D.W. Loveland. Near-horn prolog. In Proc. of the 4th International Conference on Logic Programming, Melbourne, Australia, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....program formulae with multiple heads , i.e. having the form (Head 1 : Headn ) Body: where each Head i is a positive literal. We shall see how the connective acts as a form of disjunction. LO is from this point of view on a similar line of extensions of Prolog like those described in [20, 27], which also allow multiple heads with a disjunctive reading. However, there is one crucial distinction to be made, in that our use of disjunction will be quite more restrictive than the one adopted in the approaches above, although it is also 1 going to be interestingly related to them: the ....

....a corresponding disjunctive theorem of LO encoding the given substitution. This is what entitles us to refer to the LO disjunction as constructive restriction of classical disjunction. By contrast, indefinite answer substitutions will characterize extensions of Prolog such as those described in [20, 27], which allow, like LO, disjunctive heads of clauses but assume full classical disjunction. Let us consider for example the following theorem of Classical Logic (the example is taken from [20] p(a) p(b) c 9x p(x) In languages like those of [20, 27] this query yields the indefinite answer ....

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D.W. Loveland. Near-horn prolog. In Proc. of the 4th International Conference on Logic Programming, Melbourne, Australia, 1987.

....program clauses with multiple heads , i.e. having the form (Head 1 : Headn ) Body: where each Head i is a positive literal. We shall see how the connective acts as a form of disjunction. LO is from this point of view on a similar line of extensions of Prolog like those described in [15, 21], which also allow multiple heads with a disjunctive reading. However, there is one crucial distinction to be made, in that our use 1 of disjunction will be quite more restrictive than the one adopted in the approaches above, although it is also going to be interestingly related to them: the ....

....a corresponding disjunctive theorem of LO encoding the given substitution. This is what entitles us to refer to the LO disjunction as constructive restriction of classical disjunction. By contrast, indefinite answer substitutions will characterize extensions of Prolog such as those described in [15, 21], which allow, like LO, disjunctive heads of clauses but assume full classical disjunction. Let us consider for example the following theorem of Classical Logic (the example is taken from [15] p(a) p(b) c 9x p(x) In languages like those of [15, 21] this formula is provable with the ....

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D.W. Loveland. Near-horn prolog. In Proc. of the 4th International Conference on Logic Programming, Melbourne, Australia, 1987.

....that hyper tableaux can be seen as an extension of SLO resolution. Keywords: Disjunctive Logic Programming, Fixpoint Semantics, SLO, Theorem Proving 1 Introduction For disjunctive logic programs (DLPs) there are several proposals for defining interpreters, like the nearHorn Prolog family [ Loveland, 1987 ] SLI Resolution [ Lobo et al. 1992 ] SLO Resolution [ Rajasekar, 1989 ] model tree construction [ Fernandez and Minker, 91 ] or restart model elimination (RME) Baumgartner et al. 1995; Baumgartner and Furbach, 1994a ] There have also been different approaches to assign least fixpoints ....

D.W. Loveland. Near-Horn Prolog. In J.-L. Lassez, editor, Proc. of the 4th Int. Conf. on Logic Programming, pages 456--469. The MIT Press, 1987.

....logic programming. It became apparent, on the other hand, that in order to become satisfactory tools for knowledge representation, logic programming languages should be expanded to allow for better handling of incomplete information. Work in this direction was started by Minker [Min82] Loveland [Lov87] and others, who investigated the possibility of expanding logic programs by disjunctive information. In [GL90, PW89, Gel92b] extensions of logic programming by classical (or strong) negation and epistemic operators were suggested. Unlike traditional nonmonotonic formalisms, these extensions ....

....smallest (in a sense of set theoretic inclusion) subset S of Lit such that (i) for any rule L 0 or : or L k L k 1 : Lm from Pi, if L k 1 ; Lm 2 S, then for some i, 0 i k, L i 2 S; ii) if S contains a pair of complementary literals, then S = Lit. 14 Independently, Loveland [Lov87] considered extensions to Horn logic programs which he called near Horn logic programs. His main concern was efficient implementation [Lov87, SL88] 15 For positive disjunctive logic programs this definition is similar to Minker s [Min82] original definition. For a precise relationship see ....

