Revisiting Epistemic Specifications

Re visiting Epistemic Specifications Mirosław T ruszczy ´ nski Department of Computer Science Univ ersity of K entuck y Lexington , KY 40506, USA mirek@cs.uky. edu In hono r of Michael Gelfond on his 65th b irthday! Abstract. In 1 991, Michael Gelfond introduce d the langu age of ep istemic speci- fications. The go al was to de v elop tools for modeling prob lems that require so me form of meta-reason ing, that is, reasoning ov er multiple possible worlds. De- spite their relev an ce t o kno wledge representation, epistemic specifications hav e recei ved relati vely little attention so far . In this p aper , we revisit the formalism of epistemic specification. W e of fer a ne w definition of the formalism, propose se v- eral semantics (o ne of wh ich, under syntactic restrictions we assume, turns out to be equi v alent t o t he original semantics by Gelfond), deri ve some complexity re- sults and, finally , sho w the effecti veness of the formalism for mod eling problems requiring meta-reas oning considered recently by Faber and W oltran. All these results sho w that epistemic specifications deserve much more attention that has been afford ed to them so far . 1 Intr oduction Early 1990s were marked by several major developments in knowledge re presentation and nonm onoton ic reasonin g. One of the mo st important among them was the intro - duction o f d isjunctive logic pr ograms with classical ne gation b y M ichael G elfond a nd Vladimir Lifschitz [1]. The languag e of the formalism allowed for rules H 1 ∨ . . . ∨ H k ← B 1 , . . . , B m , n ot B m +1 , . . . , not B n , where H i and B i are classical literals, that is, ato ms and classical or str ong n egations ( ¬ ) of ato ms. In the paper, we will write “strong” rather than “classical” negation, as it reflects more accur ately the role an d th e behavior of the operato r . The answer - set se- mantics for prog rams consisting of such rules, introduced in the same paper, gene ralized the stable-mod el semantics of normal logic progr ams propo sed a couple of y ears earlier also by Gelfond an d L ifschitz [2] . The p roposed extensions of the langu age of nor- mal logic p rogram s were motiv a ted by knowledge repr esentation co nsiderations. W ith two negation oper ators it was straigh tforward to distinguish between P being false by default (there is no justification for adopting P ), an d P being str ongly false (there is evidence for ¬ P ). The form er would be written as not P while the latter as ¬ P . An d with the disjunction in the head of rules one co uld mode l “indefinite” rules which, when applied, provid e partial infor mation only (o ne of th e alternatives in the he ad hold s, b ut no prefere nce to any of them is gi ven) . Soon after disjuncti ve lo gic program s with strong negation were intr oduced , Michael Gelfond proposed an a dditional important e xtension, this time with a mo dal o perator [3]. He c alled th e resu lting f ormalism th e lang uage o f ep istemic specific ations . Th e motiv a tion came again from knowledge representatio n. The goal was to provide means for the “co rrect representa tion of incomp lete information in the pr esence of multiple extensions” [3]. Surprisingly , d espite their evident relev ance to the th eory of non monoto nic reason- ing as well as to the practice of knowledge representation, epistemic s pecifications have received relati vely little attention so far . T his state of a ff airs may soo n ch ange. Recen t work by F aber an d W oltran o n meta -r ea soning with answer - set prog rammin g [4] shows the need for langua ges, in which o ne co uld expr ess pr operties ho lding a cross all answer sets of a progr am, s omethin g Michael Gelfond foresaw already tw o decad es ago. Our goal in this paper is to revisit the fo rmalism of epistemic specificatio ns an d show that they deserve a seco nd lo ok, in fact, a p lace in th e f orefro nt of knowledge rep- resentation resear ch. W e will estab lish a general semantic framework for the formalism, and identify in it the precise location of Ge lfond’ s epistemic specifications. W e will de- riv e se veral c omplexity results. W e will also sh ow that the original idea o f Gelfon d to use a mod al op erator to model “what is known to a reasoner” has a br oader sco pe of applicability . In par ticular, we will sh ow that it can also be u sed in co mbination with the classical logic. Complexity results presented in th is p aper provid e an add itional motivation to study epistemic specifications. Even though pro grams with strong negation often look “more natural” as t hey more directly alig n with the n atural language description of knowledge specifications, the exten sion of the language o f normal lo gic progra ms with the stro ng negation op erator does not ac tually incre ase the expressive power o f the forma lism. This poin t was made already by Gelfo nd and Lifschitz, who o bserved th at there is a simple and concise w ay to compile the strong negation away . On the oth er hand, the extension allowing the d isjunction o perator in the head s of rules is a n essential one . As the comp lexity results show [5,6], the class of prob lems tha t can be repr esented by means of disjun ctiv e logic pr ograms is strictly larger (assuming no collapse o f the polyno mial hierarchy ) than the class of p roblems that can be modeled by normal log ic progr ams. In the same vein, extension by the m odal operator along the lines prop osed by Gelfond is essential, too. It does lead to an addition al jump in the complexity . 2 Epistemic Specifications T o m otiv ate e pistemic specification s, Gelfond discussed the following e xample. A cer- tain college has these rules to determine the eligibility of a student for a scholarship: 1. Students with high GP A are eligible 2. Students from unde rrepresen ted gr oups and with f air GP A are eligible 3. Students with low GP A are not eligible 4. When these rules are insufficient to dete rmine eligibility , the student sho uld be interviewed by the scholarship committee. Gelfond argu ed that there is no simple way to repr esent these ru les as a disjun cti ve logic progr am with stron g n egation. The re is no problem with the first thr ee rules. They ar e modeled co rrectly by the following three logic program rules ( in the language with bo th the default and strong negation operato rs): 1. el ig ibl e ( X ) ← highGP A ( X ) 2. el ig ibl e ( X ) ← u nder re p ( X ) , fairGP A ( X ) 3. ¬ el ig ibl e ( X ) ← lowGP A ( X ) . The problem is with the fourth rule, as it has a clear meta-reasonin g fla vor . It should apply wh en the possible world