Model Generation and State Generation for Disjunctive Logic. Programs. University of Tubingen. Sand 13, D { Tubingen, Germany

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1 Model Generation and State Generation for Disjunctive Logic rograms Dietmar Seipel 1, Jack Minker 2, Carolina Ruiz 2 1 Department of Computer Science. University of Tubingen. Sand 13, D { Tubingen, Germany seipel@informatik.uni-tuebingen.de 2 Institute for Advanced Computer Studies and Department of Computer Science. University of Maryland. College ark, MD U.S.A. fminker, cruizcg@cs.umd.edu Abstract This paper investigates two xpoint approaches for minimal model reasoning with disjunctive logic programs. The rst one, called model generation [4], is based on an operator T dened on sets of Herbrand interpretations, whose least xpoint is logically equivalent to the set of minimal Herbrand models of the program. The second approach, called state generation [12], uses a xpoint operator T s based on hyperresolution. It operates on disjunctive Herbrand states and its least xpoint is the set of logical consequences of, the so{called minimal model state of the program. We establish a useful relationship between hyperresolution by T s and model generation by T. Then we investigate the problem of continuity of the two operators T s and T. It is known that the operator T s is continuous [12], and so it reaches its least xpoint in at most! steps. On the other hand, the question of whether T is continuous has been open. We show by a counterexample that T is not continuous. Nevertheless, we prove that it converges towards its least xpoint in at most! steps too, as follows from the relationship that we show exists between hyperresolution and model generation. We dene an iterative version of T that computes the perfect model semantics of stratied disjunctive logic programs. On each stratum of the program, this operator converges in at most! steps. Model generations for the stable semantics and the partial stable (and so the well{ founded) semantics are respectively achieved by using this iterative operator together with the evidential transformation [3] and the 3-S transformation [16]. 1 Introduction The semantics of a disjunctive logic program has been characterized by its set of minimal Herbrand models or, equivalently, by the collection of all positive disjunctions that hold in every minimal Herbrand model of the program (see [11]). This collection is called the minimal model state of the program. The last two authors acknowledge the support of the National Science Foundation under grant number IRI

2 These equivalent semantic denitions gave rise to two alternative ways of computing the meaning of a program. The rst one, denoted here by model generation, relies on a xpoint operator T M that operates on sets of Herbrand interpretations and whose least xpoint is the set of minimal Herbrand models of the program. This operator was originally introduced by Fernandez and Minker in [4] (see also [2, 5]) for the case of disjunctive logic programs without function symbols. The second approach, developed by Minker and Rajasekar [12], is based on a xpoint operator T s dened on sets of positive disjunctions called states. This operator uses hyperresolution to construct the model state of the program as its least xpoint. We refer to this approach as state generation. In this paper, we further investigate the nature of model and state generations and prove some useful relationships between them. In particular, we investigate the problem of continuity of the two operators T s and T M. It is known that the operator T s is continuous [12], and so it reaches its least xpoint in at most! steps. On the other hand, the question of whether T M is continuous when applied to arbitrary disjunctive logic programs (with function symbols) has been open. We argue that this problem is ill{posed as the domain of T M is not closed under least upper bounds. We then give a natural extension of T M in terms of a new operator T dened on a more suitable domain and reformulate the continuity problem for the new operator. We prove, by means of a counterexample, that T is not continuous. Nevertheless, from a relationship that we show exists between state generation by T s and model generation by T, we prove that T reaches its least xpoint in at most! steps, too. We dene an iterative version of T that computes the perfect model semantics of stratied disjunctive logic programs. On each stratum of the program, this operator converges in at most! steps. Due to the characterizations of the stable and the partial stable (and so the well{founded) semantics respectively presented in [3] and [16], this iterative operator can be used to construct the stable models and partial stable models (the well{founded model in particular) of normal disjunctive logic programs. The paper consists of the following sections: Section 2 presents some basic denitions and notation. Section 3 summarizes the main properties of state generation. Section 4 surveys model generation, denes the new operator T and shows that it is not continuos. This section also proves the existence of minimal models of disjunctive logic programs. Section 5 establishes some useful relationships between model generation and state generation. As a consequence of these relationships, it is proven that T reaches its least xpoint in at most! steps. Section 6 provides an iterated version of the operator T that constructs the perfect models of stratied disjunctive logic programs, and is used to generate the (partial) stable models of normal disjunctive logic programs. Section 7 concludes the paper. 2 Basic Denitions and Notations Given a rst order language, a disjunctive logic program consists of logical inference rules of the form A 1 _ : : : _ A k B 1 ^ : : : ^ B m ; (1) where A i, i 2 h 1; k i, and B i, i 2 h 1; mi, are (positive) atoms in the language and k; m 2 IN 0. 1 A rule is called a fact if m = 0. The set of all ground instances of the rules and facts in is 1 By IN + we denote the set f 1; 2; 3; : : : g of positive natural numbers, whereas IN 0 denotes the set f 0; 1; 2; : : : g of all natural numbers. 2

