[MT93a, ALS94, Nie95, MNR95]. All these algorithms exploit kow characterizatios of extesios of default theories i terms of sets of geeratig defaults,

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1 Miimal umber of permutatios suciet to compute all extesios a ite default theory Pawe l Cholewiski ad Miros law Truszczyski Departmet of Computer Sciece Uiversity of Ketucky Lexigto, KY pawel mirek@cs.egr.uky.edu Abstract I this paper we aalyze a algorithm for geeratig extesios of a default theory. This algorithm cosiders all permutatios (orderigs) of defaults. For each permutatio, it costructs a tetative extesio icremetally, i each step rig the rst applicable default, where the applicability of a default is deed with respect to the part of a tetative extesio costructed so far. Whe o more defaults ca be red, a a posteriori cosistecy check is performed to test whether all defaults that were red remai applicable with respect to the al theory. If so, this theory is retured as a extesio. Otherwise, the ext orderig is tried. Straightforward worst case aalysis implies that this algorithm may have to ispect all! permutatios i order to be complete (here is the umber of defaults i a iput default theory). I this paper we show that this umber ca be sigicatly reduced. Namely, we exhibit a set of b=2c 0:8 p 2 = permutatios whose ispectio guaratees that the algorithm will detect all extesios. I additio, we preset a simple algorithm to geerate all these permutatios. 1 Itroductio Default logic itroduced by Reiter [Rei80] is oe of the most widely studied omootoic formalisms. It was proposed as a kowledge represetatio mechaism, as it ofte oers cocise descriptios of various kowledge domais ad commosese reasoig situatios. However, i order to be a full-blow kowledge represetatio tool, algorithmic methods are eeded to process default theories ad compute their extesios. Several algorithms to perform these tasks were proposed recetly 1

2 [MT93a, ALS94, Nie95, MNR95]. All these algorithms exploit kow characterizatios of extesios of default theories i terms of sets of geeratig defaults, sets of justicatios, or orderig of defaults. They search through the space of the appropriate objects ad idetify those that ideed geerate extesios. I the worst case, the size of the search space of each such algorithm is at least expoetial. However, some pruig is usually possible ad results i substatial speedups several experimetal studies are ow uderway ad prelimiary results are beig reported [NS95, CMMT95]. I this paper we will aalyze oe of the basic algorithms to geerate extesios. The algorithm works by cosiderig all orderigs (permutatios) of the set of defaults of a default theory. It was itroduced i [MT93a] ad further studied i [ALS94]. We will show that by some simple modicatios, the size of the search space for q this algorithm ca be reduced from! to b=2c 2= 2 = p 0:8 2 = p (throughout the paper, we assume that the set of defaults i a iput default theory has cardiality ). We will also show that this value is best possible. Hece, the resultig algorithm costructs all extesios of a default theory by cosiderig a search space that at the start of the algorithm's executio is guarateed to have size o(2 ). To the best of our kowledge, this is the oly algorithm so far with such property. Other algorithms start with the full search space cosistig of all subsets of the set of defaults, or of all subsets of the set of justicatios. The size of the former of these search spaces is 2, ad ca be eve larger for the latter oe (although i this latter case, it ca sometimes be smaller). We will start with a brief descriptio of basic algorithms for geeratig extesios. Oly a geeral outlie of the rst two of them will be give here. The third of these algorithms, which is the mai subject of this paper, will be described i detail. Select-defaults-ad-check 1. Select a set of defaults S D. 2

3 2. Check if S is a set of geeratig defaults. If so, output the theory geerated by S as a extesio. 3. Repeat util all subsets of D are cosidered or prued. This is perhaps the most commoly studied algorithm. For its detailed descriptio see [MT93a]. I the versio give above, the algorithm ispects all 2 subsets of D. However, pruig techiques ca be icorporated ito this method to dyamically restrict the search space durig the executio of the program. For example, oce a set S of defaults is foud to be geeratig, its proper subsets ad supersets eed ot be cosidered. This follows from the fact that extesios are icomparable ad therefore their geeratig sets must be icomparable too (see [MT93a] for details). Still, the iitial search space has size 2. Select-justicatios-ad-check 1. Select a set of justicatios J j(d). 2. Fid the set of defaults S whose justicatios belog to j(d). 3. Compute the set of cosequeces E of W that ca be derived by meas of defaults i S (a default \res" if its prerequisite has bee derived earlier). 4. If all justicatios i J are cosistet with E ad every default ot i S has at least oe justicatio ot cosistet with E, the output E as a extesio. 5. Repeat util all subsets of j(d) are cosidered or prued. Theoretical basis for this algorithm ca be traced back to the cocept of a argumetatio framework [Du93, BTK93]. Recetly, Niemela ad Simmos [Nie95, NS95] preseted a detailed descriptio, aalysis ad implemetatio of the method ad performed a experimetal study. They showed that the algorithm performs 3