s ( answer sets) deter mined by th e first three r ules do not fully specify the status of eligibility o f a studen t a : neith er a ll of them contain el ig ibl e ( a ) nor all of them contain ¬ el ig ibl e ( a ) . An obvious attempt at a form alization: 4. inter v ie w ( X ) ← not el i g ib l e ( X ) , no t ¬ el ig i bl e ( X ) fails. It is just another rule to be add ed to the progr am. Thu s, when the answer-set semantics is used, th e ru le is interpreted with respect to in dividual answer sets and no t with respect to collectio ns of answer-sets, as required for this application . F o r a concrete example, let us assume that all we k now about a certain student named Mike is that Mike’ s GP A is fair or high. Clearly , we d o no t have en ough inf ormation to d etermine Mike’ s elig ibility a nd so we must interview Mike. But the progr am consisting of rules (1)-(4 ) and the statement 5. f air GP A ( mik e ) ∨ hi g hGP A ( mik e ) about Mike’ s GP A, has two answer sets : { hig hGP A ( mik e ) , e l ig i bl e ( mik e ) } { f air GP A ( mik e ) , i nter v iew ( mik e ) } . Thus, the query ? inte rv iew ( mik e ) has the answer “unknown. ” T o address the p rob- lem, Ge lfond p roposed to extend the lan guage w ith a modal op erator K and, sp eaking informa lly , interpret premises K ϕ as “ ϕ is known to the p rogram ” (the original phrase used by Gelfon d was “k nown to the re asoner”), that is, true in all answer-sets. W ith this languag e e xtension, the fourth rule can be encoded as 4 ′ . inter v iew ( X ) ← not K e l ig i bl e ( X ) , not K ¬ el ig ibl e ( X ) which, intuitively , stands f or “ interview if n either the eligibility nor the non- eligibility is known. ” The way in which Gelfo nd [3] pr oposed to form alize this intu ition is strik ingly elegant. W e will now discuss it. W e start with the sy ntax of epistemic specificatio ns . As else where in t he p aper, we restrict attention to the prop ositional case. W e assume a fixed infinite co untable set A t of atoms and the co rrespond ing language L of pro positional logic. A literal is an atom, say A , or its str ong negation ¬ A . A simple mod al atom is an expression K ϕ , where ϕ ∈ L , and a simp le modal liter al is defined accordin gly . A n epistemic pr emise is an expr ession (conjunction) E 1 , . . . , E s , n ot E s +1 , . . . , not E t , where e very E i , 1 ≤ i ≤ t , is a simple m odal literal. An epistemic rule is an expression of the form L 1 ∨ . . . ∨ L k ← L k +1 , . . . , L m , n ot L m +1 , . . . , not L n , E , where every L i , 1 ≤ i ≤ k , is a literal, and E is an ep istemic premise. Collections of epistemic rules are ep istemic p r ograms . I t is clear that (ground version s of) rules (1)-( 5) and (4 ′ ) are examp les of epistemic rules, with ru le (4 ′ ) bein g an example of an epistemic rule that actually takes advantage of the extended syntax. R ules such as a ∨ ¬ d ← b, not ¬ c, ¬ K ( d ∨ ¬ c ) ¬ a ← ¬ c, not ¬ K ( ¬ ( a ∧ c ) → b ) are also examples of epistemic r ules. W e note th at the lan guage of epistemic prog rams is only a fragment of the lang uage of epistemic specification s by Gelfon d. Howe ver, it is still expressi ve enough to cov er all examples discussed b y Gelfo nd and , mor e gen erally , a broad ran ge o f p ractical applica tions, as n atural-lan guage f ormulatio ns of do main knowledge t ypically assume a rule-ba sed pattern. W e move on to the semantics, which is in terms of world views . The definitio n of a world v iew con sists of several steps. First, let W be a consistent set of literals from L . W e regard W as a three-valued interpretation of L (we will also u se the ter m th r ee- valued possible world ), assignin g to each atom one of th e three logical values t , f and u . The interpretation extends by recursion to all formulas in L , acco rding to the f ollowing truth tables ¬ f t t f u u ∨ t u f t t t t u t u u f t u f ∧ t u f t t u f u u u f f f f f → t u f t t u f u t u u f t t t Fig. 1. Truth tables for the 3-valued lo gic of Kleene. By a three-valued possible-world str ucture we mean a non -empty family of con- sistent sets o f literals (thre e-valued p ossible worlds). Let A b e a th ree-valued possible- world struc ture an d let W b e a consistent set o f litera ls. For every form ula ϕ ∈ L , we define 1. hA , W i | = ϕ , if v W ( ϕ ) = t 2. hA , W i | = K ϕ , if for every V ∈ A , v V ( ϕ ) = t 3. hA , W i | = ¬ K ϕ , if there is V ∈ A such that v V ( ϕ ) = f . Next, for e very literal or simple moda l literal L , we define 4. hA , W i | = not L if hA , W i 6| = L . W e note that neither hA , W i | = K ϕ nor hA , W i | = ¬ K ϕ depend on W . Thu s, we will often write A | = F , whe n F is a simple modal literal or its default negation. In the next s tep, we introd uce the notion of the G-r educ t of an epistemic program. Definition 1. Let P be an epistemic pr ogram, A a three-valued p ossible-world struc- tur e and W a consistent set of liter als. The G-redu ct of P with r esp ect to hA , W i , in symbols P hA ,W i , con sists of the h eads of all ru les r ∈ P such tha t hA , W i | = α , for every conjunct α occu rring in the body of r . Let H be a set of disjunc tions of literals fro m L . A set W o f literals is clo sed with respect to H if W is consistent and contains at lea st o ne liter al in co mmon with every disjunction in H . W e d enote by Min ( H ) the family of all m inimal sets of literals th at are closed with respect to H . W ith the notation Min ( H ) in hand, we are finally ready to define the concept of a world vie w of an epistemic program P . Definition 2. A th r ee- valued possible- world structure A is a world view of an epis- temic pr ogram P if A = { W | W ∈ Min ( P hA ,W i ) } . Remark 1. The G -reduct of an epistemic program co nsists of disjunctions of literals. Thus, the concep t of a world view is well d efined. Remark 2. W e note that Gelfo nd consid ered also in consistent sets of literals as minimal sets closed u nder disjunctions. Howe ver , the only such set he allowed con sisted of all literals. Consequently , the difference be tween th e Gelfond’ s seman tics a nd th e one we described above is that some pro grams ha ve a world view in the Gelfo nd’ s approach that co nsists o f a single set of all literals, while in our approach these program s do no t have a world view . B ut in all other cases, the two semantics behave in the sam e way . Let us c onsider the gr ound prog ram, say P , cor respond ing to the scholarship elig i- bility example (rule (5), and rules (1 )-(3) and (4 ′ ), grounded with respec t to the Her- brand universe { mik e } ). The only rule in volving simp le modal literals is