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D.W. Loveland. Near-Horn PROLOG. In J.-L. Lassez, editor, Proc. 4 th International Conference on Logic Programming, pages 456--459, 1987.

....iterative deepening 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 ....

Loveland, D.W. Near-Horn Prolog. Proceedings of the Fourth International Conference on Logic Programming, Melbourne, Australia, May 1987, 456--469.

....to perform a simple goal directed search, as in the case of Prolog, since the goal may not match any single literal clause head, but only be one of the literals in a disjunctive clause head. There are two solutions that have been offered for this problem. The first was provided by Loveland [44, 45] in the form of nH Prolog. 28 This solution involves performing a partial match of a goal atom against a clause head. That is, a clause is eligible for selection if one of the literals in the head unifies with the goal atom currently being reduced. The remaining literals in the head of the ....

....possess is that they do not need to determine disjunctive entailment. All constraints in the CLP framework are tells. However, the clausal approach is of interest. Specifically, a possible solution would be to store constraints in clausal form and use some variant of the nH Prolog algorithm [44, 45]. This would work in a simple manner for base constraint entailment. For negated constraints and disjunctions the approach would be to temporarily tell those negated constraints in the clause (i.e. the antecedent constraints) Then we would ask the disjunction of the base constraints (i.e. the ....

D. W. Loveland. Near-Horn PROLOG. In Lassez [40], pages 456--469.

....in the assumption set or program. The ability to express and process indefinite information and case constructions occur in real life and mechanisms to handle such specifications are important. Attention has focused on the semantics of disjunctive logic programming [1, 2, 11] proof procedures [4, 12] and applications to disjunctive databases [9, 13, 14] One point of view of each of these topics appears in the book [3] The basic requirement of disjunctive logic programming, that disjunctive information be representable in the assumption set, appears to be at variance with the desire for the ....

....(under a reduction to clausal form) that is a natural generalization of the one usually employed in logic programming. The main new aspect of this procedure is that it has a rule for backchaining that is sensitive to the presence of disjunctive information. The restart mechanism of nH Prolog [4, 5] is used for dealing with this possibility and our procedure is, in fact, an abstract presentation of the InH Prolog procedure [6] 2 Logical preliminaries We shall work within the framework of a first order logic in this paper. The logical symbols that we shall use are , oe, 9, and 8; ....

D.W. Loveland. Near-Horn Prolog. In J. Lassez, editor, Logic Programming: Proc. of the Fourth Int'l Conf., pages 456--469. MIT Press, 1987.

....logic programming. It became apparent, on the other hand, that in order to become satisfactory tools for knowledge representation, logic programming languages should be expanded to allow for better handling of incomplete information. Work in this direction was started by Minker [Min82] Loveland [Lov87] and others, who investigated the possibility of expanding logic programs by disjunctive information. In [GL90, PW89, Gel92b] extensions of logic programming by classical (or strong) negation and epistemic operators were suggested. Unlike traditional nonmonotonic formalisms, these extensions ....

....a smallest (in a sense of set theoretic inclusion) subset S of Lit such that (i) for any rule L 0 or : or L k L k 1 : Lm from 5, if L k 1 ; Lm 2 S, then for some i, 0 i k, L i 2 S; ii) if S contains a pair of complementary literals, then S = Lit. 14 Independently, Loveland [Lov87] considered extensions to Horn logic programs which he called near Horn logic programs. His main concern was efficient implementation [Lov87, SL88] 15 For positive disjunctive logic programs this definition is similar to Minker s [Min82] original definition. For a precise relationship see ....

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D.W. Loveland. Near-Horn PROLOG. In J.-L. Lassez, editor, Proc. 4 th International Conference on Logic Programming, pages 456--459, 1987.

....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 : A n B 1 : Bm with n 1 to an atom ....

D.W. Loveland. Near-Horn Prolog. In J.-L. Lassez, editor, Proc. of the 4th Int. Conf. on Logic Programming, pages 456--469. The MIT Press, 1987.