3 denoted by gnd (). A disjunctive logic program is called a disjunctive deductive database if the program does not contain any function symbols. Two important concepts are associated with a logic program : (i) First, a subset I HB of the Herbrand base HB is called a Herbrand interpretation. The set of all Herbrand interpretations is denoted by HI. A Herbrand interpretation I is called a model of if for all ground rules of the form (1) in, it holds that f B i j i 2 h 1; mi g I implies that there exists i 2 h 1; k i such that A i 2 I. The set of all Herbrand models of is denoted by MOD, and sometimes by MOD(). (ii) Secondly, the set DHB of all ground disjunctions A 1 _ : : : _ A k, k 2 IN 0, that can be formed by atoms A i 2 HB is called the disjunctive Herbrand base of. A subset S DHB is called a disjunctive Herbrand state. For reasoning with disjunctive deductive logic programs two main approaches have been developed. The rst approach generates the set MM of all minimal Herbrand models of. A model I is called minimal if there is no smaller model I 0 of for which I 0 ( I. The second approach uses hyperresolution for deriving the set MS of all ground disjunctions C 2 DHB that are logical consequences of the logic program, i.e., the disjunctions that hold in all models of the logic program. This set is called the minimal model state of the logic program [12]. Below, we put together some important notation and results on partial orderings and lattices and on xpoint theory on complete lattices, cf. e.g. [9, 1]. Let O = h S; i, where S is a set and is a binary relation on S. O is called a partial ordering on S, if is reexive, transitive and antisymmetric. A partial ordering O is called a complete lattice, if for all subsets X S there exists a least upper bound, denoted by lub (X), and a greatest lower bound, denoted by glb(x), in S. 2 A very common example of a complete lattice that will occur in this paper is O = h 2 U ; i where 2 U is the power set of some set U, ordered by set inclusion, and least upper bounds and greatest lower bounds are given by the operations union and intersection, respectively. Certain mappings T : L! L on a complete lattice O = h L; i are of special interest: monotonic mappings which preserve the partial ordering and continuous mappings which commute with the least upper bound operator of the lattice. T is called monotonic, if x 1 x 2 implies T (x 1 ) T (x 2 ), for all elements x 1 ; x 2 2 L. A subset X L is called directed, if for every nite subset X 0 of X there exists an upper bound of X 0 in X. T is called continuous, if T (lub(x)) = lub (T (X)) for all directed subsets X L, where T (X) = f T (x) j x 2 X g: Every continuous mapping on a complete lattice is also monotonic. The converse, however, is not true. The elements of a lattice which are invariant under a mapping on the lattice are called xpoints of the mapping. That is, an element x 2 L is called a xpoint of T, if T (x) = x. A xpoint x 2 L of T is called the least xpoint of T, denoted by lfp (T ), if x x 0 for all xpoints x 0 of T. 3 Given a mapping T : L! L on a complete lattice O = h L; i we can dene its ordinal powers { corresponding to repeated applications of the mapping { as new mappings on the same lattice by using transnite recursion on in conjunction with the least upper bound operation on the lattice. The ordinal powers T " : L! L of T are dened as follows: T "0 (x) = x; 2 The notations lub (X) and glb (X) are justied since for every set X its least upper bound and its greatest lower bound are unique if they exist. 3 The notation lfp (T ) is justied since for every mapping T its least xpoint is unique if it exists. 3

4 T " (x) = T (T " 1 (x)); for a successor ordinal ; T " (x) = lub (f T " (x) j < g); for a limit ordinal : For the important special case of the bottom element? = glb(l) of O, the ordinal powers T " of T are lattice elements given by T " = T " (?): The well-known theorem of Knaster and Tarski, cf. [8] and [20], relates the xpoints of a monotonic mapping on a complete lattice to the ordinal powers: If T is monotonic, then the collection of xpoints of T forms a complete lattice and so T has a unique least xpoint lfp (T ). For any ordinal, it holds T " lfp (T ); and there exists an ordinal, such that T " 0 = lfp(t ) for all 0. The smallest such is called the closure ordinal of T. If T is continuous, then T "! = lfp(t ) (see [7]). 3 State Generation The xpoint semantics of a disjunctive logic program is based on a disjunctive consequence operator T s given in [12] (see also [10]). Denition 3.1 (Consequence Operator T s ) Let be a disjunctive logic program and let S DHB be a disjunctive Herbrand state. The disjunctive consequence operator T s : 2 DHB! 2 DHB of is dened as T s (S) = S [ f C _ C 1 _ : : : _ C m j C; C 1 ; : : :; C m 2 DHB and there is a rule C B 1 ^ : : : ^ B m 2 gnd () : 8 i 2 h 1; mi : B i _ C i 2 S g: Note that according to the above denition the result of applying the operator T s to a disjunctive Herbrand state S contains the state S as well as all disjunctions that can be derived from S and the rules in by one step of hyperresolution. Minker and Rajasekar show in [12] that the disjunctive consequence operator T s is continuous with respect to the complete lattice O = h 2 DHB ; i on disjunctive Herbrand states, and hence also monotonic. The bottom element of O is? = ;. Following the general setting given in Section 2, the ordinal powers of the disjunctive consequence operator are dened with respect to the lattice O as follows. Denition 3.2 (Ordinal owers for T s on O = h 2DHB ; i) Let be a disjunctive logic program. (i) For S DHB, the ordinal powers T s " (S) are dened by T s"0 (S) = S; T s" (S) = T [ (T s s" 1 (S)); for a successor ordinal ; " (S) = T s" (S); for a limit ordinal : T s < (ii) The ordinal powers T s " are dened by T s " = T s " (;): 4