inter v iew ( mik e ) ← not K el i g ibl e ( mik e ) , not K ¬ el ig ibl e ( mik e ) . Let A be a world view of P . Being a three- valued possible- world structu re, A 6 = ∅ . No matter what W we consid er , no minimal set closed with resp ect to P hA ,W i con- tains lo wGP A ( mik e ) and, consequently , no minimal set closed with respect to P hA ,W i contains ¬ el ig ibl e ( mik e ) . It follows that A 6| = K ¬ el ig ibl e ( mik e ) . Let us assume that A | = K el ig ibl e ( mi k e ) . Then, no reduct P hA ,W i contains inter v iew ( mik e ) . Let W = { fairGP ( mik e ) } . It follows that P hA ,W i consists only of fairGP A ( mik e ) ∨ highGP A ( mik e ) . Clear ly , W ∈ Min ( P hA ,W i ) and, consequently , W ∈ A . T hus, A 6| = K el ig ibl e ( mi k e ) , a contradiction . It must b e then that A | = not K el ig ibl e ( mik e ) and A | = n ot K ¬ el ig i bl e ( mik e ) . Let W b e an ar bitrary c onsistent set of literals. Clearly , the reduct P hA ,W i contains inter v iew ( mik e ) and fa irGP A ( mik e ) ∨ highGP A ( mik e ) . If highGP A ( mik e ) ∈ W , the reduct also contains el ig i bl e ( mik e ) . Thu s, W ∈ Min ( P hA ,W i ) if and only if W = { fairGP A ( mik e ) , i nter v iew ( mik e ) } , or W = { highGP A ( mik e ) , e l ig i bl e ( mik e ) , i nter v iew ( mik e ) } . It fo llows that if A is a world v iew for P then it consists o f these two possible worlds. Con versely , it is easy to ch eck that a possible-world structure consisting of these tw o possible worlds is a world view for P . Thus, inter v iew ( mik e ) hold s in A , and so our representatio n of the example as an epistemic program has the desired beha vior . 3 Epistemic Specifications — a Br oader Perspecti ve The discussion in the p revious section demonstra tes the usefulness of f ormalisms such as that of epistemic s pecifications for knowledge repr esentation and reasoning. W e will now present a simpler yet, in m any re spects, more gen eral f ramework for epistemic specifications. The ke y to our approac h is that we consider t he semantics giv en by two- valued interp retations (sets of ato ms), and standard two- valued possible- world struc- tures (non empty collections of tw o -valued i nterpre tations). W e also work within a rather standard version of th e lang uage of modal pr opositiona l logic and so, in particula r , we allow only for o ne n egation operator . Later in the paper we show th at epistemic speci- fications by Ge lfond can b e encod ed in a r ather dir ect way in ou r form alism. Thus, the restrictions we impose are n ot essential even though, adm ittedly , no t having two k inds of negation in the language in s ome cases may make the modeling task harder . W e start by makin g precise the syntax of the langu age we will be using. As we stated earlier, we assume a fixed infinite countab le set of atoms A t . The language we consider is determined by the set At , the modal operator K , and by the boolean connectives ⊥ (0-place) , and ∧ , ∨ , and → (binary ). The B NF expression ϕ ::= ⊥ | A | ( ϕ ∧ ϕ ) | ( ϕ ∨ ϕ ) | ( ϕ → ϕ ) | K ϕ , where A ∈ At , provid es a concise definition of a formula. The parentheses are used only to disambiguate the order of binary connectives. Whenever possible, we omit them. W e d efine the unary negation connective ¬ and the 0- place connective ⊤ as abbrevia- tions: ¬ ϕ ::= ϕ → ⊥ ⊤ ::= ¬⊥ . W e call form ulas K ϕ , where ϕ ∈ L K , mod al a toms ( simple m odal atom s th at we considered earlier and will consider be low are s pecial modal atoms with K -dep th equal to 1). W e deno te this language by L K and refer to subsets of L K as epistemic theories . W e deno te the modal-free fragment of L K by L . While we will e ventually describe th e semantics (in fact, several of them ) f or arb i- trary epistemic th eories, we start with an important special case. Due to close analogies between the concepts we define below and the corr espondin g ones d efined earlier in the context o f th e formalism o f Gelf ond, we “reuse” the terms u sed there. Specifically , by an ep istemic premise we m ean a conjun ction of simp le mo dal litera ls. Similarly , by an epistemic rule we understand an expression of the form E ∧ L 1 ∧ . . . ∧ L m → A 1 ∨ . . . ∨ A n , (1) where E is an ep istemic p remise, L i ’ s are literals (in L ) and A i ’ s are atoms. Finally , we call a co llection o f epistemic ru les an epistemic pr ogram . It will always be clear f rom the context, in which sense these terms are to be understood . W e stress that ¬ is no t a primar y conn ectiv e in th e language but a derived one ( it is a sho rthand for some particu lar formulas in volving the rule symbol). Even though under so me sema ntics we p ropose b elow this negation operator has featur es of default negation, unde r some others it d oes n ot. T hus, we selected for it the standar d n egation symbol ¬ r ather than the “loaded ” n ot . A ( two-valued) possible-world structure is any non empty family A o f subsets of At (two-valued interpretatio ns). I n the remainder of the paper, when we use terms “in ter- pretation” a nd “ possible-world stru cture” witho ut any additional m odifiers, we a lw ays mean a two-valued interpretation and a two-v alued possible-world structure. Let A be a possible-world structu re and ϕ ∈ L . W e r ecall that A | = K ϕ precisely when W | = ϕ , f or ev ery W ∈ A , and A | = ¬ K ϕ , otherwise. W e will n ow define the epistemic r edu ct of an epistem ic program with respect to a possible-world structure. Definition 3. Let P ⊆ L K be a n epistemic p r ogram and le t A be a possible-wo rld structur e. The epistemic reduct of P with r espect to A , P A in symbols, is the theory obtained fr o m P as follows: eliminate e very rule with an epistemic pr emise E su ch tha t A 6| = E ; dr op th e epistemic pr emise fr o m every r emaining rule. It is clear that P A ⊆ L , and that it consists of rules of the form L 1 ∧ . . . ∧ L m → A 1 ∨ . . . ∨ A n , (2) where L i ’ s are literals (in L ) and A i ’ s are atoms. Let P be a collection of rules (2). T hen, P is a propositional theo ry . Th us, it can be interpreted by the standard pr oposition al logic semantics. Howev er , P can also be regarded as a disjun ctiv e logic program (if we write rules from rig ht to left rather tha n from left to right). C onsequen tly , P can also be interpreted by the stable-model seman- tics [2,1] and the supported-mo del semantics [7,8,9,10]. (For no rmal logic programs, the suppo rted-mo del semantics was intro duced by Apt et al. [7]. The notion was ex- tended to disjuncti ve logic pro grams by Baral and Gelfond [8]. W e re fer to papers by Brass an d Dix [9], D efinition 2. 