have also been p roposed over the years an d have just m atured to be co mputation ally competiti ve with the orig inal logic progr am- ming embo diments of the paradigm (Denec ker, 19 98; Denecker and T er novska, 2008; East and T ru szczy ´ nski , 2006). Unlike Prolog -like logic programming , A SP is fully declarative. Neither the ord er of rules in a progr am nor the order of literals in rules have any ef fect on the semantics and only ne gligible (if any) effect on the com putation. All ASP formalisms c ome with the functionality to model de ﬁnitions a nd, most importan tly , inductive deﬁnitions, in intuitive and co ncise ways. Further, there is a growing body of works th at start ad- dressing metho ds of m odular prog ram design (Dao- T r an, Eiter , Fink, and Krennwallner , 2009; Jan hunen , Oikarin en, T o mpits, and W oltr an, 200 9) and program development and debuggin g ( Brain and V os, 20 05; Brummayer and J ¨ ar visalo, 201 0). Th ese fea- tures facilitate mode ling p roblem s in ASP , a nd make ASP an ap proach accessible to non-exper ts. Most impo rtantly , though , ASP co mes with fast software f or proc essing answer- set programs. Processing of prog rams in ASP is mo st of ten don e in two steps. The ﬁrst step consists of gr ound ing the program to its equiv alent p roposition al version. In the second step, th is proposition al program is solved by a back tracking search algo rithm that ﬁnds one or m ore of its answer sets (they represent solution s) or determines th at no answer sets (solutions) exist. The current software tools employed in each step, common ly refer red to as g r ound ers and solvers , respectively , h av e alread y reached the lev e l of perfo rmance that makes it possible to use them successfully with prog rams arising from problem s of practical impo rtance. This effecti veness of answer-set programmin g too ls is a result of a long, sus- tained and systematic effort of a large segment of the Knowledge Representation commun ity , and can be attributed to a handf ul o f crucial ideas, some of them cre- ativ ely adapted to ASP fro m other ﬁelds. Speciﬁcally , doma in r estriction was es- sential to h elp control the size of ground pr ograms. It was implemen ted in lparse , the ﬁrst ASP g round er Niemel ¨ a and Simons (19 96). The well-founded semantics V an Geld er et al. (19 91) inspired strong propagation methods implemented in the ﬁrst full-ﬂedge d ASP solver smo dels Niemel ¨ a and Simons (199 6). Pr ogram comp letion Clark (1978) provid ed a bridge to satisﬁability testing. For the class of tight program s Erdem and Lifschitz (2003), it allowed for a direct use of satisﬁability testing software in ASP , the id ea ﬁrst imp lemented in an early version of the so lver cmodels 1 . Loop formulas Lin and Zhao (2002) extended the con nection to satisﬁability t esting to arbi- 1 http://www.cs. utexas.edu/user s/tag/cmodels.html Origins of ASP 3 trary prog rams. They gave rise to such succe ssful ASP solvers as assat Lin and Zhao (2002), pb models Liu and T r uszczy ´ nski (2005) and later implem entations of cmo dels Lierler and Maratea (2004). Database techniques fo r q uery o ptimization inﬂuenced the design of the g round er for the dlv system 2 (Leone, Pfeifer, Faber , Eiter , Gottlob, Perri, and Scarcello, 2006). Impor tant advances of satisﬁability testing includin g the data stru cture of watched literals , r estarts , and conﬂict-cla use learnin g wer e inc orpor ated into the ASP solver cla sp 3 , at present the f ront-ru nner a mong ASP solvers and the winner of one track of the 20 09 SA T com petition. Some o f the credit for the advent of h igh- perfor mance ASP tools is due to the initiati ve to hold ASP grounder and so lver con - tests. The two ed itions of the contest so far (Gebser , Liu, Namasiv ay am, Neumann, Schaub, and T ruszczy ´ nski, 2007; Denecker , V enne kens, Bond, Gebser , and T ruszczy nski, 2009) focused on mod - eling and on s olver perform ance, and introduced a necessary competitiv e element into the process. The mo deling fea tures of ASP and computatio nal perform ance of ASP software ﬁnd the most impo rtant reﬂection in a growing range of successful applicatio ns of ASP . They include molecu lar biology (Gebser, Guziolowski, Iv anchev , Sch aub, Siegel, Thiele, and V eber, 2010a; Gebser, K ¨ onig, Schaub, Thiele, and V eb er, 2010b), decision sup port system for space shuttle controllers (Balduccin i, Gelfond, and Nogueira, 20 06), phylogenetic systematics (E rdem, 2011), a utomated mu sic compo sition (Boenn, Brain, V os, and Fitch, 2011), p rodu ct conﬁguratio n (Soininen and Niemel ¨ a, 1 998; Tiihonen, Soininen, Niemel ¨ a, and Sulon en, 2003; Finkel and O’Sulliv a n, 2011) and repair of web -service workﬂows (Friedrich, Fugini, Mussi, Pernici, and T agni, 2010). And so, ASP is now a d eclarative pro grammin g parad igm built o n top o f a solid theoretical foundation , with features that facilitate its u se in mo deling, with software supportin g effecti ve com putation, and with a growing list of successfu l applicatio ns to its cred it. How did it all come abo ut? T his paper is an attempt to rec onstruct our personal journey to ASP . 2 Knowled g e Repr esentation Roots of Answer -Set Pro- gramming One of the key questions for knowledge rep resentation is how to model commo nsense knowledge and h ow to au tomate commo nsense reasonin g. T he question do es not seem particularly relev a nt to ASP un derstood , as it now commonly is, as a gen eral purpo se computatio nal p aradigm for solving searc h pro blems. But in fact, knowledge rep re- sentation research was essential. First, it recog nized an d em phasized th e impo rtance of pr incipled modelin g of c ommon sense and domain kn owledge. The impact o f the modeling aspect of knowledge representatio n and reasoning is distinctly visible in the current implementa tions of ASP . They support high le vel progr amming that s eparates modeling pro blem speciﬁcatio ns from pr oblem instances, p rovide intuitive mea ns to model aggr egates, and offer direct means to mod el defaults and ind uctive deﬁnitions. Second, kn owledge r epresentation researc h, and esp ecially no nmon otonic re asoning research, provided the theoretical basis for ASP formalisms: the answer-set semantics of pro grams can be trac ed back to the semantics of d efault lo gic and auto epistemic 2 www.dbai.tuwie n.ac.at/proj/dl v/ 3 www.cs.uni- potsdam.de /clasp/ 4 V . W . Ma rek, I. Niemel ¨ a and M. Truszczy ´ nski logic, the semantics o f the logic FO(ID) ( Denecker, 200 0; Denecker and T ernovska , 2008) has it roots in the well-fou nded sem antics of non mono tonic provability opera- tors. In this section we discuss the d ev elopment of those id eas in k nowledge represen- tation that ev entually took shape of answer-set program ming. I n their celebrated 1969 paper, McCarthy and Hayes wrote [...] intelligence has two parts, which we shall call the epistemo logical and the he uristic. The epistemological part is the r epresentation of the world in su ch a form that the solution of pr ob lems follows fr om th e fa cts expr essed in the r ep r esenta tion. The heu ristic part is the mechanism that on the basis o f the informa tion solves the pr o blem and decide s what to do. W ith this p aragrap h McCarthy and Hayes ushered kn owledge representation and rea- soning into artiﬁcial intelligence and moved it to one of the most prominen t positions in the ﬁeld. Indeed , what they referred to as the epistemologica l pa rt is now und er- stood as k nowledge represen tation, while th e heu ristic p art has ev o lved into bro adly understoo d automated reason ing — a search for proofs or models. The que stion how to do knowledge representatio n and reason ing qu ickly rea ched the fo refron t of artiﬁcial intelligen ce research. M cCarthy sugg ested ﬁrst-order lo gic as the for malism for knowledge r