L. McCarty, Clausal Intuitionistic Logic II. Tableau Proof Procedure, Journal of Logic Programming 5, 93 -- 132, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....constructions described in [9] since it makes it possible to provide scope to individual, function, and predicate constants. The logic described in this paper is related to logics considered by many researchers in logic programming and, most recently, in theorem proving and type theory. See [3, 5, 7, 8, 14] for the description of closely logics applied to logic programming. Similar logics, especially higher order versions, have been used as meta languages in specifying and implementing theorem provers [2, 18, 19] The logic presented here is most closely related to the first order hereditary Harrop ....

L. McCarty, Clausal Intuitionistic Logic II. Tableau Proof Procedure, Journal of Logic Programming 5, 93 -- 132, 1988.

....in the body of clauses, it is called embedded implication also. This section concretely refers to two works which, in some way, extend intuitionistic logic programming with negation. The Theory of Clausal Intuitionistic Logic The theory of clausal intuitionistic logic was developed by McCarty in [27, 28]. In this work, an extension of Horn clause logic is considered. This extension consist of adding the next types of clauses 3 : Negation Rules Embedded Rules A ( B A ( B 1 ) B 2 ) 1) A ) B A ) B 1 ) B 2 ) 2) 3 We use a notation close to the original one. 24 Since the rules in (2) ....

.... J 0 g Then ffl J is the largest set of substates s s 0 that satisfies N [ H. ffl J is the greatest fixpoint of the transformation T (J) TN (J) J 0 Since it cannot be proved that the operator TN is continuous, the constructibility of that fixpoint cannot be concluded. In another paper [28], MacCarty proposed a proof procedure based on tableau for this theory but its effectiveness is not proved. Nevertheless, the proof procedure is sound and complete with respect to the fixpoint semantics. Intuitionistic Logic Programming Extended with Negation as Failure Bonner et al. 7] ....

L.T. McCarty. Clausal intuitionistic logic II. tableau proof procedures. Journal of Logic Programming, 5:93--132, 1988.

.... logical fragment was proved to preserve the proof theoretic properties at the basis of the logic paradigm, namely to be a uniform proof system [14] Higher order Hereditary Harrop formulas underlie the languages Prolog[15] and Isabelle [18] first order Hereditary Harrop formulas are explored in [9, 10]. In addition to embedded implications, Hereditary Harrop formulas allow universal quantifications in the bodies of clauses. The combination of these two extensions provides a rich notion of dynamic scoping, which adresses important issues in as various fields as theorem proving [6, 18, 5] ....

.... natural language processing [19, 17, 7] type inference [5, 20] software engineering [21] and modularity in logic programming [13] However, the concrete use of Hereditary Harrop formulas has too often been limited by its restriction to a top down evaluation strategy, described for instance in [16, 18, 10]. Works in natural language processing, notably Pereira s semantic interpretation [19] and in deductive databases [1] advocate evaluations based on a bottom up search. Fixpoint semantics have been provided for Hereditary Harrop formulas [12, 10] but they require infinite structures and thus ....

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L. T. McCarty. Clausal intuitionistic logic II. tableau proof procedure. Journal of Logic Programming, 5:93--132, 1988.

....knowledge. As Horn clauses are not a particularly large fragment of first order logic, it is perhaps not surprising that this class of formulae has such a relatively strong property. There have been various schemes proposed for logic programming languages which 1 are extensions of Horn clauses [2, 8, 9, 22, 23, 28, 29, 31]. Given these various extensions, it seems natural to ask whether there is a maximal class of formulae which may be used as a programming language. Moreover, there does not seem to be universally agreed criterion which may be used to determine what constitutes a logic programming language, without ....

L.T. McCarty, Clausal Intuitionistic Logic II. Tableau Proof Procedures, Journal of Logic Programming 5:2:93-132, 1988.

