D. Plaisted. Non-Horn Clause Logic Programming without Contrapositives. Journal of Automated Reasoning, 4(3):287--325, September 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....goal can be found by retrieving from the cache store all cached solutions that are instances of the goal. If subsumption of cached solutions is employed, however, unifiable solutions rather than instances must be retrieved from the cache store. Our cache di#ers in use from the cache described in [28] in which the cache may be used even when it is not complete. It is similar to an unimplemented modification developed for iterative deepening of the ET # algorithm for Datalog programs outlined in [11] When caching is used, the modifications indicated in Figure 3 are made to the search routine ....

D. Plaisted. Non-Horn Clause Logic Programming without Contrapositives. Journal of Automated Reasoning, 4(3):287--325, September 1988.

.... existence of a Prolog predicate unify that performs sound unification with the occurs check and the observation that the occurs check is unnecessary if the clause head is linear, there is an elegant method of transforming clauses to isolate parts that may require unification with the occurs check [26, 27]. Repeated occurrences of variables are replaced by new variables to make the clause head linear. Unifying the clause head with a goal can then proceed without the occurs check and will not create any circular bindings. The new variables in the transformed clause head are then unified with the ....

Plaisted, D.A. Non-Horn clause logic programming without contrapositives. Journal of Automated Reasoning 4, 3 (September 1988), 287--325.

....a reduction 2 Such representations can be overridden by annotating an input file, e.g. the clause representing symmetry in a theorem involving inequalities: X Y ) Y X) should not be represented twice. In addition, refinements to ME inference mechanisms such as Plaisted s positive refinement [Pla88] and Spencer s foothold format [Spe90] are supported that may reduce the number of potential reduction alternatives at the expense of potentially longer proofs. 3 METEOR was originally developed on the GP1000 and performance data given in [AL91] is from the version of METEOR running on a GP1000. ....

D. Plaisted. Non-Horn clause logic programming without contrapositives. Journal of Automated Reasoning, 4(3):287--325, September 1988.

....4 Comparison with Modified Problem Reduction Format Many researchers have dealt with negation in logic programming. Their work has appeared in InH Prolog [3] Selected Linear Without contrapositive Variants [5] Prolog Technology Theorem Prover [7] and Modified Problem Reduction Format (MPRF) [6] are related to our work. MPRF appears to be the most closely related system to ours in terms of deduction principles, and we chose to compare the two systems. MPRF and our inference system are similar by the fact they both have an explicit case analysis rule. MPRF system is made efficient ....

Plaisted David A., Non-Horn Clause Logic Programming Without Contrapositives, Journal of Automated Reasoning 4, p. 287-325, 1988.

....it become necessary to compile in an occurs check. In the special case where no variables are repeated in one of the terms, i.e. when the term is linear, no occurs check will be necessary for the entire unification operation. There are alternative methods to assuring sound unification. Plaisted [28] has suggested an elegant method, which we currently use in another version of PTTP, of transforming clauses to isolate parts that may require unification with the occurs check. Repeated occurrences of variables are replaced by new variables so as to make the clause head linear. Matching the ....

....search. F Prolog also performs nonclausal inference. Its implementation in Prolog rather than as a Prolog extension makes it much slower than PTTP. Other inference systems that are complete extensions of Prolog for non Horn clauses have been proposed by Eder [11] Loveland [18, 19] Plaisted [28] and a group at Technische Universitat Munchen (TUM) 3] In Eder s system, the non Horn clause PQR S is represented by the clause r; s p, q, and only the literals r and s are matched in resolution operations. Non Horn proofs involving case analysis are generated by resolution using lemmas ....

Plaisted, D.A. Non-Horn clause logic programming without contrapositives. 1987.

.... existence of a Prolog predicate unify that performs sound unification with the occurs check and the observation that the occurs check is unnecessary if the clause head is linear, there is an elegant method of transforming clauses to isolate parts that may require unification with the occurs check [12, 13]. Repeated occurrences of variables are replaced by new variables to make the clause head linear. Unifying the clause head with a goal can then proceed without the occurs check and will not create any circular bindings. The new variables in the transformed clause head are then unified with the ....

