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**Unformatted text preview: **and I be a structure such that (Pc ) (I ) and (Pc) (I ) . Then I is a proper model of P i TP (I ) v I , and
c The bottom-up xpoint iteration of TP is de ned as follows:
c TP "0 = I TP "n+1 = TP (TP "n) TP "! = tn<! TP "n :
c
? c c c c c Note that owing to the monotonicity of TP it has a least xpoint lfp(TP ) and since ORLog is function-free, we trivially have lfp(TP ) = TP "! . One of our main results is the following theorem, proved analogously to the classical case. The only subtlety is handling clause inheritance via context switch.
c c c c Theorem 3.5 Let P be a program and Pc be its closure. Then, 1. MP = MP = lfp(TP ), where MP = MP is the least proper model of P, and 2. lfp(TP ) can be computed in a nite number of bottom-up iterations.
c c c c Proof: (1). MP = ufI j I is a proper Herbrand model of Pc g
c by Theorem 3.1 67 Note that whenever I is a (proper) model of P (and hence of Pc), (Pc ) (Pc ) (I ) . From this, we can see that (I ) and MP = ufM j M is a proper model of Pcg = ufM j TP (M) v M M is proper and (Pc )
c c c c by proposition 3.4 = lfp(TP ) by the monotonicity of TP on the class of Pc satisfying the conditions in Lemma 3.1, and Knaster-Tarski xpoint theorem, proving (1) (I ) , (Pc ) (I ) g, (2). Since ORLog programs are function free, the least xpoint above can be computed in a nite number of steps, just as for Datalog, which proves (2). 2 Observation 3.1 Let P be an i-consistent program and Pc be its closure. Then MP = MP = lfp(TP ) = TP "! is the intended model of P. 2
c c c Theorem 3.5 establishes the equivalence between the declarative semantics based on intended models and the xpoint semantics based on the least xpoint of the operator TP . It remains to establish their equivalence to the proof-theoretic semantics given in Section 3.6. As in the classical case, we accomplish this by relating the stage of a ground atom A { the smallest number of iteration k such that A 2 TP "k { to the height of a proof tree for an atom more general than A.
c c and (ii) non p-goals (pred and r-goals). Note inheritance only applies to p-goals. Basis: Suppose the proof tree has height 1. Then there are two possible cases { (i) either G is a p-clause local to some object, a pred-clause or an r-clause, or it is an inherited p-clause in some object from another object where it is local. Case 1: There exists a unit clause A 2 Pc such that mgu(G A) = . Case 2: There exist unit clauses A B 2 Pc such that A is a unit p-clause of the form p m(a1 : : : ak ) 7! a ] and B is a unit i-clause of the form o p @mk ], and G is of
0 0 0 0 0 7! Proof: By induction on the height k of a proof tree. There are two cases, (i) p-goals c Theorem 3.6 (Soundness) Let Pc be a closed program, Pc be the Herbrand inb stantiation of Pc, G be an atomic p-, r- or pred-goal, and G be all the ground instances d of G. If Pc ` G is provable then 9k such that G v TP "k .
c 68 the form o m(a1 : : : ak ) 7! a]. Then the inheritance rule must be applied. Hence it must be the case that = mgu(o o ), = mgu(p ] p), % = p = o ] and = = mgu(< o a1 : : : ak a > ] < p a1 : : : ak a > %]). d Pc in c Let = for the rst case and = for the second case. Since G d both the cases, it follows from the de nition of TP that G v TP "2. Inductive step: Assume that the claim is true for any proof of height k ; 1 of a goal of the form Pc ` B . Then we have again two cases. Case (i): There must exist a clause of the form A B 2 Pc such that = mgu(G A), and = and we have a proof of height k for Pc ` G where = and the root node labelled deduction. Case (ii): There must exist a p-clause A B 2 Pc and a unit i-clause C 2 Pc such that A is of the form p m(a1 : : : ak ) 7! a ] and C is of the form o p @mk ], and G is of the form o m(a1 : : : ak ) 7! a]. We can now construct a proof tree of height k for Pc ` G such that the root node is labelled inheritance, and = mgu(o o ), = mgu(p ] p), % = fp = o g and = mgu(< o a1 : : : ak a > = ] < p a1 : : : ak a > %]). Also = and = % . d d d By inductive hypothesis, B v TP "k 1 holds. Since G = A in both d d the cases, it follows from lemma 3.1 that A v TP "k because either (i) B v d TP "k 1 , or (ii) fBd o p @mk ] g v TP "k 1 , o p @mk ] being unit clauses and d c d o p @mk ] o pdmk ] Pc . Hence G v TP "k . @ 2
0 0 0 0 0 c c 0 0 0 0 0 7! 0 0 0 0 0 c ; c c ; 0 0 7! c ; 0 0 7! 0 0 7! 0 0 7! c A 2 Pc such that either (i) = mgu(G A) and = , or (ii) there exist unit clauses A B 2 Pc such that A is a unit p-clause of the form p m(a1 : : : ak ) 7! a ], B is a unit iclause of the form o p @mk ], and G is of the form o m(a1 : : : ak ) 7! a] such that = mgu(o o ), = mgu(p ] p), % = fp = g and = mgu(< o a1 : : : ak a > =o ] < p a1 : : : ak a > %]). Then there is a proof for Pc ` A of height 1 which
0 0 0 0 0 7! 0 0 0 0 0 Proof: Again we proceed by induction on k . Basis: Suppose G 2 TPc "1. Then there must exist a unit clause Theorem 3.7 (Comple...

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