....Furthermore we discuss that hyper tableaux can be seen as an extension of SLO resolution. Keywords: Disjunctive Logic Programming, Fixpoint Semantics, SLO 1 Introduction For disjunctive logic programs (DLPs) there are several proposals for defining interpreters, like the nearHorn Prolog family [Lov87], SLI Resolution [LMR92] SLOResolution [Raj89] model tree construction [FM91] or restart model elimination (RME) BF94a, BFS95] There have also been different approaches to assign least fixpoints to DLPs, like the state semantics [MR90] a semantics based on model trees [FM91] and approaches to ....

.... 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 procedural reading of clauses is better ....

D.W. Loveland. Near-Horn Prolog. In J.-L. Lassez, editor, Proc. of the 4th Int. Conf. on Logic Programming, pages 456--469. The MIT Press, 1987.

....always been an important aspect. From there we learned that it is more difficult to prove a calculus to be answer complete than only refutationally complete. Recently, there has been considerable effort to use full first order logic instead of only Horn clauses as the basis of logic programming [23]. This (positive) disjunctive logic programming approach is investigated from various directions. From the view of theorem proving one is concerned with modifying theorem provers such that they can be used as interpreters for logic programming purposes. The nonmonotonic reasoning community is ....

D.W. Loveland. Near-Horn Prolog. In J.-L. Lassez, editor, Proc. of the 4th Int. Conf. on Logic Programming, pages 456--469. The MIT Press, 1987.

....need contrapositives. PROTEIN is a complete theorem prover for first order clause logic. It is characterized by the following features: PROTEIN is based on the PTTP implementation technique [Sti88] for model elimination [Lov69] PROTEIN offers alternative inference rules for case analysis [Lov87, BF93]. In this setting no contrapositives are needed, and hence the system is well suited as an interpreter for disjunctive logic programming. PROTEIN includes theory reasoning [Sti85, Bau92, Bau94] in a very general way. An auxiliary program can be used to derive a suitable background reasoner ....

....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.W. Loveland. Near-Horn Prolog. In J.-L. Lassez, editor, Proc. of the 4th Int. Conf. on Logic Programming, pages 456--469. The MIT Press, 1987.

....thus concludes this chain. We will treat both versions of theory model elimination in a precise way in the main part. Theorem Proving Without Contrapositives Another source for this work is the upcoming of various (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. ....

D.W. Loveland. Near-Horn Prolog. In J.-L. Lassez, editor, Proc. of the 4th Int. Conf. on Logic Programming, pages 456--469. The MIT Press, 1987.

....the consideration of the complexity of query processing, and we will sketch some of these results in the next section. Indefinite information has also been considered in the context of theorem proving and logic programming, under the rubric of non Horn programs [91, 131] and nearHorn programs [77, 78]. These are logic programs consisting of clauses of the form [A 1 (x) A n (x) B(x; y) where the A i are atoms and B is a conjunction of 9 literals. The emphasis in this work has been to devise theorem proving and query answering techniques which operate efficiently in practice, rather ....

D.W. Loveland. Near-Horn Prolog. In J-L. Lassez, editor, Logic Programming: Proceedings of the Fourth International Conference, pages 456--469. The MIT Press, 1987.

....introducing a new variable. Now, the question arises whether model elimination can be modified 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] ....

D.W. Loveland. Near-Horn Prolog. In J.-L. Lassez, editor, Proc. of the 4th Int. Conf. on Logic Programming, pages 456--469. The MIT Press, 1987.

....taxonomical knowledge as in the Krypton system [10] building in theory unification procedures and equality reasoning as by Paramodulation or E resolution. Another source for this 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: ....

D.W. Loveland, `Near-Horn Prolog', in Proc. of the 4th Int. Conf. on Logic Programming, ed., J.-L. Lassez, pp. 456--469. The MIT Press, (1987).

....system provides the basis for a proof procedure that generalizes the one usually employed in logic programming towards dealing with all of classical logic. An interesting aspect of this proof procedure is that its rule for backchaining incorporates within it the restart mechanism of nH Prolog [Lov87, Lov91] for dealing with disjunctive information in assumption sets and the mechanism with the same name of QNR Prolog [Gab85] for handling hypotheticals in goals. 2 Logical preliminaries We will work within the framework of a first order logic in this paper. The logical symbols that we assume as ....