Plaisted, D.A. Non-Horn clause logic programming without contrapositives. Journal of Automated Reasoning 4, 3 (September 1988), 287--325.

....reduce inner loop processing time. More specifically, the preprocessing reduces the penalty in inner loop speed for the use of a stronger proof procedure. This variant is actually closer in structure to systems proposed by others, notably N Prolog [3, 2] and the Simplified Problem Reduction Format [9, 10] (hereafter referenced as SPRF) than to nH Prolog, as we note later. We present the variant here, discuss some of its characteristics, particularly those distinct from nH Prolog, and establish soundness and completeness properties for this variant. The variant that we consider is Inheritance ....

....can realize on occasion. In addition, InH Prolog turns out to be a quite natural formulation of the near Horn Prolog approach to extending Prolog, an impression that is reinforced by the observation that InH Prolog has natural counterparts in a sequent style theorem proving procedure (SPRF [9, 10]) and a Prolog for hypothetical implication (N Prolog [3, 2] see Program: p(X) q(a) q(b) r(b) p(X) q(X) s(X) s(c) p(c) Query: q(X) i.e. FALSE : q(X) Deduction using weak cancellation: FALSE : q(X) # f0:p(Y)g where X = a restart : FALSE # p(Y) q(X1) # p(Y) ....

David Plaisted. Non-Horn clause logic programming without contrapositives. J. Automated Reasoning, 4:287--325, 1988.

....size instances of generic problems (e.g. the pigeon holes problem [CR79] ffl A utility to convert the problems to existing ATP formats. Currently the KIF [GF92] 3TAP [HBG94] leanTAP [BP95] METEOR [Ast92] MGTP [FHKF92] Otter [McC94] PTTP [Sti84] SETHEO [STvdK90] SPASS [WGRar] and SPRFN [Pla88] formats are supported, and the utility can easily be extended to produce any format required. ffl General guidelines outlining the requirements for ATP system evaluation. The TPTP is managed in the manner of a software product, in the sense that fixed releases are made. Each release of the TPTP ....

D.A. Plaisted. Non-Horn Clause Logic Programming Without Contrapositives. Journal of Automated Reasoning, 4(3):287--325, 1988.

....backtracking. red 0 is a shorthand for red( The tree positioning commands and the concepts of successors, predecessors and subgoals are the same as in FLEX (see Section 8.1.1) 8.1. 3 MPRT MPRT is an interactive proof editor for the modified problem reduction format as defined by Plaisted in [Pl88]. Proofs are represented as trees similiar to FLEX and ME. A discussion of the tree representation of modified problem reduction can be found in [Me90] MPRT uses three inference rules: axiom application, assumption application and splitting (see below) Note that MPRT is much simpler than the ....

Plaisted, D. A.: Non-Horn Clause Logic Programming without Contrapositives, Journal of Automated Reasoning 4 (1988), 287--325.

....a positive literal in the head. The approach from [9] completely ignores this aspect by using SLI resolution which requires all contrapositives. There are proposals for first order proof calculi using program clauses only in this procedural reading, e.g. Plaisted s problem reduction formats [16], or the nearHornProlog family introduced by Loveland and his co workers [11] These approaches introduce new calculi or proof procedures, for which efficient implementations still have to be developed. For a thorough discussion we refer to [3, 5] Our aim was to modify ME such that it can be ....

D. Plaisted. Non-Horn Clause Logic Programming Without Contrapositives. Journal of Automated Reasoning, 4:287--325, 1988.

....Head Selection Function. Unlike model elimination (ME) RME restricts in extension steps the connections into the extending clause to the head literals. We now go one step further, by introducing a head selection function. This concept is also present in Plaisted s Problem Reduction Formats [7], but not in the nearHorn Prolog family [5] 2 It is required that c is stable under lifting, which means that for any substitution s, whenever c(Q s) q]s then c(Q ) q] Definition 2.2 (Head Selection Function, 1] A head selection function f is a function that maps a clause A 1 : ....