epresentation . Th e reason s beh ind the proposal were quite app ealing. First-or der log ic is “ descriptively univ ersal” and p roved itself as the formal langu age o f mathematics. Mo reover , key reason ing task s in ﬁrst-ord er logic could b e automated, assuming one ad opted ap propr iate restrictions to escape semi- decidability of ﬁrst-order logic in its general form. Howe ver, ther e is no free lunch and it tur ned o ut that ﬁrst-order logic could not be ju st taken off the shelf an d u sed f or knowledge repr esentation with no extra effort required . The problem is that d omain knowledge is rarely com plete. More often than not, inform ation available to us has gaps. And the same is true for artiﬁcial agen ts we would like to fu nction autonom ously on o ur b ehalf. Reasoning with incomp lete knowledge is inh erently d efeasible . Dependin g o n h ow the world turns ou t to be (o r depend ing on how the ga ps in ou r k nowledge are closed), some conc lusions r eached earlier may have to be withd rawn. The mon otonicity of ﬁrst-order log ic consequen ce relation is at odds with th e non monoton icity of defeasible reaso ning and m akes mod- eling defeasible reasoning in ﬁrst-order logic difﬁcult. T o be effecti ve even when av ailab le inf ormation is incom plete, hum ans often de- velop and use defaults , that is , rules that typically work b ut in some exceptional situa- tions shou ld not be used. W e are good at learning d efaults and recog nizing situations in which the y shou ld not be used. In everyday life, it is thanks t o defaults that we are not bogge d down in the qua liﬁcation pr ob lem McCarth y ( 1977), that is, no rmally we do not check that every possible p recond ition for an action holds b efore we take it. And we naturally take a dvantage of the frame axiom McCarthy and Hayes (1969) wh en reasoning , th at is, we take it that thing s remain as they are un less they are cha nged by a n action. Moreover, we do so avoiding the difﬁculties p osed b y th e ramiﬁcation pr oblem Finger ( 1987), which is concerne d with side effects of action s. However , ﬁrst-order logic conspicuously la cks defaults in its syntactic repertoir e n or does it p ro- vide an obvious way to simu late them. It is no t at all surprising, given th at d efaults Origins of ASP 5 have a def easible ﬂa vor ab out them. Not being aware that a situation is “exceptio nal” one may a pply a default but later b e forced to withdraw the co nclusion u pon ﬁn ding out the assumption of “non-excep tionality” was wrong. Y et a nother pro blem for the use of ﬁrst-o rder logic in knowledge rep resentation comes fro m the nee d to mo del deﬁnition s, mo st no tably the in ductive ones. The way humans represent deﬁnitions has an aspec t of defea sibility tha t is related t o the closed- world assumption . I ndeed, we often deﬁn e a concept b y specif ying all its known instantiations. W e understand such a d eﬁnition a s meaning also that noth ing else is an instance o f the con cept, even though we r arely if ever say it explicitly . But the main problem with deﬁnitions lies else where. Deﬁn itions often are inductive and their correct m eaning is cap tured by th e n otion of a least ﬁxp oint. First-order log ic cannot express the notion of a least ﬁxpoint a nd so do es no t pr ovide a way to specify inductive deﬁnitions. These problems did not go unrecognized and in late 1970s researchers were seek- ing ways to a ddress them. Some pro posals called fo r extensions of ﬁrst-ord er log ic by explicit means to mod el defaults while other argued that the lang uage can stay the same but the semantics had to change. In 1980 , the Artiﬁcial In telligence Journal pub- lished a double issue dedicated to nonmon otonic reason ing, a form of reasoning based on but departing in major ways from that in ﬁrst-ord er logic. The issue contained three pap ers by McCarthy (1 980), Reiter (1980), and McDerm ott and Do yle (19 80) that launched the ﬁeld of nonmo notonic reasoning. McCarthy’ s propo sal to bend the languag e of ﬁrst-order logic to the need s of knowledge repr esentation was to ad just the semantics of ﬁrst-ord er logic a nd to base the entailm ent relation amo ng sentences in ﬁrst-order logic on minimal models only McCarthy (1980). He called the resulting formalism c ir cumscription an d demon- strated how circu mscription could be u sed in several settin gs wh ere ﬁrst-o rder log ic failed to w ork well. Reiter (19 80) e xtended the syntax of ﬁrst-order logic by defaults , inference ru les with exception s, and described formally reason ing with defaults. Re- iter was predo minantly inter ested in reasoning with normal defaults b ut his d efault logic was much mor e general. Finally , McDermo tt and Doyle propo sed a logic based on the language of m odal logic which, as they suggested, was also an attempt to model reasoning with d efaults. This last paper was found to suffer fro m min or tech- nical pro blems. T wo ye ars later, McDerm ott (1982) published another paper which corrected and extended the earlier one. These th ree p apers demonstrated that shortcom ings of ﬁrst-o rder log ic in mo del- ing incomplete kno wledge and suppor ting reasoning from these representations could be addressed without gi ving up on the logic entire ly b ut by adju sting it. They sparked a ﬂurr y o f re search activity direc ted at u nderstand ing and fo rmalizing non mono tonic reasoning . On e of the most importan t and lasting outcomes of those efforts was the au- toepistemic logic pro posed by Moore (1984, 19 85). Papers by Moore can be regarded as closing the ﬁrst phase of the nonm onoton ic r easoning as a ﬁeld of study . Identify ing nonmon otonic reasoning as a pheno menon deserving an in-dep th study was a major milestone in logic, philosoph y and artiﬁcial intelligence. The prospect of understan ding and auto mating reasoning with incomplete information , of the type we humans are so good at, excited th ese research commun ities and attracted many re- searchers to the ﬁeld. Accordingly , the ﬁrst 10 -12 years of n onmon otonic r easoning research bro ught m any fu ndamen tal results and established solid theoretical foun da- 6 V . W . Ma rek, I. Niemel ¨ a and M. Truszczy ´ nski tions for circumscription McCarthy (1 980); Lifschitz ( 1988), d efault logic (Reiter and Criscuolo, 1981; Hanks and McDermott, 1986; Marek and T ruszczy ´ nski, 1989; Pearl, 199 0), au- toepistemic log ic (Moore, 19 85; Niemel ¨ a, 19 88; Marek and T ruszczy ´ nski , 1991; Shvarts, 1990; Schwarz, 199 1) and mod al nonmonoto nic log ics in the style of McDermott and Doyle (Marek , Shv ar ts, and T ruszczy ´ nski, 1993; Schwarz, 199 2; Schwarz and T ruszczy ´ nski, 1992). Researc hers m ade pr ogress in clarifying th e r elationship between these f or- malisms Konolige (1988, 198 9); Marek and T ruszczy ´ nski (198 9b); Bidoit and Froidev a ux (1991); T ruszczy ´ nski (1991). Compu tational aspects received m uch attention, too. First c omplexity results a ppeared in late 198 0s and ea rly 1 990s Cado li and Lenzerini (1990); Marek and T ruszcz y ´ nski ( 1991); Kautz and Selman ( 1989); Gottlob ( 1992); Stillman (1 992) and early , still naive at th at time, im plementation s of auto mated rea - soning with nonm onoton ic logics were developed aro und th e same time Ethe rington (1987); Niemel ¨ a and T uo minen (1986, 19 87); Ginsberg (198 9 ). Sev eral research m ono- graphs we re pub lished in late 19 80s and early 1 990s systema tizing that p hase of nonmo noton ic r easoning research and making it accessible to outside commu nities Besnard (1989); Brewka (1991); Marek and Truszczy ´ n ski (1993 ). Expectation s broug ht up by the advent of nonmonoto nic r easoning formalisms were high. It was thought that nonmono tonic log ics w o uld f ac ilitate concise and elab- oration tolerant repre sentations of kn owledge, an d th at through the use of def