.... approaches to higher type logic programming essentially amount to allowing implications to appear within program bodies, thus permitting a kind of modular structure [28] Such embedded implications have behavioral similarities with intuitionistic implication, and as such have been investigated in [26,27]. Either an intuitionistic version of negation, or negation as failure can be added. If negation as failure is added, a three valued approach seems most natural, 17] Example 37 Suppose we have a directed graph, and we wish to write a program that can determine whether there is a path from one ....

McCarty, L. T. Clausal intuitionistic logic II. tableau proof procedures. Journal of Logic Programming 5 (1988), 93--132.

....to readers, and we felt it adequately represented the whole class of examples. 7 Embedded implications In [8] a three valued semantics was given for a logic programming language containing embedded implications and negation as failure. This built on earlier work of [13] and was related to [11, 12]. In this section we show how logic programming with embedded implications can be generalized naturally to the bilattice context in a way that, when specialized to the consistent part of Belnap s logic, coincides with the semantics of [8] In addition, once put into the bilattice framework the ....

McCarty, L. T. Clausal intuitionistic logic II. tableau proof procedures. Journal of Logic Programming 5 (1988), 93--132.

....unsuccessful paths are lost altogether. In our system, previously proved formulae are maintained explicitly and therefore can be passed to other reasoning modules. The language of the logic system presented in this paper is that of Generalized Horn Clause Intuitionistic Logic (GHCIL) McC88a, McC88b] The inference engine can be briefly characterized as a GHCIL reasoner with declarative meta level control and explicit representation of previously derived knowledge. The next three sections describe the GHCIL language, the meta level control, and the explicit representation of previously ....

....For all people P : P is considered a dedicated person if P is working under the assumption that there is some unfinished work W that is assigned to P . Thus, embedded implications can be seen as hypothetical statements. For a more detailed discussion of clausal intuitionistic logic, see [McC88a, McC88b] In our system, there are several distinguished predicates that may occur in GHCIL clauses. One of them is falsum: GHCIL clauses having falsum as their head indicate contradictory situations. Negation as failure is not suitable for our purposes because it destroys the equivalence of declarative ....

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L. T. McCarty. Clausal intuitionistic logic II. Tableau proof procedures. The Journal of Logic Programming, 5:93--132, 1988.

....these. There are some logical systems that can be used as a basis of logic programming and that contain natural notions of scope for program clauses and constants. For example, the logic of hereditary Harrop formulas, parts of which were developed independently by Gabbay and Reyle [GR84] McCarty [McC88a, McC88b], and Miller [Mil86, Mil89b, Mil90] allows for a simple stack based structuring of the runtime program and set of constants. The modal logic of Giordano, Martelli, and Rossi [GMR88] provides an interesting variation on hereditary Harrop formulas that has a different runtime structuring of ....

L. T. McCarty. Clausal intuitionistic logic II. tableau proof procedure. Journal of Logic Programming, 5:93--132, 1988.

.... generalized Horn C lause logic) is a logic programming system developed at the University of Erlangen N urnberg [Bec88, BT92, Tob94] Its language is that of generalized Horn clause intuitionistic logic (in the following abbreviated as GHCIL for a discussion of clausal intuitionistic logic see [McC88a, McC88b]) Risc programs are sets of so called GHCIL clauses. GHCIL clauses are (implicitly) universally quantified implications of the following form: 1. Each definite clause is a GHCIL clause. 2. If A is an atomic formula and B 1 ; Bn are GHCIL clauses (n 0) then A B 1 ; Bn is a ....

....As demonstrated in the example, queries that would not terminate without control clauses can terminate if control clauses are used. 7 Correctness and Completeness Results Generalized Horn Clause Logic is intuitionistically equivalent to a certain subset of McCarty s Clausal Intuitionistic Logic [McC88a, McC88b]. According to Tobermann [Tob94] the calculus of generalized Horn clauses upon which the prover of Risc is based is logically sound and complete. Since Risc performs depth first search, it is combinatorially incomplete in the same way as Prolog: it cannot effectively find a proof for a logical ....