D. Plaisted. Non-Horn Clause Logic Programming Without Contrapositives. Journal of Automated Reasoning, 4:287--325, 1988.

....as the antecedent. Further, in all conjecture clauses all positive literals are negated so that all literals are written in the antecedent of an implication. The default style is sign, i.e. the abbreviation fsetheo means fsetheo:sign. sprfn, to convert CNF problems to the SPRFN format [Pla88] tptp, to convert FOF and CNF problems to the TPTP format, substituting include instructions with the actual formulae. The default Format is tptp. ffl d Directory : Specifies the top level directory below which the output files are to be placed. If the Directory value is , then all ....

D.A. Plaisted. Non-Horn Clause Logic Programming Without Contrapositives. Journal of Automated Reasoning, 4(3):287--325, 1988.

....are not explicitly expressed in the proof procedure in Program 2.2, they may be specialised into the proof procedure by the specialisation process. One of the criticisms against model elimination is the requirement that all contrapositives of clauses must be available for deduction steps [84]. In many cases our method eliminates many unnecessary contrapositives and so also reduces the size of the search space. Examples of this can be found in Chapter 6 where all contrapositives can be deleted that have a literal that cannot contribute to a proof (information given for input ....

....without contrapositives is another candidate for our specialisation methods and it should be interesting to compare the results inferred for their proof procedure with that presented in this thesis. Other techniques for optimising model elimination, such as the Positive Refinement of Plaisted [84] reduces the need to do ancestor resolution and may make our inferred information less useful. However, ancestor resolution still has to be done (or some other mechanism has to be introduced that replaces ancestor resolution) and in many cases our analysis information still leads to a reduction in ....

D. Plaisted. Non-Horn clause logic programming without contrapositives. Journal of Automated Reasoning, 4:287--325, 1988.

....these approaches. In previous work the authors introduced the family of RME calculi as goal oriented interpreters for positive disjunctive logic programs [BF94a] more recently, we investigated variants of RME for computing answers to queries for DLPs [BFS95] RME is related to Plaisted s MPRF [Pla88] and Loveland s nearHorn Prolog [Lov87] The idea throughout these calculi is to enter a clause A 1 Delta Delta Delta Am B 1 Delta Delta Delta B n only through one of the head literals A 1 ; Am , but never through one of the body literals B 1 ; B n . Thus, a natural ....

D. Plaisted. Non-Horn Clause Logic Programming Without Contrapositives. Journal of Automated Reasoning, 4:287--325, 1988.

....a positive literal in the head. The approach from [18] completely ignores this aspect by using SLI resolution which requires all contrapositives. There are proposals for first order proof calculi using program clauses in only this procedural reading, e.g. Plaisted s problem reduction formats [27], or the near Horn Prolog family introduced by Loveland and his co workers [21] For a thorough discussion we refer to [4] These approaches introduce new calculi or proof procedures, for which efficient implementations still have to be developed. Our aim is to modify ME such that it can be used ....

....restart literals from the leaf towards the top. As a further benefit of this search order note that a definite answer will be enumerated before a non definite answer, provided it allows for a shorter proof. Ancestry RME has some similarities to Plaisted s modified problem reduction format (MPRF) [27]. Expressed in our terminology, MPRF corresponds roughly to the non strict version of Ancestry RME. As major differences we see that answer computation was not an issue in [27] and also that regularity refinements were not considered. These differences justify the need for our new proofs. Now we ....

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D. Plaisted. Non-Horn Clause Logic Programming Without Contrapositives. Journal of Automated Reasoning, 4:287--325, 1988.

....establish that the query 9 L is proved whenever it should be provable. Before stating the completeness theorem, we address several points worth noting or recalling. Regarding the result, the assertion is very close to completeness results for similar systems, in particular the SPRF procedure [Pla82, Pla88] and the N Prolog procedure [GR84, Gab85] see chapter 6) The key difference is the inclusion here of the cancellation pruning rule, which demands the use of the distinguished head within its block. As we have previously stated, this provides a strong pruning rule, although its optimal use does ....