easible inference rules like defaults it w o uld suppor t fast reasoning. Howe ver, around the time of the ﬁrst Knowledge Representation an d Reason ing Con ference, KR 1 989 in T or onto, concerns started to surface in discussions and papers. First, ther e was the issue of m ultiple belief sets, dependin g on the lo gic u sed rep- resented as extensions o r expansio ns. A prevalent interp retation of the problem was that multiple belief sets provid ed the basis for sk ep tical and brave m odes of reasoning. Skeptical reasoning m eant consid ering as co nsequen ces a reasoner was sanc tioned to draw only th ose form ulas that wer e in ev er y belief set. Brave reasonin g required a non-d eterministic commitme nt to one of the p ossible b elief sets with all its elem ents becoming con sequences o f the un derlying theory (in the nonmonoton ic logic at hand ). The ﬁrst approach was ea sy to u nderstan d an d accep t at the in tuitive level. But as a reasoning mechan ism it was rather weak as in gen eral it supported few non-trivial in- ferences. The second appr oach was undersp eciﬁed — it pr ovided n o guid elines on how to select a belief set, and it was not at all obvious h ow if at all human s per form such a selection. Bo th skeptical an d brave reasoning suffered from the fact that the re were n o p ractical pr oblems lyin g arou nd that c ould o ffer some dir ection as to how to proceed with any of these two app roaches. Second, none of the main nonm onoto nic logics seemed to pr ovide a goo d for- malization of the notion o f a default or of a defeasible consequen ce relation. Th is was q uite a sur prising and in the same time worrisome o bservation. Nonmo notonic reasoning br ough t attention to the con cept of default and soo n research ers raised the question of h ow to reason abo ut defaults rather than with d efaults (Pea rl, 199 0; Kraus, Lehman n, and Magido r, 19 90; Leh mann and Magidor, 199 2). A somewhat different version of the same q uestion asked abo ut de feasible co nsequen ce re lations, whether they can b e charac terized in terms o f in tuitiv e ly acceptab le ax ioms, and whether th ey have semantic ch aracterization s Gabbay (19 89); Makinso n (19 89). De- spite the success of circu mscription, default and autoepistemic logic s in addressing se veral pro blems of knowledge representatio n, it was not clear if or how they could Origins of ASP 7 contribute to th e q uestions above. I n fact, it still remain s an open p roblem whether any deep conne ction between these logics and th e stud ies o f abstra ct n onmon otonic inference relations exists. Next, the complexity r esults obtained at about same time Marek and T ru szczy ´ nski (1991); Eiter and Gottlo b (19 93a); Gottlob (19 92); Stillman (1992); Eiter and Gottlob (1993 b , 1995) wer e viewed as n egati ve. They dispelled any hope of h igher comp uta- tional efﬁciency of nonmono tonic reasoning. Even under the r estriction to the p ropo- sitional case, basic reasoning tasks turned o ut to be as com plex as an d in some cases ev e n mor e co mplex (assumin g poly nomial hierar chy does not collapse) th an reasoning in pr oposition al logic. Even mo re d iscouragin g results were o btained f or the general languag e. Finally , the question s of applications and implementatio ns was becom ing more and mor e urgent. There we re no pr actical artiﬁcial intelligen ce applications und er development at tha t time that r equired non mono tonic reasoning. Nonmonoto nic log- ics co ntinued to be extensiv ely studied an d d iscussed at AI and KR co nferen ces, but the b elief tha t they can have pra ctical imp act was diminishing . Th ere was a gr owing feeling that they might amount to not much more b ut a theoretical exercise. Complex- ity results notwithstan ding, the ultimate test of whether an approa ch is practical can only com e fro m experiments, as the worst-case comp lexity is on e thing but real life is anoth er . But th ere was little work on imp lementation s and one of th e main rea sons was lack of test cases who se hardne ss on e could con trol. Researchers continued to analyze “by hand” small examples arguing abou t correctn ess of their default or au- toepistemic lo gic rep resentations. T hese toy examp les were approp riate fo r the task of u nderstand ing basic reasonin g p atterns. But they were simply too easy to p rovide any meaningful insights into automated reasoning algorithms and their performance . And so the ear ly 1990s saw a growing sentiment that in order to prove itself, to m ake any lasting impact on the theory a nd pr actice o f knowledge repre sentation and, mo re gen erally , on artiﬁcial inte lligence, p ractical and efﬁcient systems fo r non- monoto nic reason ing had to be developed and their usefu lness in a broad rang e o f applications demonstrated. Despite of all the doom and gloom of tha t time, there were reasons for optimism, to o. Th e theoretical understanding of no nmon otonic logics reached the level when development of sophisticated com putational meth ods became possible. Co mplexity results were disappo inting but the co mmun ity reco gnized that they con cerned th e worst case setting only . Human experien ce tells u s that there are good reasons to think that real life does not gi ve rise to worst-case instances too often , in fact, that it rarely does. Th us, through experime nts and the focus on reasoning with structured theories one could hop e to obtain ef ﬁciency sufﬁcient for practical applica- tions. Moreover, it was high ly likely that on ce implem ented systems started showing up, they would excite the comm unity , demonstrate the p otential of nonmo notonic log- ics, and spawn competitio n which would result in improvemen ts of algorithm s and perfor mance advances. It is interesting to note that m any of the objections and cr iticisms aime d at non- monoto nic reasoning were instrume ntal in helping to identify key aspects of an swer- set prog ramming . Default lo gic did not provide an acceptable f ormalization of reason- ing abo ut defaults but inspired the answer-set semantics of log ic prog rams Bidoit and Froidev au x (1987); Gelfond and Lifschitz (1988, 1991) and helped to s olve a long -standing prob- lem of how to interpret negation in logic p rogram ming. Answer - set programm ing, 8 V . W . Ma rek, I. Niemel ¨ a and M. Truszczy ´ nski which ado pted the syntax of lo gic pro grams, as well as the an swer-set semantics, can be regarded as an implementa tion of a sign iﬁcant fragm ent of default lo gic. The lack of obviou s test cases f or experime ntation with imp lementations forc ed research ers to seek them outside of artiﬁcial intelligence and led them to th e area of graph prob- lems. Th is experience sho wed that the phen omeno n of multiple b elief sets can be turned f rom a bug to a feature, wh en r esearchers re alized th at it allows on e to mode l arbitrary search p roblems, with extensions, expansion s or answer sets, dep ending on the logic u sed, representin g problem solutio ns (Cad oli, Eiter , and Gottlob, 1997; Marek and Remmel, 2003). Howe ver important, knowledge representation was no t the only source of inspir a- tion for ASP . Inﬂuen ces o f research in several other areas of computer scien ce, such as databases, lo gic prog ramming and satisﬁability , are also easily iden tiﬁable an d must be men tioned, if only brieﬂy . On e of the key themes in research in logic program ming in the 19 70s and 19 80s was the q uest f or the m eaning of th e negation ope rator . Stan- dard logic programmin g is b uilt aroun d the idea of a single intend ed Herbrand model. A program r epresents the declar ativ e knowledge about the domain o f a prob lem to solve. Some elements of the mo del, more accurate ly , gro und ter ms the model deter- mines, represent solutions to the pro blem. All works well for H orn pro grams, with the least Herbrand