Reiters Default Logic Is a Logic of Autoepistemic Reasoning And a Good One, Too

Reiter’ s Default Lo gic Is a L ogic o f A utoepis- temic Reasoning And a Good One, T oo Mar c Denecker Departmen t of C omp uter Science K.U. Leuven Celestijnenlaan 200A B-3001 Hev erlee, Belgium V ictor W . Mar ek Mirosław T ruszczy ´ nski Departmen t of C omp uter Science University of Kentucky Lexington, KY 40506-06 33, USA Abstract: A fact apparently not observed earlier i n the literature of non monoto nic reasoning is that Reiter , in h is default logic paper, did not directly fo rmalize informal defaults. Instead, he translated a default into a certain n atural langu age proposition and provided a form alization of the latter . A few years later , Moore noted that pro po- sitions like the on e used b y Reiter are fu ndamen tally dif ferent tha n defaults and exhibit a certain autoep istemic natur e. Thus, Reiter had de veloped his default logic as a for- malization of autoepistemic propo sitions rather than of defaults. The ﬁrst goal of this paper is to sho w that some problem s of Reiter’ s default logic as a form al way to reason about in formal d efaults are directly a ttributable to the au- toepistemic nature o f default logic an d to the mismatch between informa l d efaults and the Reiter’ s formal d efaults, the latter bein g a for mal expression of the au toepistemic propo sitions Reiter used as a repre sentation of informal defaults. The second goal of our pap er is to comp are the work of Reiter an d Moor e. While each of the m a ttempted to fo rmalize au toepistemic prop ositions, the mo des of rea- soning in their respective logics were different. W e revisit Moore’ s and Reiter’ s in- tuitions and present them from the perspective of autothe or emhood , where th eories can in clude p roposition s referring to the theory ’ s own theo rems. W e then discuss the formalizatio n of this perspec ti ve in the logics of Moore and Reiter , respecti vely , using the unify ing semantic framework for d efault and autoe pistemic lo gics that we d ev el- oped ea rlier . W e argu e that Reiter’ s d efault logic is a better for malization of Mo ore’ s intuitions about autoepistemic prop ositions than Moo re’ s own autoepistemic logic. 1 Introd uction In this volume we celebra te the p ublication in 19 80 o f the special issue of the Ar- tiﬁcial In telligence Jo urnal on Nonmon otonic Reason ing that inclu ded three semi- 2 M. Denecker , V . W . Mar ek and M. T ruszczy ´ nski nal papers: Logic for Defau lt Reasoning by Reiter (198 0), Nonmo noton ic Logic I by McDermott and Doyle (1980), and Cir cumscription — a form o f nonmon otonic rea- soning by M cCarthy (1980). While the roots o f the sub ject g o ear lier in time, these papers are un i versally viewed as the main catalysts for th e emergence of nonmon o- tonic reasonin g as a distinct ﬁeld of study . Soon after the papers were published , nonmo notonic reason ing a ttracted widespread attention of r esearchers in the area o f artiﬁcial in telligence, and established itself ﬁrm ly as an integral sub- area of knowl- edge rep resentation. Over th e year s, the appeal of nonmono tonic reasoning went far beyond artiﬁcial intelligence, as many of its research challenges raised funda mental questions to philoso phers and mathematical lo gicians, and stirred su bstantial in terest in those communities. The groun dbreak ing pape r by McCarthy and Hayes (1969) about ten years before had cap tured the g rowing concern with the logica l represen tation of common sense knowledge . Attention focu sed o n th e representatio n of defaults , p roposition s that ar e true for most ob jects, that commo nly assume the form “most A ’ s are B ’ s. ” 1 Defaults arise in all applications involving common sense reaso ning an d requir e specially tai- lored fo rms of reasoning. F or instance, a default “most A ’s are B ’s” u nder suitab le circumstances should enable one to infer from the premise “ x is an A ” tha t “ x is a B . ” Th is in ference is d efeasible . Its con sequent “ x is a B ” may be false even if its premise “ x is an A ” is true . It may have to be withdrawn when new inf ormation is obtained. Providing a g eneral, fo rmal, domain indepen dent and elabor ation to lerant representatio n of defaults an d a n accoun t of what inferen ces can be rationa lly drawn from them was the artiﬁcial intelligence challenge of the time. The logics propo sed by McCarthy , R eiter, and McDer mott and Doyle were devel- oped in an attempt to formalize reaso ning wher e defaults are present. They went ab out it in different ways, howe ver . McCarthy ’ s cir cumscription extended a set o f ﬁrst-or der sentences with a second-o rder axiom asserting minimality of certain predica tes, typ i- cally of abno rmality pr edicates that c apture th e exceptions to defaults. Th is reﬂected the assumptio n that the world deviates as little a s possible f rom the “ normal” state. Circumscription has p layed a promin ent role in nonm onoton ic reasoning. I n p articu- lar , it h as been a precur sor to p reference log ics (Sho ham, 198 7) that p rovided furth er importan t insights into reasoning abou t defaults. Reiter ( 1980) and M cDermott and Doyle (19 80), on the othe r han d, f ocused on th e inference pattern “most A ’ s ar e B ’s. ” I n Reiter’ s words (Reiter, 1980, p. 82): ‘W e take it [th at is, th e default “Most birds can ﬂy” — DMT to mean something like “If an x is a bird, then in the ab sence of any inform ation to the contrary , infer that x can ﬂy . ”’ Thus, Reiter (and also M cDermott an d Doyle) quite literally e quated a d efault “most A ’ s a r e B ’s” with an in ference r ule that inv olves, besides the prem ise “ x is an A ” , an ad ditional pre mise “ther e is no inform ation to the contrary” or, more sp eciﬁcally , “ther e is n o information indica ting that “ x is not a B . ” The role of this latter premise, a con sistency con dition, is to ensur e the ratio nality of applying the default. In logic, inference r ules are me ta-logical objects th at a re not expressed in a logica l langu age. 1 In this pa per , we interpret the term “def ault” as an informal statement “most A ’ s are B ’ s” (Reiter, 1980). The term is sometimes interpreted more broadly to capture communicat ion con venti ons , frame axioms in temporal reasoning, or statemen ts s uch as “ normall y or typically , A’ s are B’ s ”. Reiter’ s Default Logic Is a Logic of Autoepistemic Reasoning 3 Reiter , M cDermott and Doyle sought to dev elop a logic in which such m eta-logical inference rules could be stated in the logic itself. They equ ipped th eir logics with a suitable modal operator (in the case of Reiter, embedded within “his” default expres- sion) to be ab le to expre ss the c onsistency condition and , in place of a default “most A ’ s ar e B ’ s” , they used the statement “if x is an A and if it is consistent (with the available information ) to assume tha t x is a B , then x is a B . ” W e w ill call this latter statement th e Reiter-McDermott-Doyle (RMD, f or shor t) p roposition associated with the default. Moore (1 985) was one of the ﬁrst, if not the ﬁrst, who realized that defaults an d their RMD pro positions a re of a different nature. Th is is how Moo re (1985, p . 76) formu lated RMD pro positions in terms of theoremho od and non-the oremho od: ‘[In the approac hes of McDerm ott and Doyle, and o f Reiter — DMT] the inference that birds can ﬂy is hand led b y having, in effect, a rule that says that, for any X, “X can ﬂy ” is a theorem if “X is a bird” is a theorem and “X cannot ﬂy” is not a theorem. ’ Moore then conten ded that RMD prop ositions ar e autoepistemic statements, that is, introspective statements referring to the r easoner’ s own b elief or th e theory ’ s o wn the- orems. He pointe d out fundamental d ifferences b etween the n ature of d efault prop osi- tions and auto epistemic ones and argued that the log ics dev eloped by McDerm ott and Doyle (1980) and , in th e follow-up paper, by McDermott (1 982), ar e attempts a t a logical formalizatio n of of autoepistemic statements and not of defaults. Not ﬁnding the Mc- Dermott and Doyle for malisms quite adeq uate as autoe pistemic logics, Moore (19 84, 1985) proposed an alternative, th e autoe pistemic logic . Unfortu nately , Moore did not refer to the paper by Reiter (1980) but only to those by McDermo tt and Doyle (198 0) a nd McDermo tt ( 1982), and his comments on this topic wer e not extrapolated to Reiter’ s logic. Neither d id Moore explain what could go wro ng if a d efault is replace d by its RMD prop osition. Y et, if Moor e is right then given the clo se correspon dence between Reiter’ s an d McDermott and Doyle’ s views on defaults, also R eiter’ s logic i s an attempt at a formalization of autoepistemic rather than o f default pr opositions. Moreover , if d efaults are re ally fu ndamen tally different from autoepistemic pr opositions, as Moo re claimed, it should be p ossible to ﬁnd dem onstrable de fects of Reiter’ s default logic for reason ing abou t defaults that could be attributed t o the different na ture of a default and of its R eiter’ s autoepistemic translation. Our main o bjective in Section 2 is to argue that Mo ore was righ t. W e show there two forms o f such defects that (1) the RMD prop osition is not always sound in the sense that infer ences made from it are not always rational with respect to the origi- nal defaults, and (2) the RMD propo sition is not always complete , th at is, there are sometimes rational in ferences fro m the origin al defaults that are n ot covered by this particular inference rule. In fact, both types of problems can be illustrated with exam- ples long known in the literatur e. In the remaining sections, we explain Reiter’ s default lo gic as a for malization of autoepistemic pro positions and show t hat in fact, Reiter’ s d efault logic is a better for- malization o f Mo ore’ s in tuitions than Moo re’ s own autoep istemic logic. On a for mal lev el, our in vestigations exploit the results on the unifying semantic framew ork fo r de- fault logic and au toepistemic logic that we pro posed earlier (Denecker, Ma rek, and T ruszczy ´ nski, 4 M. Denecker , V . W . Mar ek and M. T ruszczy ´ nski 2003). That work was based o n a algeb raic ﬁx point theo ry f or n onmon otone oper a- tors (Den ecker , Marek, and Truszczy ´ nski , 2 000). W e sho w th at the dif feren t dialec ts of autoepistemic reason ing stemming from our infor mal analysis can be given a prin- cipled formaliza tion using these algeb raic techniques. In o ur overview , we will stress the view on autoepistemic logic as a logic of autotheo r emhood , in which theories can include propo sitions referr ing to the theory’ s o wn theor ems. Some history . W e mentio ned th at Moo re’ s comm ents concernin g the RMD pro po- sition and the forma lisms by McDermo tt and Doyle (1 980) and McD ermott (1982) have never be en applied to Reiter’ s logic. For example, Konolige (1988), who was the ﬁrst to investigate the fo rmal link between au toepistemic reasonin g and default logic, wrote that “ the motivation and formal character of these two systems [Reiter’ s default and Moor e’s autoepistemic logics – DMT] ar e differ ent ”. This by passes the fact that Reiter , as we have seen, starts his enterpr ise of building default logic after tran slating a default into a prop osition which Moore later identiﬁed as an autoep istemic proposi- tion. There may b e se veral reasons why Moo re’ s comments ha ve ne ver b een extrapo- lated to Reiter’ s lo gic. As mentio ned before, one is th at Moor e did not r efer to the paper by Reiter ( 1980) but only to the p apers by McDermott and Doyle (198 0 ) and McDermott (19 82). In add ition, the logics o f Reiter and, respectiv ely , McDermo tt an d Doyle were q uite different; the formal con nection was not known at that time (mid 1980s) and was established on ly about ﬁ ve years later (T ruszczy ´ nski , 1991). Also au- toepistemic and d efault logics seemed to be q uite different (Marek and T ruszczy ´ nski, 1989), and eventually tur ned o ut to be d ifferent in a certain precise sen se (Go ttlob, 1995). More over , the intuitions un derlying the n onmon otonic logics of the time ha d not been so clear ly articulated, not e ven in Moo re’ s work as we will see later in th e paper, and wer e not easy to f ormalize. This was clearly d emonstrated about ten years later by Halper n (199 7 ), who reexamined the intuition s presented in the o riginal papers of default logic, autoepistemic logic and Lev esque’ s (1990) related logic of o nly know- ing an d showed gaps an d ambigu ities in these intuitions, and various non -equivalent ways in which they cou ld be formalized. As a result, the natu re of au toepistemic pro positions, its relationship to defaults and what may go wron g wh en the latter are enco ded by the ﬁrst, was n ev er well un - derstood. The rele vance of Moor e’ s claims for Reiter’ s default logic has never become generally acknowledged. Reiter’ s log ic has never been thought of and has never been truly an alyzed as a for malization of autoepistemic reasoning. The in ﬂuence o f Re- iter’ s paper h as b een so large, that ev en today , the default “most A ’ s a r e B ’s” and the statement “if x is an A and if it is consistent to assume that x is a B , then x is a B ” 2 are still considered syn onymou s in so me