4 surprisigly well. This ca be attributed mostly to some iterestig pruig techiques used i the implemetatio. However, at the begiig, the algorithm faces the search space of all subsets of the set of all justicatios. The size of this search space ca be smaller tha 2. However, it is easy to give examples of default theories with defaults, for which the set of justicatios has at least elemets ad, cosequetly, the set of all subsets of the set of justicatios has size at least 2. Fially, we will describe, this time i more detail, the algorithm which we study i this paper. The mai advatage of this algorithm is its close relatioship to some basic ituitios behid default reasoig. Namely, the idea of a default reasoig is to process defaults accordig to our curret state of kowledge. We start with W as all that is kow. We use this curret state of our kowledge to d a applicable default ad exted the theory costructed so far. We cotiue this way util o more applicable defaults remai. This method is icorporated i the algorithm give below. To break ties betwee defaults that ca be chose at a give stage we use a orderig of defaults ad always choose the rst applicable oe. Clearly, the problem with this approach is that some facts derived later i the process may \block" defaults used earlier. Therefore a al cosistecy check is eeded (step 5, below). Select-orderig-ad-check Iput: A ite default theory (D; W ) with jdj = Output: A list of all extesios of (D; W ). 1. Select a permutatio d 1 ; : : : ; d of D. 2. Mark all defaults i D available. 3. Set S := W. 4. Repeat util o loger possible: d the smallest i such that d i is marked available ad is applicable with respect to S (that is, the prerequisite of d i is 4

5 a cosequece of S ad every justicatio of d i is cosistet with S). Mark d i used ad add its cosequet to S. 5. If every justicatio of every used default is cosistet with S, output C(S) as a extesio. 6. Repeat all these steps util all permutatios are used. The followig result was proved i [MT93a]. Theorem 1.1 The algorithm Select-orderig-ad-check correctly ds all extesios of a give default theory. Despite its simplicity, Select-orderig-ad-check algorithm is computatioally complex. It requires ispectig all! permutatios of the set of defaults. I such form it caot be competitive with the two algorithms described before. I [MT93a], it is show that it suces to cosider oly 2 permutatios to guaratee that all extesios of a default theory will be foud (i other words, to guaratee the completeess of the method). I the ext sectio, we will show that there is a set of permutatios X of size b=2c 0:8 2 = p that also guaratees the completeess of the select-orderigad-check method. We will describe a algorithm to ecietly costruct the set X ad we will show that X caot i geeral be further reduced without sacricig the completeess property (although usig pruig techiques, for some default theories it may be possible to d all extesios by cosiderig oly a fractio of permutatios i the set X ). 2 Results We start by recallig a property of the algorithm Select-orderig-ad-check proved i [MT93a]. It ivolves the otio of the set of geeratig defaults. For the deitio 5

6 of this cocept, as well as of other cocepts from default logic the reader is referred to [MT93a]. Theorem 2.1 Let (D; W ) be a ite default theory. Let d 1 ; : : : ; d be a permutatio of defaults i D. If for some k, 0 k, fd 1 ; : : : d k g is the set of geeratig defaults for a extesio S of (D; W ), the the executio of the steps (2) - (5) of Select-orderig-ad-check algorithm will produce ad output S. 2 This theorem implies a corollary which is the basis for the results of our paper. Corollary 2.2 Let (D; W ) be a default theory ad let jdj =. Let X be ay set of permutatios such that () each subset of D appears as a prex (i some orderig) i at least oe permutatio from X. The, the algorithm Select-orderig-ad-check restricted to the set of permutatios X correctly ds all extesios of (D; W ). Proof: It follows from Theorem 1.1 that ay output of the algorithm Select-orderigad-check (ad, hece, also of its modied versio) is a extesio. To complete the proof we eed to show that every extesio of (D; W ) will be foud by the modied Select-orderig-ad-check algorithm. To this ed, let us cosider a extesio S of (D; W ). Let G D be the set of geeratig defaults for S. By coditio (), there is a permutatio i X such that the defaults i G form a prex of. By Theorem 2.1, S will be the output produced by the Select-orderig-ad-check algorithm after it processes the permutatio. 2 Corollary 2.2 yields a method to improve the performace of the algorithm Selectorderig-ad-check. Namely, we eed to d (ad use i the algorithm) the miimum size set of permutatios A satisfyig coditio (). 6