mod el of a Horn prog ram as the natural choice for the intended mod el. But the negation operato r , being in grained in the way humans de scribe knowledge, cannot be av o ided. The lo gic progr amming community recognized this and the nega- tion was an element of Pr olog, an im plementation of logic prog ramming , rig ht f rom the very beginning . And so, th e question aro se f or a d eclarative (as opposed to the proced ural) accou nt of its semantics. Subsequen t studies identiﬁed a non-classical nature of the negation operato r . This nonmo noton ic aspect of the n egation operator in log ic prog ramming was also a com - plicating factor in the effort to ﬁnd a single intend ed mo del of logic pr ograms with negation. It became clear that to succeed on e either had to restrict the class of pro - grams or to move to th e three-valued settings. The ﬁrst line of researc h resulted in an im portant class o f stra tiﬁed p rogram s (Apt, Blair , and W alker, 1988), th e secon d one led Fitting (198 5 ) an d Kunen (19 87) to the Kripk e-Kleene mod el and, later o n, V an Geld er, Ross, and Schlipf (1991) to the well-founded model. In the hind sight, the con nection to knowledge repr esentation and nonm onoton ic reasoning should have bee n quite evident. Ho wev e r , th e knowledge rep resentation and log ic pr ogramm ing communities had little overlap at the time . And so it was no t before the work by Bidoit and Fro idev aux (19 87) an d Gelfon d (19 87) that the con- nection was mad e explicit an d the n exploited . That work demo nstrated tha t intuitive constraints on an intended model cannot be reconciled with the requiremen t of its uniquen ess. I n oth er words, with negation in the syntax , we m ust a ccept the reality of multiple intend ed models. Th e connection between logic programming and knowl- edge r epresentation , especially , default and autoep istemic logics was imp ortant. On the one han d, it showed that logic programmin g ca n provide syntax for an interest- ing non -trivial fragment of these log ics, an d d rew attention of resear chers attemp ting implementatio ns o f nonmon otonic reason ing systems. On th e other hand, it led to the notion of a stable mo del o f a logic prog ram with negatio n. It also r einforc ed the importan ce o f th e key question how to ad apt the phen omeno n of multiple inte nded models for problem s s o lving. Origins of ASP 9 The work in da tabases pr ovided a link between query lan guages an d logic pro- grammin g. One of the ou tcomes o f this work was DA T ALOG , a fr agment of logic progr amming without function symbols, pro posed as a query languag e. Th e database research resulted in im portan t the oretical studies c oncern ing comp lexity , expressi ve power and connection o f DA T ALOG to the SQL q uery language (Cado li et al., 19 97). D A T ALOG was implem ented, for instance as a par t of DB2 datab ase manag ement system. DA T ALOG introduced an important distinction between extension al and in- tentional d atabase com ponen ts. Extension al database is th e co llection o f tables that are stored in the datab ase, the corresp onding relatio n n ames known as extensional p red- icate symbo ls. The inten sional d atabase is a collectio n of inten tional tab les deﬁned by DA T ALOG queries. In time th is distinction was ad opted b y answer-set pr ogram- ming as a way to sep arate pro blem speciﬁcation fro m data. The d atabase com munity also considered extensions o f DA T ALOG with the n egation c onnective. Because of the sema ntics of the resulting lan guage, mu ltiplicity of an swers in DA T ALO G ¬ was a pr oblem, as it was in a more ge neral setting of arbitr ary prog rams with n egation. Therefo re, D A T ALOG ¬ never tur ned into a p ractical d atabase qu ery lang uage (al- though , its stratiﬁed versio n could very well be used to this en d). Howe ver, it was certainly an interesting fr agment of lo gic pro grammin g. And ev e n thou gh its expr es- si ve power was much lower than that of g eneral pro grams, 4 there was ho pe th at fast tools to p rocess D A T ALOG ¬ can be developed. Jumpin g ahead, we note her e that it was DA T ALOG ¬ that was eventually adopted as th e ba sic lang uage of answer-set progr amming. 3 T owards Answer -Set Programming at the University of K entucky Having outlined so me of th e key ideas beh ind the emergen ce of answer-set pro gram- ming, we now move o n to a more persona l accou nt o f research ideas that eventually resulted in the formulation of the an swer-set prog ramming paradig m. In this sec- tion, V ictor Marek and Mirek T ruszczyn ski, discuss the ev olu tion of their understand- ing o f non mono tonic log ics an d how they could be used for c omputatio n th at led to their pap er Stab le logic pr ogramming — an alterna tive logic pr ogramming paradigm Marek and T ru szczy ´ nski (1 999). A closely in tertwined story o f Ilkk a Niemel ¨ a, fol- lows in the sub sequent section. As th e two acco unts are stro ngly person al and n eces- sarily q uite sub jectiv e, fo r th e mo st part they are g iv en in the ﬁrst person. An d so , in this section “we” and us refers t o V ictor and M irek, just as “I” in the ne x t one to I lkka. In mid 19 80s, one of us, V ictor, started to study n onmon otonic logic s following a suggestion f rom W ito ld Lip ski, h is for mer Ph.D. studen t and clo se collab orator . Lip- ski drew V ictor’ s attention to Reiter’ s p apers on closed -world assumption and default logic Reiter (1 978, 198 0). In 1984 , V ictor attended the ﬁr st Nonmo notonic Reason- ing W o rkshop at Mohonk , NY , and came back convinced abo ut the importance of problem s th at were discussed ther e. In the following year, he attracted Mirek to the progr am of the study of math ematical foundation s of nonmon otonic reasoning. 4 It has to be noted though that the expressi ve power of genera l programs with function symbols and nega tion goe s well beyond what could be accepted as computabl e under all reasonable semantics (Schlipf, 1995; Marek, Nerode, and Remmel, 1994). 10 V . W . Ma rek, I. Niemel ¨ a and M. Truszczy ´ nski In 19 88 Micha el Gelfond visited us in Lexingto n and in his pre sentation talked about the use of autoepistemic logic Moore ( 1985) to provide a semantics to logic progr ams. At the time we were already study ing auto epistemic logic, insp ired b y talks V ictor attended at M ohonk an d b y Mo ore’ s p aper on autoepistemic lo gic in the Artiﬁcial Intelligence Journ al Moore ( 1985). W e kne w by then that stable sets of formu las of moda l logic, intro duced by Stalnaker (1980) and shown to be essential for autoepistemic lo gic, c an b e con structed b y an iterated induc ti ve deﬁnition fr om their modal-f ree part Marek (19 89). W e also r ealized the importan ce of a simple norm al form for autoepistemic theories introduce d by Konolige (1988 ). Thus, we were excited to see that logic prog rams can be understoo d as some simple autoepistemic theories thanks to Gelfond’ s interpretatio n Ge lfond (1987). Soon there- after , we also r ealized th at logic program s could b e interpreted also as default logic theories and that the mean ing of logic prog rams induced on them by default logic ex- tensions is the same as that ind uced by auto epistemic expansions Marek and T ruszczy ´ nski (1989 b ) . It is impo rtant to note that default logic was ﬁrst used to assign the m eaning to logic progra ms by Bidoit and Froid ev aux (19 87), but we did not know ab out their work at the time. Bidoit and Froidevaux effecti vely deﬁn ed th e stable model seman- tics f or logic pro grams. They did so ind irectly and with explicit r eferences to default extensions. T he direct deﬁn ition o f stable models in logic pro grammin g terms came about one year later in the celebrated paper by Gelfond and Lifschitz (1988). What becam e apparen t to us soo n after Gelfond’ s visit was that