5 Example 3.3 (Disjunctive Transitive Closure) Consider the disjunctive logic program = f path(x; Y ) arc(x; Z) ^ path(z; Y ); path(x; Y ) arc(x; Y ); arc(a; b) _ arc(a; c); arc(b; d); arc(c; d) g; that consists of the classical transitive closure rules and some disjunctive facts for the arc{relation, cf. Figure 1. a Q 3 Q sq b c Q QQ Qs 3 d Figure 1: Graph with disjunctive arcs The ordinal powers S n = T s " n = T s "n (;), n 1, are given by S 1 = f arc(a; b) _ arc(a; c); arc(b; d); arc(c; d) g; S 2 = S 1 [ f arc(a; b) _ path(a; c); path(a; b) _ arc(a; c); path(b; d); path(c; d) g; S 3 = S 2 [ f path(a; b) _ path(a; c); path(a; d) _ arc(a; c); path(a; d) _ arc(a; b); S 4 = S 3 [ f path(a; d) g; S n = S 4 ; for all n 4: path(a; d) _ path(a; c); path(a; d) _ path(a; b) g; Thus, T s "! = T s " 4: This example shows that T s facts. can also derive denite facts from disjunctive A disjunction C 0 is called a sub-disjunction of another disjunction C if every atom appearing in C 0 also appears in C. C 0 is called a proper sub-disjunction of C if C 0 6= C and C 0 is a sub-disjunction of C. For a disjunctive Herbrand state S, let can (S) = f C 2 S j 6 9C 0 2 S : C 0 is a proper sub-disjunction of C g; exp (S) = f C 2 DHB j 9C 0 2 S : C 0 is a sub-disjunction of C g: can (S) and exp (S) are respectively the canonization and the expansion of S, and it holds that can (S) S exp (S). Two disjunctive Herbrand states S 1 and S 2 are called equivalent if exp (S 1 ) = exp (S 2 ). This is denoted by S 1 S 2. The minimal model state MS is equivalent to the least xpoint of the disjunctive consequence operator T s, and it can be derived as MS = exp (T s "!), as was proven by Minker and Rajasekar in [12]. 5

6 Theorem 3.4 (Characterization of MS [12]) Let be a disjunctive logic program. Then MS lfp(t s ) = T s " ; where is the closure ordinal of T s on O = h 2DHB ; i: 4 Model Generation The model generation approach constructs the minimal Herbrand models of a given logic program. For a denite logic program (without disjunctions) and a given Herbrand interpretation I the classical consequence operator T of van Emden and Kowalski [21] computes the Herbrand interpretation J that consists of the head atoms of all rules in gnd (), such that the bodies of the rules are satised by I. The unique minimal Herbrand model of the program is precisely the least xpoint of this operator. For disjunctive logic programs, model generation deals with sets of Herbrand interpretations. For conciseness, we abbreviate the set of Herbrand interpretations as coin (collection of interpretations). We use the following two operations min and exp for a coin I: min (I) = f I 2 I j 6 9J 2 I : J ( I g; exp (I) = f I 2 2 HI j 9J 2 I : J I g: Note that we use the operator exp for states as well as for coins, but it will be clear from the context to which case we are refering. A coin I is called canonical if it does not contain two dierent Herbrand interpretations I; J such that I ( J, i.e. if I = min (I). A coin I is called expanded if for each Herbrand interpretation I 2 I it also contains all Herbrand interpretations that are supersets of I, i.e. if I = exp (I). E.g. for the Herbrand base HB = f a; b; c g and the coin I = f fag; fa; bg; fb; cg g; we get min (I) = f fag; fb; cg g and exp (I) = I [ f fa; cg; fa; b; cg g: The following consequence operator T generalizes the operator T of van Emden and Kowalski to the case of disjunctive logic programs. T maps coins to coins. Denition 4.1 (Consequence Operators T and T M ) Let be a disjunctive logic program. (i) The consequence operator operates on sets I 2 2 HI (ii) The consequence operator operates on sets I 2 2 HI T : 2 HI! 2 HI of Herbrand interpretations: [ T (I) = MOD(T(I)): s I2I T M : 2HI! 2 HI of Herbrand interpretations: T M (I) = min (T (I)): 6