D.W. Loveland. Near-Horn Prolog. In J. Lassez, editor, Logic Programming: Proc. of the Fourth Int'l Conf., pages 456--469. MIT Press, 1987.

....of the proof procedure with respect to computing answers are given. Key Words: Disjunctive Logic Programming, SLDresolution, Proof Procedure. 1. Introduction The success of Prolog motivates people to use full firstorder logic instead of only Horn clauses as the basis of logic programming (Loveland 1987). Positive disjunctive logic programming is one of the efforts in this consideration, it extends Horn clause programming by allowing more than one atom to occur in the head of a program clause. The declarative semantics of such programs is defined by the set of all logic consequences of the ....

....The new proof procedure then is based on the extended CH transformation. It works just like the SLD resolution except an extra checking. As a result it preserves many important properties enjoyed by SLD resolution. In contrast with existing procedures in (Baumgartner, Furbach, Stolzenburg 1997; Loveland 1987; 1991) it dispenses with the ancestor cancellation completely and therefore speeds up the inner loop. In addition, its conceptual simplicity makes it easy to be implemented. A possible implementation would consist of the SLD resolution together with a special propose theorem prover (for ....

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Loveland, D. 1987. Near-Horn Prolog. In Lassez, J.-L., ed., Proc. of the 4th Int. Conf. on Logic Programming, 456--469. The MIT Press.

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) D.W. Loveland. Near-Horn Prolog. In J.-L. Lassez, editor,Proc. of the 4th Int. Conf. on Logic Programming, pages 456--469.MIT Press, 1987.

....programs, i.e. programs that allow clauses with disjunctions of atoms as a consequent. Among the studies are those of Minker, Rajasekar, and Lobo [2, 9] that provide a 1 model theoretic semantics for disjunctive logic programs and a fixpoint characterization of the same semantics. Also, Loveland [5, 6] introduced a Unit near Horn Prolog (UnH Prolog) procedure tailored to the logic programming paradigm and also addressing the pragmatic issue of computation speed and effectiveness. Loveland and Reed [7] subsequently introduced the Inheritance near Horn Prolog (InH Prolog) variant that is simpler, ....

D.W. Loveland. Near-Horn Prolog. In J. Lassez, editor, Logic Programming: Proc. of the Fourth Int'l Conf., pages 456--469. MIT Press, 1987.

....assumption set or program. The ability to express and process indefinite information and case constructions occur in real life and mechanisms to handle such specifications are important. Attention has focused on the semantics of disjunctive logic programming [Pry90, GRS91, LMR89] proof procedures [Lov87, RLS91] and applications to disjunctive databases [Min82, RT88, YH85] One point of view of each of these topics appears in the book [LMR92] The basic requirement of disjunctive logic programming, that disjunctive information be representable in the assumption set, appears to be at variance with the ....

....(under a reduction to clausal form) that is a natural generalization of the one usually employed in logic programming. The main new aspect of this procedure is that it has a rule for backchaining that is sensitive to the presence of disjunctive information. The restart mechanism of nH Prolog [Lov87, Lov91] is used for dealing with this possibility and, in fact, our procedure is, in a sense, an abstract presentation of the InH Prolog procedure [LR91] 2 Logical Preliminaries We shall work within the framework of a first order logic in this paper. The logical symbols that we shall use are , ....

D.W. Loveland. Near-Horn Prolog. In J. Lassez, editor, Logic Programming: Proc. of the Fourth Int'l Conf., pages 456--469. MIT Press, 1987.

....Prolog project at Duke University developed out of the basic goal of extending Prolog to disjunctive logic programs while retaining as much of the clarity and procedural simplicity of Prolog as possible. The approach taken to achieve this goal was to combine Prolog with case analysis reasoning. In [Lov87, Lov91], Loveland developed the procedure now known as Unit near Horn Prolog (UnH Prolog) originally called Progressive nH Prolog ) as well as the conceptually simpler but incomplete variant Naive near Horn Prolog (NnH Prolog) Following these, Loveland and Reed [LR91] developed the Inheritance ....