despite both au- toepistemic expansions a nd default extensions in ducing the same semantics on logic progr ams, it was just serendip idity an d not the r esult o f the inher ent equiv alen ce of the two logics. In fact, we no ticed th at ther e was a d eep mismatch between Moo re’ s autoepistemic logic with the seman tics of expan sions an d Reiter’ s default lo gic with the sema ntics of extension s. In the sam e time, w e discovered a fo rm of default lo gic, to be mo re pre cise, an alternativ e seman tics of default logic, which was the perfect match for that of expan sions for auto epistemic logic Marek and T ruszczy ´ nski (1989 ) . This research culminated about 15 years later with a paper we co-autho red with Marc Denecker that provided a deﬁnitive account of the relatio nship b etween default and au - toepistemic logics (Denec ker , Marek, and T ruszczy ´ nski, 2003) and resolved prob lems and ﬂaws of a n ear lier attem pt at exp laining the relationship due to Konolige (198 8 ). Another paper in this volume ( Denecker, Mar ek, and T ru szczynski, 201 1) discusses the informal basis for that work and summarizes all the ke y results. The relationship between de fault and auto epistemic logic was of only marginal importan ce f or the later emergence of answer-set p rogram ming. But ano ther re sult inspired by Gelfo nd’ s visit turne d out to be essential. In our study o f auto epistemic logic we wanted to establish the complexity of the existence of expansions. W e ob- tained the result by showing th at the p roblem o f the existence o f a stable mo del o f a logic program is NP-complete and, by doing so, we obtained the same complexity for the p roblem o f th e existence of expansion s of autoepistemic th eories o f so me simp le form but still rich enoug h to capture log ic p rogram s under Gelfond’ s interpr etation Marek and T ru szczy ´ nski (19 91). The result for autoepistemic lo gic did n ot turn ou t to be particularly sig niﬁcant as the class of autoepistemic th eories it pertained to was narrow . And it was soon supplanted by a gen eral result due to Gottlob (1992), who p roved the existence of the expansions pro blem to be Σ P 2 -complete. But it was an en tirely different matter with Origins of ASP 11 the complexity result concerning the e x istence of stable models of progr ams! First, our proof reduced a com binatorial problem, that of the existence of a kernel in a directed grap h, to the existence o f stable model of a suitably de ﬁned pr ogram. This was a stro ng ind ication that stable seman tics may , in principle, lead to a g eneral purpo se form alism f or solving com binatorial and, more generally , search problems. Of course we did not f ully realize it at the time. Second , it w a s quite clear to us, especially after the ﬁrst KR conference in T or onto in May 1989 th at the success o f nonmo noton ic logics can c ome on ly with implementation s. Many p articipants of the confere nce ( we recall David Poo le and Matt Ginsberg b eing especially vocal) called for workin g systems. Since by then we understoo d the complexity of stab le-model computatio n, we asked tw o University of K entuc ky st udents Elizab eth and Eric Free- man to design and implem ent an algo rithm to compute stable models of propositional progr ams. Th ey succeeded albeit with limits — the im plementatio n co uld process progr ams with a bout 20 variables o nly . Still, th eirs was most likely the ﬁr st work ing implementatio n of stable-mod el com putation. Unfor tunately , with th e M.S. degrees under their belts, Eric and Elizabeth left the University of Kentucky . For about three years after this ﬁrst dab into impleme nting reasoning systems based on a nonmo noton ic logic, our attention w as focu sed on more theoretical studies and on the work o n a mon ograp h on mathematical found ations of nonm onoton ic rea- soning based on the paradig m of context-de penden t reasoning . However , the matter of implem entations had con stantly been on the backs o f our minds and in 1992 we decided to give the matter an other try . As we felt we und erstood d efault logic we ll and as it was co mmon ly viewed as the nonm onoton ic logic o f th e fu ture, in 199 2 we started the project, D efault Reasoning System DeReS. W e aimed at implementing rea- soning in th e un restricted langu age of pro positional default log ic. W e also started a side pro ject to DeReS, the TheoryBase project, aimed at developing a s o ftware system generating default theor ies to be u sed for testing DeReS. The time was right as two promising studen ts, Pawel Cholewinski and A rtur Mikitiuk, joined the University of Kentucky to pursue doctor ate degrees in compu ter science. As is common in such circumstan ces, we were look ing f or a sponsor of this re- search and foun d one in the US Army Research Of ﬁce (US AR O), which was willing to sup port th is work . A colleague of ours, Jurek Jaro mczyk, also at the Uni versity of Kentucky , coined the term De ReS, a pu n on an o ld polish word “d eresz” presently rarely used and meaning a stallion, quite appr opriate for the pro ject to be conducted in Lexington, “the world cap ital of the horse. ” In the p roposal to US ARO we pro mised to in vestigate basic reasoning problems of default logic: 1. Computin g of extensions 2. Skeptical reaso ning with default th eories — testing if a fo rmula belong s to all extensions of an in put default theory 3. Brav e reason ing with default the ories — testing if a formu la belong s to som e extension of an i nput default theory . The b asic co mputation al device was b acktrack ing search for a b asis of an extension of a ﬁnite default th eory ( D , W ) . This was ba sed on two o bservations due to Reiter: that while default extensions of a ﬁnite de fault theo ry are inﬁnite, they are ﬁnitely generated ; and that the genera tors are all for mulas of W an d the consequent formulas 12 V . W . Ma rek, I. Niemel ¨ a and M. Truszczy ´ nski of some defaults from D . W e also em ployed ideas such as relaxed stratiﬁcation of defaults Cholewi ´ nski (1995); Lifschitz and T urner (1994) for prunin g the search space and relev an ce grap hs for simplifying provability . W e also thought it was impo rtant to have the nonmono tonic reaso ning comm unity accept the challen ge of developing implemen tations of automated nonmon otonic rea- soning. Our propo sal to US ARO contain ed a request for funding of a retreat dedicated to knowledge rep resentation, nonm onoto nic r easoning an d log ic pro grammin g. The key goals for the retreat were: 1. T o stimulate app lications of nonmo noton ic formalisms and imp lementation s of automated reasoning systems based on nonmon otonic logics 2. T o pro mote the pr oject to c reate a public domain library of b enchmar k pro blems in nonmon otonic reasoning . W e held the workshop in Shakerto wn , KY , in October 1994 . Over 30 leading re- searchers in nonm onoto nic reasoning p articipated in talks and we p resented there early prototy pes of DeReS and Theory Base. Imp ortantly , we heard then for th e ﬁr st time from Ilk ka Niemel ¨ a about the work on systems to p erform non mono tonic reason ing in the languag e of logic p rogram s in his g roup at the Helsinki Un iv e rsity of T echno l- ogy . The meeting h elped to elevate the importanc e of implementatio ns of nonmo no- tonic reason ing systems and their applica tions. It evidenced ﬁrst advances in the ar ea of implem entations, as well a s in the ar ea of ben chmark s, essential as so far most problem s consider ed as benchmark s were toy prob lems such as “T weety” and “Nixon Diamond. ” The DeReS system was not d esigned with any speciﬁc application s in mind. At the time we believed that, since default logic could mode l se veral aspects of com- monsense reaso ning, once DeReS became av ailable, many artiﬁcial intelligence and knowledge repre sentation