K. R. Apt and M. H. van Emden. Contributions to the theory of logic programming. Journal of the ACM, 29(3):842--862, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... The central component of existing logic programming systems is a refutation procedure, which is based on the resolution rule created by Robinson [21] The first such refutation procedure, called SLD resolution, was introduced by Kowalski [13, 31] and further formalized by Apt and Van Emden [1]. SLD resolution is only suitable for positive logic programs, i.e. programs without negation. Clark [8] extended SLD resolution to SLDNF resolution by introducing the negation as finite failure rule, which is used to infer negative information. SLDNF resolution is suitable for general logic ....

K. R. Apt, M. H. Van Emden, Contributions to the theory of logic programming, J. ACM 29(3):841-862 (1982).

....logic is even weaker. But the least model can be directly expressed by an inductive definition. 4 By the Skolem Lowenheim Theorem, no set of first order axioms can even fix the cardinality of its models, let al..one fix a single model. Negation as Failure is investigated by Apt and van Emden [3] and Lloyd [29] Essentially, they develop the theory of inductive definitions so as to distinguish divergent computations from finite failures. Negation goes beyond monotone inductive definitions: with negated subgoals, the function # above may not be monotone. However, perhaps the database can ....

....returns 3. Since there is no higher order unification, when apply is rewritten, the first argument must be normalisable to a function or a # expression. The user can define higher order functions: map(F, map(F, H T] apply(F, H] map(F,T) The value of map(lambda( X] X X) [1,2,3]) is [1,4,9] The justification of higher order functions requires extending the theory of narrowing to allow function variables in rewrite rules. This appears straightforward if function variables are not allowed in goals. The full incorporation of function variables would require higher order ....

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Apt, K. R., van Emden, M. H., Contributions to the theory of logic programming,

....within an algebraic framework allows us to eliminate inessential details speci c to a particular logic, simplify arguments and nd common principles underlying di erent nonmonotonic formalisms. The roots of this algebraic approach can be traced back to studies of semantics of logic programs [vEK76, AvE82, Fit85, Prz90] and of applications of lattices and bilattices in knowledge representation [Gin88] Exploiting the concept of a bilattice and relying on some general properties of operators on lattices and bilattices, Fitting proposed an elegant algebraic treatment of all major 2 , 3 and 4 valued semantics of ....

K.R. Apt and M.H. van Emden. Contributions to the theory of logic programming. Journal of the ACM, 29(3):841-862, 1982.

....Such sequents are the basis of the bunched logic programming language BLP. A denotational semantics for BLP (in the absence of ) may be given within BI s elementary resource semantics by giving a reconstruction of the Kripke style least fixed point semantics for intuitionistic logic programming [14, 1, 32, 40, 2]. We sketch the key steps, for simplicity in a purely propositional setting, as follows: Define a commutative monoid P = P; e; v) of programs as worlds, in which P is the set of hereditary Harrop bunches, is and its unit e is ;m , and Q v P just in case, for some P , Q P ; P ....

K.R. Apt and M.H. van Emden. Contributions to the theory of logic programming. J. ACM, 29(3):841--862, 1982.

....set. Progress through this state space may be made by simpli cation steps applied to either the goal set or the disagreement set. In any given case, these steps must be relativized to a particular program P. The notion of a P derivation [35] that generalizes SLD derivations described in [6] for rst order Horn clause logic makes this idea precise. De nition 3.1. Let P be a program and let G and be symbols for sets of goal formulas and substitutions, respectively. Further, let D be a symbol for a disagreement set or the special value F. Finally, let MATCH be a function on ....

K. R. Apt and M. H. van Emden. Contributions to the theory of logic programming. Journal of the ACM, 29(3):841-862, 1982. 48

....fashion to produce all the uni ers of the original set. In general, such a tree may be in nite and may also include nonterminating branches. Notice however that it must always be nitely branching. The matching tree is similar in structure to SLD trees used in conjunction with (pure) Prolog [3]. It is in fact possible to merge these two trees together to get one tree that describes the search space for our higher order language. This can be done through the notion of a P derivation [31] 14 Let MATCH be a function on exible rigid disagreement pairs that produces the set of imitation ....

K. R. Apt and M. H. van Emden. Contributions to the theory of logic programming. Journal of the ACM, 29(3):841-862, 1982.

....Such sequents are the basis of the bunched logic programming language BLP. A denotational semantics for BLP (in the absence of ) may be given within BI s elementary resource semantics by giving a reconstruction of the Kripke style least fixed point semantics for intuitionistic logic programming [13, 1, 33, 41, 2]. We sketch the key steps, for simplicity in a purely propositional setting, as follows: In general, G contains what Prolog calls logical variables , which are existentially quantified, and we seek substitution instances of G which are consequences of P . 22 . Define a commutative monoid P ....