7 For each I 2 I, T (I) contains all Herbrand interpretations J which extend I and at the same time satisfy all ground rules of whose bodies are satised by I. Thus, from each interpretation several interpretations may be derived. Furthermore, the result T (I) is expanded. The operator T M was originally introduced by Fernandez and Minker [4] (see also [2, 5]) to compute the minimal Herbrand models of a disjunctive deductive database (i.e. a disjunctive logic program with no function symbols). In this case, due to the fact that the Herbrand base of a database is nite, coins are nite sets of nite interpretations and so have the property that T (I) = exp (T M(I)). 4 Example 4.2 (Consequence Operator T M ) Consider the disjunctive deductive database of Example 3.3. (i) For the Herbrand interpretation I = ; the rules whose bodies are satised by I are precisely the facts of. Thus, T M (f;g) = I 1 = fi 1 ; J 1 g; is the set of minimal Herbrand interpretations of the facts of, where I 1 = f arc(a; b); arc(b; d); arc(c; d) g; J 1 = f arc(a; c); arc(b; d); arc(c; d) g: (ii) For the Herbrand interpretations in I 1 the bodies of some ground instances of the second rule are satised. Thus, T M extends I 1; J 1. We get T M(I 1) = I 2 = fi 2 ; J 2 g; where I 2 = I 1 [ f path(a; b); path(b; d); path(c; d) g; J 2 = J 1 [ f path(a; c); path(b; d); path(c; d) g: Fernandez and Minker (cf. [5]) investigated some of the properties of the consequence operator T M based on the following subsumption relation v dened for coins I; J 2 2HI : I v J i 8 J 2 J : 9 I 2 I : I J: E.g. for I = f fag; fa; bg; fb; cg g and J = f fag; fcg g we get J v I: The pair O = h 2 HI ; vi however is only a quasi-ordering, since on 2 HI the relation v is reexive and transitive, but not antisymmetric. To overcome this, one can work with equivalence classes of coins as follows: Two coins I; J 2 2 HI are called equivalent with respect to the quasiordering v, denoted by I v J, if they subsume each other, i.e. I v J i I v J and J v I: Fernandez and Minker then restricted the domain of T M to O min = h 2 HI min ; vi; where 2HI min consists of all canonical coins. This subdomain does form a partial ordering. For disjunctive deductive databases, this partial ordering is also complete since the Herbrand base of a such a database is nite. Based on this, they proved the monotonicity of T M on O min and the following characterization of the set MM of minimal Herbrand models of in terms of the consequence operator T M and its ordinal powers T M " = T M " (f ; g) with respect to O min. 4 For an arbitrary disjunctive logic program, a coin can be an innite set and it can contain innite interpretations. Thus, this property may not hold for, since there may exist some I 2 T (I) for which there is no minimal interpretation in T (I) contained in I. 7

8 Theorem 4.3 (Characterization of MM [5]) Let be a disjunctive deductive database. Then where is the closure ordinal of T M MM = lfp (T M ) = T M " ; on O min = h 2 HI min ; vi: Since the Herbrand base of a disjunctive deductive database is nite, the operator T M its least xpoint in a nite number of iterations. reaches Example 4.4 (Disjunctive Transitive Closure) For the disjunctive logic program of Example 3.3 all ordinal powers T M " n = f I n ; J n g, n 1, consist of two Herbrand interpretations, where I 1 ; J 1 and I 2 ; J 2 have been given in Example 4.2 and I 3 = I 2 [ f path(a; d) g; J 3 = J 2 [ f path(a; d) g: It holds I n = I 3 and J n = J 3, for all n 3. Thus, the set of minimal Herbrand models of is given by MM = f I 3 ; J 3 g. In principle, one can apply the operator T M to disjunctive logic programs containing function symbols. The question of whether or not the operator is continuous in this extended context has been open. Notice however that, in the extended context, the partial ordering O min = h 2 HI min ; vi is not a complete lattice (the least upper bound of a collection X of canonical coins may not exist). Hence the question of continuity in this subdomain is ill{posed. We reformulate the continuity problem in a more appropriate domain O exp = h 2 HI exp ; vi; where 2 HI exp consists of all expanded coins. It is easy to show that in this subdomain of expanded coins the relation v reduces to superset inclusion, since for all I; J 2 2 HI it holds I v J i exp (I) exp (J ): This shows that O exp = h 2 HI exp ; vi is a complete lattice, and that the least upper bound and the greatest lower bound of a set X 2 HI exp of \ expanded coins are given [ by lub exp (X) = I; glb exp (X) = I2X The bottom element of O exp is the coin? exp = HI consisting of all Herbrand interpretations. Note that both the intersection and the union of a set X of expanded coins is an expanded coin. 5 We remark that the construction of O exp is similar to the concept of free!-completion of partial orderings, cf. [23]. In the domain O exp we can work with the operator T that produces expanded coins. For every coin I it holds I v T (I); since every Herbrand interpretation J 2 T (I) is derived by extending an Herbrand interpretation I 2 I, where I J. Moreover, the operator T is monotonic on O = h 2 HI ; vi (and hence in O exp ): I2X I v J implies T (I) v T (J ): Following the general setting given in Section 2, the ordinal powers of T to the complete lattice O exp as follows. 5 This does not hold for canonical coins. I: are dened with respect 8

