J.W. Lloyd: Foundations of Logic Programming, 2nd Editon, Springer-Verlag, Berlin, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....may in turn be mapped to a graphical representation in G KARL. So G KARL provides the same expressive power as L KARL. L KARL enriches Horn logics with negation (normal programs) by additional modeling primitives. So the expressive power of L KARL equals the expressive power of normal programs [Llo87]. 5 Tools and Applications The hypermedia based MIKE tool [Neu93] supports the entire development process in MIKE by different editors. Among others it provides graphical editors for the static, functional, and dynamic view to the model of expertise. Up to now for elementary inference actions ....

J.W. Lloyd: Foundations of Logic Programming, 2nd Editon, Springer-Verlag, Berlin, 1987.

....state variables of dynamic logic store the contents of knowledge roles. A one to one correspondence must exist between the decomposition hierarchies at both layers. The logical language L KARL used to describe the domain, the inference layer, and their connection has a Herbrand model semantics [Llo87]. KARL allows stratified negation under the closed world assumption using the minimal (i.e. perfect) Herbrand model as semantics [Prz88] Constraints check this model for correctness. In contrast to Prolog, the evaluation of these clauses is set oriented [Ull88] not one but all instantiations of ....

J.W. Lloyd: Foundations of Logic Programming, 2nd Editon, Springer-Verlag, Berlin, 1987.

.... stable model of the program (DB ; IC ) They are obtained through the query Q obtained from Q by replacing, for p ##,every positive literal p( s)by p( s; t and every negative literal #p( s)by p( s; f ) Now Q can be transformed into a query program (Q )by a standard transformation [30, 1]. This query program will be run in combination with (DB ; IC ) Example 11. For the query Q(y) #zBook(kafka; y ; z ) we generate Q (y) #zBook(kafka;y;z;t ) that is transformed into the query program clause Answer(y) # Book(kafka; y ; z ; t ) 6 Computing from the Program ....

Lloyd, J.W. ########### ## ##### ###########. Springer Verlag, 1987.

....Fig. 7. The hyperbolic ontology view. 20 translation: e.g. more complex Frame logic expressions like O[A V:C[AA VV] can also be translated. The output of this stage are generalized logic programs. After this translation the output has to be translated further to normal logic programs [Lloyd, 1987] via Table 3. Principle of Translating Frame Logic to Predicate Logic Frame Logic Meaning Predicate Logic C 1 : C 2 class C 1 is a subclass of C 2 sub(C 1 , C 2 ) O : C O is an instance of class C isa(O,C) C 1 [A= C 2 ] for the instances of C 1 an attribute A is defined, ....

J.W. Lloyd: Foundations of Logic Programming, 2nd Edition. Springer-Verlag, 1987

....s such that s fl a 2 v a . This is the case if and only if U j= s 2 (a 2V v a ) i.e. if and only if U j= s 0 2 v a and s 1 fl a = hs 0 : Since h is the identity function, we can take s 0 to be a itself and s 1 to be f(a) where f denotes the realizer of the identity axiom as defined e.g. in [5], XII, 1.5. Hence s = ha; f(a)i: We need to determine t in such a way that t fl T rans(v a ) 8z8R Gamma z 2 v a R z Theta v a 8x 2 z 9y R(x; y) 9v 2 v a Gamma 8x 2 z 9y 2 v R(x; y) 8y 2 v 9x 2 z R(x; y) Delta Delta : 40 L. CROSILLA AND M. RATHJEN Clearly t has to be a ....

M. Beeson: Foundations of Constructive Mathematics, (Springer Verlag, Berlin, 1985).

....type. It is itself potentially a generalized inductive definition and amounts therefore to the introduction of a potentially impredicative type. A number of results were known on the proof theoretic strength of Martin Lof type theories. Many of the earliest results are presented in Beeson [Be 85] For example, using an embedding argument it is shown there that intuitionistic type theory without universes has the complexity of Department of Mathematics, Uppsala University, Uppsala, Sweden y Department of Mathematics, The Ohio State University, Columbus, Ohio 43210, USA z The second ....