parts o f th e nonm onoton ic reason - ing comm unity . Y et, in fact, they are quite d ifferent and , more imp ortantly , a log ical representatio n of the second is un satisfactory for rea soning about the ﬁrst. 2 Or its propositi onal version “if A and if it is consiste nt to assume B , then B ” . Reiter’ s Default Logic Is a Logic of Autoepistemic Reasoning 5 2 Reiter’ s Defaults Are Not Defaults But A utoepistemic Statements Our g oal below is to justify th e claim in the title of the section. T o avoid co nfusion , we emp hasize that by a defa ult we m ean an in formal expression of the type most A ’ s ar e B ’ s . In Reiter’ s approach (similarly in that of McDe rmott and Doyle), the default is ﬁrst translated into an RMD pr oposition if x is an A an d if it is co nsistent with th e available information to assume th at x is a B , then x is a B , whic h is then expressed by a Reiter’ s default e xpress ion in d efault logic: A ( x ) : M B ( x ) B ( x ) . T o explain the sectio n title, let us assume a setting in which a hu man exp ert has knowledge about a domain that consists of proposition s and de faults. In th e approach of Reiter (the sam e app lies to McDer mott and Doyle), the expert builds a knowledge base T by including in T formal representa tions of the proposition s (given as fo rmulas in the languag e o f classical logic) and of RMD propo sitions o f the d efaults (given by the correspon ding Reiter’ s default expressions). The presence of Reiter’ s default expressions in T means that T contain s pro positions referr ing to its own inform ation content, i.e., to what is consistent with T , o r du ally to what T en tails or does not entail. Moore (1985) called s uch reﬂexive p roposition s autoep istemic and argued that they st atemen ts could be phrased in terms of theorems and non-theo rems of T . Reiter de veloped a default expression as a formal expression of the RMD propo si- tion r ather than of the default itself (the same h olds for McDermo tt and Doyle). T his is why th is logic expression d oes not capture the full infor mal conten t of the d efault. When considere d mo re closely , it indeed becom es ap parent that a default and its R MD propo sition ar e no t equivalent or even related in a strict logical sense. A straightfor- ward possible-world analysis reveals th is. The default might b e true in the actual world (say 9 5% of the A ’ s ar e B ’ s) but if there is just on e x that is an A and n ot a B , an d for which T has no e vidence that it is not a B , the RMD p ropo sition is false in th is world and x is a witness o f this. T hus, it is obvious th at in ma ny applications where a default holds, its RMD proposition does not. Conversely , the default might not hold in the actu al world ( few A ’ s are in fact B ’ s) yet th e expert k nows all x ’ s that are n ot B ’ s, in which case the RMD pro position is true. A fund amental difference pointed out by Moo re between defaults and autoe pis- temic proposition s, is that the la tter are naturally nonmono tonic but inferen ce rules used f or reasonin g with th em are not defeasible . For example, extending the knowl- edge base T containing an RMD p roposition with new in formatio n, e.g., th at some x is not a B , may indeed ha ve a nonmo noton ic effect and delete som e previous inferen ces, e.g., that x is a B . The initial inf erence of x is a B , r esulted in a fact that was false. Howe ver , that inference was not defeasible. The essential pr operty of a d efeasible in- ference is that it may de riv e a false con clusion fro m premises that are true in the actual world. For instance, the infe rence from most A ’s ar e B ’s and x is a n A that x is a B is defeasible as its consequent may be false while the premises are true. In the c ontext of our example above the theory , say T , entailed the false fact tha t x is a B from the premises (i) the RMD pro position, (ii) x is an A an d (iii) T c ontained no evidence that x is no t a B . It was not def easible since one of its pr emises was false. Indeed , the 6 M. Denecker , V . W . Mar ek and M. T ruszczy ´ nski RMD propo sition was false and x was a witness. Th e inf erence ru les app lied are no t defeasible (th ey ar e, essentially , the introduction of conjunction and m odus pon ens). T o sum up, an inf erence from a knowledge b ase inv olving a n RMD pro position m ay be false b ut only if the RMD prop osition itself is f alse. T o emp hasize further conseq uences of e quating defaults and RMD pr opositions we will lo ok at well-known examples from the literature . First, we turn our attention to the question whether ther e are cases when apply ing the RMD proposition leads to in- ferences that do not seem rational (lack of “soundness” with respect to understood in- formally “rationality”). Th e Nixon Diamond e xamp le by Reiter and Criscuolo (1981) and reasoning pro blems with related inheritance networks illustrate the prob lems that arise. Example 1 Richard M. Nixon , the 37th president of the United States, was a Re- publican and a Qua ker . Most Republicans are hawks while most Quakers are doves (paciﬁsts). Nobod y is a h awk and a dove. Some peo ple are neither hawks nor doves. Encodin g the Reiter-McDermott-Doyle propo sition of these defaults in d efault logic , we obtain the following theory: Repu b l ican ( N ixon ) ∧ Quak er ( N ixon ) ∀ x ( ¬ Dov e ( x ) ∨ ¬ Quak er ( x )) Repu b l ican ( x ) : M H awk ( x ) H awk ( x ) Quak e r ( x ) : M D ov e ( x ) D ov e ( x ) . In default logic , this the ory gives rise to two extensions. In o ne of them Nixon is believed to be a hawk and n ot a dove, in the other one, a dove and not a hawk. But is this rational? As we mentio ned above, th e use of an RMD-propo sition is rational when it is expected to hold for most x , and hen ce, in ab sence of infor mation, it is likely to hold for some speciﬁc x . But in the case of Nixo n, we k now in advance that at least one of the two “Nixon ” instan ces of the RMD pr oposition s has to be wr ong. As to which o ne is wrong , without fu rther informatio n one cou ld as well throw a coin. Moreover, it is not u nlikely that th ey a re bo th wrong and that in fact, Nixon is neither dove nor hawk. And in fact, it seems mo re ratio nal not to app ly any of the defaults, leading to a situation where it is not known whether Nixo n is a dove, a hawk or neither . Th e ratio nale of using the RMD propo sition as a su bstitute fo r th e de fault does not hold for Nixon or any other republican quaker for that ma tter . ✷ Example 2 Let us assume now th at all quakers are rep ublicans. In this case, the default that most q uakers (say 95%) are d oves is more speciﬁc than and overrules the default that most repu blicans ( say 95%) are h awks. It is rational here to give priority to the quaker default, leadin g to the d efeasible con clusion that Nixo n is a d ove. Howe ver , th is conclusion cann ot be derived from the RMD prop ositions because their consistency pr emise “it is consistent to assume that x is a dove ( r espectively a ha wk)” is too general to take such information into account. ✷ Such scenario s we re studied in the context o f inh eritance hierarch ies (T ouretzky, 1986). T o reason correctly on this sort of applications using Reiter’ s logic, the consis- tency condition of the RMD pro positions has to be tweaked to take the hierarc hy into account a nd giv e priority to the q uaker d efault. For example, we can refor mulate the Reiter’ s Default Logic Is a Logic of Autoepistemic Reasoning 7 RMD prop osition o f the default “most republicans ar e hawks” as “if x is known to be a republican and it is co nsistent to assume that he is a ha wk and it is co nsistent to assume that he is not a quaker , then x is a hawk” , which takes additional informatio n into account. Such mod iﬁed rules can of course be rep resented in default logic. Af ter all, the logic was developed f or representing (defeasible) in ference rules. But, as in the examples above, they cann ot be inferr ed from the RMD proposition s. An d the inference s that can be drawn fr om the RMD propositions are n ot always the ratio nal ones. The next prob lem that arises is of a com plementar y natur e and c oncerns (lack of) completen ess with respect to “rational” in ferences. Are there cases where ra tional albeit defeasible inferen ces can be drawn fr om d efaults that cannot be inferred fr om RMD prop ositions? As suggested above by our gen eral d iscussion, the an swer is indeed p ositi ve. After all, the RMD p roposition expresses only a single and quite speciﬁc type of inference that might be associated with a default. Example 3 As an illustration , let us consider the defaults most Swedes ar e blond and most Japanese ha ve black h air . Nobod y is both Swede an d Japanese, or has both blon d and black hair . If we learn know th at Boris is a Swede or a Japan ese then , given that he can not be both Swede and Japanese, it seems ratio nal to co nclude defeasib ly that Boris’ s hair is blo nd or black. In othe r words, defaults can (sometimes) be combin ed and together giv e rise to defeasible inference rules like: Boris is Swede or J apane se: M Boris’ s hair is blond or black Boris’ s hair is blond or black . If all we know is that Boris is Swede or Japa nese, the co nclusion o f this r ule can not be drawn from the two original RMD propo sitions f or the s imple reason that for each, one of their premises is not satisﬁed: it is not known that Boris is a Swede, and neither is it known that he is Japanese. For instance, in the lo gic o f Reiter, the two defaults would be encoded as S w e de ( x ) : M B l ond ( x ) B l ond ( x ) and J apanese ( x ) : M B l ack ( x ) B l a c k ( x ) . If we only know S wede ( B or is ) ∨ J apane s e ( B or is ) , then neither S w ede ( B ori s ) nor J apanese ( B ori s ) can be established. Ther efore, the premises of neither rule are established and no inference can be made. Even more, if we accept Reiter’ s logic as a logic of autoep istemic pr opositions, th ese co nclusions should not be drawn from these expressions. ✷ This example shows a clear case of a desired defeasible inference that can not be drawn fro m the rules e xpr essed in the two R MD proposition s. A default expression in Reiter’ s logic that would do the job has to encode explicitly th e combined inf erence rule: S w e de ( x ) ∨ J apanese ( x ) : M ( B l ond ( x ) ∨ B l ack ( x )) B l ond ( x ) ∨ B l ack ( x ) . This e xpr esses an inference rule wh ich is not deriv able from the orig inal RMD propo- sitions in the logics of Reiter , McDer mott, Doyle, o r Moore. Default logic does not support such reasonin g un less the combined inference rule is explicitly encode d as well. 8 M. Denecker , V . W . Mar ek and M. T ruszczy ´ nski Example 4 Assume that we now ﬁnd o ut that Boris has b lack ha ir . Giv en that h e is Japanese or Swed e, and gi ven the defaults fo r both , it seems rationa l to assume that he is Japan ese. Can we i nfer this fro m the combined inference rules expressed above and giv en that no body can be blond and b lack, or Swede a nd Japan ese? The an swer is n o and, consequently , yet another infer ence rule sh ould be added to obtain this infer ence. ✷ Problems of the se kind were reported many times in the NMR literatu re an d promp ted attempts to “improve” Reiter’ s default logic so as to cap ture add itional de- feasible inferen ces o f the info rmal d efault. This is, howe ver, a difﬁcult en terprise, as i t starts from a logic whose semantical apparatus is developed for a very speciﬁc for m of reasoning , namely autoepistemic r easoning . And while at th e fo rmal level the result- ing logics (Brewka, 1 991; Schaub, 19 92; Lukaszewicz , 19 88; Mikitiuk and T ruszczy ´ nski, 1995) capture some asp ects of d efaults that Reiter’ s logic does not, also they formal- ize a small fragmen t only of what a d efault represen ts an d, certain ly , non e h as e volved into a method of reasoning about defaults. In th e same time, theor ies in these log ics entail formulas th at cannot be justiﬁed fr om th e point of view of default logic as an autoepistemic logic. T o summarize, an RMD propo sition expresses one defeasible in ference ru le as- sociated with a default. It often derives r ational assumption s from the d efault but not alw ays, and it may easily miss some useful and natural defeasible infer ences. T he RMD p roposition is autoepistemic in n ature; Reiter’ s orig inal default log ic is therefo re a for malism for au toepistemic reasoning . As a logic in which infere nce r ules can b e expressed, default logic is quite useful for reason ing on defaults. Th e price to be paid is that the human expert is responsible for expressing t he d esired defeasible inference rules stemming from the defaults and for ﬁ ne-tu ning the consistency cond itions of the inference rules in case of c onﬂicting defaults. T his may re quire substantial effort and leads to a methodo logy that is no t elaboration tolerant. While our discussion shows that in general, RMD proposition s an d Reiter’ s de- faults do not a lign well with th e informal concep t of a default of the form most A ’ s are B ’s , ther e are o ther non monoto nic rea soning patterns th at are c orrectly ex- pressed thro ugh Reiter’ s d efaults. In p articular, patterns such as com munication con- ventions, database or information storag e co n ventions and policy rules in the typolog y of McCarthy (198 6), can be expressed well by true autoep istemic propositions and, consequen tly , are correctly fo rmalized in Reiter’ s logic. E.g., the convention that an airport customs d atabase explicitly co ntains the nationality of only non- American pas- sengers, is correctly speciﬁed by the Reiter default : M N ational i ty ( x ) = U S A N ational ity ( x ) = U S A . Similarly , the po licy rule that th e dep artmental meetings are norm ally h eld on W e dnes- days at noon , is corr ectly formalized by : M T ime ( meeti ng ) = ” W ed, noon ” T ime ( meeting ) = ” W ed, noon ” . In spite of such examples, th e fact remain s that default logic is n ot a lo gic of de- faults. Ar e there other logics that could be regarded as such? There have been several Reiter’ s Default Logic Is a Logic of Autoepistemic Reasoning 9 interesting attempts at form alizing defaults most A ’ s a r e B ’s . Most importan t of them focused on defaults as con ditional assertions an d on a bstract nonmon otonic conse- quence relation s ( Makinson, 1989; Lehm ann, 19 89; Pearl, 1990; Krau s, Lehmann, and Magidor, 1990; Lehman n and Magidor, 19 92). Th is r esearch direction resulted in elegant math- ematical theo ries and d eep insights into the natu re of some forms of n onmon otonic reasoning . Howe ver, it is not directly related to our effort h ere. Thu s, rather than to discuss it we refer to the papers we cited. Instead, in the remainder of the paper , we focus on the second objective identiﬁed in th e in troductio n. Th at is, we provide an informa l basis to auto epistemic reason- ing, we place Reiter’ s default logic ﬁrmly among dialects of autoepistemic reasoning , and show that Reiter’ s lo gic was a watershed point that pinp ointed one o f the most fundam ental and most imp ortant forms of autoepistemic reasoning. 