4, and Inoue and Sakama [ 10], Sec tion 5, for more de- tails). W e wr ite M ( P ) , S T ( P ) and S P ( P ) for the sets of mo dels, stab le models and supported mod els of P , resp ectiv ely . An impor tant obser vation is th at each of these semantics giv es rise to the correspond ing n otion of an epistemic extension. Definition 4. Let P ⊆ L K be an epistemic pr ogram. A p ossible-world structur e A is an epistemic mod el ( r espec tively , an epistemic stable model , or an epistemic sup ported model ) of P , if A = M ( P A ) (r espe ctively , A = S T ( P A ) or A = S P ( P A ) ). It is clear that Definition 4 can easily b e adjusted also to other s emantics of pro posi- tional theories and progr ams. W e briefly mention two su ch semantics in th e last section of the paper . W e will n ow show that ep istemic prog rams with th e semantics of epistem ic sta- ble models ca n p rovide an adeq uate repr esentation to the scholarship e ligibility exam- ple fo r Mike. The av ailab le inf ormation can be rep resented by the following pro gram P ( mi k e ) ⊆ L K : 1. el ig ibl e ( mik e ) ∧ nel i g ib l e ( mik e ) → ⊥ 2. f air GP A ( mik e ) ∨ hi g hGP A ( mik e ) 3. highGP A ( mik e ) → e l ig i bl e ( mik e ) 4. under r ep ( mik e ) ∧ fairGP A ( mik e ) → el ig ibl e ( mi k e ) 5. lowGP A ( mik e ) → nel ig i bl e ( mik e ) 6. ¬ K el ig i bl e ( mik e ) , ¬ K nel i g ibl e ( mik e ) → i nterv iew ( mik e ) . W e use the pr edicate neligible to model the strong negation of th e predicate el ig i bl e that appears in the represen tation in ter ms of epistemic p rogra ms by Gelfo nd (thus, in particular, th e pr esence of the first clause, which p recludes the f acts el ig ibl e ( mik e ) an d nel ig ibl e ( mik e ) to b e tru e tog ether). T his extension of the languag e and an extra r ule in the represen tation is the price we pay for eliminating one negation operator . Let A co nsist of the interpretatio ns W 1 = { fairGP A ( mik e ) , inter v iew ( mik e ) } W 2 = { highGP A ( mik e ) , eli g ib l e ( mik e ) , inter v iew ( mik e ) } . Then the reduct [ P ( mik e )] A consists of ru les ( 1)-(5) , wh ich ar e unaffected by the reduct oper ation, and of the fact interv iew ( mik e ) , r esulting from rule (6) when the reduct oper ation is per formed (a s in log ic p rogram ming, when a rule h as the empty antecedent, we d rop the implication symbol f rom the no tation). One can ch eck that A = { W 1 , W 2 } = S T ([ P ( mik e )] A ) . Thu s, A is an epistemic stab le mod el of P (in fact, the only on e). Clearly , inte rv iew ( mi k e ) hold s in the mo del (as we would expec t it to), as it ho lds in e ach of its possible-world s. W e note that in this particular case, the semantics of epistemic supported models yields exactly the same solution. 4 Complexity W e will now study the complexity of reasoning with epistemic (stable, supported ) mod- els. W e provid e details for the case of epistemic stable m odels, an d only present the results fo r th e oth er two seman tics, as the tec hniques to prove them are very similar to those we develop fo r the case of epistemic stable models. First, we no te that e pistemic stable models o f an epistemic p rogra m P can be rep- resented by partitions o f the set of all modal a toms of P . This is impo rtant as a priori the size o f p ossible-world structu res on e n eeds to con sider as candidates for e pistemic stable models may be expo nential in th e size o f a progra m. Thus, to obtain good co m- plexity boun ds alternative polynom ial-size representations of epistemic stab le mo dels are needed. Let P ⊆ L K be an epistemic pr ogram a nd ( Φ, Ψ ) b e the set of mod al atoms of P (all these modal ato ms are, in fact, simple). W e write P | Φ,Ψ for the progr am obtained from P by eliminating ev ery rule whose epistemic prem ise contains a conjun ct K ψ , where K ψ ∈ Ψ , or a conjunct ¬ K ϕ , where K ϕ ∈ Φ (these r ules are “‘blocked” by ( Φ, Ψ ) ), and b y eliminating the epistemic premise from e very other rule of P . Proposition 1. Let P ⊆ L K be an epistemic p r ogram. If a possible-world structure A is a n epistemic stable model of P , then ther e is a p artition ( Φ, Ψ ) of the set of mod al atoms of P such that 1. S T ( P | Φ,Ψ ) 6 = ∅ 2. F or every K ϕ ∈ Φ , ϕ holds in every stable model of P | Φ,Ψ 3. F or every K ψ ∈ Ψ , ψ d oes not hold in at least one stable model of P | Φ,Ψ . Con versely , if there ar e such partitions, P h as epistemic stable models. It fo llows that epistemic stable models can be represented by partitions ( Φ, Ψ ) sat- isfying conditions (1)-(3 ) from the proposition above. W e ob serve that deciding whethe r a partition ( Φ, Ψ ) satisfies co nditions (1)-(3 ) from Proposition 1, can be accomplished by polyn omially many calls to an Σ P 2 -oracle and, if we restrict attention to non-disjunctive e pistemic programs, by p olyno mially m any calls to an NP -o racle. Remark 3. If we adjust Proposition 1 by replacing the term “stable” with the term “sup - ported, ” and replacing S T () with S P () , we obtain a characterization of epistemic sup- ported mod els. Similar ly , omitting the term “stable, ” and replacing S T () w ith M () yields a characterization of epistemic models. In ea ch case, one can d ecide whether a partition ( Φ, Ψ ) satisfies c ondition s (1)-(3) by poly nomially many calls to an NP -oracle (this claim is e vid ent f or the case o f ep istemic mo dels; fo r the case o f ep istemic sup - ported models, it f ollows from the fact that supp orted m odels semantics d oes not g et harder when we allow disjunctions in the heads or rules). Theorem 1. The pr oblem to decide whether a no n-disjunctive epistemic pr ogram ha s an epistemic stable model is Σ P 2 -complete. Proof: Our comm ents above imply that the p roblem is in the cla ss Σ P 2 . Let F = ∃ Y ∀ Z Θ , where Θ is a DNF formula. The pro blem to d ecide whether F is tr ue is Σ P 2 - complete. W e will r educe it to the pro blem in que stion and, con sequently , demonstrate its Σ P 2 -hardn ess. T o this end, we construct an epistemic p rogram Q ⊆ L K by in cluding into Q th e following clauses (atoms w , y ′ , y ∈ Y , and z ′ , z ∈ Z are fresh): 1. K y → y ; and K y ′ → y ′ , for ev ery y ∈ Y 2. y ∧ y ′ → ; a nd ¬ y ∧ ¬ y ′ → , for ev ery y ∈ Y 3. ¬ z ′ → z ; and ¬ z → z ′ , for z ∈ Z 4. σ ( u 1 ) ∧ . . . ∧ σ ( u k ) → w , where u 1 ∧ . . . ∧ u k is a d isjunct of Θ , and σ ( ¬ a ) = a ′ and σ ( a ) = a , for e very a ∈ Y ∪ Z 5. ¬ K w → . Let us assume th at A is an epistemic stable model o f Q . In p articular, A 6 = ∅ . It must b e that A | = K w (o therwise, Q A has no stable mo dels, tha t is, A = ∅ ). Let us define