....refutation in which the goal atom is selected for reduction cancellation. The subtree with root sequent FALSE # h 2 corresponds to the restart block in the linear refutation. 6. 2 SPRF The first system we will consider is the Simplified Problem Reduction Format (SPRF) developed by Plaisted [Pla82, Pla88]. The SPRF is a Gentzenstyle axiom system that employs sequents of the form Gamma L where Gamma is a sequence of atoms and L is an atom. The axioms for the system are of the form Gamma L for L 2 Gamma. The input set is expected to be in clausal form with negative facts positively ....

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D. Plaisted. Non-Horn clause logic programming without contrapositives. J. Automated Reasoning, 4:287--325, 1988.

....the following output formats: the MGTP format [FHKF92] the Otter .in format [McC90] the set of support and inference rules to be used are also specified, and are included in the output file) the PTTP format [Sti84] the SETHEO . lop format [STvdK90] the SPRFN format [Pla88] the TPTP format, substituting include instructions with the actual clauses. It is simple to add new formatting capabilities to the tptp2X utility. The tptp2X utility can also be used to rearrange the clauses and literals in a problem. This facilitates testing the sensitivity of an ATP ....

D.A. Plaisted. Non-Horn Clause Logic Programming without Contrapositives. Journal of Automated Reasoning, 4(3):287--325, 1988.

....as an intermediate language. Hence, it should be straightforward to use model elimination and PTTP in the context of non Horn logic programming. Indeed this possibility is discussed in various papers; however it is discarded by some authors because of the necessity to use contrapositives (e.g. [Lov91, Pla88]) The argument is given by Plaisted [Pla88] explicitly: In general, however, we feel that the need for contrapositives makes it difficult to view model elimination as a true programming language in the style of Prolog, since the user has less control over the search. Suppose, for example we are ....

....straightforward to use model elimination and PTTP in the context of non Horn logic programming. Indeed this possibility is discussed in various papers; however it is discarded by some authors because of the necessity to use contrapositives (e.g. Lov91, Pla88] The argument is given by Plaisted [Pla88] explicitly: In general, however, we feel that the need for contrapositives makes it difficult to view model elimination as a true programming language in the style of Prolog, since the user has less control over the search. Suppose, for example we are given an input clause 1 prove(and(X ; Y ) ....

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D. Plaisted. Non-Horn Clause Logic Programming Without Contrapositives. Journal of Automated Reasoning, 4:287--325, 1988.

....by additional features, like theory handling or case analysis. This is done in PROTEIN. This research was sponsored by DFG within the Schwerpunktprogramm Deduktion . The case analysis style of reasoning came up with various (non theory) calculi which do not need all contrapositives ([Lov87, Pla88]) A detailed comparison of those calculi can be found in [RL92] In [BF93] we have made a small change to model elimination which also avoids contrapositives and has some distinguished features. This modification of model elimination is called restart model elimination; its distinguished feature ....

D. Plaisted. Non-Horn Clause Logic Programming Without Contrapositives. Journal of Automated Reasoning, 4:287--325, 1988.

....as the antecedent. Further, in all conjecture clauses all positive literals are negated so that all literals are written in the antecedent of an implication. The default style is sign, i.e. the abbreviation fsetheo means fsetheo:sign. sprfn, to convert CNF problems to the SPRFN format [Pla88] thinker, to convert FOF problems to the THINKER format [Pel82, Pel87] tptp, to convert FOF and CNF problems to the TPTP format, substituting include instructions with the actual formulae. waldmeister, to convert CNF problems to the Waldmeister format [HBF96] The default ....

D.A. Plaisted. Non-Horn Clause Logic Programming Without Contrapositives. Journal of Automated Reasoning, 4(3):287--325, 1988.