3 Studies of Relationships Between Default Logic and A utoepistemic Logic K onolige (198 8 ) was the ﬁrst to in vestigate a formal link between default an d au- toepistemic logic. He pr oposed the following tran slation Kon fro m default log ic to autoepistemic logic: α : M β 1 , . . . , β n γ 7→ K α ∧ ¬ K ¬ β 1 ∧ · · · ∧ ¬ K ¬ β n → γ and argued that Kon was equiv a lence preser ving in th e sense that default extensions of the default theory were exactly the autoe pistemic expansions o f its t ranslation . This translation is intuitively appea ling, essentially expressing formally the RMD proposi- tion o f the de fault in modal logic, an d it indeed plays an importan t role in the story . Nev ertheless, it tur ned out th at this translation was only p artially correct (K onolig e, 1989). Later, Gottlo b (1 995) pr esented a correct tr anslation from default logic to au- toepistemic logic but also proved that no modula r tr anslation exists. The latter result showed that these two lo gics ar e essentially d ifferent in some importan t aspect. As a result, the auto epistemic natur e o f default lo gic, which Moor e had implicitly po inted at, and his implicit criticism o n d efault logic as a log ic of defaults were never widely acknowledged. But Reiter’ s logic is just that — a logic of autoepistemic r easoning. More over , in many respects it is a better logic of autoepistemic reaso ning than the on e by Moore. Our goal no w is to reconsider the intuitio ns of autoepistemic reasoning, to distinguish between different d ialects of it and to d ev elop princip led for malizations fo r these di- alects. In particular, we relate Reiter’ s and Moore’ s logics, and explain in wh at sense Reiter’ s logic is better than Moor e’ s. Ou r discussion uses the form al results we dev el- oped in an earlier pap er (Denecker et al., 2003). There we used the algeb raic ﬁxpoint theory for arbitr ary lattice operator s (Den ecker et al., 2000) to deﬁne fou r different semantics of default logic and of autoepistemic logic. This theory can be summariz ed as follows. A com plete lattice h L , ≤i induc es a com plete bilattice h L 2 , ≤ p i , where ≤ p is th e precision or der on L 2 deﬁned as follows: ( x, y ) ≤ p ( u, v ) if x ≤ u and v ≤ y . T u ples ( x, x ) ar e called exact. For any ≤ p -mono tone operator A : L 2 → L 2 that is 10 M. Denecker , V . W . Mar ek and M. T ruszczy ´ nski A : L 2 → L 2 Kripke-Kleene least ﬁxpoint O A : L → L O A ( x ) = A 1 ( x, x ) Suppor ted ﬁxpo ints S A : L → L S A ( x ) = lfp ( A 1 ( · , x )) Stable ﬁxpoints S A : L 2 → L 2 S A ( x, y ) = ( S A ( y ) , S A ( x )) W e ll-found ed least ﬁxpoint T able 1: L attice operators and the correspond ing semantics symmetric , that is, A ( x, y ) = ( u, v ) if and only if A ( y , x ) = ( v , u ) , we can deﬁne three derived operator s. These four operator s i dentif y four dif ferent types of ﬁxpoints or least ﬁxpoints (when the derived ope rator is monotone ). Th ey are summarized in T able 1 (where the o perator A 1 ( · , · ) used to deﬁne O A is the p rojection o f A o n the ﬁrst coordin ate). By assumption, A is a ≤ p -mono tone ope rator on L 2 and its ≤ p -least ﬁxpoint is called the Kripke-Kleene ﬁxpo int of A . Fixp oints of th e operator O A correspo nd to exact ﬁxp oints of A ( x is a ﬁxp oint of O A if and only if ( x, x ) is a ﬁxp oint of A ) and are called supported ﬁxpoints of A . T he operator S A is an anti-mon otone op- erator on L . Its ﬁxpo ints yield exact ﬁxp oints of A (if x is a ﬁx point of S A then ( x, x ) is a ﬁxpoint of A ). They are called stable ﬁxpo ints o f th e operato r A . It is clear that stable ﬁx points are supported . The operator S A is a ≤ p -mono tone operator on L 2 and its ≤ p -least ﬁxpoint is called the well-found ed ﬁxpo int of A (ﬁxpo ints of S A are also ﬁxp oints of A ). Th e nam es of these ﬁx points reﬂect the well-known se- mantics of logic prog ramming , wher e th ey were ﬁrst studied by means of operato rs on lattices. T aking Fitting’ s four-valued immediate conseq uence op erator (Fitting, 1985) for A , we proved ( Denecker et al., 2000) that the four different types of ﬁx point correspo nd to four well-known sema ntics of log ic programmin g: Kripke-Klee ne se- mantics (Fitting, 19 85), suppo rted mode l semantics (Clark, 1978), stable semantics Gelfond and Lifschitz (1988) and well-foun ded semantics (V an Gelder et al., 199 1). This elegant p icture extends to default lo gic and au toepistemic logic Denecker et al. (2003). In th at paper, we iden tiﬁed the semantic op erator E ∆ for a default the ory ∆ , and the semantic oper ator D T for an auto epistemic th eory T . Both oper ators whe re deﬁned on the bilattice of possible-world sets, which we introduce f ormally in the fol- lowing section. Just as fo r logic programming , each oper ator d etermines three deriv ed operator s and so, fo r each logic we ob tain four types of ﬁxp oints, ea ch induc ing a semantics. Some o f these semantics turn ed o ut to correspon d to semantics prop osed earlier; other sem antics were new . Impor tantly , it turned out that the oper ators E ∆ and D Kon (∆) are id entical. Hen ce, Konolige’ s map ping tu rned o ut to be equivalence preserving for each of the four types of seman tics! T able 2 summarizes the results. The ﬁrst two lines a lign the theories and the co rrespond ing ope rators. The last four lines describe the matching sema ntics (the ne w semantics for au toepistemic and de- fault lo gics obtained from this o perator-based appr oach Denecker et al. (2000) are in bold font). From this purely mathem atical point of vie w K onolige ’ s intuition seems basically right. His mappin g failed to establish a corresp ondenc e between Reiter extensions and Moore expansions on ly because they ar e on different levels in the hierarc hy of the semantics. O nce we correctly align the dialects, his transform ation works perfectly . Con versely , we a lso proved that the s tanda rd method to eliminate nested modalities in Reiter’ s Default Logic Is a Logic of Autoepistemic Reasoning 11 default theory ∆ K o n − → autoepistemic theory T semantic operato r E ∆ K o n − → semantic operator D T KK-ext ension K o n − → KK-e xtensio n (Deneck er et al., 1998) W e ak extensions K o n − → Moore e xpansio ns (Marek and T ruszczy ´ nski, 1989) (M oore, 1984) Reiter extensions K o n − → Stable extensions (Reite r , 1980) W e ll-found ed e xtension K o n − → W ell-f ounded extension (Baral and Subrahmani an , 1991) T able 2: The align ment of default and autoepistemic logics the modal logic S5 can be used to translate any a utoepistemic logic theory T into a default theory that is equiv alent to T u nder each of the four semantics. While the non-mo dularity result by Go ttlob (1995) had shown that default logic and autoepistemic logic are essentially dif ferent logics, our results s umma rized abov e unmistakenly point out that default and autoepistemic log ics a re tightly co nnected logical systems. They sugg est that the four seman tics formalize different dialects of auto epistemic reasoning an d that Reiter and Moor e formalized different dialects. Therefo re, in the rest of the paper , we will vie w Reiter’ s logic si mply as a fragment of modal logic, as identiﬁed by K onolige ’ s mapping. 4 F ormalizing A utoepistemic Reasoning — an Inf or - mal Perspecti ve In our p aper ( Denecker et al., 20 03) we developed a p urely algeb raic, abstract study of semantics. The study iden tiﬁed the (non monoto ne) oper ators of autoepistemic and default logic theories, and a pplied the dif feren t notion s o f ﬁxpoints to them. What that paper was missing was an account of wh at these ﬁx point constru ctions mean at the informa l le vel and how the different d ialects in the fr amew ork dif fer . Being as clear as possible about the informal semantics of autoepistemic theories is ess ential, as it is there where problems with formal accounts start. This is the gap that we close in the rest of this p aper . T o this end we ﬁrst retur n to the o riginal co ncern o f Reiter, and of McDerm ott and Doyle. Let us suppose th at we have incomp lete knowledge ab out the a ctual world, represented in, say , a ﬁrst order theory T , and that we know that mo st A ’ s are B ’ s. Following the Reiter , Mc Dermott and Doyle approach, we would li ke to assert the following proposition: If for some x , T | = A ( x ) and B ( x ) is co nsistent with T (that is, T 6| = ¬ B ( x ) ), the n B ( x ) . In fact, we would like to express this statemen t in the log ic and, moreover, to add this prop osition, with its referen ces to wh at T entails o r does not entail, to T itself. 12 M. Denecker , V . W . Mar ek and M. T ruszczy ´ nski What we obtain is a theo ry T that r efers to its own theorem s. In this view th en, modal literals K ϕ in an autoepistem ic theor y T = { . . . F [ K ϕ ] . . . } are to be inter- preted informa lly as statem ents T | = ϕ , and th e th eory T itself as having the f orm T = { . . . F [ T | = ϕ ] . . . } , em phasizing the intuition of the self-referential natu re of autoepistemic theories. This view reﬂects what seems to us the m ost precise intu ition that Moo re p roposed : to view autoepistemic pro positions as inferen ce rules. Specializin g the discussion above t o the autoep istemic formula K α 1 ∧ · · · ∧ K α n ∧ ¬ K β 1 ∧ · · · ∧ ¬ K β m → γ (1) we can write it (infor mally) as: T | = α 1 ∧ · · · ∧ T | = α n ∧ T 6| = β 1 ∧ · · · ∧ T 6| = β m → γ , and under stand it (informally) as an infer ence rule: if α 1 , . . . , α n are theorems and β 1 , . . . , β m are not theorems (2) then γ holds. which is consistent with Mo ore’ s (19 85, p. 76) position we cited ear lier . Alterna- ti vely , K ϕ can be read as “ ϕ can be derived, or pr oven ” ( again, from the theory itself), which amounts at the in formal le vel just to a d ifferent wording . W e will refer to th is notion of theorem and derivation as autotheo r em and au toderivation , respectively . According ly , we will call the basic Moo re’ s perspective as that of autotheo r emhood . The autotheor emhood view can be seen as a special case of a mo re gener ic view , also propo sed by Moo re, b ased on autoepistemic agents . I n th is view which , inci- dentally , is the r eason b ehind the name auto epistemic logic , an autoepistemic theory is seen as a set of in trospective pro positions, belie ved by the agent, about the actua l world and his own beliefs about it. The crucial assumption is the one which Levesque (1990) dubbed later the All I Know As sumptio n: the assump tion t hat all that is known by the agent is gr ou nded in his theo ry , in th e sense that it belongs to it or can be de- riv ed from it. In the case of the au totheore mhood vie w , the a gent is nothing else than a per soniﬁcation of the theo ry itself, and what it kn ows is what it entails. W e d iscuss alternative instances of this agent-b ased view in the next section. But let us now fo cus on developing the auto theoremh ood perspectiv e. W e regard it as a more precise intuition that is mo re am enable to formalizatio n despite the fact that self-refer ence, which is evidently p resent in the n otion of autoth eoremh ood, is a notorio usly complex phenomeno n. It plague d, albeit in a different form, the theo ry of truth in philosop hical logic with millennia-o ld paradoxes (T arski, 1983; Kr ipke, 1975; Barwise and Etchemend y , 1987). Th e best k nown example is the famous liar parado x: “This sentence is false. ” An autoepistemic theory that is clearly reminiscent of this parado x is: T liar = {¬ K P → P } . In the autotheor emhood view , this theor y states that if it does no t entail P then P holds. Howe ver, if P is not entailed , th en we have an argument for P , and if P is Reiter’ s Default Logic Is a Logic of Autoepistemic Reasoning 13 entailed, the uniqu e pro position o f the theory is trivially satisﬁed; n o a rgument f or P can be constru cted. T his is mutatis mutandis the argument for the in consistency of the liar sentence. In view of the difﬁculties that self-r eference has po sed to the development o f the theor y of truth, it would be naiv e to hope that a crisp, unequiv ocal formalizatio n of auto epistemic logic existed. Moore (1 985, p. 82) explained the difﬁculty of deﬁning the seman tics f or au - toepistemic inference rules ( 2) as follows. When the inferenc e rules are mono tonic, that is, when m = 0 , ‘once a form ula h as been generated at a given stage, it rema ins in the gen- erated set o f formu las at e very sub sequent stage. [...] The prob lem with attempting to follo w this pattern with nonmono tonic inferen ce rules [that is, when m > 0 (note o f the autho rs)] is th at we cann ot d raw non mono- tonic inferenc es reliably at any particular stage, since something inferred at a later stage may in validate them. ’ T o put it differently , the problem is that when a rule (2) is applied to derive γ at some stage when all α i ’ s ha ve been inf erred to be theorems and n one of th e β j ’ s has been derived, later inferences may deri ve so me β j and hence, in validate the deriv ation of γ . In suc h case, Moore argu es, all w e c an do is to characterize the desired result as the solution of a ﬁxpoin t equation instead of com puting it by a ﬁxpoint construction : ‘Lacking such an iterative structure , n onmon otonic systems often use nonco nstructive “ﬁxed po int” d eﬁnitions, which do not d irectly yield al- gorithms f or enumera ting the “derivable” formulas, but do deﬁne sets of formu las that respect the intent of the non monoto nic inf erence rules. ’ This was an extre mely clear an d com pelling repr esentation of intuitions behin d no t only the Moore ’ s o wn autoepistemic logic, but also behind the formalisms of McDer- mott and Doyle, and of Reiter , too , for that matter . It is useful no w to lo ok at these ide as from a mo re formal p oint of vie w . L et us consider a mod al theor y T over some vocabulary Σ . Let T consist of “infe rence rules” of the for m (2), wh ere for simplicity we assume that all fo rmulas α i , β j , γ are objective (that is , contain no moda l operato r). 3 The inference processes that Moo re had in mind are syntac tic in nature a nd are deriv ations of formulas. Y et, it is straightfo rward to cast these inference processes in semantical terms. Let W be the set of all Σ -interpretatio ns. A state o f belief is r epresented as a set B ⊆ W o f po ssible world s. 4 Intuitively , e ach elem ent w ∈ B rep resents a possible world , a state of affairs that satisﬁes the a gent’ s beliefs. A world w 6∈ B repr esents an impossible world , a state of affairs that violates at least on e propo sition of th e agent. Gi ven a set B rep resenting the w orlds held possible by an agent, the following, standard, d eﬁnition formalize s wh ich (mod al) f ormulas the agen t believes (or kn ows — we do not distinguish between these two modalities in our discussion). Deﬁnition 1 W e deﬁne the satisﬁa bility r e lation B , w | = ϕ as in the mod al logic S5 by the standard recurs ive rules of pr oposition al satisfactio n augmen ted with on e addition al rule: B , w | = K ϕ if for every v ∈ B , B , v | = ϕ. 3 Our appro ach works equally well for arbitrary modal theories. 4 A possible -world set is a specia l Kripke structure in which the accessibilit y relat ion is tota l. 14 M. Denecker , V . W . Mar ek and M. T ruszczy ´ nski W e then deﬁne B | = K ϕ ( ϕ is believed or known in state B ) if for every w ∈ B , B , w | = ϕ . This deﬁnition extends the standar d deﬁnition of truth in the sense that if ϕ is an ob - jectiv e formu la then B , w | = ϕ if and on ly if w | = ϕ . W e deﬁne Th ( B ) = { ϕ | B | = K ϕ } and Th obj ( B ) the restriction of Th ( B ) to objective form ulas. Th ese sets repre- sent all modal formu las and all ob jectiv e formulas, respectiv ely , known in the state of belief B . It is n atural to o rder belief states according to “how muc h” they believ e or know . For two belief states B 1 and B 2 , we deﬁne B 1 ≤ k B 2 if Th obj ( B 1 ) ⊆ Th obj ( B 2 ) or , equiv alently , if B 2 ⊆ B 1 . The ordering ≤ k is often called the knowledge ordering. W e obser ve that B 1 ≤ k B 2 does not entail Th ( B 1 ) ⊆ Th ( B 2 ) , due to the non mono- tonicity of moda l literals ¬ K ϕ expressing ignoran ce, some of wh ich m ay be true in B 1 and false in B 2 . W e can see Moore’ s infer ence processes as sequences ( B i ) λ i =0 of possible-world sets such that B 0 = W , th e po ssible-world set o f m aximum ign orance in which only tautologies are k nown. In each der iv a tion step B i → B i +1 , some worlds w ∈ B i might be found to be impossible and elimina ted in B i +1 ; othe r worlds w 6∈ B i might be established to be possible and added to B i +1 . T his process is described thro ugh Moore’ s sema ntic operator D T , which maps a possible-world set B to the possible- world set { w | B , w | = T } . For theories co nsisting of form ulas (1), D T ( B i ) is exactly the set of all possible worlds that satisfy the conc lusions γ o f all inference rules that are “activ e” in B i , that is, for which B i | = K α j , 1 ≤ j ≤ n , and B i 6| = K β j , 1 ≤ j ≤ m . Let us c ome back to M oore’ s claims. The no nmon otonicity of the infer ence rules (2), o r mo re precisely , formu las (1) is due to the negativ e conditions ¬ K β j ( β i not known, n ot proved, not a theorem ). So let us assume that m = 0 for all inf erence rules in T . 5 One can show that under this ass ump tion D T is a monoton e op erator with respect to ⊆ : if B 1 ⊆ B 2 , then D T ( B 1 ) ⊆ D T ( B 2 ) . This can b e rep hrased in terms of knowledge ordering : if B 1 ≤ k B 2 , th en D T ( B 1 ) ≤ k D T ( B 2 ) . In oth er words, the operator D T is also mo notone in terms of the knowledge o rdering ≤ k . Moore ’ s inference process ( B i ) λ i =0 is now an incr ea sing sequence in the kn owledge order ≤ k . It yields a least ﬁxpoint B T in the knowledge order (equiv alently , the greatest ﬁxpoint of D T in the subset o rder ⊆ ) . Every o ther ﬁx point of D T contains mo re knowledge than B T . The ﬁxpoint B T is th e intended b elief state associa ted with the theor y T o f monoto nic infer ence rules. In the gener al case of nonm onoto nic infer ence rules ( m > 0 , f or som e ru les), the operator D T may n ot be mon otone. The inference process con structed with D T may oscillate and never reach a ﬁxpoint, o r may reach an u nintend ed ﬁxpo int d ue to the fact that it may der i ve that a world is impossible on the b asis of an assumptio n ¬ K β i which is later with drawn. In such case, stated Moore , all we can do is to focus on possible-world sets th at “respect the intent of the nonmo notonic infere nce ru les” as expressed by a ﬁxp oint eq uation associated to T , rather th an being the result of a ﬁ xpoint construction . I n this way Moore arrived at his seman tics o f auto epistemic logic, summarize d in the fo llowing deﬁnition. 5 For arbitra ry theories T , the correspon ding assumption is that the re are no modal literal s K ϕ occurring positi vely in T . Reiter’ s Default Logic Is a Logic of Autoepistemic Reasoning 15 Deﬁnition 2 An auto epistemic expansion of a mo dal theory T over Σ is a possible- world set B ⊆ W such that B = D T ( B ) . W e agree with Moo re that the con dition of bein g a ﬁxpoin t of D T is a n ecessary condition for a be lief state to be a possible-world model of T . However , it is obvi- ously not a su fﬁcient on e, at least n ot in the auto theoremh ood view o n T . T his is obvious, as this semantics does not coincide with Moore’ s o w n ideas on the semantics of monoto nic i nfer ence rules. A cou nterexample is the following theory: T = { K P → P } . This theo ry consists of a unique m onoto nic infere nce r ule, alb eit a rather useless one as it says “ if P is a theor em then P ho lds ”. Acc ording to Moore’ s account of monotonic inference rules, the intended possible-world model of this theory is W = { ∅ , { P }} (we assume th at Σ = { P } ). Y et, T has two a utoepistemic expansion s, th e seco nd being the self-supporte d possible- world set {{ P }} . It is worth noting that th is theory is related to yet anothe r famous p roblematic statement in the theory of truth, namely the truth sayer : “This sentence is true. ” The truth value of this statement can be consistently assumed to be true, or equ ally well, to be false. The refore, in Krip ke’ s ( 1975) three-valued truth theor y , the truth value of the truth sayer is undetermined u . In case of the related autoepistemic theory { K P → P } , also Moor e’ s seman tics d oes not d etermine whether P is known or not. But in the au totheore mhood view , it is clear that P shou ld not b e known and this transpir es from Moore ’ s own explanations on monoto nic infer ence rules. 6 W e come back to the issue of self-suppo rted expansions in Section 5, where we explore alternative p erspectives on a utoepistemic propositions, in which such self-suppor ted expansions might be acceptable. The main que stion then is: Can we improve Moor e’ s method to build in ference processes in the pr esence of nonmono tonic inferen ce ru les in T ? I n this respect, the situation has chang ed si nce 1 984. The algebraic ﬁxpoint theory for nonmo noton e lat- tice operato rs (Denecker et al. , 2000), which we developed and then used to build the unifyin g sem antic framework f or default and autoe pistemic logics (Den ecker et al., 2003), gives us new tools f or d eﬁning ﬁxpo int construction s and ﬁxp oint equation s which can be applied to Moore’ s problem. W e illustrate now these tools in an in formal way and refer to the se intuitions later when we introduce major concepts for a formal treatment. Let us consider the theory: T = { P , ¬ K P → Q, K Q → Q } . Inform ally , the theory expresses that P ho lds, that if P is not a theorem then Q hold s, and that if Q is a th eorem, then Q hold s. Intuitively , it is clear wha t the mod el of this theory shou ld be: P is a theor em, hen ce the secon d for mula cannot b e used to der i ve 6 There does not seem to be an analogous strong argument why the truth sayer sentence should be fal se. Y et , Fitting (1997) proposed a reﬁne ment of Kripke’ s theory of truth in which truth is minimi zed and the truth saye r statement is false . For th is, he used the same wel l-founded ﬁxpoint con struction that we wil l use belo w to obtain a semantics that minimizes knowle dge for aut otheoremhood theories. 16 M. Denecker , V . W . Mar ek and M. T ruszczy ´ nski Q and neither can the tr uth say er prop osition K Q → Q . Therefo re, th e intend ed possible-world set is B T = { { P } , { P , Q }} , that is, P is entailed, Q is unknown. It is easily veriﬁed that B T is a ﬁxpoint of D T . Y et D T has a seco nd, u nintende d ﬁxpoint {{ P , Q }} which contain s more kno wledge than B T . This is a prob lem as i t is this un intended ﬁxpoint that is obtaine d by iterating D T starting with W . Th e reason for this m istake is that the second, nonmono tonic inferen ce ru le applies in the initial stage B 0 = W wh en ¬ K P holds. Later, when P is derived, the conclusion th at Q is a theorem continues to reprod uce itself thro ugh the third truth sayer rule. The pro blem above is that at eac h step and for ea ch world w an a ssumption is made of whether w is possible or impossible. Each such an assumption might be right or wrong. T hese assumptio ns are re vised by iterated application o f D T . In the context of monoto nic inf erence rules, the on ly wrong ass ump tions that might be made d uring the monoto nic ﬁxpo int constru ction starting in W are that some world is possible, while in fact it turns out to b e im possible. But these wrong assumptions can nev er lead to an erron eous application of an inf erence r ule: if a co ndition K ϕ of an inferen ce r ule holds when w is assumed to b e possible, th en it will still hold when w turns out to b e impossible. But in th e context of nonmonoton ic infe rence rules, an inference rule may ﬁre due to an erron eous assumption an d its conclusion m ight be maintained thro ugh a circu lar argument in all later iteration s. I n our scen ario, it is the initial assumption that worlds in wh ich P is false are p ossible that lead to the assump tion that worlds in which Q is false are im possible, and th is assumptio n is later rep roduce d by a circular reasoning using the third truth sayer proposition for Q . The solution to this prob lem is very simple: never make a ny unju stiﬁed a ssumption about th e status of a world . Start witho ut any assumption ab out the status of any worlds and only a ssign a speciﬁc status when certain. W e will elabor ate this idea in two steps. In the ﬁrst s tep, we illustrate this idea for a simpliﬁcation T ′ of T , in which the third axiom K Q → Q h as been deleted. 