7 We start by derivig a lower boud for such a set of permutatios. Let us cosider the set of propositioal variables fp 1 ; : : : ; p g. Dee defaults d i by For a subset P of fp 1 ; : : : ; p g dee ' P d i = : Mp i p i : = W p2p :p. Let k = b=2c + 1. It is easy to check that the default theory (fd i : 1 i g; f' P : jp j = kg) has exactly extesios, each correspodig to a subset of the set of defaults of size b=2c. Hece, i order to guaratee the completeess of the algorithm Select-orderig-ad-check, the set of permutatios it uses must have at least b=2c elemets. b=2c Notice that the default theory described above has its objective part, W, of size expoetial i the cardiality of the set of defaults. This raises a iterestig questio: is there a default theory of size (measured as the total umber of occurreces of propositioal letters) polyomial i ad havig b=2c extesios? We have oly partial results o this subject. Namely, assume that = rt for some itegers r ad t. Partitio the set of propositioal atoms fp 1 ; : : : ; p g ito r disjoit subsets Q 1 ; : : : ; Q r, each of size t. For each of these disjoit sets Q i of atoms apply the costructio described above. Deote by (D Q i; W Q i) the resultig default theory. Clearly, each (D Q i; W Q i) has t bt=2c extesios ad the size of WQ i is t bt=2c+1 (bt=2c + 1). Next, dee D = S r i=1 D Q i ad W = S r i=1 W Q i. Sice the sets Q i are disjoit, it follows that the umber of extesios of (D; W ) is ad the size of W is t bt=2c! r! t (bt=2c + 1)r: bt=2c + 1 t Notice ow that if t is selected so that bt=2c, the the size of W is O(2 ) ad the umber of extesios of (D; W ) is 2 c, where c < 1 ad c! 1. Hece, our costructio yields a default theory with defaults, the total size polyomial i, ad with the umber of extesios that is close to the desired boud of b=2c 0:8 2 = p. 7

8 I the remaiig part of the paper we will costruct the set of b=2c permutatios satisfyig the coditio (). We will study this issue i abstract, combiatorial terms. A collectio X of permutatio of f1; : : : ; g is called complete if each subset of f1; : : : ; g appears as a prex (i some orderig) i at least oe permutatios from X. Our goal is to costruct a complete set of permutatios of f1; : : : ; g of size b=2c. We will rst provide a simple proof that for every the miimum size of a com- plete set of permutatios is ideed b=2c. Although our proof is iductive ad, hece, implies a method to costruct a miimum complete set, this method caot be directly icorporated ito Select-orderig-ad-check algorithm. I the last part of the paper, we will adapt a techique from [Ku73] to describe a alterative method, ad show that it ca be used to improve the Select-orderig-ad-check algorithm. We start with a simple techical lemma { a corollary to the celebrated Hall's theorem [Hal35], see also [Bol78]. Lemma 2.3 Let G = (V [ U; E) be a bipartite graph with vertex classes V ad U i which all the vertices i V have the same degree, ad all the vertices i U have the same degree. If juj jv j, the the maximum matchig i G covers all vertices i V. Proof: Let us deote jv j = V, juj = U, ad let k V ; k U be the degrees of vertices i V ad U respectively. Sice V U ad V k V = U k U, we have k V k U. By Hall's theorem, G has a matchig which covers V if ad oly if for ay J V, j (J)j jjj ( (J) deotes the set of vertices i U, adjacet to at least oe vertex of J). Suppose that there is a subset J V for which j (J)j < jjj. There are at least k V jjj edges icidet to (J). Sice j (J)j < jjj, there must be at least oe vertex i (J) which has degree at least k V + 1. But all vertices i U have degree k U k V, so a cotradictio follows. 2 We will use ow Lemma 2.3 to show that the miimum size of a complete set of permutatios of elemets is b=2c. 8