....(non theory) calculi which do not need all contrapositives 2 . Near Horn Prolog ( Loveland, 1987, Loveland, 1991] is motivated from the background of logic programming, and needs contrapositives for each positive literal in a clause only; The (modified) simplified problem reduction format [Plaisted, 1988] is more a theorem proving system and needs only one single contrapositive per clause. A detailed comparison of these calculi and of another one, N Prolog, can be found in [Reed and Loveland, 1992] In [Baumgartner and Furbach, 1993] we have made 2 In systems with contrapositives n procedural ....

....n procedural counterparts (contrapositives) L i L 1 Delta Delta Delta L i Gamma1 L i 1 Delta Delta Delta L n for a clause L 1 Delta Delta Delta L n have to be considered. Each contrapositive represents a different entry point during the proof search into the clause. See [Plaisted, 1988] for a motivation for the avoidance of contrapositives. 1 Introduction 4 a small change to model elimination which also avoids contrapositives and allows for a PTTP implementation. The common idea behind all these calculi is to carry out case analysis wrt. the positive literals in a disjunctive ....

D. Plaisted. Non-Horn Clause Logic Programming Without Contrapositives. Journal of Automated Reasoning, 4:287--325, 1988.

....ffl Arbitrary size instances of generic problems (e.g. the pigeon holes problem [CR79] ffl A utility to convert the problems to existing ATP formats. Currently the KIF [GF92] 3TAP [HBG94] leanTAP [BP95] METEOR [Ast92] MGTP [FHKF92] Otter [McC94] PTTP [Sti84] SETHEO [STvdK90] and SPRFN [Pla88] formats are supported, and the utility can easily be extended to produce any format required. ffl General guidelines outlining the requirements for ATP system evaluation. The TPTP is managed in the manner of a software product, in the sense that fixed releases are made. Each release of the TPTP ....

D.A. Plaisted. Non-Horn Clause Logic Programming Without Contrapositives. Journal of Automated Reasoning, 4(3):287--325, 1988.

....transforming, and generating TPTP problem files. In particular, it: ffl Converts from the TPTP format to formats used by existing ATP systems. The system formats available are KIF [31] leanTAP [8] 3TAP [33] METEOR [6] MGTP [29] Otter [66] PTTP [105] SETHEO [93] SPASS [120, 119] SPRFN [81], and TPTP (substituting include instructions with the actual clauses) ffl Applies various transformations to the clauses of TPTP problems. The transformations available are to reverse the order of the literals or clauses, to randomly reorder the clauses and literals, to remove selected ....

D.A. Plaisted. Non-Horn Clause Logic Programming Without Contrapositives. Journal of Automated Reasoning, 4(3):287--325, 1988.

....as the antecedent. Further, in all conjecture clauses all positive literals are negated so that all literals are written in the antecedent of an implication. The default style is sign, i.e. the abbreviation f setheo means f setheo:sign. sprfn, to convert CNF problems to the SPRFN format [Pla88] thinker, to convert FOF problems to the THINKER format [Pel82, Pel87] tptp, to convert FOF and CNF problems to the TPTP format, substituting include instructions with the actual formulae. waldmeister, to convert CNF problems to the Waldmeister format [HBF96] The default Format ....

D.A. Plaisted. Non-Horn Clause Logic Programming Without Contrapositives. Journal of Automated Reasoning, 4(3):287--325, 1988.

....literal in the head. The approach from [Lobo et al. 1992] completely ignores this aspect by using SLI resolution which requires all contrapositives. There are proposals for first order proof calculi using program clauses only in this procedural reading, e.g. Plaisted s problem reduction formats [Plaisted, 1988] , or the near Horn Prolog family introduced by Loveland and his co workers [Loveland, 1991] These approaches introduce new calculi or proof procedures, for which efficient implementations still have to be developed. For a thorough discussion we refer to [Baumgartner and Furbach, 1994a] Our ....

D. Plaisted. Non-Horn Clause Logic Programming Without Contrapositives. Journal of Automated Reasoning, 4:287--325, 1988.

