s the set of onsequenes. Skeptil onsequenes re more roust in the sense tht they hold in ll possile relities desried y defult theory. All its desirle p

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1 Skeptil Rtionl Extensions Artur Mikitiuk nd Miros lw Truszzynski University of Kentuky, Deprtment of Computer Siene, Lexington, KY 40506{0046, frtur Astrt. In this pper we propose version of defult logi with the following two properties (1) defults with mutully inonsistent justi- tions re never used together in onstruting set of defult onsequenes of theory (2) the resoning formlized y our logi is relted to the trditionl skeptil mode of defult resoning. Our logi is sed on the onept of skeptil rtionl extension. We give hrteriztion results for skeptil rtionl extensions nd n lgorithm to ompute them. We present some omplexity results. Our min gol is to hrterize ses when the lss of skeptil rtionl extensions is losed under intersetion. In the se of norml defult theories our logi oinides with the stndrd skeptil resoning with extensions. In the se of seminorml defult theories our formlism provides desription of the stndrd skeptil resoning with rtionl extensions. 1 Introdution In this pper we investigte version of defult logi with the following two min properties. First, defults with mutully inonsistent justitions re never used together in onstruting set of defult onsequenes of theory. This hs implitions for the dequy of our system to hndle situtions with disjuntive informtion. Seond, the resoning formlized y our logi is losely relted to the trditionl skeptil mode of defult resoning. In the se of norml defult theories it oinides with the stndrd skeptil resoning with extensions. In the se of seminorml defult theories our formlism provides desription of the stndrd skeptil resoning with rtionl extensions. Our logi is dened y mens of xpoint onstrution nd not s the intersetion of extensions, s is usully the se with the skeptil resoning. Hene, our results provide xpoint desription of the stndrd skeptil resoning from norml defult theories nd, in the se of rtionl extensions, from seminorml defult theories. Defult logi, introdued y Reiter [10], is one of the most extensively studied nonmonotoni systems. Severl reent reserh monogrphs oer omprehensive presenttion of theoretil nd prtil spets of defult logi [1, 3, 6]. Defult logi ssigns to defult theory olletion of theories lled extensions. Extensions model ll possile relities" desried y defult theory nd re used s the sis for two modes of resoning rve nd skeptil. In the rve mode, n ritrrily seleted extension denes the set of onsequenes for defult theory. In the skeptil one, the intersetion of ll extensions serves

2 s the set of onsequenes. Skeptil onsequenes re more roust in the sense tht they hold in ll possile relities desried y defult theory. All its desirle properties notwithstnding, there re situtions where defult logi of Reiter is not esily pplile. In prtiulr, defult logi does not hndle well inomplete informtion given in the form of disjuntive luses [9, 2, 4, 7]. To remedy this, severl moditions of defult logi were proposed disjuntive defult logi [4], umultive defult logi [2], onstrined defult logi [11] nd rtionl defult logi [7]. The rst system introdues new disjuntion opertor to hndle "eetive" disjuntion. The ltter three tke into ount, in one wy or nother, the requirement tht defults with mutully inonsistent justitions must not e used in the onstrution of the sme extension. Not surprisingly then, they re somewht relted. Connetions etween umultive defult logi nd onstrined defult logi re studied in [11]. Reltions etween onstrined defult logi nd rtionl defult logi re disussed in [8]. In this pper we ontinue our investigtion of rtionl defult logi introdued in [7]. The key ide ehind the onept of rtionl extension of defult theory (D W ) is tht of mximl set of defults in D tive with respet to theories W nd S. The olletion of ll suh sets (it is