resear chers would use it in their work . And we simply regarded broadly understoo d knowledge represen tation prob lems as the main applica- tion area for DeReS. W ork ing on DeReS immediately broug ht up to o ur attention the question of test- ing and perfo rmance evaluation. In the sum mer o f 1988, Mirek atten ded a meeting on comb inatorics whe re Donald Knuth talked abou t the pr oblem of testing gr aph al- gorithms and his propo sal how to do it rig ht. Knu th was of the opinio n that testing algorithm s on rand omly gene rated graph s is in sufﬁcient and, in fact, o ften irre lev ant. Graphs arising in r eal-life settings ra rely resemble grap hs generated at r andom from some prob abilistic mo del. T o ad dress th e problem , Knu th d ev eloped a sof tware sy s- tem, Stanfo rd Graph Base, providin g a mech anism for creating collec tions of graph s that could be then used in pro jects dev elo ping graph algorithm s. Graph s produced by the Stanford Gra phBase were mostly ge nerated fr om re al-life objects such as m aps, dictionaries, novels and images. Some wer e based on rath er obscure sources such as sporting ev ents in Australia. The do cumentatio n was superb (the boo k by Knuth on the Stanfo rd G raphBase is still av ailable). Th e Stanford GraphBase was free and its u se was not restricted. Fro m o ur pe rspective, two aspec ts we re essential. First, the Stanf ord Gr aphBase p rovided a uniq ue id entiﬁer to every graph it cr eated a nd so experiments cou ld b e d escribed in a way allowing o thers to rep eat them literally and perfor m comp arisons on iden tical sets o f graphs. Second , the Stanford GraphBase Origins of ASP 13 supported creating families of examp les similar but inc reasing in size, th us allowing to test scalability of algorithms being developed. In retrospect, the moment we started talkin g about testing our implementatio ns of default logic was the deﬁning m oment on our path towards th e answer-set prog ram- ming pa radigm. Based on o ur com plexity result co ncernin g the existence o f stab le models and its implication for default logic, we knew that all NP-complete g raph problem s could be red uced to the p roblem of the existence of extensions. T he r educ- tions expr essed instances of graph pro blems as default th eories. Thu s, in o rder to get a family of default the ories, similar but gr owing in size, we needed to select an NP- complete p roblem o n g raphs (say , the hamilto nian cycle problem ), g enerate a family of gra phs, and generate for e ach graph in th e family the correspon ding default th e- ory . These th eories could be u sed to test a lgorithms for com puting extensions. Th is realization gave rise to the TheoryBase, a software system generating default theor ies based on reduction s of gra ph prob lems to th e existence o f the extension pr oblem and developed o n top o f the Stan ford Gr aphBase, which served as the source of graph s. The Theo ryBase p rovided default theo ries b ased o n six well-known gra ph pro blems: the existence of k -colorings, Ha miltonian cycles, kernels, in depend ent sets of size at least k , and vertex covers of size at most k . As the Stanford G raphBase provided an unlimited supply of gr aphs, the Theo ryBase o ffered an unlimited sup ply of default theories. W e will recall here the TheoryBase en coding of the existence of a k -color ing pr ob- lem as i t shows that already then some funda mental aspects of the meth odolo gy of rep- resenting search problems as default theories started to emerge. Let G = ( V , E ) be an undirected graph with the set of vertices V = { v 1 , . . . , v n } . Let C = { c 1 , . . . , c k } be a set of colors. T o express the proper ty that vertex v is co lored with c , we in troduced propo sitional ato ms c l rd ( v , c ) . For each vertex v i , i = 1 , . . . , n , an d for eac h colo r c j , j = 1 , . . . , k , we deﬁned the default rule col or ( v i , c j ) = : ¬ cl r ( v i , c 1 ) , ..., ¬ cl r ( v i , c j − 1 ) , ¬ cl r ( v i , c j +1 ) , ..., ¬ cl r ( v i , c k ) cl r ( v i , c j ) . The set of default r ules { col or ( v i , c j ) : j = 1 , . . . , k } mod els a constrain t that vertex v i obtains exactly one color . The default theory ( D 0 , ∅ ) , wher e D 0 = { c ol or ( v i , c j ) : i = 1 , . . . , n, j = 1 , . . . , k } , has k n extensions co rrespon ding to all possible co lorings ( not nece ssarily pr o per ) of the vertices of G . Thu s, the de fault th eory ( D 0 , ∅ ) deﬁnes the basic spa ce of objec ts within which we need to search for solutions. In the present-day answer-set pro gram- ming implementations choice or cardinality ru les, which of fer much more con cise representatio ns, are used for that pu rpose. Next, our Theo ryBase encodin g im posed constraints to eliminate those co lorings tha t are not proper . T o this e nd, we used ad- ditional d efault rules, which we called killing defaults, and which now are typically modeled by logic p rogra m rules with the empty head. T o describe them we used a new propositional v ariable F and deﬁned lo c al ( e, c ) = cl rd ( x, c ) ∧ cl r d ( y , c ) : ¬ F F , 14 V . W . Ma rek, I. Niemel ¨ a and M. Truszczy ´ nski for ea ch edge e = ( x, y ) of th e grap h and for each color c . Each default lo c al ( e, c ) “kills” all co lor assignmen ts wh ich give co lor c to b oth ends of edge e . It is easy to check (and it also follows f rom n ow well-known m ore g eneral results) that defaults of the for m lo c al ( e, c ) “kill” all no n-pro per coloring s and leave precisely tho se tha t are pro per . T his two-step modeling m ethodo logy , in which we ﬁrst deﬁne the space of objects that con tains all s o lutions, and then impo se co nstraints to weed away those that fail some problem speciﬁcations, constitutes th e main way by which search pro blems are modeled in ASP . The key lesson for u s from the Th eoryBase pro ject was that combina torial p rob- lems can be represented as default theories and that constructing these representations is easy . It w as th en for the ﬁrst time that we sen sed tha t pro grams ﬁnding exten- sions o f de fault theories could be used as genera l purp ose prob lem solving tools. It also lead us, in ou r internal discussions to think ing about “second-or der” ﬂavor of default logic, given th e way it was used for comp utation. Ind eed, in all theo ries we developed for th e Th eoryBase, extensions rather than their single elements rep- resented solutions. In other words, the main reasoning task did not seem to be that of skeptical or br av e reasoning (does a form ula follow skeptically o r bravely from a default theory) but co mputing en tir e extension s. W e talked abo ut th is seco nd- order ﬂav o r w hen pr esenting our paper on DeReS at the KR con ference in 199 6 (Cholewi ´ nski, Marek, and T r uszczy ´ nski , 1 996). At that tim e, we knew we were clo s- ing in on a new declar ativ e problem-solving para digm based on nonm onoto nic lo gics. A prob lem for us was, h owe ver, a fairly p oor perf ormance of DeReS. T he default extensions ar e closed und er con sequence. This m eans tha t pro cessing of default the- ories requires testing provability of prerequ isites and justiﬁcations of defaults. This turned out to b e a major pr oblem affecting the processing time of ou r impleme nta- tions. It is not surprising at all in vie w of the complexity results of Gottlob (199 2) and Stillman (1992). Speciﬁca lly , e x istence of extensions is a Σ P 2 -complete problem. There is, o f course, an easy case o f pr ovability when all formulas in a default theory are co njunctio ns of literals only . No w the problem with the provability of premises disappea rs. Ho wev er, De ReS organ ized its search fo r solutions by lo oking for sets of generating d efaults, inh eriting this appro ach fr om the case o f general default theories, rather than for literals generating an e xten