K.R. Apt and M.H. van Emden. Contributions to the theory of logic programming. J. ACM, 29(3):841--862, 1982.

....which such least models exist. For these classes TP effectively specifies a model generation procedure. Examples are the class of definite programs (see e.g. 11] where the correspondence between the least model of P and the least fixed point of TP can be shown by lattice theoretic arguments [1], or the class of acceptable programs, where the just mentioned correspondence can be shown using metric methods [5] It turned out that the computation of the least model for a program from one of the just mentioned classes can be performed by a recursive network of binary threshold units if the ....

K.R. Apt and M.H. Van Emden. Contributions to the Theory of Logic Programming. Journal of the ACM, 29, pp. 841--862, 1982.

....within an algebraic framework allows us to eliminate inessential details specific to a particular logic, simplify arguments and find common principles underlying different nonmonotonic formalisms. The roots of this algebraic approach can be traced back to studies of semantics of logic programs [vEK76, AvE82, Fit85, Prz90] and of applications of lattices and bilattices in knowledge representation [Gin88] Exploiting the concept of a bilattice and relying on some general properties of operators on lattices and bilattices, Fitting proposed an elegant algebraic treatment of all major 2 , 3 and 4 valued semantics of ....

K.R. Apt and M.H. van Emden. Contributions to the theory of logic programming. Journal of the ACM, 29(3):841--862, 1982.

....is that the value of A in the minimal model of p B C; B q ; C r g depends only on the minimum of q and r. Quantitative deduction is not applicable in situations where, for example, A should have a higher value for q = 0:5 and r = 1 than for q = 0:5 and r = 0:5. The method followed in [1, 3, 5] is to associate with each rule set P a mapping TP from interpretations to interpretations and to show that xpoints of TP are models of P . Then various mathematical results about TP can be used to discover properties of models. Here we follow the same method. First a reminder of the de nition ....

....a monotone function, for any rule set P . That is, I 1 I 2 implies TP (I 1 ) TP (I 2 ) It is well known that monotonicity implies that the least xpoint lfp(T P ) of TP , namely fI : TP (I) Ig exists and is equal to fI : TP (I) Ig; and dually for greatest xpoints (see for example [1], or [5] A useful connection between models and xpoints is established by Theorem 2.6, which was rst stated and proved for the qualitative case in [3] It makes just as much sense, and is just as true, in the quantitative case. For every rule set P and for every I BP , P is true in I ....

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Apt, K. R. and van Emden, M. H., Contributions to the Theory of Logic Programming, J. Assoc. Comput. Mach. 29:841-862 (1982).

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K. R. Apt and M. H. van Emden. Contributions to the theory of logic programming. Journal of the ACM, 29(3):842--862, 1982.

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Apt, K.R. and M.H. van Emden (1982). Contributions to the Theory of Logic Programming. Journal of the ACM 29(3), 841-862.

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K. R. Apt and M. H. van Emden, Contribution to the theory of logic programming, JACM, 29 (1982), pp. 841-862.

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K. R. Apt and M. H. van Emden. Contributions to the theory of logic programming. Journal of the ACM, 29(3):841--862, 1982.

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K. R. Apt and M. H. van Emden, Contribution to the theory of logic programming, JACM, 29 (1982), pp. 841-862.

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K.R. Apt, M.H. van Emden. Contributions to the Theory of Logic Programming. Journal of the ACM 29(3), 1984, pp. 841-862

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Apt, K. R., and van Emden, M. H. Contributions to the theory of logic programming. J. ACM 29, 3 (1982) 841 -- 862.

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K.R. Apt and M.H. van Emden. Contributions to the theory of logic programming. Journal of the ACM, 29(3):841-862, 1982.

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K. R. Apt, M. H. V. Emden, Contributions to the theory of logic programming, Journal of the Association of Computing Machinery 29 (3) (1982) 841-862.

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K.R. Apt and M.H. van Emden. Contributions to the theory of logic programming. Journal of the ACM, 29(3):841--862, 1982.

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K. R. Apt and M. H. van Emden. Contributions to the theory of logic programming. Journal of the ACM, 29(3):841-862, 1982.

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K. R. Apt and M. H. van Emden. Contributions to the theory of logic programming. Journal of the ACM, 29(3):841-862, 1982.

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K. R. Apt and M. H. van Emden, Contributions to the Theory of Logic Programming, J. ACM July 1982, vol. 29, no. 3, pp. 841-862.

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K.R. Apt, M.H. van Emden, Contributions to the Theory of Logic Programming, J. ACM 29 (1982) 841-862

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K.R. Apt and M.H. van Emden. Contributions to the theory of logic programming. Journal of the ACM, 29(3): pp. 841-862, 1982.

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