lwys nonempty) is denoted y MA(D W S). Intuitively, it ontins every group of defults the resoner n selet to justify tht S is rtionl extension of (D W ) (if none works, S is not rtionl extension). Tht is, S is rtionl extension if S n e derived from W y mens of some set of defults A 2 MA(D W S). In this pper, we strengthen the requirements for hypothetil ontext S to e rtionl extension. As result we otin new xpoint onstrution nd new lss of extensions skeptil rtionl extensions. (The word "extension" is eing used here in roder sense. A rtionl or skeptil rtionl extension of defult theory is not, in generl, n extension of the theory in Reiter's sense see Exmples 1 nd 2.) Speilly, for theory S to e skeptil rtionl extension, S must e extly the set of formuls tht n e derived from W y mens of every set of defults A 2 MA(D W S). In other words, S onsists of those formuls the resoner n justify no mtter whih element from MA(D W S) is seleted for resoning. This motivtes the term skeptil used to designte these extensions. The lss of skeptil rtionl extensions hs severl desirle properties. For mny defult theories, it ontins lest element with respet to inlusion. In suh se, this lest skeptil rtionl extension n e used s forml model of skeptil defult resoning (sometimes identil with nd sometimes dierent from the trditionl model of skeptil defult resoning). In this pper we investigte properties of skeptil rtionl extensions. We restrit ourselves to the propositionl se only. We give hrteriztion results for skeptil rtionl extensions nd n lgorithm to ompute them. We present some omplexity results. Our min gol is to hrterize ses when the lss of skeptil rtionl extensions is losed under intersetion. We otin the strongest results for norml nd seminorml defult theories. We show tht the intersetion of ll rtionl extensions of seminorml defult theory is its lest skeptil rtionl extension. In prtiulr, it mens tht the intersetion of ll extensions

3 of norml defult theory is, in ft, its lest skeptil rtionl extension. 2 Denitions nd Exmples Let L e lnguge of propositionl logi. A defult is ny expression of the form M 1 M k where, i, 1 i k nd re propositionl formuls from L. The formul is lled the prerequisite of d, p(d) in symols. The formuls i, 1 i k, re lled the justitions of d. The set of justitions is denoted y j(d). Finlly, the formul is lled the onsequent of d nd is denoted (d). For olletion D of defults y p(d), j(d) nd (D) we denote, respetively, the sets of ll prerequisites, justitions nd onsequents of the defults in D. A defult of the form M ( M(^), resp.) is lled norml (seminorml, resp.). A defult theory is pir (D W ), where D is set of defults nd W is set of propositionl formuls. A defult theory (D W ) is norml (seminorml, resp.) if ll defults in D re norml (seminorml, resp.). A defult theory (D W ) is nite if oth D nd W re nite. For set D of defults nd for propositionl theory S, we dene nd D S = M 1 M k 2 D nd S 6` i 1 i k Mon(D) = p(d) (d) d 2 D Given set of inferene rules A, y Cn A () we men the onsequene opertor of the forml proof system onsisting of propositionl lulus nd the rules in A (it is dened for ll theories in the lnguge). The key notion of (stndrd) defult logi is the notion of n extension 1. A theory S is n extension for defult theory (D W ) if S = Cn DS (W ). For detiled presenttion of defult logi the reder is referred to [6]. In [7] we introdued the notions of n tive set of defults nd rtionl extension of defult theory. A set A of defults is tive with respet to sets of formuls W nd S if it stises the following onditions AS1 j(a) =, or j(a) [ S is onsistent, AS2 p(a) Cn AS (W ). The set of ll susets of set of defults D whih re tive with respet to W nd S will e denoted y A(D W S). Oserve, tht is tive with respet to every W nd S. Hene, A(D W S) is lwys nonempty. By the Kurtowski-Zorn Lemm, every A 2 A(D W S) is 1 Our denition is dierent from ut equivlent to the originl denition of Reiter [10].