1. Initially , no world is known to be possible or impossible. At this stage, the truth value of the u nique modal literal K P in T ′ cannot be established. Y et, some things are clear . First, all worlds in which P is false, that is, ∅ and { Q } , are certain ly impossible since they vio late the ﬁrst formula, P , of T ′ . Second, the world { P, Q } is deﬁnitely possible since no matter whe ther P is a theorem or not, this world satisﬁes the two formu las o f T ′ . All this can be established with out makin g a single unsafe assumption . Thus, the on ly world ab out wh ich we are unc ertain at this stage is the world { P } in which Q is false. Due to the secon d axiom, this world is p ossible if P is known and impossible otherwise. 2. In the next pass, we ﬁrst use the knowledge that we gained in the previous step to re- ev aluate th e mod al literal K P . In particular, it c an be seen that P is true in all possible w orlds and in the last remain ing world of u nknown status, { P } . This suf ﬁces to establish that P is a theorem , that is, that K P is true. W ith this n ewly g ained inform ation, we can establish the status of the last world and see that { P } satisﬁes th e two axioms of T . Hence this world is possible. The construc tion stops here . T he next pass w ill no t ch ange anything , an d we obtain the po ssible-world set B T ′ = {{ P } , { P , Q }} . No w , let u s add the third axiom K Q → Q b ack and consider the full theory T . Reiter’ s Default Logic Is a Logic of Autoepistemic Reasoning 17 1. The ﬁrst step of the construction is identical to the one ab ove an d de termines the status for all worlds except { P } : { P, Q } is possible, and ∅ and { Q } are impo ssible. 2. As befor e, in the second pass, K P can be established to be tru e. The second modal literal K Q in T canno t be estab lished yet since its tr uth depen ds o n the status o f the world { P } . T he literal would be false if { P } is possible, and true o therwise. Thus the truth of the third axiom in { P } is still undetermined . W e are blocked here. 3. But there is a way o ut of the deadlo ck. So far , the method s to deter mine whether a world is p ossible or impossible were pe rfectly symmetrical. The solutio n lies in breaking this symmetry . In T , we ha ve a truth-sayers axiom: it is consistent to a ssume that Q is a theor em, and also to assume th at Q is n ot a theore m. In semantical ter ms, both assumptions on world { P } are co nsistent: if this world is cho sen po ssible, then K Q is false and a ll axiom s are satisﬁed in { P } ; if it is ch osen im possible, then K Q is true. Since we want to interp ret the mo dal operator as a theoremh ood modality , it is clear what assumption to make: th at Q is not a theo rem. W e should make the assumption o f igno rance an d take it that the world is possible (and Q is not a th eorem). Thus, we obtain again the possible world model B T = {{ P } , { P, Q }} . From these two examples, we can extract the con cepts necessary to fo rmalize the above informal reasoning pro cesses. At each step, we have partial info rmation about the status of worlds th at was gain ed so far . This n aturally for malizes as a 3-valued set o f worlds. W e call such a set a partial p ossible-world set . Formally , a p artial possible-world set B is a f unction B : W → { t , f , u } , where W is the collection o f all inte rpretation s. Standard , total po ssible-world sets can be vie wed as special cases, where t he only two v alues in the ran ge of the fun ction are t and f . In the context of a partial possible world B , we call a world w certainly possible if B ( w ) = t and po tentially possible if B ( w ) = t or u . Like wise, w e call a world w certainly impossible if B ( w ) = f and p otentially impossible if B ( w ) = f or u . If B ( w ) = u , w is po tentially p ossible and potentially impossible. W e deﬁne C P ( B ) as the set of certainly possible worlds of B , P P ( B ) as the set of p otentially possible worlds, and likewise, C I ( B ) and P I ( B ) as the sets of certainly impo ssible, respectively potentially impossible worlds of B . At each in ference step B i → B i +1 , we evaluated the p roposition s of T in one or more u nknown worlds w , given the partial information a vailable in B i . When all propo sitions of T turned out to be tru e in w , w was derived to be possible; if som e ev aluated to false, w was inferr ed to be impossible. T o cap ture this formally , we need a thr ee-valued truth fun ction to ev aluate theo ries in the context of a world w , the o ne we are examinin g, and a partial possible-world set B . The value of this truth fun ction on a th eory T , de noted as | T | B ,w , is selected from { t , f , u } . There are som e obvious proper ties that this fu nction should satis fy . 1. The th ree-valued truth function sho uld co incide with the standard (implicit) truth function for modal logic in total po ssible-world sets. In p articular, when B is a total possible-world set, that is, B has n o unk nown worlds, then | T | B ,w should be true precisely when B , w | = T (and false, otherwise). 18 M. Denecker , V . W . Mar ek and M. T ruszczy ´ nski 2. Th e three-valued truth function shou ld b e mono tone with respect to the pr e cision of the partial po ssible-world sets. A mor e pr ecise partial possible-world set is one with fewer (with respect to inclusion) unknown worlds. The in tuition pre sented in (2) ca n be form alized as follows. W e deﬁn e B ≤ p B ′ if B ( w ) ≤ p B ′ ( w ) , wh ere the latter ( partial) orde r ≤ p on tru th values is the one generated by u ≤ p t and u ≤ p f . A th ree-valued truth f unction | T | B ,w is monotone in B if B ′ ≤ p B ′′ implies that | T | B ′ ,w ≤ p | T | B ′′ ,w . In p articular, if | T | B ,w is mo notone in B and B is a to tal po ssible- world set such that B ′ ≤ p B , then | T | B ′ ,w = t implies that B , w | = T , and | T | B ′ ,w = f implies that B , w 6| = T . Designing such a three-valued truth function is routine, th e prob lem is that the re is more than one sen sible solution. On e appr oach, origin ally p roposed by De necker et al. (1998), extends Kleene’ s (1952) three-valued truth evaluation to modal logic. Deﬁnition 3 F o r a formula ϕ , world w ∈ W and partial possible- world set B , we deﬁne | ϕ | B ,w using the standard Kleene truth evaluation rules of thr e e-valued logic augmen ted with on e additional rule: | K ϕ | B ,w =      f if | ϕ | B ,w ′ = f , for some w ′ such that B ( w ′ ) = t t if | ϕ | B ,w ′ = t , for all w ′ such that B ( w ′ ) = t or u u otherwise. F or a theory T , we deﬁne | T | B ,w in the standar d way of thr ee-valued logic: | T | B ,w =    f if | ϕ | B ,w = f , for some ϕ ∈ T t if | ϕ | B ,w = t , for all ϕ ∈ T u otherwise. T o illustrate th e use of this truth fun ction, let us evaluate the form ula K ϕ , wher e ϕ is ob jectiv e, in the co ntext of a partial p ossible-world set B and an arb itrary world w . W e h ave | K ϕ | B ,w = t if P P ( B ) | = K ϕ , that is, if all po tentially po ssible worlds satisfy ϕ . Likewise, we h ave | K ϕ | B ,w = f if C P ( B ) 6| = K ϕ , that is, at least one certainly p ossible world vio lates ϕ . L et B be a m ore pr ecise total p ossible world set; that is, B ≤ p B or equ i valently , P P ( B ) ⊇ B ⊇ C P ( B ) . Then, obviously , if K ϕ holds true in B , the formu la is true in B , and if K ϕ is false in B then it is false in B as well. In general this truth function is conservative (that is, ≤ p -mono tone) in the sense that if a f ormula evaluates to true or false in some partial possible-world set, then it has the same tru th value in every more p recise po ssible-world set thus, in par ticular , in e very total possible-world set B such that B ≤ p B . It is easy to see (and it was proven formally by Denecker et al. (2003)) that this truth function satisﬁes the two desider ata listed abov e. W e also note that this is not the only reasonab le w ay in which the three -valued truth fun ction can be deﬁned. W e will come back on this topic in Section 4.5. W e now revie w th e fram ew ork of semantics of autoep istemic r easoning we in- troduced in ou r study of the relationship between the default logic o f Reiter a nd th e autoepistemic logic of Moore (Denecker et al., 200 3). W e listed th ese seman tics in Reiter’ s Default Logic Is a Logic of Autoepistemic Reasoning 19 the previous section. All seman tics in the framework require that a (pa rtial) possible- world mo del B of an autoep istemic theory be justiﬁed by some ty pe of an inference process: B 0 → B 1 → . . . → B n = B . At each step i , mod al liter als K ϕ appearin g in T are ev aluated in B i . Whe n su ch literals are d erived to be tru e or false, this might lead to further inferen ces in B i +1 . T aking the semantic point of view , we understan d an inference here as a step in which some worlds of undeterm ined statu s are derived to b e possible and some o thers are derived im possible. Dialects of au toepistemic logic, and so of default logic, too, dif fer from each other in th e n ature o f the d eriv ation step B i → B i +1 , and in initial assumptions B 0 they make. Some dialects make no initial ass ump tions at all; in some others making certain initial “guesses” is allowed. In this way , we obtain autoe pistemic log ics of different degrees of gr ound edness . In th e following section s, we describe inferenc e processes underly ing each of the four seman tics in the framew ork described in Section 3. Finally , we link the above concepts with the algebraic lattice theor etic concepts sketched in the previous sectio n an d used in the seman tic fram ew ork of Denec ker et al. (2003). The re, the different semantics of an auto epistemic theor y T emerged as dif- ferent types of ﬁx points of a ≤ p -mono tone o perator D T on the bilattice co nsisting of arbitrary pa irs ( B , B ′ ) of possible-world sets. The partial possible-world sets B correspo nd to the consistent pairs ( P P ( B ) , C P ( B )) in this bilattice; a pair ( B, B ′ ) is consistent if B ⊇ B ′ , that is, certainly possible w orlds are potentially po ssible. Inco n- sistent pairs gi ve rise to possible-world sets that in addition to tru th values t , f and u require the four th one i for “inco nsistency”. The Kleene truth functio n deﬁned above can be extended easily to a four-v alued truth function on the full bilattice. The operator D T on that bilattice was then deﬁned as follows: D T ( B ) = B ′ , if for every w ∈ W , B ′ ( w ) = | T | B ,w . W e obser ve that this oper ator maps partial po ssible-world sets into p artial po ssible- world sets and that it coincides with Moore’ s deri vation op erator D T when applied on total possible-world sets. In the seq uel, we will often repr esent a partial possible-world set B in its bilattice representatio n, as the pair ( P P ( B ) , C P ( B )) of resp ectiv ely pote ntially possible and certainly possible worlds. For example, the least precise par tial possible-world set ⊥ p for Σ = { P , Q } will be written as ( {∅ , { P } , { Q } , { P , Q }} , ∅ ) : all worlds ar e potentially possible; no world is certainly possible. W e will now discuss the four seman tics d iscussed above that deﬁne different di- alects of autoepistemic reasoning . 4.1 The Kripke-Kleene semantics This semantics is a direct for malization of the discussion above. W e are given a ﬁnite modal theory T (we adopt the ass ump tion of ﬁniteness to simplify pre sentation, b ut it can be omitted). A Kripke-Kleene infer enc e pr ocess is a sequen ce B 0 → . . . → B n 20 M. Denecker , V . W . Mar ek and M. T ruszczy ´ nski of partial possible-world sets such that: 1. B 0 is the totally u nknown partial possible- world set. That is, for every w ∈ W , B 0 ( w ) = u . W e de note this p artial possible-world set by ⊥ p . This ch oice o f the starting point indicates that Kripke-Kleen e infer ence pro cess do es n ot make any initial assumptions. 