....as they are solved and avoid repeatedly solving them. These techniques generally only work for Horn clauses and not for rst order logic. Plaisted showed that only the MESON and Model Elimination strategies [9] with unit lemmas and caching [1] the simple and modi ed problem reduction formats [15, 16] and the Hyperlinking strategy [8] are polynomial. However, the rst two of these strategies (with caching) are not complete for rstorder logic; the second two are ineOEcient on nonHorn clauses, and have not been extended to equality. He also notes that none of the resolution strategies are ....

D. Plaisted. NonHorn clause logic programming without contrapositives. Journal of Automated Reasoning 4 (1988) pp. 287-325.

....in such a way that the above natural contrapositives suffice. Indeed, this is achieved in restart model elimination (there are also several other calculi achieving this, e.g. Loveland s NearHorn Prolog [Loveland, 1987] Gabbay s N Prolog [Gabbay, 1985] and Plaisted s problem reduction formats [Plaisted, 1988] ) In restart model elimination, for non definite clauses, like A B :C :D only the following contrapositives have to be considered: A :B ; C ; D B :A; C ; D 1 Taken from [Plaisted, 1988] It is even possible to restrict to one of those (cf. the selection function below) Of course ....

.... [Loveland, 1987] Gabbay s N Prolog [Gabbay, 1985] and Plaisted s problem reduction formats [Plaisted, 1988] In restart model elimination, for non definite clauses, like A B :C :D only the following contrapositives have to be considered: A :B ; C ; D B :A; C ; D 1 Taken from [Plaisted, 1988] . It is even possible to restrict to one of those (cf. the selection function below) Of course there is a price to pay. In restart model elimination an additional inference rule is needed, the restart rule, which allows to begin a completely new proof once a negative body literal, such as :B ....

D. Plaisted. Non-Horn Clause Logic Programming Without Contrapositives. Journal of Automated Reasoning, 4:287--325, 1988.

....of disjunctive program clauses. We now go one step further, by introducing a head selection function. This is a means to distinguish one single head literal, which is then the only one allowed to use for an extension step (this concept is also present in Plaisted s Problem Reduction Formats [ Plaisted, 1988 ] but not for the nearHorn Prolog family) In our example restart model elimination refutation from Figure 2 we used the clause P Q for an extension step at a leaf literal :Q . Now, if the P literal is to be the distinguished literal, then this extension step would be impossible. This is a ....

D. Plaisted. Non-Horn Clause Logic Programming Without Contrapositives. Journal of Automated Reasoning, 4:287--325, 1988.

....work is the upcoming of various (nontheory) calculi which do not need all contrapositives 2 . Near Horn Prolog ( 16] is motivated from the background of logic programming, and needs contrapositives for each positive literal in a clause only. The (modified) simplified problem reduction format [20] is more 1 Universitat Koblenz, Institut fur Informatik, Rheinau 1, 56075 Koblenz, Germany, E mail: peter infko.uni koblenz.de 2 In systems with contrapositives n procedural counterparts (contrapositives) L i : Gamma L 1 ; Delta Delta Delta ; L i Gamma1 ; L i 1 ; Delta Delta Delta ; L ....

....n procedural counterparts (contrapositives) L i : Gamma L 1 ; Delta Delta Delta ; L i Gamma1 ; L i 1 ; Delta Delta Delta ; L n for a clause L 1 Delta Delta Delta L n have to be used. Each contrapositive represents a different entry point during the proof search into the clause. See [20] for a motivation for the avoidance of contrapositives. a theorem proving system and needs only one single contrapositive per clause. A detailed comparison of these calculi and of another one, N Prolog, can be found in [21] In [7] we have made a small change to model elimination which also ....

D. Plaisted, `Non-Horn Clause Logic Programming Without Contrapositives ', Journal of Automated Reasoning, 4, 287--325, (1988).

....which works within the original proof system, is to proceed as if L rules are not required in the proof being constructed and, when they are determined to be actually necessary, to attempt to insert them at an appropriate point in the proof. The modified problem reduction format of Plaisted [Pla88] is based on this approach and on noting that the use of assumption formulas of the form (B 1 : B n ) oe A where A is atomic can be driven, even in this context, by the atomic formula for which proof is sought. 6 Conclusion We have examined the applicability of the notion of uniform ....