4 ontined in mximl (with respet to inlusion) element of A(D W S) (see [7]). Dene MA(D W S) to e the set of ll mximl elements in A(D W S). In [7], we dened S to e rtionl extension for defult theory (D W ) if S = Cn AS (W ) for some A 2 MA(D W S). We will now dene the notion of skeptil rtionl extension. Denition 1. A theory S is skeptil rtionl extension for defult theory (D W ) if S = Cn AS (W ) 2 A2MA(DWS) We will illustrte the notions dened ove with severl exmples. The rst exmple exhiits defult theory whih does not hve n extension or rtionl extension ut hs skeptil rtionl extension. In ll exmples,, nd d stnd for distint propositionl toms. Exmple 1. Let D = f M g. The defult theory (D ) hs no extension nd no rtionl extension. On the other hnd, S = Cn(f _ g) is its skeptil rtionl extension. Indeed, we hve nd M MA(D S) = M M A2MA(DS) The defult theory (f M g ) is lssil exmple of theory without extensions. More generlly, defult theory ontining the defult M, where Cn AS () = Cn(fg) Cn(fg) = Cn(f _ g) = S is n tom tht does not pper in ny other defult or formul, does not hve n extension. Hene, the ft tht Cn(f _ g) is skeptil rtionl extension of the defult theory of Exmple 1 my seem ounterintuitive. However, the mening of the defults in D is if (, resp.) is possile, then onlude (, resp.). In the ontext of Cn(f _ g), ny of the two defults n re (ut not together). Hene, no mtter wht is the hoie, _ follows. The next exmple shows tht there re lso defult theories whih hve extensions ut do not hve skeptil rtionl extensions. Exmple 2. Let us onsider the defult theory (D W ), where W = f _ g nd D = M M d M( _ d) ^ d This theory hs unique extension Cn(f_ dg). We proved in [7] tht (D W ) does not hve rtionl extensions. Assume tht S is skeptil rtionl extension for (D W ). Then _ 2 S. If ^ d =2 S then M MA(D W S) = M( _ d) ^ d M d M( _ d) ^ d 2

5 T Thus, A2MA(DWS) (W ) = L nd S 6= L (euse ^d 62 S). So, ssume CnAS tht ^ d 2 S. Then MA(D W S) = M M d nd T A2MA(DWS) CnAS (W ) = Cn(f dg) 6= S (euse ^d =2 Cn(f dg)). Hene, (D W ) does not hve skeptil rtionl extensions. 2 One of the properties we re espeilly interested in here is losure under intersetion of the fmily of skeptil rtionl extensions. The following exmple presents defult theory for whih the fmily of skeptil rtionl extensions is losed under intersetion. This theory is norml. We will lter show tht this property holds for every norml defult theory with nite numer of extensions. Exmple 3. Let W = nd M D = M M Let S 1 = Cn(f g), S 2 = Cn(f g) nd S = S 1 S 2 = Cn(fg). Then MA(D W S 1 ) = M nd M M MA(D W S 2 ) = M M M MA(D W S) = M M Clerly, S 1 nd S 2 re extensions, rtionl extensions nd skeptil rtionl extensions for (D W ) nd S is lso skeptil rtionl extension for (D W ). 2 For some defult theories the fmily of their skeptil rtionl extensions is not losed under nite intersetion. Exmple 4. Let W = nd M Md D = M( _ ) Md ^ Md M Let S 1 = Cn(fg), S 2 = Cn(fg) nd S = S 1 S 2 = Cn(f _ g). Then MA(D W S 1 ) = M Md M Md MA(D W S 2 ) = M( _ ) Md ^ M( _ ) Md ^ Md M nd MA(D W S) = MA(D W S 2 ). It is esy to see tht S 1 nd S 2 re skeptil rtionl extensions for (D W ) while S is not. This defult theory does not hve lest skeptil rtionl extension. Let us note tht S 1 nd S 2 re lso rtionl extensions for (D W ). Let S 3 = Cn(f g). We hve MA(D W S 3 ) = MA(D W S 1 ). Hene, S 3 is lso rtionl extension of (D W ). Finlly, it is esy to see tht S 3 is the only (Reiter's) extension for (D W ). 2