A = { y ∈ Y | A | = K y } , and B = { y ∈ Y | A | = K y ′ } . It follo ws that Q A consists of the following rules: 1. y , for y ∈ A , an d y ′ , for y ∈ B 2. y ∧ y ′ → ; a nd ¬ y ∧ ¬ y ′ → , for ev ery y ∈ Y 3. ¬ z ′ → z ; and ¬ z → z ′ , for z ∈ Z 4. σ ( u 1 ) ∧ . . . ∧ σ ( u k ) → w , where u 1 ∧ . . . ∧ u k is a d isjunct of Θ , and σ ( ¬ a ) = a ′ and σ ( a ) = a , for e very a ∈ Y ∪ Z . Since A = S T ( Q A ) an d A 6 = ∅ , B = Y \ A (due to clauses of type (2)). I t is clear that the p rogram Q A has stable models and that they are of the form A ∪ { y ′ | y ∈ Y \ A } ∪ D ∪ { z ′ | z ∈ Z \ D } , if that set does not imply w throu gh a rule o f typ e (4), or A ∪ { y ′ | y ∈ Y \ A } ∪ D ∪ { z ′ | z ∈ Z \ D } ∪ { w } , otherwise, wher e D is any subset of Z . As A | = K w , there are no stable models o f the first typ e. Th us, the family of stable models of Q A consists of all sets A ∪ { y ′ | y ∈ Y \ A } ∪ D ∪ { z ′ | z ∈ Z \ D } ∪ { w } , where D is an arbitrar y sub set of Z . It follows that for ev ery D ⊆ Z , the set A ∪ { y ′ | y ∈ Y \ A } ∪ D ∪ { z ′ | z ∈ Z \ D } satisfies the bo dy of at least one rule of ty pe ( 4). By the construction, fo r every D ⊆ Z , th e valuation o f Y ∪ Z deter mined by A and D satisfies the corresp onding disjunct in Θ an d so, also Θ . In o ther w ords, ∃ Y ∀ Z Θ is true. Con versely , let ∃ Y ∀ Z Θ be true. Let A b e a sub set o f Y such that Θ | Y / A holds for ev ery truth assignment of Z (b y Θ | Y / A , we m ean th e formula o btained b y simp lifying the formu la Q with respect to the truth assignment of Y determined by A ). Let A consist of a ll sets of the form A ∪ { y ′ | y ∈ Y \ A } ∪ D ∪ { z ′ | z ∈ Z \ D } ∪ { w } , where D ⊆ Z . It f ollows that Q A consists of clauses (1 )-(4) above, with B = Y \ A . Sin ce ∀ Z Θ | A/ Y holds, it fo llows that A is precisely the set of stable m odels of Q A . Thus, A is an epistemic stable model of Q . ✷ In the general case, the complexity goes one level u p. Theorem 2. The pr oblem to d ecide wheth er an epistemic pr ogram P ⊆ L K has an epistemic stable model is Σ P 3 -complete. Proof: Th e mem bership f ollows from the earlier re marks. T o prove the hardness part, we con sider a QBF fo rmula F = ∃ X ∀ Y ∃ Z Θ , where Θ is a 3-CNF fo rmula. For ea ch atom x ∈ X ( y ∈ Y and z ∈ Z , respectively), we introdu ce a fresh atom x ′ ( y ′ and z ′ , respectively). Finally , we introd uce three additional fresh atoms, w , f and g . W e now construct a disjuncti ve epistemic progra m Q by including into it the fol- lowing clauses: 1. K x → x ; and K x ′ → x ′ , for ev ery x ∈ X 2. x ∧ x ′ → ; and ¬ x ∧ ¬ x ′ → , for ev ery x ∈ X 3. ¬ g → f ; and ¬ f → g 4. f → y ∨ y ′ ; and f → z ∨ z ′ , for ev ery y ∈ Y and z ∈ Z 5. f ∧ w → z ; and f ∧ w → z ′ , for ev ery z ∈ Z 6. f ∧ σ ( u 1 ) ∧ σ ( u 2 ) ∧ σ ( u 3 ) → w , f or ev ery clause C = u 1 ∨ u 2 ∨ u 3 of Θ , where σ ( a ) = a ′ and σ ( ¬ a ) = a , for every a ∈ X ∪ Y ∪ Z 7. f ∧ ¬ w → w 8. ¬ K ¬ w → Let us assume that ∃ X ∀ Y ∃ Z Θ is true. Let A ⊆ X describe the tru th assignm ent o n X so that ∀ Y ∃ Z Φ X/ A holds (we d efine Φ X/ A in the proof o f the p revious result). W e will show that Q has an epistemic stable mod el A = { A ∪ { a ′ | a ∈ X \ A } ∪ { g } } . Clearly , K x , x ∈ A , a nd K x ′ , x ∈ X \ A , a re tru e in A . Also, K ¬ w is tru e in A . All other moda l atoms in Q are false in A . Thu s, Q A consists of rules x , for x ∈ A , x ′ , for x ∈ X \ A and of rules ( 2)-(7 ) above. Let M be a stable mode l of Q A containing f . It fo llows that w ∈ M and so, Z ∪ Z ′ ⊆ M . Moreover, the Gelfond -Lifschitz r educt of Q A with respect to M co nsists o f r ules x , fo r x ∈ A , x ′ , for x ∈ X \ A , all ¬ -free constraints of type (2), ru le f , and r ules (4)-(6) above, and M is a minimal mo del o f this prog ram. Let B = Y ∩ M . By the min imality o f M , M = A ∪ { x ′ | x ∈ X \ A } ∪ B ∪ { y ′ | y ∈ Y \ B } ∪ Z ∪ Z ′ ∪ { f , w } . Since ∀ Y ∃ Z Φ X/ A holds, ∃ Z Φ X/ A,Y /B holds, too. Thus, let D ⊆ Z be a subset o f Z such that Φ X/ A,Y /B ,Z/D is tr ue. I t follows that M ′ = A ∪ { x ′ | x ∈ X \ A } ∪ B ∪ { y ′ | y ∈ Y \ B } ∪ D ∪ { z ′ | z ∈ Z \ D } ∪ { f } is also a model of the Gelfond-Lifschitz reduct of Q A with respect to M , con tradicting the minimality of M . Thus, if M is an a nswer set of Q A , it must contain g . Conseque ntly , it do es not contain f and so no rules of type (4 )-(7) contribute to it. It fo llows that M = A ∪ { a ′ | a ∈ X \ A } ∪ { g } and, as it indeed is an answer set of Q A , A = S T ( Q A ) . Thus, A is a epistemic stable model, as claimed. Con versely , let as assume that Q has a n ep istemic stable m odel, say , A . It must be that A | = K ¬ w (otherwise, Q A contains a contradiction and has no stable m odels). L et us define A = { x ∈ X | A | = K x } an d B = { x ∈ X | A | = K x ′ } . It fo llows that Q A consists of the clauses: 1. x , for x ∈ A an d x ′ , for x ∈ B 2. x ∧ x ′ → ; and ¬ x ∧ ¬ x ′ → , for ev ery x ∈ X 3. ¬ g → f ; and ¬ f → g 4. f → y ∨ y ′ ; and f → z ∨ z ′ , for ev ery y ∈ Y and z ∈ Z 5. f ∧ w → z ; and f ∧ w → z ′ , for ev ery z ∈ Z 6. f ∧ σ ( u 1 ) ∧ σ ( u 2 ) ∧ σ ( u 3 ) → w , f or ev ery clause C = u 1 ∨ u 2 ∨ u 3 of Φ , wher e σ ( a ) = a ′ and σ ( ¬ a ) = a , for every a ∈ X ∪ Y ∪ Z . 7. f , ¬ w → w W e h av e that A is precisely the set o f stable mo dels of this pr ogram. Since A 6 = ∅ , B = X \ A . If M is a stable model o f Q A and co ntains f , then it contain s w . But then, as M ∈ A , A 6| = K ¬ w , a co ntradictio n. It follo ws th at there is no stable model containing f . T hat is, the progr am consisting of the follo win g rules has n o stable m odel: 1. x , for x ∈ A an d x ′ , for x ∈ X \ A 2. y ∨ y ′ ; and z ∨ z ′ , for ev ery y ∈ Y a nd z ∈ Z 3. w → z ; and w → z ′ , for ev ery z ∈ Z 4. σ ( u 1 ) ∧ σ ( u 2 ) ∧ σ ( u 3 ) → w , for every clause C = u 1 ∨ u 2 ∨ u 3 of Θ , where σ ( a ) = a ′ and σ ( ¬ a ) = a , for every a ∈ X ∪ Y ∪ Z . 5. ¬ w → w But then , the fo rmula ∀ Y ∃ Z Θ | X/ A is true a nd, consequ ently , the formu la ∃ X ∀ Y ∃ Z Θ is true, too. ✷ For th e other two epistemic semantics, Remark 1 implies that the p roblem of the existence of an epistemic model (epistemic supported mod el) is in the c lass Σ P 2 . The Σ P 2 -hardn ess of the p roblem can be proved by similar techniqu es as those we used f or the case of epistemic stable models. Thus, we have the following result. Theorem 3. The pr oblem to d ecide wheth er an epistemic pr ogram P ⊆ L K has an epistemic model (epistemic supported model, r espectively ) is Σ P 2 -complete. 