2. For each i = 0 , . . . , n − 1 , there is a set o f worlds U such that for every w ∈ U , B i ( w ) = u , | T | B i ,w 6 = u and B i +1 ( w ) = | T | B i ,w , an d for e very w / ∈ U , B i ( w ) = B i +1 ( w ) . Thu s, in each step of the deri vation the s tatus of the w orld s tha t are certainly possible and certainly impossible does not chang e. All that can chang e is the status of some worlds of unknown status ( worlds, that ar e po tentially p ossible and p otentially impossible). T his set is deno ted b y U ab ove. I t is not necessary tha t U contain s all worlds that are un known in B i . In the derivation, w orld s in U become certainly possible or certainly impo ssible, depe nding on how the theo ry T ev aluates in them. If for such a potentially p ossible world w ∈ U , | T | B i ,w = t , w becom es certainly possible. I f | T | B i ,w = f , w becomes certainly impossible. Other wise, the status o f w does not change. As such a deriv ation starts from the least precise, hence assump tion- free, partial possible-world set ⊥ p , all these deriv ations are assumption-free. 3. Th e halting cond ition: n o more infere nces can be made once we reach the state B n . Here this means that for each unknown w ∈ W , | T | B n ,w = u . The pro cess term inates. This precise deﬁn ition form alizes an d genera lizes the inform al co nstruction we presented in the previous section. When applied to the theory we consider ed there, T ′ = { P, ¬ K P → Q } , one Kripke-Klee ne inference proce ss th at migh t be pro duced is (we re present her e worlds, or interpretation s, as sets of atoms they satisfy , and partial possible-world sets B as pairs ( P P ( B ) , C P ( B )) ): ⊥ p → B 1 = ( {∅ , { P } , { P , Q } } , ∅ ) { Q } certainly impossible → B 2 = ( {{ P } , { P , Q }} , ∅ ) ∅ certainly impossible → B 3 = ( {{ P } , { P , Q }} , {{ P , Q }} ) { P , Q } certa inly possible → B 4 = ( {{ P } , { P , Q }} , {{ P } , { P, Q }} ) { P } certain ly possible . The ﬁrst deriv ation can b e made since | P ∧ ( ¬ K P → P ) | ⊥ p ,w = f , for w = { Q } (in fact, f or every w , in which P is false). Th e second der iv a tion is justiﬁed similarly as the ﬁrst o ne. The third de riv ation fo llows as | P ∧ ( ¬ K P → Q ) | B 2 ,w = t , for w = { P, Q } , an d the f orth on e as | P ∧ ( ¬ K P → Q ) | B 3 ,w = t , for w = { P } . Let us explain o ne m ore detail of the last of these claims. Here, | P | B 3 ,w = t h olds becau se P holds in w = { P } . Mo reover , | K P | B 3 ,w = t as P h olds in e very world th at is potentially possible in B 3 . Thu s, |¬ K P | B 3 ,w = f a nd so indeed , |¬ K P → Q | B 3 ,w = t . The shortest derivation sequence that corr esponds exactly to the infor mal co n- struction of the previous section is: ⊥ p → ( {{ P } , { P , Q }} , {{ P , Q }} ) → ( {{ P } , { P , Q }} , {{ P } , { P, Q }} ) . The fact t hat there may be multiple Kripke-Kleene infer ences pro cesses is not a prob - lem as all of them end in the same partial possible-world. Reiter’ s Default Logic Is a Logic of Autoepistemic Reasoning 21 Proposition 1 F or every modal theory T , all Kripke-Kleene i nference pr ocesses con- ver ge to th e same p artial possible- world set, which is the ≤ p -least ﬁxp oint of the operator D T . W e call th is special p artial possible-world set the Kripke-Kleene extension of the modal theory T . While the Kripke-Kleene con struction is an intuitively sound co nstruction, it has an obvious disadvantage: in general, its termina ting partial belief state may no t match the intend ed belief state even if T consists of “mon otonic” inferen ce rules (no negated modal ato ms in th e antec edents of f ormulas o f th e fo rm (1) ). An examp le where this happen s i s the truth sayer theo ry: T = { K P → P } . It consists of a single mono tonic inferen ce rule, an d its inten ded total p ossible-world set is {{∅ , { P }} , which in the cur rent ( P P , C P ) notation correspo nds to ( {{∅ , { P }} , { {∅ , { P }} ) . Howe ver , the one and on ly Kripke-Kleene constru ction is ⊥ p → ( {∅ , { P }} , {{ P }} ) . Then the co nstruction halts. No more Krip ke-Kleene infe rences on th e status of worlds can be made and the intended possible-world set is not reached. W e conclud e with a historical note. T he name Kripke-Kleene semantics was used for the ﬁrst time in the context o f the semantics of logic program s by Fitting (1985). Fitting built o n ideas in an earlier work by Klee ne (1952), and on Kripke’ s (19 75) theory of truth, where Kripke discussed how to handle the liar parad ox. 4.2 Moore ’ s autoepistemic logic Moore’ s autoep istemic logic has a simple formaliza tion in our fram ew ork. A possible- world set B is an autoepistemic expan sion o f T if th ere is a on e-step deriv ation for it: B 0 → B 1 , where B 0 = B 1 = B . Clearly , here we allow the inferenc e p rocess to make initial assumptions. Moreover, in th e deriv ation step B 0 → B 1 we simply verify that we made no incorr ect assumption s and that n o add itional infer ences can be drawn. The inference (more accurately here, the veriﬁcation) process w orks as follows: 1. A world w is derived to be possible if B 0 , w | = T . 2. A world w is derived to be impo ssible if B 0 , w 6| = T . Thus, for mally , B 1 = { w | B 0 , w | = T } = D T ( B 0 ) . Consequen tly , the limits of this deriv ation proc ess are indeed precisely the ﬁx points of the Mo ore’ s o perator D T (we stress that we talk here only about total possible-world sets). Since D T coincides with D T on total possible- world sets, all a utoepistemic ex- pansions are ﬁxpoints of D T . Thus, we hav e the following result. 22 M. Denecker , V . W . Mar ek and M. T ruszczy ´ nski Proposition 2 The Kripke-Kleene e xtension is less p r ecise than any oth er autoepis- temic expansion of T . If the Kripke-Kleene extension is total, then it is the un ique autoepistemic expansion of T . The weakn ess of Moore’ s logic from the point o f view of modeling th e auto theo- remhoo d vie w has been argued above. In Section 5 , we w ill discuss another interp re- tation of autoepistemic logic in which his semantics may be more adequ ate. 4.3 The well-fou nded knowledge deriv ation The pro blem with the Krip ke-Kleene deriv ation is that it treats igno rance and knowl- edge in the same way . Ign orance is reﬂected b y the presence of possible worlds. Knowledge is reﬂected by th e presence of im possible worlds. In the Kr ipke-Kleene deriv ation, both possible and imp ossible worlds a re derived in a symmetr ic way , by ev aluating the theory T in the context of a w orld w , given the partial knowledge B . What we would like to do is to impose ignora nce as a default. That a world is possible should not have to b e d erived. A world shou ld be p ossible u nless we c an show that it is impossible. In other words, we n eed to impose a principle of maximiz- ing ignorance , or equivalently , minimizing knowledge . Un der such a principle, it is obvious that the possible-world set {{ P }} cann ot be a model of the t ruth sayer theory T = { K P → P } . It does n ot min imize knowledge while the other candidate for a model, the possible-world set {∅ , { P }} , d oes. T o reﬁne the Kripke-Kleene con struction of knowledge, we need an additio nal deriv ation step that allows us to in troduce the assumption of ignoran ce. In tuitively , in such a deriv ation step, we con sider a set U o f unknown w orlds, which ar e turned into certainly possible worlds to maximize ignorance . Formally , a well-foun ded in fer ence pr ocess is a deriv ation process B 0 → . . . → B n that satisﬁes the same conditio ns as a Kripke-Klee ne inference proc ess except that some derivation steps B i → B i +1 may also be justiﬁed as f ollows (by the maximize- ignorance principle): MI: Th ere is a set U o f w orlds such that B i +1 ( w ) = B i ( w ) fo r all w 6∈ U and for all w ∈ U , B i ( w ) = u , B i +1 ( w ) = t and | T | B i +1 ,w = t . In other word s, in suc h a step we pick a set U of unknown words, assume that they are ce rtainly possible, and verify that this assumption was justiﬁed, tha t is, unde r the increased lev el o f ignoran ce, all of them turn out to be certainly possible. T o put it yet differently , we select a set U o f unknown worlds, fo r which it is co nsistent to a ssume that they are certainly possible, an d we turn them into certainly possible worlds (increasing our ignorance). By analogy with the notion of an unf ounded set of atoms (V an Gelder et al., 1991), we call the set of worlds U , with respect to which the maximize-ig noran ce princip le applies at the partial belief state B i , an unfounded set for B i . W e also n ote that the halting conditio n of a well- found ed infere nce process is stronger than that for a Kr ipke-Kleene process. This means that for each unknown world w of B n , | T | B n ,w = u and in ad dition, B n does n ot allow a MI infe rence step, that is, it has no non-emp ty unfounded s et. There are two properties of well-f ound ed infere nce p rocesses tha t are worth n ot- ing. Reiter’ s Default Logic Is a Logic of Autoepistemic Reasoning 23 Proposition 3 All well-founded infer ence pr ocesses co n ver ge to the same (partial) possible-world set. This pr operty gives r ise to the well-foun ded extension of the modal th eory T de- ﬁned as the limit o f an y well-foun ded inf erence proce ss. This lim it can be shown to coincide with the well-fou nded ﬁxpo int of D T , that is, the ≤ p -least ﬁxpoint of the operator S D T deﬁned in the previous section. Another importan t property concern s theories with no positive occurrences of the modal operator (for instance, th eories co nsisting of formulas (1 ) with no modal literals ¬ K β j in the anteceden t). Proposition 4 If T contain s only negative occurr e nces o f th e mo dal o perator , then the well-founded e xtension is the ≤ k -least ﬁxpoint of D T . This pro perty shows that the well-found ed extension semantics has all ke y proper- ties of th e desired semantics of sets of “monoto nic inference rules. ” Let us revisit the truth sayer theory : T = { K P → P } . The Kripke-Kleene construction is ⊥ p → ( {∅ , { P }} , {{ P }} ) . The inf erence that { P } is possible is also sanctioned und er the r ules of the well- found ed inference process. Howev er, while th ere is no Kr ipke-Kleene deriv ation that applies now , th e maxim ize-ignor ance pr inciple d oes apply and the well-fou nded in fer- ence process can co ntinue. Namely , in the belief state gi ven by ( {∅ , { P }} , {{ P }} ) , there is one world of u nknown status (neither certain ly impossible, no r certainly po s- sible): ∅ . T aking U = {∅ } and ap plying the max imize-igno rance princ iple to U , we see that the well- found ed inferenc e process extends and yields ( {∅ , { P }} , {∅ , { P }} ) . This possible-world set is to tal and so, necessarily , the limit o f the proc ess. Th us, this (total) possible-world set { ∅ , { P }} is th e well- found ed extension o f th e theory { K P → P } . The well-founded extension is total not only for monotonic theories. For instance, let us consider the theory : T = { K P ↔ Q } or equiv alently , { K P → Q, ¬ K P → ¬ Q } . Intuitively , there is no thing known ab out P , hence Q sho uld be false. The un ique Kripke-Kleene in ference process e nds wh ere it starts, that is, w ith ⊥ p . Inde ed, when K P is un known, no certainly possible or c ertainly impossible world s can be derived. Howe ver , the p ossible-world set U = {∅ , { P }} is un found ed with respect to ⊥ p . Indeed , if bo th worlds are assumed po ssible, K P e valuates to false, and b oth worlds satisfy T . Thus, in the well-fo unded der iv a tion we can establish that an d th en, in the next two steps, we can derive the impo ssibility of the two remain ing unkn own worlds, ﬁrst of { Q } and then of { P, Q } . Th is yields the following well-f ounded in ference process: ⊥ p → B 1 = ( {∅ , { P } , { Q } , { P , Q }} , {∅ , { P }} ) → B 2 = ( {∅ , { P } , { P , Q }} , {∅ , { P }} ) → B 3 = ( {∅ , { P }} , { ∅ , { P }} ) . 24 M. Denecker , V . W . Mar ek and M. T ruszczy ´ nski In other cases, th e well-fo unded extension is a partial p ossible-world set. An ex- ample is the theory: {¬ K P → Q , ¬ K Q → P }} . In this case, ther e is on ly one well-foun ded inference pr ocess, which der iv es that { P , Q } is a certainly possible world and derives no certain ly impossible worlds. Tha t is, the well-foun ded e xtension is: ( { ∅ , { P } , { Q } , { P , Q }} , {{ P , Q } } ) . 4.4 Stable possible-world sets W e r ecall th at a partial po ssible-world set B correspo nds to the pair of to tal po ssible- worlds sets: ( P P ( B ) , C P ( B )) , wher e P P ( B ) is the set of potentially possible worlds and C P ( B ) is the set of certainly po ssible worlds. W e now deﬁn e a stable d erivation for a po ssible-world set B as a sequence of partial belief states of the form: ( W , B ) → ( P P 1 , B ) → . . . → ( P P n − 1 , B ) → ( P P n , B ) , where: 1. P P n = B 2. F or e very i = 0 , . . . , n − 1 , and f or every w ∈ P P i \ P P i +1 , | T | ( P P i ,B ) ,w = f . That is, som e worlds w in wh ich T is false with respect to B i = ( P P i , B ) become certainly impossible and are removed from P P i to form P P i +1 . 3. Halting condition: fo r ev ery w ∈ P P n , | T | ( P P n ,B ) ,w = t or u . If a total belief set B has a stable deriv ation then we call B a stab le extension . This concept captures the idea of the Reiter’ s extension of a default theory . W e recall that an infer ence r ule (1) ev aluates to false in world w with r espect to ( P P i , B ) if w 6| = γ , P P i | = K α i , fo r all i , 0 ≤ i ≤ n , and B 6| = K β j , fo r all j , 0 ≤ j ≤ m . W e see h ere an asymmetric treatme nt of prerequ isites α i and justiﬁcations β j which are ev a luated in two different po ssible world sets. The same feature shows up , not co incidentally , in Reiter ’ s d eﬁnition of extension of a d efault theory . The intuition un derlying a stable deriv ation comes fro m a different imp lementation of the idea that igno rance does n ot need to be justiﬁed an d that only k nowledge must be justiﬁed. In a partial possible- world set B , the compone nt sets P P ( B ) and C P ( B ) have different roles. Since P P ( B ) determin es the certainly impossible worlds, this is the possible- world set that determin es wh at is deﬁnitely k nown. On the other hand th e set C P ( B ) of certain ly possible worlds d etermines what is deﬁnitely not known by B . A stab le d eriv ation for B is a justiﬁcation for each impo ssible world of B (each world is initially po tentially possible but e ventually determin ed n ot to be in B , th at is, determin ed imp ossible in B ). The key poin t is that this justiﬁcation may use th e assumption of the igno rance in B . By ﬁx ing C P ( B i ) to be B , it takes th e igno rance in B f or granted . What is justiﬁed in a stab le inferen ce process is the impossible worlds of B , not the possible worlds. W e saw abov e that the theory {¬ K P → Q , ¬ K Q → P } Reiter’ s Default Logic Is a Logic of Autoepistemic Reasoning 25 has a partial well-foun ded exten sion. It turns out that it has two stable extensions {{ P } , { P , Q }} and {{ Q } , { P, Q }} . For instance, the following stable derivation re- constructs B = { { P } , { P , Q }} . Note that in any partial p ossible-world set ( · , B ) (that is, wher e the worlds of B a re certain ly po ssible), K Q e valuates to false. I n all such c ases, T ev aluates to false in a ny world in which P is false. Hence we have the following v ery short stable deriv ation: ( W , B ) → ( B , B ) . W e now hav e two key resu lts. The ﬁrst one links up well-fou nded an d stable extensions. Proposition 5 If the well-foun ded extension is a total po ssible-world set, it is the unique stable extension. The second resu lt shows th at indeed , the K onolige’ s translation works if the se- mantics of default logic of Reiter and the au toepistemic lo gic of Moore are cor rectly aligned. Here we state the result for the most important case of default extensions and stable extensions, b ut it extends, as we noted earlier , to all semantics we considered. Proposition 6 F or every d efault theo ry ∆ , B is a n extension o f ∆ if a nd on ly B is a stable extension of Kon (∆) . 