6 We onlude this setion with n lterntive hrteriztion of tive sets. Proposition 2. A set A of defults is tive with respet to sets of formuls W nd S if nd only if it stises AS1 nd the following ondition AS2 0 p(a) Cn Mon(A) (W ). 2 3 Generl Properties In this setion we present some results (Theorems 8 nd 9, Corollry 10) providing suient onditions for the intersetion of skeptil rtionl extensions to e skeptil rtionl extension too. These results will e used in Setions 5 nd 6. We strt with severl uxiliry oservtions. (Simple proofs of Lemms 3, 4 nd 6 re omitted due to spe restrition.) Lemm 3. Let (D W ) e defult theory. Let S e set of formuls nd let A 2 A(D W S). Then A 2 A(D W T ) for every theory T suh tht A stises AS1 for T. 2 Lemm 4. Let (D W ) e defult theory. Let S nd T e theories suh tht S T. If A 2 MA(D W S) nd A 2 A(D W T ), then A 2 MA(D W T ). 2 Lemm 5. Let (D W ) e defult theory nd let S = T k i=1 S i (k 1), where eh theory S i is losed under propositionl provility. Then MA(D W S) S k i=1 MA(D W S i ). Proof. Let A 2 MA(D W S). Then A stises AS1 for S. Thus, j(a) = or j(a) [ S is onsistent. If j(a) = then for every i (1 i k), A stises AS1 for S i. Let us ssume now tht j(a) [ S is onsistent. We hve We will show tht Cn(j(A) [ S) = Cn(j(A) [ Cn( k i=1 k i=1 (j(a) [ S i )) = S i ) = Cn( k i=1 k i=1 (j(a) [ S i )) (1) Cn(j(A) [ S i ) (2) Clerly, the left-hnd side of (2) is ontined in the right-hnd T side. So, we need k to prove only the onverse inlusion. Consider formul ' 2 Cn(j(A)[S i=1 i). For every i (1 i k), ' is provle from j(a) [ S i. By the Comptness Theorem, for every i, there is nite suset S 0 i of S i suh tht ' is provle from j(a) [ S 0 i. Let ' i e the onjuntion of ll formuls from S 0 i (1 i k). Then ' is provle from j(a) [ f' i g. Sine S i is losed under propositionl onsequenes, T ' i 2 S i. Consequently, T T k ' 1 ' k 2 S i=1 i. Let T v e vlution k stisfying (j(a) [ S k i=1 i). Sine (j(a) [ S k i=1 i) = j(a) [ S i=1 i, it follows

7 tht v stises j(a) nd v stises ' 1 ' k. Hene, v stises j(a) [ f' i g for some T i (1 i k). Sine ' is provle from j(a) [ f' i g, v stises '. Thus, k ' 2 Cn( (j(a) [ S i=1 i)). It follows from (1) nd (2) tht if j(a) [ S is onsistent then for some i (1 i k) j(a) [ S i is onsistent. Hene, in oth ses (j(a) =, or j(a) [ S is onsistent) A stises AS1 for some S i (1 i k). By Lemm 3, A 2 A(D W S i ) for some i (1 i k). Sine S S i, then y Lemm 4, A 2 MA(D W S i ) for some i (1 i k) nd we re done. 2 We will denote y GD(D S) the set of generting defults from D with respet to S, tht is, GD(D S) = M1 M k 2 D S ` nd S 6` i 1 i k Lemm 6. Let theory S e n extension of defult theory (D W ) nd let A 2 A(D W S). Then A GD(D S). In prtiulr, if GD(D S) 2 A(D W S) then MA(D W S) = fgd(d S)g. 2 Exmple 4 indites tht the notions of n extension, rtionl extension nd skeptil rtionl extension re, in generl, dierent. However, under some onditions they oinide. One suh sitution is desried in our rst theorem (the proof is omitted due to spe restrition). Theorem 7. Let (D W ) e defult theory nd let S e propositionl theory suh tht MA(D W S) = fgd(d S)g. Then S is n extension of (D W ) if nd only if S is rtionl extension of (D W ) if nd only if S is skeptil rtionl extension of (D W ). 2 The next severl results desrie onditions whih gurntee tht the intersetion of skeptil rtionl extensions is lso skeptil rtionl extension. Theorem 8. Let fs i i 2 Ig e set of skeptil rtionl extensions for defult theory (D W ), let S = T i2i S i nd MA(D W S) = S i2i MA(D W S i ). Then S is skeptil rtionl extension for (D W ). Proof. We hve i2i A2MA(DWS) Cn AS (W ) = A2MA(DWS i) Cn Mon(A) (W ) = i2i A2MA(DWS) A2MA(DWS i) Cn Mon(A) (W ) = Cn AS i (W ) = S i = S Theorem 9. Let fs i i 2 Ig e set of extensions for defult theory (D W ) suh tht for every i 2 I, MA(D W S i ) = fgd(d S i )g. Let S = T i2i S i nd MA(D W S) S i2i MA(D W S i ). Then S is skeptil rtionl extension for (D W ). i2i 2