sion. And that was s till a problem. There are typically many more rules in a default theory than atoms in the lang uage. At th e Interna tional Joint Conf erence an d Sym posium o n L ogic Programm ing in 1996, Ilkka and his stude nt Patrik Simons presented the ﬁrst report on their smod- els system Niem el ¨ a and Simons (199 6). But it seems fair to say that on ly a similar presentation and a dem o Ilk ka gave at the Lo gic Programm ing and Non -Mono tonic Reasoning Con ference in 1997 , in Da gstuhl, mad e th e com munity really take n otice. The lparse/smodels constituted a major con ceptual br eakthro ugh an d handled nicely all the traps DeReS did no t a void. First, lparse/smodels focused on the right fragment of de fault logic, logic prog ramming with the stable-mo del semantics. Next, it o rga- nized search for a stable model by look ing for atoms that form it. Finally , it suppor ted progr ams with variables and separated, as was the standar d in logic progr amming and databases, a progra m (a pr oblem speciﬁcation ) from an extension al database (an in- stance of the problem) . The work by Niemel ¨ a had us focus o ur thinking abo ut n onmo notonic log ics as computatio nal devices on the narrower but all- importan t case of logic pro grams. W e Origins of ASP 15 formu lated our ideas about the second- order ﬂa vor of p roblem solving with non mono - tonic logics a nd contrasted them with the traditio nal Pro log-style interpretation of logic programm ing. W e stated our in itial tho ughts on the meth odolog y of prob lem solving tha t exp loited ou r idea s o f m odeling comb inatorial p roblems th at we used in the Theo ryBase pr oject, as well as th e notion of p rogram -data separation that came from the database commu nity and was, as we just mentioned, already used in our ﬁeld by Niemel ¨ a. These ideas f ormed the ba ckbon e of our p aper on a n alter native way logic progr amming could be used for solving search pro blems Marek and Truszczy ´ n ski (1999). 4 T owards Answer -Set Pr ogram ming at the Helsin ki Univ ersity of T echn ology In this section Ilk ka Niemel ¨ a discusses the d ev elopments at the Helsinki Un iv er sity of T echno logy th at led to the paper Logic Pr ograms with Sta ble Mo del S emantics a s a Con straint Pr ogramming P a radigm Niemel ¨ a (1 999). Similarly as in the previous section, th e accou nt is very personal and qu ite su bjective. Hen ce, in this section ”I ” refers to Ilkka. I g ot expo sed to nonmono tonic reason ing whe n I joined th e group of Professor Leo Ojala at th e Helsinki University of T ech nology in 19 85. The gro up was studying speciﬁcation and veriﬁcatio n techniq ues of distributed systems. One of th e the mes was speciﬁcation of distributed systems using mo dal, in p articular, temp oral and dy- namic logics. The g roup h ad got interested in the solutions o f the frame pr ob lem based on no nmon otonic log ics when loo king for compa ct an d comp utationally e fﬁ- cient log ic-based speciﬁcation techn iques for d istributed systems. My r ole as a new research assistant in the group was to examine autoepistemic logic by Moore, non- monoto nic m odal logics by McDerm ott an d Doyle, and default logic by Reiter fr om this perspective. There was a need for too l support and to gether with a doctoral stud ent Heikki T uomin en we developed a system that we called the Helsinki Logic Machine, “an experimental reasoning system designed to provide assist ance need ed for applica- tion orien ted r esearch in log ic” (Niemel ¨ a and T uominen, 1986, 19 87). The system included too ls for theorem proving , mode l synthesis, model ch ecking, formula ma - nipulation fo r mo dal, temporal, e pistemic, deon tic, dy namic, an d n onmon otonic log- ics. I t was written in Quintus Prolog and contained implementations, for instance, for Reiter’ s default log ic, McDer mott and Doyle style nonmon otonic modal logic, an d autoepistemic logic in th e pr o positiona l ca se based on the literature an d some own work (E theringto n , 198 7; Mc Dermott and Doyle, 19 80; Niemel ¨ a, 1988). While no n- monoto nic reasoning was a side- track in th e Helsink i Logic Ma chine, it seems that it was one of the ear liest working no nmono tonic reaso ning systems althoug h we were not very well aw ar e of this at the time. The work and, in p articular, t he difﬁculties in d ev e loping efﬁcient tools led m e to further in vestigations to gain a deeper understanding of algorith mic issues and related complexity questions (Niemel ¨ a, 19 88; Niemel ¨ a, 198 8, 1990; Niemel ¨ a, 199 2). Similar questions were stud ied b y o thers a nd in the early 90s resu lts explain ing the alg orith- mic difﬁculties star ted em erging. These results sh owed that ke y reasonin g tasks in 16 V . W . Ma rek, I. Niemel ¨ a and M. Truszczy ´ nski major no nmon otonic logics are co mplete for th e secon d level of the p olynom ial hier- archy Cadoli and Lenzerini (199 0); Gottlob (1992); Stillman (19 92); Niemel ¨ a (1992). This indicated that these n onmo notonic lo gics have two o rthogonal sources of co m- plexity th at we called classical reason ing and con ﬂict resolution. Orthogo nality means that e ven if we assume that classical reasoning can be do ne ef ﬁcien tly , nonmonoto nic reasoning still remains NP-hard (unless the polyn omial hierarchy collap ses). These results made me to foc us more on conﬂict reso lution to develop techniq ues for pruning the s earch space of poten tial e x pansions/extension s. One approach was to develop co mpact characterizatio ns of expansion s/extensions captur ing their ke y ingre- dients. For autoep istemic logic I developed s u ch a cha racterization based on the id ea that expansions can be cap tured in terms of the mod al sub formu las in the premises and classical reasoning and exploited the idea in a d ecision p rocedur e for autoepistemic logic N iemel ¨ a (1988). T og ether with Jussi Rintan en w e also showed that if o ne lim- its th e theo ry in suc h a way that c onﬂict resolutio n is easy by r equiring stratiﬁcatio n, then ef ﬁcient reasoning is possible by furth er restriction s af fecting the other source of complexity Niemel ¨ a and Rintanen (199 2). The characterization based on modal subformulas gene ralizes also to default lo gic where extensions can be captured using justiﬁcations in the rules and leads to an inter- esting way of o rganizing the search for expan sions/extensions as a binary sear ch tr ee very similar to that in the DPLL algorithm for SA T Niemel ¨ a (19 94, 19 95). Furthe r prunin g techniques can be in tegrated to cu t substan tial parts of the potential search space for expan sions/extensions an d exploit, for instan ce, stratiﬁed pa rts of the rule set. My initial but very unsystematic e xperimen tation gave prom ising results. In 19 94 enco uraged and challenged by the Shakerto wn W o rkshop organ ized by V ictor and Mirek, I d ecided to restrict to a simple su bclass of de fault theories, that is, logic progr ams with the stable mod el semantics. For this subclass classical r easoning is essentially limited to Horn clauses and can be do ne efﬁciently in linear time u sing technique s propo sed by Dowling and Gallier in the 198 0s Dowling and Gallier (1 984). I had no par ticular ap plication in mind. The go al was to stud y whether the co nﬂict resolution techniques I had de velop ed for autoepistemic and default l ogic w o uld scale up so that it would be possible to handle very larg e sets of rules which meant at that time thousands or ev en tens of tho usands of rules. At that time Patrik Simons join ed my group and started work ing on a C++ imp le- mentation of the general algorithm tailored to logic programs. Patrik had excellent in- sights into the key implementation issues from very early on and the ﬁrst version w as released in 1995 Niemel ¨ a and Simons ( 1995). The C++ implementation was