9 Denition 4.5 (Ordinal owers for T Let be a disjunctive logic program. (i) For I 2 2 HI exp on O exp = h 2 HI exp ; vi), the ordinal powers T " (I) are dened by T T T "0 (I) = I; " (I) = T \ " (I) = < (T T " 1 (I)); for a successor ordinal ; " (I); for a limit ordinal : (ii) The ordinal powers T " are dened by T " = T " (? exp ): For disjunctive deductive databases it can be shown that the respective ordinal powers of the operators T and T M are equivalent, i.e. T " v T M " : It holds T " = exp (T M " ) and T M " = min (T " ): We show now by means of a counterexample that the operator T is not continuous. Example 4.6 (Non{continuity of Model Generation through T ) Consider the disjunctive logic program = f a 0 a(x) ^ a(y ) ^ di (X; Y ); a(x) a(y ); d(f(c)) d(c) g and the complete lattice O exp = h 2 HI exp ; i: Consider the following Herbrand interpretation for the predicate symbol di : I di = f di (f m (c); f n (c)) j m; n 2 IN 0 ; m 6= n g: This Herbrand interpretation I di says that for di (X; Y ) to be true, it must hold that X and Y denote dierent terms. Let a n denote the atom a(f n (c)), for all n 2 IN +, and let I a = fa n jn 2 IN + g. We are interested in the following expanded coins I n, where all I 2 I n contain the Herbrand interpretation I di : I n = exp (f fa m g [ I di j m n g); for n 2 IN + : The set X = f I n j n 2 IN + g of coins forms a chain, since it holds I n I k ; for all n k: Thus, X is a directed set of coins. The least upper bound of X is given by lub exp (X) = \ n2in + I n = f I HB j I \ I a is innite and I di I g; i.e. it consists exactly of those Herbrand interpretations I, which contain innitely many of the atoms a n and also contain the whole Herbrand interpretation I di. Thus, each of these Herbrand interpretations I contains at least two dierent atoms a n = a(f n (c)) and a k = a(f k (c)) and the corresponding atom di (f n (c); f k (c)). This implies that the rst rule of the logic program will extend I with the atom a 0, and that the second rule of the logic program will extend I with all other atoms a m 2 I a. Thus, it holds T (lub exp (X)) = exp (f fa 0 g [ I a [ I di g): 9

10 On the other hand it holds T (I n ) = exp (f I a [ I di g); for all n 2 IN + ; i.e. all coins T (I n), n 2 IN +, are identical. Thus, \ lub exp (f T (I n) j n 2 IN + g) = T n2in + (I n) = exp (f I a [ I di g): This shows that the operator T does not commute with lub exp, i.e. T This implies also that T to O = h 2 HI ; vi. (lub exp (X)) 6= lub exp (f T (I n ) j n 2 IN + g): is not continuous with respect to O exp = h 2 HI exp ; vi nor with respect In the next section we show that even though T is not continuous it reaches its least xpoint in at most! steps. Furthermore, based on the operator T, we are able to generalize the characterization of the minimal Herbrand model semantics that Fernandez and Minker obtained for disjunctive deductive databases to arbitrary disjunctive logic programs, namely that MM v lfp (T ) = T "! (see Theorem 5.5). 2 The following lemma is used in the next section. It shows that the set of minimal Herbrand models of a disjunctive logic program is equivalent to the set of all Herbrand models of the logic program with respect to the quasi-ordering v. In other words, this means that each Herbrand model of the logic program contains a minimal Herbrand model of the program. This result is not trivial since, for an arbitrary disjunctive logic program, it could have been the case that there were an innitely decreasing chain M 1 ) M 2 ) M 3 ) : : : of models of. Lemma 4.7 (Existence of Minimal Models) Let be a disjunctive logic program. Then MM v MOD: roof: For each M 2 MOD we want to show that there exists some I 2 MM such that I M. Let I M = f I 2 MOD j I M g. We prove below that each chain K I 6 M has a lower bound I in I M. Using Zorn's lemma, this implies that the coin I M contains a minimal element. Hence, MM v MOD: The statement MOD v MM is trivial, since MM MOD: Now let K I M be a chain of Herbrand models. \ Obviously, the Herbrand interpretation I = is a lower bound of K and I M. We want to show that I is also a model of, i.e. I 2 MOD. Consider a rule A 1 _ : : : _ A k B 1 ^ : : : ^ B m 2 gnd () I2K such that I j= B 1 ^ : : : ^ B m. Since K is a chain, we get that 6 i.e. for each I; J 2 K it either holds I J or J I. J = f I \ f A 1 ; : : :; A k g j I 2 I g I 10