8 Proof. Every S i is skeptil rtionl extension for (D W ) (Theorem 7). By Theorem 8, to prove the ssertion it sues to show tht S i2i MA(D W S i ) MA(D W S). Let A 2 MA(D W S i ) for some i 2 I. Hene, A = GD(D S i ). Sine S S i, A stises AS1 for S. By Lemm 3, A 2 A(D W S). There is B 2 MA(D W S) suh tht A B. Aording to the ssumption, B 2 MA(D W S j ) for some j 2 I, tht is, B = GD(D S j ). Sine A = GD(D S i ), we hve GD(D S i ) GD(D S j ). Thus, Cn(W [ (GD(D S i ))) Cn(W [ (GD(D S j ))). By Theorem 3.57 in [6], we hve S i = Cn(W [(GD(D S i ))) nd S j = Cn(W [ (GD(D S j ))), so we get S i S j. Sine S i nd S j re extensions of the sme defult theory, S i = S j nd A = B. Hene, A 2 MA(D W S). 2 Lemm 5 nd Theorem 9 imply the following orollry. Corollry 10. Let S 1 S k (k 1) e extensions of defult theory (D W ) suh tht for every i (1 i k), MA(D W S i ) = fgd(d S i )g. Then S = T k i=1 S i is skeptil rtionl extension for (D W ). 2 Oserve tht even though in Theorem 9 nd Corollry 10 we ssume tht sets S i re extensions, y Theorem 7 every S i is lso rtionl extension nd skeptil rtionl extension for (D W ). 4 Algorithmi Issues Proposition 11. Let S e skeptil rtionl extension for defult theory (D W ) suh tht D is nite. Then S = Cn(W [ f' 1 ' k g), where every ' i = V (A i ) for some A i 2 MA(D W S). Proof. Sine D is nite, MA(D W S) is nite, s well. Let us ssume tht MA(D W S) = fa 1 A k g. For eh A i 2 MA(D W S) dene ' i = V (A i ) (sine eh A i is nite, ' i is well-dened). Sine A i 2 A(D W S), Cn (Ai)S (W ) = Cn(W [ (A i )) = Cn(W [ f' i g). Hene, S = k i=1 Cn (Ai)S (W ) = k i=1 Cn(W [ f' i g) = Cn(W [ f' 1 ' k g) 2 If (D W ) is nite then the numer of sets of the form Cn(W [ f' 1 ' k g), where every ' i is of the form V (A) for some A D, is lso nite. For every suh set S, one n ompute MA(D W S) nd hek whether S = T A2MA(DWS) CnAS (W ). Thus, we hve the following lgorithm for omputing skeptil rtionl extensions. 1. For every A D, ompute ' A = V (A) (' = >). Let = f' A A Dg. 2. For every () ompute = W, () for every A D, verify whether A 2 MA(D W W [ f g), () ompute ' = W A2MA(DWW[f g) ' A,

9 (d) hek whether W [f'g ` nd W [f g ` ' if so, output Cn(W [f g) s skeptil rtionl extension for (D W ). The following exmple shows tht there re defult theories (D W ) nd sets S suh tht the size of MA(D W S) is exponentil in the size of D. It follows tht n lgorithm for verifying whether S is skeptil rtionl extension of (D W ) must hve in the worst se n exponentil omplexity. Exmple 5. Let us onsider the defult theory (D W ) where Mp1 D = Mp 1 Mp n Mp n p 1 p 1 p n p 1 p n re distint propositionl toms nd W =. Then MA(D W Cn()) hs 2 n elements, eh of them otined y seleting extly one defult from eh pir Mpi p i Mpi p i. 2 The omplexity of resoning with skeptil rtionl extensions in the generl se remins n open prolem. p n 5 Seminorml Defult Theories In this setion we study skeptil rtionl extensions of seminorml defult theories. We show tht every seminorml defult theory hs lest skeptil rtionl extension nd tht it oinides with the intersetion of ll rtionl extensions. Our rst min result of this setion shows tht every skeptil rtionl extension of seminorml defult theory n e represented s the intersetion of ertin numer (possily innitely mny) of rtionl extensions. Theorem 12. For every skeptil rtionl extension S of seminorml defult theory (D W ) there is set fs i i 2 Ig of rtionl extensions for (D W ) suh tht S = T i2i S i. Proof. Let S e skeptil rtionl extension for (D W ). Consequently, we hve S = T A2MA(DWS) CnAS (W ). Let MA(D W S) = fa i i 2 Ig nd let us denote S i = Cn (Ai)S (W ) (i 2 I). Then S = T i2i S i. We will show tht eh S i is rtionl extension for (D W ). Sine A i 2 MA(D W S), A i stises AS1 for S. Hene, j(a i ) = or j(a i ) [ S is onsistent. If j(a i ) = then A i stises AS1 for S i. If j(a i ) [ S is onsistent then, sine W S, j(a i ) [ W is onsistent. Sine ll defults in A i re seminorml, j(a i ) implies (A i ). It follows tht j(a i ) [ (A i ) [ W is onsistent. Sine A i 2 MA(D W S), Cn (Ai)S (W ) = Cn(W [ (A i )). Hene, S i = Cn(W [ (A i )). It follows tht j(a i ) [ S i is onsistent. Consequently, A i stises AS1 for S i. Thus, in oth ses (j(a i ) =, or j(a i ) [ S is onsistent) A i stises AS1 for S i. By Lemm 3, A i 2 A(D W S i ). Sine S S i, then y Lemm 4, A i 2 MA(D W S i ). It follows tht (A i ) Si = Mon(A i ) = (A i ) S. Sine