....if the list c 1 ; c n comprises all the parameters of t, then (c 1 ; c n ) t is a closed application term without any classification variables. For later use we state a consequence of the Abstraction Lemma which can be derived in the same way as for the calculus (cf. F 75] Be 85] VI2.7) 1.6 Corollary (Recursion Theorem) 8f9g8x 1 : 8x n g(x 1 ; x n ) f(g; x 1 ; x n ) 1.7 Corollary T 0 can be interpreted in T f 0 . Proof : The interpretation consists in successively replacing all those comprehension constants c OE which do not belong to T f 0 ....

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M. Beeson: Foundations of Constructive Mathematics, Springer Verlag, Berlin 1985.

....can be seen by noting that if the left hand side of an instantiated activity constraint is satis ed by an assignment, it is satis ed by all its supersets as well. A monotonic operator has a unique least xpoint that can be computed by iteratively applying the operator, starting from the empty set [4]. As a solution must not contain unjusti ed activevariables, it should be such a least xpoint, denoted by lfp(# # # ) This ensures that all the assignments in a solution are justi ed by the instantiated activity constraints. ######### ## Given a DCSP # = ## # ## # # # ## # ## # alegal ....

J. Lloyd. ########### ## ##### ###########. Springer-Verlag, 2nd edition, 1987.

....that Godel refers to is, from a classical point of view, equivalent to weak Konig s lemma, to be dealt with in section 7. Brouwer was able to prove the fan theorem using a principle of bar induction which he felt was justified by a constructive interpretation of the terms involved (cf. Beeson [1985], Troelstra and van Dalen [1988] In a posthumous paper, C. Spector [1962] used a generalization of this principle to justify a computational scheme which he dubbed bar recursion. With this scheme Spector was able to provide a functional interpretation of full second order arithmetic, that is, ....

....arithmetic proves a formula , the witnessing functions can be taken to be recursively continuous, in a sense defined, independently, by Kreisel [op. cit. and Kleene [1959a] cf. also Feferman [1993] For an interesting extension of bar recursion to transfinite types, see Friedrich [1985]. Godel s Functional Interpretation 367 natural way of interpreting the set variables of PA 2 (CA) with function variables of type 1. Recall the axiom of choice (AC ) 8x 9y (x; y) 9f 8x (x; f(x) We will denote the scheme in which the variables x and y are restricted to be of type oe ....

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Foundations of Constructive Mathematics, Springer-Verlag, Berlin. E. Bishop

....t minus x 1 ; x n and such that T 0 t # t (x 1 ; x n ) t: x 1 ; x n ) t is written for t . 5 The most important consequence of the Abstraction Lemma is the Recursion Theorem. It can be derived in the same way as for the calculus (cf. F 75] F 79] Be 85] VI.2.7) Actually, one can prove a uniform version of the Recursion Theorem (with a recursion operator) in the applicative fragment of T 0 . Corollary 2.2 (Recursion Theorem) 8f9g8x 1 : 8x n g(x 1 ; x n ) f(g; x 1 ; x n ) 2.2 Formulation of the monotone inductive ....

M. Beeson: Foundations of Constructive Mathematics, Springer Verlag, Berlin 1985.

....type. It is itself potentially a generalized inductive definition and amounts therefore to the introduction of a potentially impredicative type. A number of results were known on the proof theoretic strength of Martin Lof type theories. Many of the earliest results are presented in Beeson [Be 85] For example, using an embedding argument it is shown there that intuitionistic type theory without universes has the complexity of true arithmetic. Aczel in [A 77] showed that intuitionistic type theory with one universe ML 1 , but without well ordering types, has the same proof theoretic ....

....if the list c 1 ; c n comprises all the parameters of t, then (c 1 ; c n ) t is a closed application term without any classification variables. For later use we state a consequence of the Abstraction Lemma which can be derived in the same way as for the calculus (cf. F 75] Be 85] VI2.7) Corollary 1.6 (Recursion Theorem) 8f9g8x 1 : 8x n g(x 1 ; x n ) f(g; x 1 ; x n ) Corollary 1.7 T 0 can be interpreted in T f 0 . Proof : The interpretation consists in successively replacing all those comprehension constants c OE which do not belong to T f 0 ....