called smodels a nd it computed stable mod els for groun d n ormal pro grams. It gave sur- prising good re sults immediately a nd c ould handle pr ograms with a few tho usand groun d rules. Challeng e benchmar ks were comb inatorial problems, mainly colorabil- ity and Hamiltonian cycles, an idea that I learnt from Mirek and V ictor in Shakertown. For such ha rd pr oblems the perf orman ce of smo dels was sub stantially be tter th an state-of-the- art tools such as the SLG system Chen and W a rren (199 6). When developing benchm arks f or evaluating novel algorithm ic ideas and imp le- mentation techniques we s oon r ealized that workin g with groun d p rogra ms is too c um- bersome. In practice, f or pro ducing large enoug h interesting grou nd progr ams for benchm arking we needed to write separ ate pr ograms in some other lan guage to gen - erate gro und logic progr ams. This to ok consider able time for each benchm ark family Origins of ASP 17 and was quite inﬂexible and error-prone. W e realized that in ord er to attr act users and to be able to attack real application s we need ed to suppor t logic p rogram rules with variables. For h andling rules with variables we decided to em ploy a two level arch itecture. The ﬁr st phase was concerne d with g r ound ing , a pro cess to gen erate a set of g roun d instances of th e rules in the progra m so that stable mod els are preserved. Actual stable-mode l com putation was taking place in th e second model sear ch phase o n th e progr am grounded in the previous one. Th e idea was to have a separation of concern , that is, be able to e xploit advanced database and othe r such techniqu es in the ﬁrst phase and novel search an d prunin g techniqu es in th e o ther in such a w ay that both steps c ould be d ev eloped r elativ ely indepen dently . W e released the ﬁrst such system in 1996 Niemel ¨ a and Simons (1996). This was a major step forward in attrac ting users and getting closer to applica- tions. Such a system suppo rting rules with variables en abled co mpact and mo dular encodin gs of pro blems withou t any further host languag e. I t was now also possible to separate the problem speciﬁcation and the data providing the instance to be solved. W ork ing with the system and studying potential applications made me realize that logic pro grammin g with the stable model sem antics is very d ifferent from tradition al logic pr ogramm ing imp lemented in various Pr olog systems. The se systems are an- swering queries by SLD resolution and producin g answer substitutions as results. But we were using logic programs more like in a con straint pro grammin g approach where rules are seen as constraints on a solution set (stable model) of t he program and where a solution is not an answer substitution b ut a stable model, that is, a v aluation that sat- isﬁes all the rules. This is like in constraint satisfaction problems where a solution is a variable assignment satis fying all the constra ints. I wro te down these ideas in a paper Logic Pr ograms with Stable Model Semantics as a Constraint Pr ogramming P ar adigm which was ﬁrst presen ted in a workshop on Comp utational As pects of Nonmono tonic Reasoning in 19 98 Niemel ¨ a (19 98) and th en app eared as an extende d journal version in 19 99 N iemel ¨ a (1999). The pap er em phasized, in particular, th e knowledge r epre- sentation advantages of log ic programs as a constraint satisfaction frame work : “Logic program ming with the stable mod el semantics is p ut fo rward as an interesting constraint programmin g paradigm. It is shown that the paradigm embed s classical logical satisﬁability but seems to provide a more expressi ve framework from a kno wledge repr esentation point of view . ” In 199 8 we put m ore and mor e em phasis o n p otential app lications and, in partic- ular , on prod uct conﬁgur ation. This mad e us realize that a mor e efﬁcient grou nder supportin g an extended modeling langu age is need ed. At th at po int another student, T om mi Sy rj ¨ anen, with excellen t imp lementation skills and insight on language design, joined the gr oup and work o n a new groun der, lp arse , started. The go al was to en- force a tig hter typin g of the variables in the rules to facilitate th e app lication of mo re advanced database techniques for gro unding and the in tegration of built-in predicates and functio ns, for instanc e, for arithmetic. W e also rea lized th at fo r many ap plications no rmal log ic p rogram s were inade- quate not allowing comp act and intuitive encod ings. This led to the introd uction of new languag e constru cts: (i) c hoice ru les fo r enco ding choices instead of recursive odd 18 V . W . Ma rek, I. Niemel ¨ a and M. Truszczy ´ nski loops needed in norm al pr ograms an d ( ii) cardinality and weight constraints for typ- ical conditio ns need ed in many practical applicatio ns Soinin en and Niemel ¨ a (1 998); Niemel ¨ a et al. (19 99). In or der to fully exploit th e extension s comp utationally Patrik Simons developed techniqu es to p rovide built-in suppo rt for them also in the model search phase in the version 2 of smodels Simons (199 9 ) . So in 1999 when Vladimir Lifschitz coined the term answer -set program ming, the system that we had with lparse as the gro under and smodels version 2 as the model search eng ine o ffered q uite pr omising per forman ce. For example, for propo - sitional satisﬁability the perfo rmance o f smo dels co mpared nicely to the best SA T solvers a t that time (bef ore mo re efﬁcient co nﬂict driven clause learning solvers like zchaff em erged). Moreover , very interesting serious ap plication work started. For example, at the Helsinki Univ er sity of T ec hnolog y we cooperated with the product data m anagemen t gr oup o n au tomated p roduc t con ﬁguration which eventually led to a spin-off com pany V arian tum ( http ://www.vari antum.com/ ). M oreover , in V ienna the dlv pro ject for ha ndling disjunctive p rogram s had started a cou ple y ears earlier and had already made promising progress. 5 Conclusions Now , mo re than 12 years since ASP b ecame a recog nizable paradigm of search pro b- lem solv ing, we see th at the efforts of r esearchers in various dom ains: artiﬁcial intel- ligence, knowledge rep resentation, n onmo notonic reasoning , satisﬁability an d o thers resulted in a programming for malism that is being used in a variety of areas, but prin- cipally in those where the modelers face the issues o f d efaults, frame ax ioms an d other nonm onoton ic phenom ena. The experience o f ASP pro gramm ers shows that these phen omena can be naturally incorpo rated into the practice of mod eling real-life problem s. W e believe ASP is her e to stay . It provides a venue f or pro blem mo deling, p rob- lem descrip tion and p roblem solving . This does n ot mean that the pro cess of devel- oping ASP is ﬁnished. Certainly new extensions o f ASP will emerge in the future. Additional desiderata inc lude: software en gineerin g tools for testin g correctness of implementatio n, integrated de velopm ent en vironm ents and othe r tools that will speed up th e process of the use of ASP in nor mal p rogra mming prac tice. Better groun ders and better so lvers a ble to w ork with incremental g roun ding only will certainly emerge. Similarly , ne w app lication domains will surface and br ing ne w generations of in vesti- gators and, more importantly , users for ASP . Ackno wledgments The work o f the second author was partially supported b y the Academy of Finlan d (project 12239 9). T he work of the third auth or was p artially suppor ted by th e NSF grant IIS-0913 459. Origins of ASP 19 Refer ences K. Apt, H. A. Blair, and A. W alker . T oward s a theory of declarativ e knowledg e. In J. Minker , editor , F oundations of deductive databases and logic pr ogramming , pages 89–142 . 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