5 Modeling with Epistemic Programs W e will now present se veral prob lems which illu strate the advantages offered by the lan- guage of epistemic pro grams we de veloped in the pre vious tw o sections. Whenever we use predicate programs, we under stand that their semantics is that of the correspon ding groun d progr ams. First, we co nsider two graph pro blems r elated to the e xistence of Hamilton ian cy- cles. Let G be a directed g raph. An edge in G is critical if it belongs to every hamilto- nian cycle in G . The follo wing problems are of interest: 1. Giv en a directed graph G , find the set of all critical edges of G 2. Giv en a directed graph G , and integers p an d k , find a set R of no more than p new edges such that G ∪ R has no more than k critical edges. Let H C ( v tx, e dg e ) be any stan dard ASP encoding of the Hamilto nian cycle pr ob- lem, i n which p redicates vtx and edge r epresent G , and a predicate hc rep resents edges of a can didate hamiltonian cycle. W e a ssume the ru les o f H C ( v tx, e dg e ) are written from left to right so t hat they can be re garded as elements of L . Then, simply adding to H C ( v tx, edg e ) the rule: K hc ( X , Y ) → critic al ( X , Y ) yields a correct repre sentation of the first problem. W e write H C cr ( v tx, e dg e ) to denote this prog ram. Also, for a directed graph G = ( V , E ) , we defin e D = { v tx ( v ) | v ∈ V } ∪ { edg e ( v , w ) | ( v , w ) ∈ E } . W e have the follo w ing result. Theorem 4. Let G = ( V , E ) be a directed graph. If H C cr ( v tx, e dg e ) ∪ D has no epistemic stable mod els, th en every edge in G is critical (trivially). Otherwise, the ep is- temic pr ogram H C cr ( v tx, e dg e ) ∪ D has a uniqu e ep istemic stable mo del A and the set { ( v , w ) | A | = cr itical ( u , v ) } is th e set of critical edges in G . Proof (Sketch): Le t H be the gr oundin g of H C cr ( v tx, e dg e ) ∪ D . If H ha s no epistemic stable models, it follows tha t th e “n on-epistemic” part H ′ of H has n o stable models (as no atom of th e fo rm c riti cal ( x, y ) appears in it). As H ′ encodes th e e xistence o f a hamiltonia n cycle in G , it follows that G ha s no Hamilton ian cycles. Thus, tri vially , ev ery edg e of G belo ngs to ev ery Hamiltonian cycle of G and so, every e dge of G is critical. Thus, let us assume that A is an epistemic stable model of H . Also, let S b e the set of all stable models of H ′ (they correspon d to Hamiltonian c ycles of G ; each mo del contains, in particular, atoms of the f orm hc ( x, y ) , where ( x, y ) ranges over the edge s of the correspond ing Hamilton ian cycle). T he red uct H A consists o f H ′ (non- epistemic part o f H is u naffected by the red uct o peration ) and o f C ′ , a set of so me facts of the form cr itical ( x, y ) . Thus, the stable models of th e redu ct are o f the fo rm M ∪ C ′ , wher e M ∈ S . That is, A = { M ∪ C ′ | M ∈ S } . L et us denote by C the set of the atom s cri tical ( x, y ) , wher e ( x, y ) belong s to e very hamiltonia n cycle o f G (is critical) . One can compute now that H A = H ′ ∪ C . Since A = S T ( H A ) , A = { M ∪ C | M ∈ S } . Thus, H C cr ( v tx, e dg e ) ∪ D has a unique epistemic stable model, a s claim ed. It also follows that the set { ( v , w ) | A | = cr itical ( u , v ) } is the set of critical edg es in G . ✷ T o r epresent the second p roblem, we p roceed as follows. First, we “select” new edges to be added to the graph and impose constraints th at guarantee that all new e dges are indeed new , and that no mo re than p new e dges are selected (we use here lparse syntax for brevity; the constraint can be encod ed s trictly in the langua ge L K ). v tx ( X ) ∧ v tx ( Y ) → new E dg e ( X , Y ) new E dg e ( X , Y ) ∧ edg e ( X , Y ) → ⊥ ( p + 1) { new E dg e ( X , Y ) : v tx ( X ) , v tx ( Y ) } → ⊥ . Next, we define the set of edges of the extended g raph, using a predicate edg eE G : edg e ( X , Y ) → e dg eE G ( X , Y ) new E dg e ( X , Y ) → edg eE G ( X , Y ) Finally , we define critical ed ges and impose a co nstraint on their num ber (again, ex- ploiting the lparse syntax for brevity sake): edg eE G ( X , Y ) ∧ K hc ( X , Y ) → cri tical ( X , Y ) ( k + 1) { cr itica l ( X, Y ) : edg eE G ( X , Y ) } → ⊥ . W e define Q to consist of all these r ules tog ether with all the rules of the pro gram H C ( v tx, edg eE G ) . W e now h av e the follo wing theorem. The proof is similar to th at above and so we omit it. Theorem 5. Let G be a directed graph. Th er e is an extension of G with no mor e than p new edges so that the r esulting graph has no more than k critical edges if and only if the pr ogram Q ∪ D has an epistemic stable mode l. For another example we consider the unique model problem : gi ven a CNF form ula F , the goa l is to d ecide wheth er F ha s a uniqu e minimal m odel. The uniq ue model problem w as a lso considered by Faber and W o ltran [4]. W e will sho w tw o en codings of the problem by mean s of epistemic pr ograms. The first on e uses the semantics of epistemic mo dels and is especially direct. The other on e uses th e semantics of ep istemic stable models. Let F be a pr oposition al th eory consisting of con straints L 1 ∧ . . . ∧ L k → ⊥ , where L i ’ s are liter als. Any prop ositional theory can be rewritten into an e quiv alent theory of such f orm. W e d enote by F K the formula obta ined fro m F by replacing every atom x with the modal atom K x . Theorem 6. F or every theory F ⊆ L consisting of constraints, F has a lea st mode l if and only if the epistemic pr ogram F ∪ F K has an epistemic model. Proof: Let us assume th at F has a least model. W e d efine A to consist of all mod els of F , and we deno te the least model of F by M . W e will show that A is an ep istemic model of F ∪ F K . Clearly , for every x ∈ M , A | = K x . Similarly , for every x 6∈ M , A | = ¬ K x . Thus, [ F K ] A = ∅ . Con sequently , [ F ∪ F K ] A = F an d so, A is p recisely the set of all models of [ F ∪ F K ] A . Thus, A is an epistemic model. Con versely , let A b e an epistemic mo del o f F ∪ F K . It f ollows that [ F K ] A = ∅ (otherwise, [ F ∪ F K ] A contains ⊥ and A would have to be empty , contradicting the definition o f an epistem ic mod el). Thus, [ F ∪ F K ] A = F and consequently , A is the set of all models of F . L et M = { x ∈ At | A | = K x } and let a 1 ∧ . . . ∧ a m ∧ ¬ b 1 ∧ . . . ∧ ¬ b n → ⊥ (3) be a rule in F . The n, K a 1 ∧ . . . ∧ K a m ∧ ¬ K b 1 ∧ . . . ∧ ¬ K b n → ⊥ is a rule in F K . As [ F K ] A = ∅ , A 6| = K a 1 ∧ . . . ∧ K a m ∧ ¬ K b 1 ∧ . . . ∧ ¬ K b n . Thus, for some i , 1 ≤ i ≤ m , A 6| = K a i , or for some j , 1 ≤ j ≤ n , A | = K b j . In the first case, a i / ∈ M , in the latter , b j ∈ M . I n either case, M is a model of r ule (3). It follows that M is a mo del of F . L et M ′ be a m odel of F . Th en M ′ ∈ A an d, by the definition of M , M ⊆ M ′ . Thus, M is a least model of F . ✷ Next, we will encod e th e same prob lem as an epistemic progr am unde r the epistemic stable model sema ntics. The idea is quite similar . W e only need to add rules to ge nerate all candidate models. Theorem 7. F or every theory F ⊆ L consisting of constraints, F has a lea st mode l if and only if the epistemic pr ogram F ∪ F K ∪ {¬ x → x ′ | x ∈ At } ∪ { ¬ x ′ → x | x ∈ A t } has an epistemic stable model. W e n ote that an even simp ler enco ding can be obtained if we use lparse ch oice rules. In th is case, we ca n replace {¬ x → x ′ | x ∈ At } ∪ { ¬ x ′ → x | x ∈ At } with {{ x } | x ∈ At } . 