10 S i = Cn (Ai)S (W ), we hve S i = Cn (Ai)S i (W ). Hene, S i is rtionl extension of (D W ). 2 In [6] tehnique for onstruting n extension of defult theory from n ordering of defults ws presented nd thoroughly studied. In [7] we dpted this tehnique to the se of rtionl extensions. We will use some properties of this onstrution in the proof of the seond min result of this setion. The reder is referred to [6, 7] for detils. We ssume tht the set of the toms of our lnguge L is denumerle. Consequently, the set of ll defults over the lnguge L is denumerle. Let (D W ) e defult theory nd well-ordering of D. We dene n ordinl. For every ordinl < we dene set of defults AD nd defult d. We lso dene set of defults AD. We proeed s follows If the sets AD, <, hve een dened ut hs not een dened then 1. If there is no defult d 2 D n S < AD suh tht () j(d) = or W [ ( S < AD ) [ j( S < AD ) [ j(d) is onsistent, nd () W [ ( S < AD ) ` p(d), then =. 2. Otherwise, dene d to e the -lest defult d 2 D n S < AD suh tht the onditions () nd () ove hold. Then set AD = S < AD [ fd g. When the onstrution termintes, put AD = S < AD. The theory Cn(W [ (AD)) will e lled generted y the well-ordering. We will need the following property of this onstrution. Theorem 13. (extended version of [7]) Let (D W ) e seminorml defult theory nd let e well-ordering of D. Then T = Cn(W [ (AD)) is rtionl extension for (D W ). Moreover, AD 2 MA(D W T). 2 It follows from this theorem tht every seminorml defult theory hs rtionl extension. In the proof of our next result we will lso need the following proposition. Proposition 14. (extended version of [7]) Let (D W ) e defult theory nd let S nd T e rtionl extensions of (D W ) suh tht S = Cn AS (W ) for some A 2 MA(D W S), T = Cn BT (W ) for some B 2 MA(D W T ) nd A B. Then A = B nd S = T. 2 Now we re redy to present the seond min result of this setion. Theorem 15. The intersetion of ll rtionl extensions of seminorml defult theory is the lest skeptil rtionl extension for this theory. Proof. Let fs i i 2 Ig e the set of ll rtionl extensions of seminorml defult theory (D W ) nd let S = T i2i S i. By Theorem 12, we need only to prove tht S is skeptil rtionl extension for (D W ). Let A 2 MA(D W S). Let us onsider ny well-ordering of D in whih the defults in A preede ll other defults. Assume lso tht the defults of A

11 re ordered y ording to the order in whih their orresponding monotoni inferene rules re pplied in the proess of omputing Cn AS (W ). It is esy to see tht A AD. Sine the theory (D W ) is seminorml, the theory generted y, Cn(W [ (AD)), is rtionl extension for (D W ) (Theorem 13), tht is, Cn(W [ (AD)) = S i for some i 2 I. Moreover, 2 MA(D W S i ). AD stises AS1 for S i. Hene, j(ad) = or j(ad) [ S i is onsistent. If j(ad) = then = (every defult in D hs justition). Sine A AD =, A = AD. If j(ad) [ S i is onsistent then, sine S S i, j(ad) [ S is onsistent. By Lemm 3, AD 2 A(D W S). By the mximlity of A, we get A = AD. Hene, in oth ses (j(ad) =, or j(ad) [ S i is onsistent) A S = AD, tht is, A 2 MA(D W S i ) for some i 2 I. Thus, MA(D W S) i2i MA(D W S i ). Moreover, we proved tht for every A 2 MA(D W S) there is i 2 I suh tht Cn AS (W ) = Cn(W [ (A)) = S i nd A 2 MA(D W S i ) (3) Thus, Cn AS (W ) = A2MA(DWS) i2i 0 S i for some I 0 I. We will show tht I 0 = I, tht is, tht for every i 2 I, there is B 2 MA(D W S) suh tht S i = Cn BS (W ). Sine S i is rtionl extension for (D W ), there is B 2 MA(D W S i ) suh tht S i = Cn BS i (W ). Sine S S i, then y Lemm 3, B 2 A(D W S). Hene, there is C 2 MA(D W S) suh tht B C. By (3), there is j 2 I suh tht C 2 MA(D W S j ) nd Cn CS (W ) = S j. It is esy to see tht C Sj = Mon(C) = C S. Hene, S j = Cn CS j (W ). By Proposition 14, B = C, tht is, B 2 MA(D W S). Moreover, B S = Mon(B) = B Si. Thus, S i = Cn BS i (W ) = Cn BS (W ). Hene, we hve shown tht I 0 = I, tht is, T A2MA(DWS) CnAS (W ) = S. Thus, S is skeptil rtionl extension for (D W ). 