[Article contains additional citation context not shown here]

M. Beeson: Foundations of Constructive Mathematics, Springer Verlag, Berlin 1985.

....( match ( end temperature symptoms indicate refine ( Data abstract Abstract match Solution refine Solution class data illness Domain knowledge Fig. 3. The model of expertise of heuristic classification in KARL. a mapping via Horn clauses extended by stratified negation (cf. [Llo87], Prz88] in MIKE. CRLM and MIKE are both methods of developing expert systems. 5 Therefore, it is not without interest to ask how both methods view the difference between expert systems and ordinary software programs. The question of how an approach characterizes an expert system is not only ....

J.W. Lloyd: Foundations of Logic Programming, 2nd Editon, Springer-Verlag, Berlin, 1987.

....is taken into account for L KARL, in fact, only Herbrand models are admitted. This does not create a significant restriction because every set of formulae can be transformed into a logically equivalent set of clauses and for every set of clauses S holds: S has a model iff S has a Herbrand model [Llo87]. A Herbrand model H is a set of ground positive literals. These literals are regarded to be true according to the given Herbrand model H. A negative literal is true if the corresponding positive literal is not an element of H. Similar to predicate logic with equality, it is required that such a ....

J.W. Lloyd: Foundations of Logic Programming, 2nd Editon, Springer-Verlag, Berlin, 1987.

.... performed, but it does not catch the complete translation: e.g. more complex Frame logic expressions like O[A V:C[AA VV] can also be translated. The output of this stage are generalized logic programs. After this translation the output has to be translated further to normal logic programs [Llo87] via a Lloyd Topor transformation [LT84] The entire translation process is surveyed in Figure 5. As result we obtain a normal logic program. Standard techniques from deductive databases are applicable to realize the last stage: the bottom up fixpoint evaluation procedure. Because we allow ....

J.W. Lloyd: Foundations of Logic Programming, 2nd Edition. Springer-Verlag, Berlin, 1987

....parenthood relation par. Then the program defines a derived relation describing ancestors, and the query asks for the ancestors of john. A well known strategy for evaluating logic programs is the bottom up strategy. It serves to define the least fixed point semantics, and is known to be complete [Lloyd 84] While the strategy is reasonably efficient when the query does not contain instantiated variables [Bancilhon 85] this example shows that it is very inefficient when bindings for some variables are given in the query. The reason is that it computes the complete anc relation and then applies ....

"Foundations of Logic Programming," J. W. Lloyd, Springer-Verlag, 1984.

....The specialized integrity constraints produced by our lazy negation as failure procedure appear quite similar although they are produced, so to speak, as a by product of checking the entire database. The simplification methods has been adapted for deductive databases by [18] also described in [17]) This includes a simple bottom up processing of the database rules starting from the updated predicates in order to identify an upper limit for the part of the database (including integrity constraints) that can be affected by the update, and thus needs to be processed in order to check that ....

J.W.Lloyd: Foundations of Logic Programming, 2nd Edition, Springer-Verlag 1987.

....the contents of knowledge roles. A one to one correspondence must exist between the decomposition hierarchies at both layers. 3.5. 2 The Formal and Operational View The logical language L KARL used to describe the domain, the inference layer, and their connection has a Herbrand model semantics [Llo87]. KARL allows stratified negation under the closed world assumption using the minimal (i.e. perfect) Herbrand model as semantics [Prz88] Constraints check this model for correctness. In contrast to Prolog, the evaluation of these clauses is set oriented [Ull88] not one but all instantiations of ....

J.W. Lloyd: Foundations of Logic Programming, 2nd Editon, Springer-Verlag, Berlin, 1987.

....the abducible hypotheses to take the non ground form of 9X(A(X) C(X) where A is a conjunction of abducible atoms and C is a set of constraints defined over the CLP (arithmetic) domain. 2.1. Preliminaries We will assume some familiarity with basic concepts of logic programming as found in [38]. We present here some of the basic preliminary material from the theory of CLP on which the ACLP framework builds. The required material from ALP is developed as needed in the section. Further details on ALP and CLP can be found in the recent surveys of [23] and [22] respectively. We will adopt ....