11 is a chain, too. Since all elements of J have a nite cardinality between 0 and k, there exists an element J 0 2 J with minimal cardinality. Since J is a chain, it holds 8J 2 J : J 0 J: Consider the Herbrand interpretation I 0 2 K, such that J 0 = I 0 \ f A 1 ; : : :; A k g. Since I j= B 1 ^ : : :^ B m and I I 0, it holds I 0 j= B 1 ^ : : :^ B m. From I 0 2 MOD we get I 0 j= A 1 _ : : :_ A k, i.e. J 0 = I 0 \ f A 1 ; : : :; A k g 6= ;. Since all elements of J are supersets of J 0, we get 8I 2 K : J 0 I: This shows that J 0 I, i.e. I j= A 1 _ : : : _ A k. Thus, I is a Herbrand model of contained in I M. 2 5 Relationships between State Generation and Model Generation For disjunctive logic programs the minimal model state MS and the set MM of minimal Herbrand models of are dual concepts, i.e., they can be derived from each other. According to [10], it holds where the dualization operations are dened by MM = MM(MS ); MS = MS(MM ); MS(I) = fc 2 DHB j 8I 2 I : I j= C g; MM(S) = min (fi 2 HI j 8C 2 S : I j= C g): This resembles the duality between the conjunctive and the disjunctive normal form of boolean formulas, since MS represents the conjunction of its disjunctions, whereas MM represents the disjunction of the conjunctions formed by its models. This duality relates the least xpoints MS and MM ( v MOD ) of the consequence operators T s and T, respectively (see Figure 2). In the following we also compare the intermediate results of the respective xpoint iterations. Corollary 5.3 establishes the relationship between the ordinal powers T s " n and T " n. Lemma 5.1 (Connection between T s and T [17]) Let be a disjunctive logic program, let I be a coin, and let S DHB be a disjunctive Herbrand state, such that I MOD(S). Then T (I) MOD(T s(s)); T(S) s MS(T (I)): 11

12 T s " 0 = ;? T s " n T s T s? T s "! MS MOD - MS MOD - MS T T " 0 =? exp T?? T " n T "! v MM Figure 2: Comparing Fixpoint Computations roof: (i) Assume M is a Herbrand interpretation, such that We will show that M 2 MOD(T s (S)): M 2 T (I): There is some I 2 I, such that M 2 MOD(T s (I)). Thus, I M. Since I MOD(S), we get I 2 MOD(S). Since I is a model of S, we get that M is a model of S. For each C 2 T s(s)ns, there is a rule C 0 B 1 ^ : : : ^ B m 2 gnd () and there are facts B i _ C i 2 S; such that C = C 0 _ C 1 _ : : :_ C m : If I j= C 1 _ : : : _ C m, then M j= C. Otherwise, since I j= B i _C i, 1 i m, we get I j= B 1^: : :^B m. Thus, C 0 2 T s(i). Since M 2 MOD(T s(i)), we get that M j= C0 : Thus, again M j= C. Summarizing, M is a model of T s (S) n S. This shows that M 2 MOD(T s (S)), i.e. (ii) Application of MS to (i) yields Since T s(s) MS(MOD(T s (S))); we get T (I) MOD(T(S)): s MS(MOD(T(S))) s MS(T (I)): T(S) s MS(T (I)): 2 From this connection we can derive some relationships between state generation and model generation, i.e., between the ordinal powers T s " of the operator T s on disjunctive Herbrand states and the ordinal powers T " of the operator T on sets of Herbrand interpretations. 12

13 Theorem 5.2 (State Generation vs. Model Generation) Let be a disjunctive logic program, let S DHB be a disjunctive Herbrand state of and let I be a coin, such that I MOD(S) and S MS(I). For all ordinals it holds roof: T T s " (I) MOD(T s" (S)); " (S) MS(T " (I)): (i) The rst set inclusion is shown by induction on. = 0 : The induction basis is shown by T "0 (I) = I MOD(S) = MOD(T s"0 (S)):! + 1 : Using the induction assumption for and the monotonicity of T we get T "+1 (I) = T (T " (I)) T (MOD(T s" (S))): Let I 0 = MOD(T s " (S)) and S 0 = T s " (S). Then I 0 = MOD(S 0 ). Applying the rst set inclusion of Lemma 5.1 to S 0 and I 0 yields T (I0 ) MOD(T s(s0 )); i.e., T (MOD(T s " (S))) MOD(T s (T s " (S)))) = MOD(T s "+1 (S)): By transitivity we get: T "+1 (I) MOD(T s"+1 (S)): a limit ordinal: Intersecting the set inclusions for all < yields \ \ " (I) MOD(T s" (S)): Since T < T we get that \ < < T " (I) = T " (I); and MOD(T s < " (S)) = MOD( [ (ii) If we apply MS to the rst set inclusion we get < T " (I) MOD(T s" (S)): T s " (S)) = MOD(T s " (S)); MS(MOD(T s " (S))) MS(T " (I)): By chaining with T s " (S) MS(MOD(T s " (S))) we get the second set inclusion. 2 Corollary 5.3 (State Generation vs. Model Generation) Let be a disjunctive logic program. For all ordinals it holds T " MOD(T s " ); T s " MS(T " ): 13