2 Corollry 16. Every seminorml defult theory hs skeptil rtionl extension. 2 Exmple 2 shows tht Corollry 16 is not true for generl defult theories. Theorem 15 shows tht the intersetion of ll rtionl extensions is skeptil rtionl extension. This is not true for n ritrry fmily of rtionl extensions of seminorml defult theory, even if the theory is nite (f. Theorem 20). Exmple 6. Let M( ^ ) D = M( ^ ) M( ^ ) The defult theory (D ) is lssil exmple of seminorml defult theory without extensions. This theory hs three rtionl extensions S 1 = Cn(fg), S 2 = Cn(fg) nd S 3 = Cn(fg). Aording to Theorem 15, their intersetion S = S 1 S 2 S 3 = Cn(f g) is skeptil rtionl extension for (D ).

12 However, the intersetions of ny two rtionl extensions, S 12 = S 1 S 2 = Cn(f _ g), S 13 = S 1 S 3 = Cn(f _ g) nd S 23 = S 2 S 3 = Cn(f _ g), re not skeptil rtionl extensions. Indeed, it is esy to see tht for ny i j (1 i < j 3), MA(D S ij ) = M( ^ ) Hene, A2MA(DS ij) M( ^ ) M( ^ ) Cn AS ij () = Cn(f g) = S 6= Sij Let us lso oserve tht none of S 1, S 2, S 3 is skeptil rtionl extension. 2 Theorem 15 implies the following orollry. Corollry 17. A formul ' elongs to ll skeptil rtionl extensions of seminorml defult theory (D W ) if nd only if ' elongs to ll rtionl extensions of (D W ). 2 The omplexity of resoning with rtionl extensions ws studied in [7]. In prtiulr, we proved tht the prolem of deiding whether formul elongs to ll rtionl extensions of nite defult theory is P 2 -omplete. It remins P 2 -omplete even under the restrition to the lss of norml defult theories. Sine every norml defult theory is seminorml, we otin the following result. Corollry 18. The prolem IN-ALL Given nite seminorml defult theory (D W ) nd formul ', deide if ' is in ll skeptil rtionl extensions of (D W ), is P 2 -omplete. 2 The omplexity of the prolem of deiding whether formul elongs to t lest one skeptil rtionl extension of seminorml defult theory remins open. The rgument we used for the prolem IN-ALL does not work here. 6 Norml Defult Theories The results otined in the previous setion for seminorml defult theories lerly extend to the se of norml defult theories. In this se, however, we n still strengthen some of them. We strt this setion with simple proposition. Proposition 19. Let S e n extension of norml defult theory (D W ). Then MA(D W S) = fgd(d S)g. Proof. If S is inonsistent then MA(D W S) = fg nd GD(D S) =, so the ssertion holds. Assume now tht S is onsistent. Aording to Lemm 6, it is suient to prove tht GD(D S) 2 A(D W S) nd this ft is proven in the proof of Theorem 3.1 in [7]. 2

13 The min result of this setion shows tht nite intersetions of extensions (or rtionl extensions - for norml defult theories these notions oinide, see [7]) re skeptil rtionl extensions. In prtiulr, (rtionl) extensions re lso skeptil rtionl extensions for norml defult theories (unlike in the se of seminorml ones). Theorem 20. Let (D W ) e norml defult theory. 1. Let S 1,,S k (k 1) e extensions of (D W ). Then S = T k i=1 S i is skeptil rtionl extension for (D W ). 2. Every skeptil rtionl extension of (D W ) n e represented s the intersetion of ertin numer (possily innitely mny) of extensions. Proof. The rst ssertion follows from Proposition 19 nd Corollry 10. The seond ssertion follows from Theorem 12 nd from the ft tht for norml defult theories extensions nd rtionl extensions oinide. 