J.W. Lloyd Foundations of Logic Programming. Springer Verlag, 1987. 46

....can be justified via Sigma Omega INDN . Corollary 3.16. CZFA IND RDC can be interpreted in Sigma 1 1 AC. Proof. This follows from the fact that the fixed point axioms of c ID 1 can be emulated in Sigma 1 1 AC by interpreting the fixed points as Sigma 1 1 sets (cf. 1] and [8]) 3.4. Lower bounds. Lower bounds for the theories CZFA Sigma 1 IND and CZFA IND can be established by interpreting suitable intuitionistic theories RA i ff of the ramified hierarchy up to level ff in them. For the definition of the theories RA i ff see [15] chapter II, 1.2. We ....

M. Beeson: Foundations of Constructive Mathematics, (Springer Verlag, Berlin 1985).

....[23] of full classical linear that is generated by the sequents of the logic of programs and goals. Whilst this construction yields a completeness theorem, it does not give an adequate account of the process of computation, such as that given by the least fixed point constructions of, for example, [16], 17] The approach of Corradini and Montanari [4] promises to be valuable in our setting. Whilst the permutability analysis seems to indicate that uniform provability provides a basis for (linear) logic programming, it can be asked to what extent logic programming is characterized by this, or a ....

J. Lloyd Foundations of Logic Programming. Springer-Verlag, 1984.

....Section 7 briefly discusses the implementation issue, pointing out some motivations for the use of the double meta representation also from the language implementation viewpoint. 2 Meta Representations LOG syntax is mostly the usual syntax of logic programming languages (cf. for instance [9]) and will be skipped here, except for those parts concerning the metarepresentations. We start with an ordinary Horn clause language and we conservatively extend it to one in which every syntactic entity is named by ground terms of the language. Precisely, each LOG syntactic object has two ....

....q,r. q. r. p= p : q,r , q , r ] yes. f(g(a) b,C) t= f ,A, b , C ] A = g(a) The second goal succeeds provided the meta variable A is instantiated to g(a) 2.4 Semantics The main differences in the semantics of LOG w.r.t. the standard case (as described for instance in [9]) are due to the presence of the double arrow operators. As regards the declarative semantics of LOG, first a privileged interpretation domain resulting from suitable modifications to the classical Herbrand universe is defined then the privileged interpretation of = is given as a relation over ....

J.W. Lloyd: Foundations of Logic Programming. Springer Verlag, 2nd ed., 1987.

....(shortly SFL) together with its declarative and operational semantics. Section 6, the core of the paper, presents our soundness and completeness results. Finally, section 7 summarizes our conclusions. The reader is assumed to be familiar with the basic theory of logic programming [Apt 90, Lloyd 87] and term rewriting systems [De Jo 90, Klop 90] 2 Algebraic preliminaries Our approach to functional logic programming relies on some algebraic notions which were exposed in [Go al 91] In this section we recall them and we also include some extensions needed for supporting 2 ALGEBRAIC ....

J.W.Lloyd: Foundations of Logic Programming, 2nd edition, Springer Verlag, 1987.

....head of the clause containing the cut. Our programs uses cuts only to help the Prolog system economize memory usage; it runs correctly even if all cuts are removed. Thus, when reading our program, the reader can safely ignore the cuts. See [CM, R, SS] for an introduction to the cut predicate and [Llo] for a possibility of defining its semantics. 2.8. Syntax Used in the Proof Any line that is highlighted in thisspecialfont and appears in display style , i.e. likethis is part of our program from now on. To begin with, we declare a few operators which will allow us to write the rest of the ....

J.W. Lloyd: Foundations of Logic Programming. Second, Extended Edition. Springer --Verlag (1987).

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Foundations of Object-OrientedLanguages, pages 60#90. LNCS 489, Springer Verlag, 1991.

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J.W. Lloyd: Foundations of Logic Programming, 2nd Editon, Springer-Verlag, Berlin, 1987.

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J.W. Lloyd: Foundations of Logic Programming, 2nd Editon, Springer-Verlag, Berlin, 1987.

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