Tabled Evaluation with Delaying for General Logic Programs

The semantics of predicate logic as a programming language. Every logic program has a natural stratiication and an iterated least xed point model. Magic sets and bottom-up evaluation of well-founded models. Query restricted bottom-up evaluation of normal logic programs. Completeness of a top-down query evaluation procedure for stratiied databases. 42 LPM(PP(S j) J)(B) for some j < i and for every B A 2 HB Prog , M(B A) = LPM(PP(S j) J)(B A) for some j < i such that A is a subgoal of S j. By inductive hypothesis, every S j (j < i) is a symmetric system and so every LPM(PP(S j) J) is a symmetric interpretation. Thus M is well deened and is also a symmetric interpretation. Furthermore it can be veriied that M = LPM(PProg J). The program P(S i) can be obtained by deleting rules from Prog. Sometimes a deleted rule is replaced by another rule, but there is no innnite chain of the replacement relationship. Let (A :) be an arbitrary pair in S i. Let G A 1 and G A 2 be arbitrary rules in Prog, where for some ordinals k 1 and k 2 , G 1 is the X-rule of a subgoal A corresponding to k 1 and G 2 is the X-rule of a subgoal A corresponding to k 2 in S i. G A 1 is said to be a parent of G A 2 , denoted by G A 1 > G A 2 , if (k 2) = G 2 hk 1 i or (k 2) = G 2 hk 1 ; Hi for some atom H. There is no innnite descending chain of the relation > since G 2 has either fewer literals that are not delayed or fewer literals that are delayed. The number of literals in G 1 and G 2 is bounded by the maximum number of literals in a rule body in a nite program P. Every rule in Prog that is not in P(S i) corresponds to an X-rule of a subgoal that is disposed when S j+1 is derived from S j for some j + 1 < i. By inductive hypothesis, LPM(PP(S j+1)