David A. Plaisted. Non-Horn clause logic programming without contrapositives. Journal of Automated Reasoning, 4(3):287--325, 1988.

....literal in the head. The approach from [Lobo et al. 1992] completely ignores this aspect by using SLI resolution which requires all contrapositives. There are proposals for first order proof calculi using program clauses only in this procedural reading, e.g. Plaisted s problem reduction formats [Plaisted, 1988] , or the nearHornProlog family introduced by Loveland and his co workers [Loveland, 1991] These approaches introduce new calculi or proof procedures, for which efficient implementations still have to be developed. For a thorough discussion we refer to [Baumgartner and Furbach, 1994a] Our aim ....

D. Plaisted. Non-Horn Clause Logic Programming Without Contrapositives. Journal of Automated Reasoning, 4:287--325, 1988.

....1989 ] Hence, it should be straightforward to use model elimination and PTTP in the context of non Horn logic programming. Indeed this possibility is discussed in various papers; however it is discarded by some authors because of the necessity to use contrapositives (e.g. Loveland, 1991, Plaisted, 1988 ] The argument is given by Plaisted [ Plaisted, 1988 ] explicitly: In general, however, we feel that the need for contrapositives makes it difficult to view model elimination as a true programming language in the style of Prolog, since the user has less control over the search. Suppose, for ....

....model elimination and PTTP in the context of non Horn logic programming. Indeed this possibility is discussed in various papers; however it is discarded by some authors because of the necessity to use contrapositives (e.g. Loveland, 1991, Plaisted, 1988 ] The argument is given by Plaisted [ Plaisted, 1988 ] explicitly: In general, however, we feel that the need for contrapositives makes it difficult to view model elimination as a true programming language in the style of Prolog, since the user has less control over the search. Suppose, for example we are given an input clause 1 prove(and(X ; Y ....

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D. Plaisted. Non-Horn Clause Logic Programming Without Contrapositives. Journal of Automated Reasoning, 4:287--325, 1988.

....these tests are not a comprehensive suite, they are varied and take into account several language constructs offered by description logics. We were also interested in the performance of some other theorem provers on these tests, and we ran these tests on otter[McC90] clin[LP92, Lee90] and sprfn[Pla88] as well. We compared their performance with those of the the description logics based on reports in [HKNP92] It is interesting to note that not all theorem provers were able to derive all of the inferences described by these tests. 3.1 Subsumption Checking in First Order Clause Sets Concepts ....

David A. Plaisted. Non-Horn clause logic programming without contrapositives. J. Automated Reasoning, 4:287--325, 1988.

....possibly more powerful; a disadvantage is that the results may appear less spectacular, especially in the early stages. We have concentrated on the latter approach. In this endeavor, we have developed a number of provers over the recent years, including the modified problem reduction format of [Pla88] and its extensions, the clause linking method of [LP92] and clause linking with semantics [CP94a] among others. We have emphasized first order logic without equality. These provers have become increasingly more powerful, each able to solve a considerable range of problems out of reach of its ....

D. Plaisted. Non-Horn clause logic programmingwithout contrapositives. Journal of Automated Reasoning, 4:287--325, 1988.

....We would like to find a back chaining strategy that is efficient on Horn clauses, based on unification, and still always permits instances to be deleted. Some such strategies exist; they are the MESON strategy, model elimination [Lov78] and the simple and modified problem reduction formats [Pla82, Pla88], all with caching. However, of these, either caching of unit lemmas and subgoals is not complete for first order logic, as for the first two, or the strategies have propositional inefficiencies for non Horn problems, as for the second two. For the MESON strategy and model elimination, if caching ....

....MESON strategy, unit lemmas result in behavior like that of model elimination with unit lemmas. The MESON strategy with unit lemmas and caching has behavior like model elimination with unit lemmas and caching. 5. 16 Problem reduction formats The simplified and modified problem reduction formats [Pla82, Pla88] simulate Prolog s back chaining mechanism, but are complete for first order logic. The simplified problem reduction format [Pla82] without caching is much the same as model elimination, for Horn clauses. The simplified problem reduction format generates formulae called decompositions. For Horn ....