McCarty, L. T.: Clausal Intuitionistic Logic II. Tableau Proof Procedures. The Journal of Logic Programming, 5(2):93--132, June 1988.

....best known extension of pure Horn clause logic within the logic programming paradigm permits negation in goals, using the notion of negation as failure. However, the idea of using implications and universal quantifiers and, in fact, arbitrary logical connectives in goals has also been advocated [GR84a, LT84, McC88a, McC88b, Mil89b, MNPS91]. There is a wide spectrum of logical languages between those given by Horn clause logic and full quantificational logic, especially if the derivability relation to be used is also open to choice. An obvious question that arises in this situation is whether some of these languages provide more ....

L. Thorne McCarty. Clausal intuitionistic logic II. Tableau proof procedures. Journal of Logic Programming, 5(2):93--132, 1988.

....tedious and we therefore do not undertake these in this paper. We conclude the paper with a brief discussion of the manner in which the procedure described here is actually being 1 Proof procedures have been presented together with proofs of correctness for closely related formula classes in [10] and [13] The structure of the procedure in [10] differs from the one considered here. Further, the discussions in [10] seem largely to note the constraints that must be placed on substitution terms without detailing simple methods for ensuring the satisfaction of these constraints. The procedure ....

....in this paper. We conclude the paper with a brief discussion of the manner in which the procedure described here is actually being 1 Proof procedures have been presented together with proofs of correctness for closely related formula classes in [10] and [13] The structure of the procedure in [10] differs from the one considered here. Further, the discussions in [10] seem largely to note the constraints that must be placed on substitution terms without detailing simple methods for ensuring the satisfaction of these constraints. The procedure in [13] has a similar structure to our procedure ....

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L. Thorne McCarty. Clausal intuitionistic logic II. Tableau proof procedures. Journal of Logic Programming, 5:93--132, 1988.

....2 A Logic for Scoping Clauses and Constants Various extensions to the foundation of logic programming have been proposed to provide scoping constructs for program clauses and constants. We shall base our language on a logic similar to N Prolog [8] the intuitionistic clausal system of [4,13,14], and the hereditary Harrop formulas of [15,16,17] Since a simple modification of the latter logic is the logic we consider here, we refer to it simply as hH 0 . We briefly describe hH 0 from an operational pointof view below. The reader interested in proof theoretic semantics should refer ....

L. Thorne McCarty. Clausal intuitionistic logic II. tableau proof procedure. The Journal of Logic Programming, 5:93--132, 1988.

.... generalized Horn C lause logic) is a logic programming system developed at the University of Erlangen N urnberg [Bec88, BT92, Tob94] Its language is that of generalized Horn clause intuitionistic logic (in the following abbreviated as GHCIL for a discussion of clausal intuitionistic logic see [McC88a, McC88b]) Risc programs are sets of so called GHCIL clauses. GHCIL clauses are (implicitly) universally quantified implications of the following form: 1. Each definite clause is a GHCIL clause. 2. If A is an atomic formula and B 1 ; Bn are GHCIL clauses (n 0) then A B 1 ; Bn is a ....

....c] F NaN F NaN fflffl app( c] c] F NaN F NaN fflffl 2 Fig. 5. Computation tree for Example 9 (without meta control) 7 Correctness and Completeness Results Generalized Horn Clause Logic is intuitionistically equivalent to a certain subset of McCarty s Clausal Intuitionistic Logic [McC88a, McC88b]. According to Tobermann [Tob94] the calculus of generalized Horn clauses upon which the prover of Risc is based is logically sound and complete. Since Risc performs depth first search, it is combinatorially incomplete in the same way as Prolog: it cannot effectively find a proof for a logical ....

McCarty, L. T.: Clausal Intuitionistic Logic II. Tableau Proof Procedures. The Journal of Logic Programming, 5(2):93--132, June 1988.

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L.T. McCarty, Clausal Intuitionistic Logic II. Tableau Proof Procedures, Journal of Logic Programming 5:2:93-132, 1988.

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