2 Corollry 21. Let (D W ) e norml defult theory with nite numer of extensions. Then the fmily of ll skeptil rtionl extensions of (D W ) is losed under intersetion. In prtiulr, the fmily of ll skeptil rtionl extensions of nite norml defult theory is losed under intersetion. 2 The question whether the intersetion of n ritrry olletion of extensions of norml defult theory is skeptil rtionl extension remins open. However, Theorem 15 nd the ft tht for norml defult theories extensions oinide with rtionl extensions imply weker result. Theorem 22. The intersetion of ll extensions of norml defult theory is the lest skeptil rtionl extension for this theory. 2 The following exmple shows tht Theorem 22 is not true for seminorml defult theories. Exmple 7. Let D = M M M( ^ ) The defult theory (D ) hs one extension S 1 = Cn(f g). This theory hs two rtionl extensions S 1 nd S 2 = Cn(f g). It hs lso two skeptil rtionl extensions S 1 nd S 3 = Cn(f _g). Hene, S 1 is the intersetion of ll extensions while S 3 is the lest skeptil rtionl extension (whih, ording to Theorem 15, oinides with the intersetion of ll rtionl extensions) nd S 1 6= S 3. 2 Theorems 20 nd 22 imply the following orollry. Corollry 23. A formul ' elongs to some (resp. ll) skeptil rtionl extension(s) of norml defult theory (D W ) if nd only if ' elongs to some (resp. ll) extension(s) of (D W ). 2

14 Using Corollry 23 nd the results from [5] on the omplexity of the prolems IN-SOME nd IN-ALL for extensions of norml defult theories we get the following omplexity result. Corollry 24. The prolem IN-SOME Given nite norml defult theory (D W ) nd formul ', deide if ' is in some skeptil rtionl extension of (D W ), is P 2 -omplete. The prolem IN-ALL Given nite norml defult theory (D W ) nd formul ', deide if ' is in ll skeptil rtionl extensions of (D W ), is P 2 -omplete. 2 7 Conlusions In this pper we proposed new version of defult logi. It is sed on the onept of skeptil rtionl extension. We showed tht in the se of norml defult theories our version of defult logi oinides with the stndrd skeptil resoning with extensions. In the se of seminorml defult theories it oinides with the stndrd skeptil resoning with rtionl extensions. We presented some generl properties of skeptil rtionl extensions, n lgorithm to ompute them nd some omplexity results. However, the omplexity of resoning with skeptil rtionl extensions from ritrry defult theories is n open prolem. Referenes 1. P. Besnrd. An introdution to defult logi. Springer-Verlg, Berlin, G. Brewk. Cumultive defult logi in defense of nonmonotoni inferene rules. Artiil Intelligene, 50183{205, G. Brewk. Nonmonotoni resoning logil foundtions of ommonsense. Cmridge University Press, Cmridge, UK, M. Gelfond, V. Lifshitz, H. Przymusinsk, nd M. Truszzynski. Disjuntive defults. In Seond interntionl onferene on priniples of knowledge representtion nd resoning, KR '91, Cmridge, MA, G. Gottlo. Complexity results for nonmonotoni logis. Journl of Logi nd Computtion, 2397{425, W. Mrek nd M. Truszzynski. Nonmonotoni logis ontext-dependent resoning. Berlin Springer-Verlg, A. Mikitiuk nd M. Truszzynski. Rtionl defult logi nd disjuntive logi progrmming. In A. Nerode nd L. Pereir, editors, Logi progrmming nd nonmonotoni resoning, pges 283{299. MIT Press, A. Mikitiuk nd M. Truszzynski. Constrined nd rtionl defult logi. In preprtion, D. Poole. Wht the lottery prdox tells us out defult resoning. In Proeedings of the 2nd onferene on priniples of knowledge representtion nd resoning, KR '89, pges 333{340, Sn Mteo, CA., Morgn Kufmnn. 10. R. Reiter. A logi for defult resoning. Artiil Intelligene, 1381{132, T. Shu. Considertions on Defult Logis. PhD thesis, Tehnishen Hohshule Drmstdt, 1992.