9 Propositio 2.4 Let N = f1; 2; : : : ; g ad X be a miimum cardiality of a complete set of permutatios of N. The jx j =! : b=2c Proof: Sice N = f1; 2; : : : ; g cotais b=2c dieret subsets of cardiality b=2c ad o two dieret subsets of the same cardiality ca be prexes of the same permutatio, jx j b=2c. We have to show that, i fact, a complete set of permutatios of size b=2c exists. I the proof we will show how to costruct such a set of permutatios. Let A be a array with b=2c rows ad colums. We will ll the etries i A with itegers from f1; 2; : : : ; g so that each row of A will cotai a permutatio of N ad, for every subset Y of f1; 2; : : : ; g, there will be a row r(y ) i A i which elemets of Y appear i the rst jy j positios. First, cosider all subsets Y 1 ; Y 2 ; : : : ; Y l of N of cardiality equal to d=2e. Clearly, l = d=2e = positios of row i. b=2c. Isert the elemets of Yi, i ay order, i the rst d=2e The partially lled matrix A has the property that every subset of N of cardiality d=2e appears i the rst d=2e positios of some row of A. Cosider a iteger j, 1 j d=2e 1. Assume that the elemets of A are arraged so that every subset of N of cardiality k, where j + 1 k d=2e, appears i the rst k positios i some row of A. We will show how to rearrage the elemets of A so that the same holds for every subset of N of cardiality k, where j k d=2e. Let A 1 ; A 2 ; : : : ; A l, l = j+1, be all the subsets of N of cardiality j + 1, ad let B 1 ; B 2 ; : : : ; B m, m = j, be all the subsets of N of cardiality j. Cosider the bipartite graph G = (U [ V; E), where U = fa 1 ; A 2 ; : : : ; A l g, V = fb 1 ; B 2 ; : : : ; B m g ad fa i ; B j g 2 E if ad oly if B j A i. Clearly all vertices i U have degree j + 1, ad all vertices i V have degree j. Sice j j + 1, Lemma 2.3 implies that G cotais a matchig, say M, coverig V. 9

10 Figure 1: Fidig permutatios coverig all subsets of size 2 for = 6. For each edge fa i ; B j g 2 M d the row i A which cotais elemets of A i i its rst j + 1 positios. Next, permute these elemets so that the elemets of B j occur i the rst j positios. Sice M is a matchig, o row of A will be permuted twice ad the whole operatio is well-deed. Observe also that permutig rows of A i this fashio preserves the property that every subset of N of cardiality k, j + 1 k d=2e, appears as a prex i some row of A. This operatio oly forces the same property to hold for sets of cardiality j. I Figure 1 we illustrate this costructio for = 6 ad j = 2. Vertices o the right correspod to all subsets of size 2. Vertices o the left correspod to permutatios coverig all subsets of size 3. The edges of the matchig (show i bold) idicate the permutatios which eed to be modied i order to guaratee that all subsets of size 2 are covered, as well. The results of the modicatios are show i the leftmost colum. By applyig the above procedure for j = d=2e 1; d=2e 2; : : : ; 1, we obtai a rearragemet of the matrix A such that every subset of N of cardiality k, 1 k d=2e, appears as a prex i some row of A. 10

11 Notice that the last b=2c positios i each row are ot lled i. For each row, ll i these remaiig positios with the elemets of N which do ot appear i this row yet. Clearly, every subset of N of cardiality b=2c appears i some row of A as its sux. Proceedig as before, we ca rearrage the last b=2c positios i each row so that every subset of N of cardiality k, 1 k b=2c, appears as a sux i some row of A. This, however, simply meas that each subset of N of cardiality k, d=2e < k is a prex i some row of A. Hece, after all these steps, A cotais i its rows b=2c permutatios of N ad every subset of N appears as a prex i some row of A. 2 The proof of Propositio 2.4 implies a method to geerate a miimum size complete set of permutatios. The straightforward implemetatio of this method is based o Hopcroft-Karp method for dig matchigs. It leads to a complicated algorithm which rus i (2 p ) space ad requires ( ) time per permutatio. Sice it costructs all permutatios \at oce", if added to Select-orderig-adcheck algorithm it would result i impractical space requiremets. A much better approach is ot to compute all the permutatios at oce but to geerate oe permutatio at a time. This way oe ca geerate all the eeded permutatios i O() space ad O() time per permutatio. A possible way of implemetig this task ca be based o the followig costructio described i [Ku73] pp We cosider a set P of paths which start i poit (0; 0) ad ed i poit (; r), where r 0. Each path cosists of segmets, with the ith segmet joiig the poit (i 1; j) with (i; j +1) or (i; j 1) (the latter beig allowed oly if j 1). Hece, the path ever goes through grid poits (i; j) with egative j. There are exactly such paths ad they correspod to all permutatios of f1; 2; : : : ; g such that every subset of f1; 2; : : : ; g appears as prex of at least oe of the permutatios. b=2c For each path p from P we costruct a permutatio of f1; 2; : : : ; g as follows. We use three lists L 1 ; L 2 ad L 3 which are iitially empty. For i = 1; 2; : : : ;, if the i-th step of p goes up, we put umber i ito list L 2 ; if the step goes dow, we put i ito 11

