R. Stark. A direct proof for the completeness of SLD-resolution. In Borger, H. Kleine Buning, and M.M. Richter, editors, Computation Theory and Logic 89, volume 440 of Lecture Notes in Computer Science, pages 382--383. Springer-Verlag, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....this model. The least term model of Clark [6] or C semantics of Falaschi et al. 8] is another natural candidate for the declarative semantics, and in fact it has been successfuly used in the probably most elegant and compact proof of the strong completeness of the SLD resolution due to Stark [12]. However, it shares with the least Herbrand model the same deficiencies. The last choice is the S semantics proposed by Falaschi et al. in [7] This semantics provides a precise match with the procedural interpretation of logic programs. So it captures completely the procedural behaviour of the ....

R. Stark. A direct proof for the completeness of SLD-resolution. In Borger, H. Kleine Buning, and M.M. Richter, editors, Computation Theory and Logic 89, volume 440 of Lecture Notes in Computer Science, pages 382--383. Springer-Verlag, 1990.

....the assumptions of the theorem both the Soundess Theorem 3.8 and the Strong Completeness Theorem 3.9 remain valid. For the completeness theorem this is not obvious, since it usually relies on the Lifting Lemma which not does not hold now. However, the admirably short and elegant proof of Stark [Sta90] does not use the Lifting Lemma and carries through. Consequently, the proof of Theorem 3.10 carries through, as well. 2 To apply this theorem let us return to the QUICKSORT program. We deal here with its correctly typed version QUICKSORT T obtained by using APPEND T instead of APPEND and in ....

R. Stark. A direct proof for the completeness of SLD-resolution. In E. Borger, H. Kleine Buning, and M.M. Richter, editors, Computation Theory and Logic 89, Lecture Notes in Computer Science 440, pages 382--383. Springer-Verlag, 1990.

....obvious, since it usually relies on the Krzysztof R. Apt 81 Lifting Lemma which does not hold now. Indeed, the query 1 2 admits a successful LD derivation, whereas all the LD derivations of its more general version X Y end in an error. However, the admirably short and elegant proof of Stark [35] does not use the Lifting Lemma and carries through. Consequently, the proof of Theorem 3.15 carries through as well. 2 Quicksort To apply this theorem reconsider the QUICKSORT program. We deal here with its correctly typed version QUICKSORT T, obtained by using APPEND T instead of APPEND and ....

R. Stark. A direct proof for the completeness of SLD-resolution. In Borger, H. Kleine Buning, and M.M. Richter, editors, Computer Science Logic 89, Lecture Notes in Computer Science 440, pages 382--383. Springer-Verlag, 1990.

....3.1. Soundness) Suppose that there exists a successful SLD derivation of P [ fQg with the computed answer substitution . Then P j= Q . 2 4. Completeness In this section we establish a completeness result. To this end we adjust the proof of strong completeness of SLD resolution due to Stark [Sta90]. We begin by introducing the following 7 concept. Definition 4.1. A finite tree whose nodes are atoms, is called an implication tree w.r.t. P if for each of its nodes A with the children B 1 ; B n , the clause A B 1 ; B n is in inst(P ) We say that an atom has an implication ....

R. Stark. A direct proof for the completeness of SLD-resolution. In E. Borger, H. Kleine Buning, and M.M. Richter, editors, Computer Science Logic 89, Lecture Notes in Computer Science 440, pages 382--383. Springer-Verlag, 1990.

....by P the set of formulas f8 C j C 2 Pg. Theorem 4.1 (Completeness of Minimal Logic for Positive Unconstrained Programs) Let P be a positive unconstrained program and F be an atom, or a conjunction of atoms, or a disjunction of atoms. If P c F , then P m F . Theorem 4. 1 is established in [14] for definite i.e. non disjunctive positive unconstrained programs. Related results are given in [13, 10] 4.4 Procedural Semantics The following Theorem shows that minimal logic derivations and SLD resolution proofs of atoms and conjunctions of atoms from positive, definite, and ....

.... deduction style weakening of classical logic s proof theory which precludes refutation proofs, has been shown to be sufficient for generalized programs corresponding to deductive databases and disjunctive logic programs, and to formalize SLD resolution proofs, thus generalizing a former result [14]. In order to provide logic programs and deductive databases with a model theory conforming to the rejection of refutation proofs, a nonclassical model theory allowing local inconsistencies has been proposed. Relying on this model theory, a notion of intended model for a generalization of logic ....

R. F. Stark. A Direct Proof for the Completeness of SLD-Resolution. Computer Science Logic. 382-383, Springer-Verlag LNCS 440, 1990.

....this model. The least term model of Clark [7] or C semantics of Falaschi et al. 9] is another natural candidate for the declarative semantics, and in fact it has been successfully used in the probably most elegant and compact proof of the strong completeness of the SLD resolution due to Stark [16]. However, it shares with the least Herbrand model the same deficiencies. The last choice is the S semantics proposed by Falaschi et al. in [8] This semantics provides a precise match with the procedural interpretation of logic programs. So it captures completely the procedural behaviour of the ....

R. Stark. A direct proof for the completeness of SLD-resolution. In Borger, H. Kleine Buning, and M.M. Richter, editors, Computation Theory and Logic 89, Lecture Notes in Computer Science 440, pages 382--383. Springer-Verlag, 1990.

....application of c . Since P is an unconstrained program, it contains no denials. Hence, A G = E G (G ) G E c . By hypothesis, E G contains no applications of c . A G can therefore be replaced in D F by E G . The result follows by induction. Theorem 4 is established in [13] for definite i.e. non disjunctive positive unconstrained programs. Related results are given in [12, 9] 4.4 Procedural Semantics The following Theorem, the proof of which is easy [1] shows that minimal logic derivations and SLD resolution proofs of atoms and conjunctions of atoms from ....