[Article contains additional citation context not shown here]

D. Plaisted. Non-Horn clause logic programming without contrapositives. Journal of Automated Reasoning, 4:287--325, 1988.

....these tests are not a comprehensive suite, they are varied and take into account several language constructs offered by concept languages. We were also interested in the performance of some other theorem provers on these tests, and we ran these tests on otter[McC90] clin[LP92, Lee90] and sprfn[Pla88] as well. We compared their performance with those of the the concept languages based on reports in [HKNP92] It is interesting to note that not all theorem provers were able to derive all of the inferences described by these tests. 3.1 Subsumption Checking in First Order Clause Sets Concepts ....

David A. Plaisted. Non-Horn clause logic programming without contrapositives. J. Automated Reasoning, 4:287--325, 1988.

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David A. Plaisted. Non-Horn clause logic programming without contrapositives. J. Automated Reasoning, 4:287--325, 1988.

....for several language constructs offered by concept languages. They have been used before to compare some terminological systems [HKNP92] We were also interested in the performance of some other theorem provers on these tests, and we ran these tests on otter[McC90] clin[LP92, Lee90] and sprfn[Pla88] as well. We compared their performance with those of the the terminological systems as reported by [HKNP92] It is interesting to note that some theorem provers were not able to derive all of the inferences described by these tests. 3.1 Subsumption Checking with the Theorem Prover Suppose ....

David A. Plaisted. Non-Horn clause logic programming without contrapositives. J. Automated Reasoning, 4:287--325, 1988.

....mechanism for non Horn clauses is not well controlled because of a case analysis rule that is hard to control. We did implement this prover, but did not obtain good results for some reason. Still, such a prover can be quite efficient for Horn clause problems. The modified problem reduction format [Pla88] continued with the philosophy that the important issue in theorem proving is subgoaling. This strategy is similar to the simplified problem reduction format, but has a better control mechanism for case analysis. This was implemented in the sprfn theorem prover which still performs well on Horn ....

D. Plaisted. Non-Horn clause logic programming without contrapositives. Journal of Automated Reasoning, 4:287--325, 1988.

....these tests are not a comprehensive suite, they are varied and take into account several language constructs offered by concept languages. We were also interested in the performance of some other theorem provers on these tests, and we ran these tests on otter[McC90] clin[LP92, Lee90] and sprfn[Pla88] as well. We compared their performance with those of the the Concept Languages based on reports in [HKNP92] It is interesting to note that not all theorem provers were able to derive all of the inferences described by these tests. 3.1 Subsumption Checking in First Order Clause Sets Concepts ....

David A. Plaisted. Non-Horn clause logic programming without contrapositives. J. Automated Reasoning, 4:287--325, 1988.

....since matings are constructed and considered one by one. However, since the number of links per literal may be much larger than two, the number of cases may be much larger than in the Davis Putnam procedure for propositional calculus, it appears. Even the modified problem reduction format prover [Pla88] has duplication by combination on non Horn problems. Another motivation of our work is to justify the claim that a theorem proving method may benefit from applying the low level heuristics a human mathematician uses to obtain a proof. By low level heuristics, we mean such methods as preferring ....

....to neural networks. The similarities with semantic networks in artificial intelligence, and with connection graph methods in theorem proving, are closer. CLIN has been implemented in Prolog. Its effectiveness has been shown by comparing it with other major provers, such as OTTER[McC89] sprfn [Pla88] and PTTP[Sti88] CLIN performs better than the other provers for propositional and near propositional problems. For other problems, CLIN can prove more of them than the other provers. This shows that duplication of instances of clauses is a real problem for theorem proving, and that an ....

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D. Plaisted. Non-Horn clause logic programming without contrapositives. Journal of Automated Reasoning, 4:287--325, 1988.

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