12 list L 1 ad move the curretly largest elemet of list L 2 ito list L 3. The resultig permutatio is equal to the cocateatio of the al cotets of L 1 ; L 2 ad L 3, each list i icreasig order (see [Ku73]). All the paths from P ca be geerated recursively. This leads to a algorithm which searches a biary tree T. This tree T has depth ad b=2c leaves. Each path from P correspods to a root-to-leaf path i T. Therefore, the whole algorithm ca be implemeted to ru i O() space ad will require O(1) time per each geerated path from P. Sice the algorithm covertig each path ito a permutatio rus i liear time, this approach requires O() time per each permutatio. This method of geeratig permutatios from X ca easily be icorporated ito the algorithm Select-orderig-ad-check. Namely, after geeratig a permutatio from X, the steps (2) - (5) of the algorithm Select-orderig-ad-check should be executed. The, the ext permutatio from X should be geerated, the steps (2) - (5) executed, ad the whole process should be repeated util all permutatios from X are cosidered. 3 Coclusios I this paper we preseted a algorithm that computes all extesios of a default theory after searchig through the search space of permutatios (orderigs) of defaults of size 0:8 2 = p. This is the rst algorithm by date, guarateeig that all extesios will be foud, ad cosiderig i the worst case sigicatly less cadidates tha 2. Versios of the algorithm Select-orderig-ad-check ad the correspodig aalogues of Theorem 1.1 are kow for other objects cosidered as models of belief sets determied by a default theory: ratioal extesios ad costrait extesios [MT93b, MT95], as well. The mai result of this paper holds for these other cases, too. That is, the set of permutatios costructed i the previous sectio guaratees the completeess of the select-orderig-ad-check method for computig weak, ratioal 12

13 ad costrait extesios. It is also iterestig to ote that the methods used i this paper ca be geeralized. Let t be a iteger such that 0 t. Modifyig the argumets used i the paper oe ca produce a set of permutatios of the size t such that every subset of f1; : : : ; g of cardiality ot greater tha t appears as a prex i at least oe of the permutatios i the set. This yields a algorithm to compute extesios of a default theory if bouds o the umber of geeratig defaults are kow. Refereces [ALS94] G. Atoiou, E. Lagetepe, ad V. Sperscheider. New proofs i default logic theory. Aals of Mathematics ad Articial Itelligece, 12:215{ 230, [Bol78] B. Bollobas. Extremal Graph Theory. Academic Press, [BTK93] A. Bodareko, F. Toi, ad R.A. Kowalski. A assumptio-based framework for o-mootoic reasoig. I A. Nerode ad L. Pereira, editors, Logic programmig ad o-mootoic reasoig. Proceedigs of the Secod Iteratioal Workshop, pages 171{189. MIT Press, [CMMT95] P. Cholewiski, W. Marek, A. Mikitiuk, ad M. Truszczyski. Default reasoig system a implemetatio of default reasoig. I preparatio., [Du93] P.M. Dug. O the acceptability of argumets ad its fudametal role i omootoic reasoig ad logic programmig (exteded abstract). I Proceedigs Iteratioal Joit Coferece o Articial Itelligece, pages 852{859, Los Altos, CA, Morga Kaufma. To appear i Articial Itelligece. [Hal35] P. Hall. O represetatives of subsets. J. of Lodo Math. Soc., 10:26{30, [Ku73] D. E. Kuth. The Art of Computer Programmig. Addiso-Wesley, [MNR95] [MT93a] W. Marek, A. Nerode, ad J. B. Remmel. Rule systems, well-orderigs ad forward chaiig. Submitted for publicatio., W. Marek ad M. Truszczyski. Nomootoic logics; cotext-depedet reasoig. Berli: Spriger-Verlag,

14 [MT93b] A. Mikitiuk ad M. Truszczyski. Ratioal default logic ad disjuctive logic programmig. I A. Nerode ad L. Pereira, editors, Logic programmig ad o-mootoic reasoig. Proceedigs of the Secod Iteratioal Workshop, pages 283{299. MIT Press, [MT95] [Nie95] A. Mikitiuk ad M. Truszczyski. Costraied ad ratioal default logics. I Proceedigs of IJCAI-95. Morga Kaufma, I. Niemela. Towards eciet default reasoig. I Proceedigs of IJCAI- 95, pages 312{318. Morga Kaufma, [NS95] I. Niemela ad P. Simmos. Evaluatig a algorithm for default reasoig. I Proceedigs of the IJCAI-95 Workshop o Applicatios ad Implemetatios of Nomootomic Reasoigs Systems, [Rei80] R. Reiter. A logic for default reasoig. Articial Itelligece, 13:81{132,