R. F. Stark. A Direct Proof for the Completeness of SLDResolution. Computer Science Logic. E. Borger, H. Kleine-Buning and M.M. Richter, eds. 382-383, Springer-Verlag LNCS 440, 1990.

....in # i a positive literal A and there exists a variant C of a clause of P in which no variables occur from # 0 # 1 # i or # i , and # i 1 is derived from # i using A, the input clause C and the most general unifier # i 1 . The following lifting lemma for pre implication trees was used in [30] for a short proof of the completeness of SLD resolution for definite programs. In the next section we will use it to lift down ESLDNFS implication trees to ESLDNFS proofs. Lemma 38 (Lifting lemma for pre implication trees) If # is the goal L 1 , Lm , # is a substitution, T 1 , ....

R. F. Stark. A direct proof for the completeness of SLD-resolution. In E. Borger, H. Kleine Buning, and M. M. Richter, editors, Computer Science Logic, selected papers from CSL '89, pages 382--383. Springer-Verlag, Lecture Notes in Computer Science 440, 1990.

....# SUB, then # # t## # R M . Since we want that M is a model of the clauses of P , each positive literal, which can be derived from negative, true literals in M using clauses from P , has to be true in M, too. To make this more clear we need the notion of an implication tree (cf. 2] and [10]) Definition 6.5 Implication trees (w.r.t. to P and M) are generated as follows: If # # t # # R M , then R( # t ) is an implication tree for R( # t ) If F j is an implication tree for L j (1 # j # n) and L 1 # . # L n # A is an instance of a clause of P , then A(F 1 , ....

....t # # R M . # Lemma 6.7 Assume that # 0 is an arbitrary node of T and T (# 0 ) # , G 0 , # 0 #. Assume that # is a substitution such that every literal of G 0 # 0 # has an implication tree. Then there exists an answer # n such that # 0 ## # n and G 0 # n # G 0 # 0 #. Proof. See also [10]. Let n be the total number of literals in the implication trees for the literals in G 0 # 0 #. By induction on i # n, we show that there exists a branch # 0 , # i in T and sequences G 0 , G i , # 0 , # i , # 0 , # i such that the following conditions are ....

R. F. Stark. A direct proof for the completeness of SLD-resolution. In E. Borger, H. Kleine Buning, and M. M. Richter, editors, Computer Science Logic, selected papers from CSL '89, pages 382--383. Springer-Verlag, Lecture Notes in Computer Science 440, 1990.

....to note that implication trees are not computations of an ideal logic programming machine like ESLDNF proofs. They are only a tool for proving properties of computations. Using implication trees one can give a very short proof for the completeness of SLD resolution for definite programs (see Stark [15]) Closed implication trees were first introduced by Apt, Blair and Walker in [1] Definition 8 Let L be a literal and P be a program. An implication tree for L with respect to P of rank k is a finite tree T whose nodes are literals and whose root is L such that (a ) if A is a positive node of T ....

R. F. Stark. A direct proof for the completeness of SLD-resolution. In E. Borger, H. Kleine Buning, and M. M. Richter, editors, Computer Science Logic, selected papers from CSL '89, pages 382--383. Springer-Verlag, Lecture Notes in Computer Science 440, 1990.

....of three valued logic. Its proof is a generalization of the proof of Kunen in [20] The proof which we give below is more direct and it does not use three valued logic. It is based on the completeness proofs in [16] and [26] In some sense Lemma 4. 2 above is a generalization of Theorem 3 of [25] and Theorem 4.5 below is a strong generalization of Lemma 2 of [25] Theorem 4.5 Let P be a program and (L 1 , L r ) be a goal in C (P ) which is not in N(P ) Let V be a set of variables which does not contain variables from (L 1 , L r ) Then there exists a term model M such ....

....of Kunen in [20] The proof which we give below is more direct and it does not use three valued logic. It is based on the completeness proofs in [16] and [26] In some sense Lemma 4.2 above is a generalization of Theorem 3 of [25] and Theorem 4. 5 below is a strong generalization of Lemma 2 of [25]. Theorem 4.5 Let P be a program and (L 1 , L r ) be a goal in C (P ) which is not in N(P ) Let V be a set of variables which does not contain variables from (L 1 , L r ) Then there exists a term model M such that (1) M = pcomp(P ) 2) M # = #(#L 1 # . # #L r ) ....

R. F. Stark. A direct proof for the completeness of SLD-resolution. In E. Borger, H. Kleine Buning, and M. M. Richter, editors, Computer Science Logic, selected papers from CSL '89, pages 382--383. Springer-Verlag, Lecture Notes in Computer Science 440, 1990.

....to note that implication trees are not computations of an ideal logic programming machine like ESLDNF proofs. They are only a tool for proving properties of computations. Using implication trees one can give a very short proof for the completeness of SLD resolution for definite programs (see Stark [15]) Closed implication trees were first introduced by Apt, Blair and Walker in [1] Definition 8 Let L be a literal and P be a program. An implication tree for L with respect to P of rank k is a finite tree T whose nodes are literals and whose root is L such that (a ) if A is a positive node of T ....

R. F. Stark. A direct proof for the completeness of SLD-resolution. In E. Borger, H. Kleine Buning, and M. M. Richter, editors, Computer Science Logic CSL '89, pages 382--383. Springer-Verlag, LNCS 440, 1990.

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