6 Connection to Gelfond ’ s Epistemic Programs W e will n ow return to the original for malism of epistemic specification s pro posed by Gelfond [3] (u nder th e restriction to epistemic programs we discussed here). W e will show t hat it can be exp ressed in a rather direct way in terms of our epistemic program s in the two-valued setting an d under the epistemic supported -model sema ntics. The red uction we are abo ut to d escribe is similar to the well-known one used to eliminate the “strong ” negation from disju nctiv e logic program s with stro ng negation. In par ticular , it requires an exten sion to the lan guage L . Specifically , for every atom x ∈ At we intro duce a f resh atom x ′ and we denote the extended language by L ′ . Th e intended role of x ′ is to represent in L ′ the literal ¬ x fro m L . Building on this idea, we assign to each set W of literals in L the set W ′ = ( W ∩ At ) ∪ { x ′ | ¬ x ∈ W } . In th is way , sets of literals f rom L (in par ticular, three-valued interpretations of L ) are represented as sets of atoms from L ′ (two-valued inte rpretation s of L ′ ). W e n ow n ote that th e truth and falsity of a form ula form L un der a three- valued interpretatio n can be expressed as the truth and falsity of cer tain f ormulas fr om L ′ in the two-valued settin g. The following r esult is well known. Proposition 2. F or e very formula ϕ ∈ L there ar e formulas ϕ − , ϕ + ∈ L ′ such tha t fo r every set of liter als W (in L ) 1. v W ( ϕ ) = t if and o nly if u W ′ ( ϕ + ) = t 2. v W ( ϕ ) = f if an d only if u W ′ ( ϕ − ) = f Mor eover , the fo rmulas ϕ − and ϕ + can be constructed in po lynomial time with r espect to the size of ϕ . Proof: This a folklo re result. W e p rovide a sk etch of a proo f for the completeness sake. W e define ϕ + and ϕ − by recursively as fo llows: 1. x + = x and x − = ¬ x ′ , if x ∈ At 2. ( ¬ ϕ ) + = ¬ ϕ − and ( ¬ ϕ ) − = ¬ ϕ + 3. ( ϕ ∨ ψ ) + = ϕ + ∨ ψ + and ( ϕ ∨ ψ ) − = ϕ − ∨ ψ − ; the case of the conju nction is dealt with analog ously 4. ( ϕ → ψ ) + = ϕ − → ψ + and ( ϕ → ψ ) − = ϕ + → ψ − . One can check that formu las ϕ + and ϕ − defined in this way satisfy the assertion. ✷ W e will no w define the tran sformation σ th at a llows us to eliminate strong negation. First, for a literal L ∈ L , we n ow defin e σ ( L ) =  x if L = x x ′ if L = ¬ x Furthermo re, if E is a simp le modal literal or its default negation, we define σ ( E ) =        K ϕ + if E = K ϕ ¬ K ϕ − if E = ¬ K ϕ ¬ K ϕ + if E = not K ϕ K ϕ − if E = not ¬ K ϕ and for an epistemic premise E = E 1 , . . . , E t (where each E i is a simple modal literal or its default negation) we set σ ( E ) = σ ( E 1 ) ∧ . . . ∧ σ ( E t ) . Next, if r is an epistemic rule L 1 ∨ . . . ∨ L k ← F 1 , . . . , F m , n ot F m +1 , . . . , not F n , E we define σ ( r ) = σ ( E ) ∧ σ ( F 1 ) ∧ . . . ∧ σ ( F m ) ∧ ¬ σ ( F m +1 ) ∧ . . . ∧ ¬ σ ( F n ) → σ ( L 1 ) ∨ . . . ∨ σ ( L k ) . Finally , for an epistemic program P , we set σ ( P ) = { σ ( r ) | r ∈ P } ) ∪ { x ∧ x ′ → ⊥ } . W e note th at σ ( P ) is indeed an epistemic program in the langu age L K (accord ing to our definition of epistemic programs). The role of t he rules x ∧ x ′ → ⊥ is to ensure that sets fo rming epistem ic (stable, sup ported) mod els of σ ( P ) correspo nd to consistent sets of literals (the only type of set of literals allowed in world views). Giv en a three -valued possible structure A , we define A ′ = { W ′ | W ∈ A} , and we regard A ′ as a two-valued possible-world structur e. W e now have the f ollowing theorem. Theorem 8. Let P b e an epistemic pr ogram according to Gelfon d. Then a three-valued possible-world structure A is a world view of P if and only if a two-valued possible- world structur e A ′ is an epistemic suppo rted model of σ ( P ) . Proof (Sketch): L et P be an ep istemic prog ram according to Gelfond , A a po ssible- world struc ture and W a set of literals. W e first observe that the G- reduct P hA ,W i can be descr ibed as the result o f a certain two-step pr ocess. Namely , we define the ep istemic r ed uct o f P with respect to A to be the d isjunctive log ic program P A obtained from P by removing ev ery rule whose epistemic prem ise E satisfies A 6| = E , and b y removing the epistemic p remise from every other ru le in P . This con struction is the thre e-valued counterp art to the one we employ in ou r a pproach . It is clear that the e pistemic r educt of P with respect to A , with som e abuse o f n otation we will denote it b y P A , is a disjunctive log ic program with strong negation. Let Q be a d isjunctive pr ogram with stro ng negation and W a set of literals. By the supp-reduct of Q with resp ect to W , R sp ( Q, W ) , w e mean the set o f the heads of all rules who se bodies are satisfied by W (which in the thr ee-valued setting means that ev ery literal in the b ody not in the scope of not is in W , a nd ev ery literal in the b ody in the scope of not is n ot in W ). A consistent set W of litera ls is a sup ported answer set of Q if W ∈ Min ( R sp ( Q, W )) (this is a n atural extension of the definition o f a supported m odel [7,8] to the case of disjuncti ve logic prog rams with stron g n egation; again, we do not regard inconsistent sets of literals as suppo rted answer sets ). Clearly , P hA ,W i = R sp ( P A , W ) . Thus, A is a world view of P accor ding to the definition by Gelfond if and only if A is a co llection of all supported answer sets of P A . W e also n ote that by Proposition 2, if E is an ep istemic p remise, then A | = E if and only if A ′ | = σ ( E ) . It follows that σ ( P A ) = σ ( P ) A ′ . In oth er words, constru cting the epistemic reduct of P with respect to A and then translating the resulting disjunctive logic prog ram with strong negation in to the corr espondin g disjunctive log ic prog ram without strong n egation yields the s ame result as first translating th e epistemic progra m (in the Gelfond’ s