4.5 Discussion W e have obtained a framew ork with fou r different semantics. This fra mew ork is pa - rameterized by the truth functio n. W e have conce ntrated on the Kleene tru th function but o ther viable cho ices exist. On e is super-valuation (van Fraassen, 1 966) which deﬁnes | T | B ,w in term s of the ev aluation of T in all possible world sets B ≥ P B approx imated by B . In particular , | T | B ,w = M in ≤ p {| T | B ,w | B ≤ p B } . In this way we obtain another instance of the fram ew ork, the family of ultimate se- mantics (Denecker et al., 2004). For many theories, th e correspon ding seman tics of the two families coincide but ultim ate sem antics are sometimes m ore p recise. An example is the theo ry { K P ∨ ¬ K P → P } . It’ s Kripke-Kleene and well-fo unded extension is the p artial possible world set ( {∅ , { P }} , {{ P }} ) and there are n o stable extensions. But t he premise K P ∨ ¬ K P is a propositional tau tology , mak ing | T | ⊥ p ,w true if w | = P and false other wise. As a consequ ence, the ultimate Kripke-Kleene, well-foun ded and uniq ue stable e xtension is {{ P }} . For a scientist interested in the form al study of the informal semantics of a certain type of (info rmal) pro positions th is div ersity is troubling . In deed, wha t is th en the nature of au toepistemic reason ing, and which of the semantics that we deﬁned an d that can b e d eﬁned by mean s of other truth fu nctions is the “corr ect” one? It is necessary to bring some order to this div ersity . In th e au totheor emhood v iew , the form al semantics should capture the infor mation content of an autoepistemic theory T tha t contain s pro positions referrin g to T ’ s o wn informa tion co ntent; the semantics shou ld determine whether a world is possible or 26 M. Denecker , V . W . Mar ek and M. T ruszczy ´ nski impossible, or equivalently , wheth er a fo rmula is or is not entailed b y T . As we saw , Moo re’ s semantics of expansion s an d the Kripke-Kleene extension semantics are arguably less suited in the case of m onoton ic infer ence rules with cyclic dependencies (cf. the truth sayer theo ry). This leaves us with four contend ers on ly: the we ll-found ed and the stable extension semantics and their ultimate v ersions. All employ a techn ique to m aximize igno rance and co rrectly han dle auto epistemic theories with mono tonic inference rules. Which of these semantics is to be preferred ? Let us ﬁrst conside r the choice of the truth fu nction. The semantics based on the Kle ene truth fu nction a nd the one s indu ced by super-valuation make different trade-offs: the higher precision of the ultimate semantics, which is g ood, co mes at the price of higher co mplexity o f reasoning , wh ich is bad (Denecker et al., 2004). When there is a trade-off b etween different desired character istics, there is per deﬁ- nition no best solu tion. Y et, when loo king clo ser , th e qu estion of the choice between these two truth functions turns out to be largely academic and without much practical relev ance. There are classes of autoep istemic theo ries for which the Kleene and the super-valuation truth fun ctions coin cide, and hence, so do the seman tics they induce. Denecker et al. (2004, Prop osition 6 .14) provide an example of such a class. Even more impo rtantly , the seman tics induced by Kleene’ s truth functio n and by sup er- valuation differ only wh en case-based reasoning o n modal litera ls is n ecessary to make certain inference s. Except for our own artiﬁcial examp les intro duced to illustrate th e formal difference between both semantics (Denecker et al., 2 004), we are not aware of any reasonab le autoepistemic or default theor y in the literatur e where suc h re asoning would be necessary . They may exist, but if they do, they will con stitute an insignif- icant fring e. The take-home message here is that in a ll practical applicatio ns tha t we are aware of, the K leene truth fu nction sufﬁces and there is no need to pay f or the in- creased complexity of super -valuation. T his limits th e num ber o f semantics still in the runnin g to only two. Of the remainin g two, the most faithful f ormalization of the a u- totheorem hood view seems to be the well-f ounde d extension sema ntics. As we v iew a theory as a set of inference rules, the constru ction o f the well-fou nded extension formalizes the process of the app lication of the in ference ru les more dir ectly than the construction of the stable extension semantics. Nev ertheless, there are some com monsense arguments fo r n ot overemphasizing the differences b etween these semantics. First, we should keep in min d that th eories of inter est are tho se that ar e developed by human exper ts, and hence, are mean ing- ful to them. What are the meanin gful theo ries in th e au totheorem hood ? Not ev ery syntactically corr ect modal theory makes sense in this view . “Paradox ical” theories such as th e liar theor y T liar can simply n ot be ascrib ed an inform ation content in a consistent manner and are no t a sen sible theory in the autoth eoremho od vie w . For theories T viewed as sets of inference rules, the inference process associated with the theory sho uld be ab le to determine the po ssibility of ea ch world an d h ence, for each propo sition, wh ether it is a theorem or n ot of T . In particular, this is the ca se when the well-f ounde d extension is to tal. W e view th eories with theorems that are subject to ambiguity and speculation with suspicion. And so, methodologies based on the au- totheorem hood vie w will naturally tend to produ ce theories with a total well- found ed extension. From a practical point of view , the presence of a u nique, constructible state of belief for an au toepistemic theory is a great advantage. For instance, u n- less the po lynomial hierar chy collapses, for such the ories the task to construct th e Reiter’ s Default Logic Is a Logic of Autoepistemic Reasoning 27 well-foun ded extension and s o, also the unique stable expansion, is easier than that of computin g a stable expansion of an ar bitrary th eory or to d etermine that non e exists. Further, for such theor ies, skeptical and cred ulous reason ing (with r espect to stable extension) coincid e and are easi er, again assuming that the polyn omial hierarch y do es not collapse, than they are in the general case. For all the reason s ab ove, a hu man expert using autoep istemic logic in the autothe- oremho od view , will be n aturally inclined to build an auto epistemic knowledge b ase with a well-founded extension th at is total. When t he well-founded sem antics indu ced by the K leene truth fu nction is tota l, the four seman tics — the two stable sema ntics and the two well-founded semantics — co incide! I t is so, in particular for the class of theories built of formulas (1) with no recur sion through negated modal literals (the so-called stratiﬁed theories (Gelfon d, 1987)). Hen ce, such a methodo logy could be enforce d by impo sing syntactical conditions. All these arguments n otwithstandin g, the fact is that ma ny de fault theories dis- cussed in the literatur e o r a rising in p ractical settings do not have a u nique well- found ed extension and that the stable and well-fo unded extension seman tics do not coincide 7 . W e have seen it above in the Nixon Diamond example. Mor e gen erally , it is the case whenever the theory includes conﬂicting d efaults and no gu idance on how to resolve conﬂicts. Such conﬂicts may arise inadverten tly for the prog rammer, in which case a goo d strategy seems to be to analyze th e conﬂicts (p otentially by study - ing the stable extensions) and to reﬁne the theory by building in conﬂict-resolu tion in the cond itions of d efault rules. Otherwise, wh en conﬂicts ar e a d eliberate dec ision of the progr ammer who inde ed do es not want to o ffer rules to resolve conﬂicts, all we can do is to accept each of the m ultiple stable extensions a s a possible model of the theory and also accept that none of them is in any way pre ferred to others. In conclusion, rath er than prono uncing a stro ng preferen ce for th e well-found ed extension o ver stable extensions or vice versa, what we w ant to point out is the attrac- ti ve featu res of theories fo r which these two semantics coincide, and advantages o f methodo logies that lead to such theor ies. 5 A utoepistemic Logics in a Broader Land scape In th is section, we u se th e newly gained in sights on th e nature of a utoepistemic reaso n- ing to clarify certain aspects of autoepistemic logic and its position in the spectrum of logics, in p articular in the families of logics of nonm onoton ic reasoning and classical modal logics. A good start for this discussion is Moor e’ s “second” view on autoepistemic logic. Later in his paper, wh en dev elopin g the expansion seman tics, Moo re rephrased his views on auto epistemic reasoning in terms of the backgro und concept of an auto epis- temic agent . Such an agent is assumed to be ideally rational and ha ve the powers of perfect intro spection. An autoepistemic theory T is viewed as a set o f propo sitions 7 Some researchers belie ve that multipl e extensi ons are needed for reason ing in the context of incomplete kno wledge. Our point of vie w is differe nt. The essence of incomple te knowledg e is that dif ferent states of af fairs are possible. Therefore, the natu ral — and standar d — represen tation of a belie f state with incomple te knowled ge is by one possible-worl d set with multiple possible worlds, and not by multipl e possible-w orld sets, which to us would reﬂect the state of m ind of an agent that does not know what to belie ve. 28 M. Denecker , V . W . Mar ek and M. T ruszczy ´ nski that are known by this agent. Mod al literals K ϕ in T now mean “I (that is, t he agent) know ϕ ” . The most impo rtant assumption , the one on which this info rmal view of autoepistemic logic largely rests, is that th e agent’ s th eory T represen ts all the agent knows (Levesque , 1 990) or , in Moor e’ s ter minolog y , wh at the age nt knows is gr ou nded in the theory . W e will call this implicit assumption the All I Know Assumption. W ithout the A ll I Know Assumption , the theo ry T would be just a list of believed introspective pr opositions. The state of belief of the ag ent might then correspo nd to any possible-world set B such that B | = K ϕ , for each ϕ ∈ T ( where B | = K ϕ if for all w ∈ B , B , w | = ϕ ). But in many such possible- world sets B , the agent would know much more than what can be deriv ed from T . In this setting, nonmo notonic inference rules such as K A ( x ) ∧ ¬ K ¬ B ( x ) → B ( x ) would not be useful for default reasoning since co nclusions drawn from them would n ot be d eriv ed from the info rmation given in T . So the pro blem is to mod el the All I Know Assumptio n in the semantics. Moo re implemented this condition by imposing that for any mod el B , if B , w | = T , then w is possible accordin g to B , i.e ., w ∈ B . Combin ing both conditions, models that satisfy the All I Know Assumption are ﬁxpoints of D T , that is Moore’ s expansions. Moore’ s e xpansion semantics does not violate the assumptions underlyin g the au- toepistemic age nt view . Ex pansions do corr espond to b elief states o f an ideally ratio - nal, fully introspective agent that believes all axioms in T and, in a sense, doe s not believe mo re that what he can justify fr om T . But the same can be said for the a utothe- oremho od view as implemen ted in the well-founded and stable extension semantics. W e may identify the theory with what the ag ent kno ws, and th e theoremhood operator with the ag ent’ s epistemic o perator K , and see the well-fou nded extension ( if it is total) or stable e xtension s as representing belief states of an agent that can be justiﬁed from T . As we stated in the pre viou s section, Moore’ s expansion sema ntics does not for- malize the autotheoremh ood view , b ut it formalizes a dialect of autoep istemic reason - ing, based on an autoepistemic agent that accepts states of belief with a weaker notion of justiﬁ cation, allowing f or s elf-supp orting states of belief. While not appro priate f or modeling default re