14 Sometimes, the set inclusions in Corollary 5.3 are strict, as the following example shows. Example 5.4 (State Generation vs. Model Generation) Given some n 2 IN +, consider the disjunctive logic program = f a b i j 1 i n g [ f b 1 _ : : : _ b n g with n rules and one fact. Let S m = T s " m and I m = T M " m: For n = 2 we get S 1 = f b 1 _ b 2 g; S 2 = f b 1 _ b 2 ; a _ b 1 ; a _ b 2 g; I 1 = f f b 1 g; f b 2 g g; I 2 = f f b 1 ; a g; f b 2 ; a g g: Thus, for the atom a we get a 2 MS(I 2 ); but a 62 S 2 : For n 2 we get the disjunctive Herbrand states S m = f b 1 _ : : : _ b n g [ f a _ b i1 _ : : : _ b i k j fi 1 ; : : :; i k g h1; ni; n m + 1 k n 1 g; for all m 2 IN + ; where hn; mi denotes the interval f n; n + 1; : : :; m g of all natural numbers between n and m. On the other hand, I 1 = f fb 1 g; : : :; fb n g g; I m = f f b i ; a g j 1 i n g; for all m 2: The least xpoint of T M is always reached after two steps: lfp (T M) = I 2; whereas the least xpoint of T s is reached after n + 1 steps: lfp(t s) = S n+1 = S n [ fag. We can show that MOD(S n ) = exp (I n [ fi b g); MS(I n ) = exp (S n [ fag); where I b = f b i j 1 i n g. This shows that a 2 MS(I n ) n S n ; inclusions in Corollary 5.3 are strict. I b 2 MOD(S n ) n I n ; i.e., the set Since the operator T s is continuous (see [12]), its ordinal powers converge towards its least xpoint in at most! steps, i.e., lfp (T s) = T s "! MS : Corollary 5.3 helps to show that the ordinal powers of the operator T converge towards its least xpoint in at most! steps, too. That is, the closure ordinal of T on O exp is!. Moreover, this least xpoint is equivalent to the set MM of the minimal Herbrand models of the program. Theorem 5.5 (Convergence of Model Generation by T ) Let be a disjunctive logic program. Then MM v lfp(t ) = T "!: roof: We will show that MM v T "! and that T "! v MM : 14

15 (i) From Corollary 5.3 we know that T "! MOD(T s "!): With T s "! MS ; we get MOD(MS ) v T "!: From Lemma 4.7 we know that MM(MS ) v MM(MS ) = MM : This shows that MOD(MS ); and from [12] we know that (ii) From and T O exp = h 2 HI exp T MM v T "!: " 0 =? exp v MM (MM ) = MM and the monotonicity of the operator T ; vi, by induction on n 2 IN 0 we get that T " n v MM ; for all n 2 IN 0 : on the partial ordering This shows that T "! = lub exp (f T " n j n 2 IN 0 g) v MM : Hence T "! v MM : Since MM is a xpoint of T, i.e., T (MM ) = MM, this implies that lfp(t ) v MM : Thus, the operator T is an example of a monotonic operator that is not continuous, but nevertheless reaches its least xpoint in at most! steps. 6 Iterative Model Generation of erfect and (artial) Stable Semantics In [5], Minker and Fernandez dened an iterative method to compute the perfect models of a stratied disjunctive deductive database. In this section we extend their method to work with arbitrary stratied disjunctive logic programs by using the operator T dened on Section 4. Given a rst order language, a normal disjunctive logic program consists of logical inference rules of the form A 1 _ : : : _ A k B 1 ^ : : : ^ B m ^ not C 1 ^ : : : ^ not C n ; (2) where A i, i 2 h 1; k i, B i, i 2 h 1; mi, and C i, i 2 h 1; ni, are (positive) atoms in the language; k; m; n 2 IN 0 ; and not is the negation-by-default operator. Given a predicate p in the language, the denition of p in is the set of all rules in whose heads (i.e. A 1 _ : : : _ A k ) contain an atom in which p appears. The denition of an atom A is taken to be the denition of the predicate symbol appearing in A. A normal disjunctive logic program is called stratied (see [13]) if it is possible to partition the set of rules of into sets f 1 ; : : :; r g, called strata, such that for every rule of the form (2) in there exists a constant c, 1 c r, such that: (1) the denition of each A i is contained in c ; (2) the denition of each B j is contained in S sc s ; and (3) the denition of each C l is contained in S s<c s. Any partition f 1 ; : : :; r g of satisfying the above conditions is called a stratication of. 2 15