....bg. The fact that a follows from each of these subcases is proven by the restart blocks. Both UnH Prolog and InH Prolog have been shown to be sound and complete systems in the sense that 9 L is a logical consequence of disjunctive logic program P if and only if there is a proof of 9 L from P [Lov87, LR91]. NnH Prolog, on the other hand, is not complete (an example which demonstrates its incompleteness can be found in [Lov91] While the description so far has mainly been at the propositional level, the procedures lift to the first order level as expected. In particular, given a non ground query ....

D.W. Loveland. Near-Horn Prolog. In J. Lassez, editor, Logic Programming: Proc. of the Fourth Int'l Conf., pages 456--469. MIT Press, 1987.

....and his students [YH85, HP88] implemented methods for computing answers to ground queries from ground disjunctive programs using the GCWA. Przymusinski [Prz86b, Prz86a] showed that SL resolution could be utilized as a procedure for dealing with disjunctive programs and the GCWA. In 1986, Loveland [Lov86, Lov87] introduced near Horn Prolog, a procedure which naturally extends SLD resolution Prolog to disjunctive programs using case analysis. Since the late 1980 s, many researchers have been addressing the semantics of disjunctive programs, including Minker and his students [LMR89, MR90, RM90, Lob90, ....

....where each head atom H j is assumed true and added as a conditional fact (subsequently subsuming the original clause) The splitting rule states that the combination of these cases suffices to show that L is a logical consequence of the original program. 46 3. 2 InH Prolog An Overview In [Lov86, Lov87, Lov91], Loveland developed the procedure now known as Unit near Horn Prolog (UnH Prolog) originally called Progressive nHProlog ) as well as the conceptually simpler but incomplete variant Naive near Horn Prolog (NnH Prolog) Following these, Loveland and Reed [LR91] developed the Inheritance ....

D.W. Loveland. Near-Horn Prolog. In J. Lassez, editor, Logic Programming: Proc. of the Fourth Int'l Conf., pages 456--469. MIT Press, 1987.

....(see [LMR92] Our focus is on proof procedures so we restrict ourselves to presenting (one version of) the Near Horn Prolog (nH Prolog) procedure mentioned earlier. This version of nH Prolog was developed by Reed and Loveland [Ree88, Ree91, LR91] based on earlier versions devised by Loveland [Lov87, Lov91]. The astute reader may recognize that the introduction of non Horn clauses such as above returns us to the full descriptive power of first order logic since every clause in a conjunctive normal form (cnf) presentation of a formula can be written as an implication of atoms. Namely, negative ....

....Another important search feature is progressive search. Although too complex to treat fully here we outline the feature. Recall that a restart block in InH Prolog always begins with the initial query FALSE. This is not true of the Unit nH Prolog variant, originally called Progressive nH Prolog [Lov87, Lov91] because it first incorporated this important search feature. However, this feature is useful for all variants. If the program clauses were processed exactly as its parent block, then it would duplicate the computation with no progress made, unless the new distinguished active head can invoke a ....

D.W. Loveland. Near-Horn Prolog. In Proceedings of the Fourth International Conference and Symposium on Logic Programming, pages 456--469. MIT Press, 1987.

.... Peter Baumgartner Delta Ulrich Furbach Universitat Koblenz Institut fur Informatik Rheinau 1 56075 Koblenz Germany E mail: fpeter,ulig informatik.uni koblenz.de For disjunctive logic programs (DLPs) there are several proposals for defining interpreters, like the nearHorn Prolog Family [ Loveland, 1987 ] SLI Resolution [ Lobo et al. 1992 ] SLOResolution [ Rajasekar, 1989 ] Model Tree construction [ Fernandez and Minker, 91 ] Restart Model Elimination [ Baumgartner et al. 1995, Baumgartner and Furbach, 1994 ] There have also been different approaches to assign least fixpoints to DLPs. ....

D.W. Loveland. Near-Horn Prolog. In J.-L. Lassez, editor, Proc. of the 4th Int. Conf. on Logic Programming, pages 456--469. The MIT Press, 1987.

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