system) into our language of epistemic programs and th en compu ting the redu ct with respect to A ′ . W e no te that there is a one-to -one corresponden ce be- tween sup ported an swer sets of P A and supp orted models of σ ( P A ) ( σ , when r estricted to program s co nsisting of rules wit hout epistemic premises, is the standard transform a- tion eliminating stron g negation and preserv ing the stable an d supp orted semantics). Consequently , there is a one-to -one corr esponden ce between supported an swer sets of P A and supported m odels o f σ ( P ) A ′ (cf. our observation above). Thus, A consists of supported answer s ets of P A if and only if A ′ consists of supported m odels of σ ( P ) A ′ . Consequently , A is a w orld v iew of P if and on ly i f A ′ is an epistemic suppo rted mod el of σ ( P ) . ✷ 7 Epistemic Models of Arbitrary Theories So far , we defined the notions of ep istemic models, epistemic stable mo dels and epis- temic supported models only for the case of epistemic programs. Ho wev er , this restric- tion is n ot e ssential. W e recall that the definition of these th ree epistemic semantics consists of two steps. The first step produ ces the red uct o f an ep istemic p rogram P with re spect to a possible-world structure, say A . This re duct hap pens to be (modulo a trivial syntactic transformation ) a standard disjunctive logic program in the language L (no modal atoms anymore). If the set o f models (respectively , stable models, supported models) of the reduct program co incides with A , A is an ep istemic m odel (respectively , epistemic stable or suppo rted mo del) o f P . Howe ver, the con cepts o f a mo del, stab le model and supp orted mode l are defined fo r a rbitrary theories in L . This is obviously well known for the semantics o f models. The stable-model sema ntics was extend ed to the full language L b y Ferrar is [11] and th e suppo rted-mo del semantics by Truszczyn- ski [12]. Thus, th ere is no r eason precludin g the extension o f the defin ition o f the cor- respond ing epistemic ty pes of models to the g eneral case. W e start b e g eneralizing the concept of the reduct. Definition 5. Let T b e an arbitrary theory in L K and let A be a po ssible-world struc- tur e. The epistemic reduct of T with r espect to A , T A in symbols, is the theory obtained fr om T by r eplacing each ma ximal modal atom K ϕ with ⊤ , if A | = K ϕ , an d with ⊥ , otherwise. W e no te that if T is an ep istemic progr am, th is notion of th e reduct does not co incide with the one we discussed before. Indeed, now n o rule is dropp ed an d no modal literals are dr opped ; rather m odal a toms are replaced with ⊤ an d ⊥ . Howe ver , the replacemen ts are executed in such a way as to ensure the same behavior . Sp ecifically , one can show that models, stable models and supported models of the two reducts coincide. Next, we generalize the concepts of the three types of epistemic models. Definition 6. Let T be a n arbitrary theo ry in L K . A possible-world structur e A is a n epistemic model (r espectively , an ep istemic stable mo del , or an ep istemic suppo rted model ) of P , if A is the set of mod els (r espectively , stable models o r s uppo rted models) of M ( P A ) . From the comments we made above, it f ollows that if T is an ep istemic pr ogram, this more g eneral definition yie lds the came notions o f epistemic m odels o f the thr ee types as the earlier one. W e note that ev en in the more gener al setting the c omplexity of reasonin g with epistemic (stable, supported) models remains unchanged. S pecifically , we ha ve the fol- lowing result. Theorem 9. The pr oblem to decide whether an epistemic theory T ⊆ L K has an epis- temic sta ble model is Σ P 3 -complete. The pr oblem to decide whether an epistemic theory T ⊆ L K has a n epistemic model ( epistemic suppo rted mo del, r espectively) is Σ P 2 - complete. Proof(Sketch) : Th e hardness part follows from our earlier results concern ing epistemic progr ams. T o p rove membe rship, we modify Pro position 1, and show a polyno mial time algorithm with a Σ P 2 oracle (NP oracle for the last two p roblems) that decid es, gi ven a propo sitional theory S and a mod al fo rmula K ϕ ( with ϕ ∈ L K and not nec essarily in L ) whether S T ( S ) | = K ϕ (respectively , M ( S ) | = K ϕ , or S P ( S ) | = K ϕ ). ✷ 8 Discussion In this pap er , we propo sed a two-valued formalism of e pistemic theor ies — subsets of the languag e of modal prop ositional logic. W e prop osed a unifor m w ay , in wh ich semantics of propo sitional theories (the classical one as well as n onmon otonic ones: stable and sup ported ) can b e exten ded to the ca se of ep istemic theories. W e sho wed that the semantics o f epistemic suppo rted mod els is closely related to the original sem an- tics o f ep istemic specifications proposed by Gelf ond. Specifically we showed that the original formalism of Gelfond ca n be expressed in a straigh tforward way b y means of epistemic progr ams in ou r sense u nder the semantics of epistemic supported m odels. Essentially all tha t is n eeded is to u se fresh symbols x ′ to represent strong negation ¬ x , and use the n egation o perator of our f ormalism, ϕ → ⊥ or , in the shorthand, ¬ ϕ , to model the default negation n ot ϕ . W e considered in more detail the three semantics mentioned above. Howe ver, o ther semantics may also yield intere sting epistemic counterpar ts. In particular , it is clear that Definition 6 can be u sed also with the m inimal mod el seman tics or with the Faber- Leone-Pfe ifer seman tics [13 ]. Each s emantics gi ves rise to an in teresting epistemic f or- malism that warrants further studies. In logic p rogram ming, eliminating strong ne gation does not result in any loss of the expressiv e power but, at least fo r th e semantics of stable models, disjunctions cannot be co mpiled away in any co ncise way (unless the p olyno mial hierarchy collapses). In the setting of ep istemic pr ograms, the situation is similar . The strong negation can be compiled away . But the availability o f disjunctions in th e heads and th e a vailability of epistemic prem ises in the bo dies of rules are essential. E ach of th ese factors separately brings the complexity one level u p. Moreover , when used together u nder the semantics of epistemic stable models they bring the complexity two levels up . Th is points to the intrinsic importan ce of having i n a kn owledge representation language means to repre- sent indefiniteness in terms of disjunctio ns, and what is known to a pro gram (th eory) — in terms of a modal operator K . 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