16 The intended meaning of a stratied disjunctive logic program is given by its collection of perfect models as dened in [13]. It is well{known (see e.g. [13]) that this collection can be constructed by induction on the strata as follows. Given a stratication f 1 ; : : :; r g of S, let M i denote the set of perfect models of the rst i strata of, i.e. of the logic program si s. By denition, the lowest stratum 1 (which may be the empty set) is free of negation{by{default and its set of perfect models is given by M 1 = MM 1. When M i has been constructed, it is used to evaluate the negated by default literals appearing in the i + 1 stratum as follows: If I 2 M i, then I i+1 is the disjunctive logic program: I i+1 = f A 1 _ : : : _ A k B 1 ^ : : : ^ B m j A 1 _ : : : _ A k fc 1 ; : : :; C n g \ I = ;g: B 1 ^ : : : ^ B m ^ not C 1 ^ : : : ^ not C n 2 i+1 and I is called the i+1 Gelfond{Lifschitz transformation S of i+1 with respect to I. The perfect models of the rst i + 1 strata are then taken to be I2M i MM I i+1 [I. The collection of perfect models of is exactly M r. It is worth noticing that this collection of models is the same independently of the particular stratication of used in the induction. For the case of function{free logic programs, Minker and Fernandez used an iterative version of their operator T M to compute this collection of perfect models [5]. We extend now this procedure to arbitrary stratied disjunctive logic programs by dening the iterative version of our operator T. Denition 6.1 (Iterative version of T ) Let be a stratied disjunctive logic program and f 1 ; : : :; r g be a stratication of. iterative version of T is dened as follows: T I T I h 1 ;:::;n+1i = [ h 1 i = min (T I 1 " h 1;; i ); I2T I h 1 ;:::;n i where h i;i i is the closure ordinal of T I ( I i [I). min (T I ( I n+1 [I) " h n+1;i i) The set of perfect models of is given by T I h 1. Notice that, due to Theorem 5.5, each of ;:::;ri the closure ordinals I i in the previous denition is at most!. Fernandez et al. [3] showed how to transform an arbitrary normal disjunctive logic program into a stratied one, denoted by E, in such a way that the perfect models of E that satisfy a given set of integrity constraints correspond to the stable models of (as dened in [14]). They called this transformation the evidential transformation. Using this characterization of stable models, the iterative version of T can be used to construct the stable semantics of. Similarly, the operator T can be used to generate the collection of partial (or 3{valued) stable models ([15]) of a normal disjunctive logic program by using a characterization of this collection of models given by Ruiz and Minker in [16]. This characterization is based on a transformation called the 3S-transformation which applied to a normal disjunctive logic program produces a constrained logic program (free of negation-by-default) 3S, whose minimal (2-valued) consistent models correspond to the partial stable models of the original program. Since the well{founded model ([22]) is a distinct partial stable model of a normal logic program (see [15]), then model generation for the well{founded semantics is also achieved using the operator T the 3S-transformation. 16 The together with

17 7 Conclusions Given a disjunctive logic program, there are two approaches for deriving the minimal model state MS and two approaches for deriving the set MM of minimal models of the database. Both sets can be derived based on hyperresolution as well as on model generation from. Using hyperresolution, MS is computed as the least xpoint of the disjunctive consequence operator T s. Using model generation, MM is computed as the least xpoint of the consequence operator T. By dualization of the minimal model state MS we get the set of minimal models of as MM(MS ). Similarly, dualization of MM yields MS as MS(MM ). As shown in this paper, both approaches converge in at most! steps. For the case of disjunctive deductive databases, these approaches have been implemented within the disjunctive deductive database engine DisLog 7, cf. [19], developed at the University of Tubingen. Experimenting with DisLog, we have observed that the eciency of each approach depends on the relation between the number n = jcan (MS )j of minimal 8 disjunctions in MS and the number m = jmm j of minimal models. We conjecture that the following may be the case. If n and m are about equal, then for each derivation it is better to use the specialized approach, i.e., to derive MS by hyperresolution and to derive MM by model generation. If n is much bigger than m, then the fastest is to use model generation for deriving MM and to derive MS from MM by dualization. If n is much smaller than m, then the fastest is to use hyperresolution for deriving MS and to derive MM from MS by dualization. Based on the theorems of Section 5 it has been shown in [18] that during a xpoint iteration with T s it is possible to switch to some iterations of model generation. Intermediate information about the size of the ordinal powers T s " n can be used to decide if this would be advantageous. However, general criteria to decide in advance which method is more ecient when faced with a particular disjunctive logic program are still to be determined. Using model generation by T, perfect, well{founded, stable and partial stable models of normal disjunctive logic programs can also be computed. For a stratied disjunctive logic program, the perfect models of can be computed by an iterative model generation applied to the strata. For a normal disjunctive logic program, the stable models and the partial stable models (the well{founded model in particular) of can be respectively computed by model generation with the evidential transformation E ([3]) and by model generation with the 3S-Transformation 3S ([16]). References [1] B.A. Davey, H.A. riestley: Introduction to Lattices and Order, Cambridge University ress, [2] J.A. Fernandez: Disjunctive Deductive Databases, h.d. Thesis, University of Maryland, [3] J.A. Fernandez, J. Lobo, J. Minker, V.S. Subrahmanian: Disjunctive L + Integrity Constrains = Stable Model Semantics, Annals of Mathematics and Articial Intelligence, vol. 8 (3-4), 1993, pp for a demo version of DisLog visit 8 a disjunction C in MS is minimal if there is no sub-disjunction C 0 of C in MS. 17

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