asoning, the semantics may work well in other d omains. I ndeed, humans som etimes do hold self-suppo rting belief s. For examp le, self-conﬁd ence, or lack of self-co nﬁdence of ten are to som e extend self-supported . Believing in o ne’ s own qu alities makes one perfor m better . And a go od perfor mance supp orts self- conﬁdenc e (and self-esteem). App lied to a scientist, this loop mig ht by represen ted by the theory consisting of the following for mulas: K ( I C anS ol v e H ardP r obl ems ) → H appy H appy → Rel axed Rel axed → I C anS ol v eH ar dP robl ems . Along similar lines, the placebo effect is a medically well-researched fact often at- tributed to self-supporting beliefs. The self-supportin g aspect under lying the placebo- effect can be described by the the ory consisting of the rules: K ( I GetB etter ) → O ptimisti c Opti mistic → I GetB etter . T aking a placeb o just ﬂips the patients into the belief that they are getting b etter . In this form of auto epistemic reasoning of an agent, self-su pportin g beliefs are justiﬁed Reiter’ s Default Logic Is a Logic of Autoepistemic Reasoning 29 and Moo re’ s expansion seman tics, difﬁcult to reconcile with the notio n o f deriv ation and theorem , may be suitable. There are yet other instances of the All I Know Assumption in the autoepistem ic agent view . For example, let us co nsider the theory T = { K P } . I n the autoth eo- remhoo d v iew , this th eory is clearly incon sistent, for there is no way this theo ry can prove P . The situation is not so clear-cut in the agent view . W e see no obvious ar- gument why the agent co uld n ot be in a state of belief in which he believes P and its conseq uences and noth ing more than th at. In fact, the logic of minima l knowledge (Halpern and Moses, 1984) intro duced as a v ariant of autoe pistemic logic accepts this state of belief for T . What our discu ssion shows is that the All I Know Assumption in Moore’ s au - toepistemic agen t view is a rathe r vague intuition, which can b e worked out in more than one way , yield ing different fo rmalization s and different dialects. It may explain why Moore built a semantics that did not satisfy his own ﬁrst intuition s (inf erence rules) and why Halp ern (1 997) could build sev eral f ormalization s for the intuitions expressed by Reiter and Moore. In contrast, the autoth eoremho od view eliminates the agent fro m th e picture and hen ce, the difﬁcult tasks to spe cify carefu lly the ke y con- cepts such as ideal r ationality , perf ect intro spection and , most of a ll, the All I Know Assumption. I nstead, it b uilds on more solid concepts of inferen ce rules, theoremhood and entailment which yields a more precise intuition. 6 Conclusions W e presented her e an analysis of infor mal fou ndation s of autoe pistemic reasoning. W e showed that ther e is principled way to arriv e at all major semantics of logic s of autoepistemic reasoning taking as the poin t o f dep arture the autotheoremh ood view o f a theory . W e see the main contributions of our work as follows. First, exten ding M oore’ s argume nts we clariﬁed th e different natur e o f defaults and autoepistem ic propositions. Look ing back at Reiter’ s in tuitions, we n ow s ee that, just as Moore had claimed about McDermo tt and Doyle, also Reiter b uilt an autoepis- temic logic and not a logic of defaults. W e showed that some lon g-standing problems with default logic c an be trac ed ba ck to pitfalls of using the autoep istemic propo sitions to encode defaults. On the o ther hand , we also showed that once we focus theor ies understoo d as consisting of au toepistemic prop ositions and adop t the autothe orem- hood per spectiv e, we a re led naturally to th e Krip ke-Kleene semantics, the sem antics of expansions by Moor e, the well-founde d semantics and the semantics of extensions by Reiter . Second, we analy zed wh at can be seen as the center of autoepistem ic lo gic: the All I Know Assumption. W e showed that this r ather f uzzy n otion leads to multiple perspectives on auto epistemic reasoning and to m ultiple dialects of the auto epistemic languag e, induced b y different notion s o f what can b e derived from (or is gr oun ded in ) a theory . On e particularly useful informal perspectiv e on autoepistemic logic goes back to Moore’ s truly insightful v iew o f autoepistemic r ules as inference rules. Th is view , which we called the autotheo remhoo d vie w , was the main focus of o ur discus- sion. In this view , theo ries “contain” th eir own entailmen t operator and “I” in the All I Know Assumption is understoo d as the the ory itself. The most faithful formal- 30 M. Denecker , V . W . Mar ek and M. T ruszczy ´ nski ization o f this vie w is the well-founded extension semantics b ut the stable-extension semantics, which extends Reiter’ s semantics to autoep istemic lo gic, coincides with the well-founded extension semantics wherev er the auto theoremh ood view seems to make sense. Th us, it was Reiter’ s default logic that for the ﬁrst time i ncor porated into the reasoning process the p rinciple of knowledge minimization , resulting in a better formalizatio n of Moo re’ s intuitions than Moore’ s o wn logic. Fifteen years ago Ha lpern (1 997) analyzed the intu itions of Reiter , McDermott and Doyle, and Moore, and sho wed that there a re alternati ve ways, in which they could be formalized . Halper n’ s work suggested that the logics prop osed by R eiter, McDermott and Doyle, and Moor e are not necessarily “determine d” by these intuitions. W e argue here that by loo king more carefully at the info rmal seman tics of those logics, they do indeed seem “predestined” and can be derived in a systematic and pr incipled way from a few basic info rmal intuitions. Ackno wledgments The work of the ﬁrst author was p artially suppo rted by FWO-Vlaanderen und er p roject G.0489 .10N. T he work of the third author was p artially sup ported by the NSF grant IIS-091 3459 . Refer ences C. Baral and V . S . Subrahmanian. Duality between alternativ e semantics of logic programs and nonmonotonic formalisms. In A. Nerode, W . Marek, and V . Subrahmanian , editors, Pr oceedings of t he 1st Internaational W orkshop on L ogic Pro gramming and Nonmonotonic Reasoning , pages 69–86. MIT Press, 1991. J. Barwise and J. Etchemendy . T he Liar: A n Essay on T ruth and Cir cularity . Oxford Uni versity Press, 1987. G. Brewka . Cumulativ e default logic: i n defense of non monotonic inference rules. Artiﬁcial Intelligence , 50(2):1 83–205, 1991. K. L. Clark. Negation as failure. In H. Gallaire and J. Minker , editors, Logic and Data Bases , pages 293–322 . Plenum Press, 1978. M. Denecker , V . W . Marek, and M.ław T ruszczy ´ nski. Approximating operators, stable oper- ators, well- founded ﬁxpoints and applications in non-mo notonic reasoning. In J. Minker , editor , Logic-based Artiﬁcial Intelligence , The Kl uwer International Series in Engineering and Computer Science, pages 127–144 . Kluwer Academic Publishers, Boston, 2000. M. Deneck er , V .W . Marek, and M. Truszczy ´ nski. Uniform semantic treatment of default and autoepistemic logics. Artiﬁcial Intelligence , 143(1):79–1 22, 2003. M. Deneck er , V . Marek, and M. Truszcz y ´ nski. Ultimate approximation and its application in nonmono tonic kno wledge representation systems. Information and Computation , 192:84– 121, 2004. M. Denecker , V . Marek, and M. Truszc zy ´ nski. Fixpoint 3-valu ed semantics for autoepistemic logic. In Pr oceedings of AAAI 1998 , pages 840–845. MIT Press, 1998. Reiter’ s Default Logic Is a Logic of Autoepistemic Reasoning 31 M. Fitting. A Kripk e-Kleene Semantics for Logic Prog rams. Jou rnal of Logic Pr ogramming , 2 (4):295–312 , 1985. M. Fit ting. A theory of tr uth that prefers falsehood. Journal of Philosophical Logic , 26:477– 500, 1997. M. Gelfond. On stratiﬁ ed autoepistemic theories. In Pr oceedings of AAAI 1987 , pages 207–211. Morga n Kaufmann, 1987. M. Gelfond and V . Li fschitz. The stable model semantics for logic programmin g. In R . A. K o walski and K. A. Bowen, editors, Pro ceedings of the International Joint Conferen ce and Symposium on Logic Pr ogr amming , pages 1070–108 0. MIT Press, 1988. G. Gottlob . T ranslating def ault logic into standard autoepistemic logic. Jou rnal of the A CM , 42 (4), 1995. J. Y . Halpern. A critical reexamination of d efault logic, a utoepistemic logic, and only knowin g. Computational Intelligence , 13(1 ):144–163, 1997. J. Y . Halpern and Y . Moses. T o wards a th eory of kno wl edge and igno rance: Preli minary repo rt. In R. Reiter , editor , Pr oceedings of the W orksh op on Non-Monotonic Reasoning , pages 12 5– 143, 1984. S. C. Kleene. Intr oduction to Metamathematics . V an No strand, 1952. K. Kono lige. On the relation between default and autoepistemic logic. Artiﬁcial Intelligence , 35:343–3 82, 1988. K. K onolige. Errata: On the relation between def ault and autoepistemic logic. Artiﬁcial Intelli- gence , 41(1):115, 1989. S. Kraus, D. Lehmann, and M. Magidor . Nonmonotonic reasoning, preferential models and cumulati ve logics. Artiﬁcial Intelligence , 44:167–20 7, 1990. S. Kripke. Outline of a theory of truth. Jo urnal of Philosophy , 72:690–712, 1975. D. L ehmann and M. Magidor . What does a conditional kno wledge base entail? Artiﬁcial Intelligence , 55:1– 60, 1992. D. J. Lehmann. W hat does a conditional kno wledge base entail? In R. J. Brachman, H. J. Lev esque, and R. Reiter , editors, Pr oceedings of the 1st Internationa l Confer ence on Princi- ples of Knowledge Repre sentation and Reasoning, KR 1989 , pages 212–222. Morgan Kauf- mann, 1989. H. J. Leve sque. All I know: a study in autoep istemic logic. Arti ﬁcial Intelligence , 42(2-3): 263–30 9, 1990. W . Lukasze wicz. Considerations on default l ogic: an alternativ e approach. Computational Intelligence , 4:1–1 6, 1988. D. Makinson. General theory of cumulati ve inference. In M. Reinfrank, J. de Kleer , M.L. Ginsberg, and E. Sande wall, editors, Nonmon otonic r easoning , volume 346 of Lectur e Notes in Computer Science , pages 1–18, Berlin-New Y ork, 1989. Springer . W . Marek and M. T ruszczy ´ nski. Relating autoepistemic and default logics. In R. J. Brachman, H. J. Levesq ue, and R. Reiter , editors, Pr oceedings of the 1st International Confer ence on Principles of Knowledge Repres entation and Reasoning, KR 1989 , pages 276–288. Morgan Kaufmann, 1989. 32 M. Denecker , V . W . Mar ek and M. T ruszczy ´ nski J. McCarthy . Circumscription - a form of nonmonoto nic reasoning. A rtiﬁcal Intell igence , 13: 27–39, 1980. J. McCarthy . Applications of circumsc ription to formalizing common-sen se kno wledge. Artiﬁ- cial Intelligence , 28(1):89 –116, 1986. J. McCarthy and P .J. Hayes. Some philosophical problems from the standpoint of artiﬁcial intelligence. In B. Meltzer and D. Michie, editors, Machine Intell igence 4 , pages 463–502 . Edinbur gh Uni versity Press, 1969. D. McDermott. Nonmonotonic Logic II: Nonmonotonic Modal T heories. Jo urnal of the ACM , 29(1):33–57 , 1982. D. McDermo tt and J. Doy le. Nonmonotonic Logic I. Artiﬁcial Intelligen ce , 13(1-2):41– 72, 1980. A. Mikitiuk and M. Truszc zy ´ nski. Constrained and rational default l ogics. In C. S. Mellish, editor , Pr oceedings of IJCAI 1995 , pages 1509–1 515. Morgan Kaufmann , 1995. R. C. Moore. Possible-world semantics f or autoepistemic logic. In R. Reiter, editor, Pro ceed- ings of the W orkshop on Non-Monotonic Reasoning , pages 344–35 4, 1984. Reprinted in: M. Ginsberg, editor , Readings on Nonmonotonic Reasoning , pages 137–142, Morgan Kauf- mann, 1990. R. C. Moore. Semantical considerations on nonmonotonic logic. Artiﬁcial Intelligence , 25(1): 75–94, 1985. J. Pearl. S ystem Z: A natural ordering of defaults wi th tractable applications to nonmonotonic reasoning. In R. P arikh, editor , Pr oceedings of the 3r d Confer ence on Theor etical Aspects of Reasoning about Knowledg e, TARK 1 990 , pages 121–135. Morgan Kaufmann, 1990. R. Reiter . A l ogic for defau lt reasoning. Artiﬁcial Intelligence , 13(1-2 ):81–132, 1980. R. Reiter and G. Criscuolo. On interacting defaults. In P . J. Hayes, editor , Pr oceedings of IJCAI 1981 , pages 270–276 . William Ka ufman, 1981. T . Schaub . On constrained default theories. In B. Neumann, editor , Pr oceedings of the 11th Eur opean Confer ence on Artiﬁcial Intelligence , pages 304– 308. Wile y and Sons, 1992. Y . Shoham. Nonmonoto nic logics: meaning and utility . In J. P . McDermott, editor , Pr oceedings of IJCAI 1987 , pages 388–393. Mor gan Kaufmann, 1987. A. T arski. The concept of truth in formalized languages. In J. Corcoran, editor, Logic, Seman- tics, Metamathematics: P apers f r om 1923-38 , pages pp. 152–278. Indianapolis: H ackett, 1983. T ranslated from T arski 1935 by J. H. W oodger . D. T ouretzk y . The mathematics of inheritance systems . Pitman, London, 1986. M. T ruszczy ´ nski. Modal interpretations of default logic. In R. Reiter and J. Mylopoulos, editors, Pr oceedings of IJCAI 1991 , pages 393– 398. Morgan Kaufmann , 1991. B. van Fraassen. Si ngular terms, truth-v alue gaps and free logic. Jou rnal of Philosophy , 63(1 7): 481–49 5, 1966. A. V an Gelder, K. A. Ross, and J. S. Schlipf. The well-founded semantics for general logic programs. Jou rnal of t he ACM , 38(3):620–650, 1991.

Reiters Default Logic Is a Logic of Autoepistemic Reasoning And a Good One, Too

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment