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# 7 completeness let pc be a closed program and g be a

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Unformatted text preview: teness) Let Pc be a closed program and G be a ground p-, r- or pred-atom. For any k, if G 2 TP "k then there is an atom A and a substitution d , such that there exists a proof tree of height k for Pc ` A, and G 2 A . c is labelled either (i) by deduction where = , or (ii) by inheritance where = respectively. It immediately implies that G = A . Inductive step: Assume that the claim holds true for k ; 1. From the de nition of 69 and = mgu(< o a1 : : : ak a > ] < p a1 : : : ak a > %]). b Since G 2 A (because G uni es with A) and G 2 TP "k , by monotonicity of TP it must be the case that TP "k 1j= B. Then by inductive hypothesis, we must have a proof for Pc ` B of height k ; 1. From this, we can then construct a proof for Pc ` A of height k from this whose root node is labelled either (i) by deduction where = and = , or (ii) by inheritance where = and = % d. respectively. It immediately implies that G 2 A 2 0 0 0 TP , if G 2 TP "k , then there are two cases. Case (i): There must exist a clause A B 2 Pc such that = mgu(G A), = , d G = A , and TP "k j= B . Case (ii): There exists a p-clause A B 2 Pc and a unit i-clause C 2 Pc such that A is of the form p m(a1 : : : ak ) 7! a ], C is of the form o p @mk ], and G is of the form o m(a1 : : : ak ) 7! a] such that = mgu(o o ), = mgu(p ] p), % = fp = g =o c c c 0 0 0 0 0 7! 0 0 c c c ; As a corollary, we have the following equivalence between the intended model semantics and the proof theory. Theorem 3.8 Let P be a program, Pc be its closure, MP be the intended model for P, and G be a ground goal. Then, we have that Pc ` G is provable i MP j= G. Proof: Pc ` G is provable () G 2 TP "! by Theorems 3.6 and 3.7 () MP j= G from Theorem 3.5 c c 2 70 Chapter 4 Inheritance Reduction as an Aid to Implementation of ORLog Language designers are often faced with situations where compromises must be made between competing requirements that are polarized in some way. Commercial viability and practicality of the system also plays an important role in the decision process that shapes its functionalities and characteristics. In the case of deductive object-oriented databases, the introduction of new features such as inheritance, object identity, encapsulation, signature, methods, etc. has increased the expressiveness, modeling capability and functionality of the database systems while the paradigm is now faced with seemingly unsurmountable complexity of the underlying semantics. The impact of this dilemma on the development of deductive object-oriented database systems has been undoubtably far reaching. This is evidenced by the multitude of opinions on an acceptable data model on which a declarative language can be built with semantic su ciency. As a consequence, there are several schools of thoughts with respect to application, theoretical rigor and implementation details of a declarative object-oriented language. This can be broadly classi ed into two groups in two orthogonal axes. (i) A predominant class of proposals attempt to capture object-oriented features in a well-known logical system such as Datalog or Prolog. In these approaches object-oriented features are captured by giving a relational interpretation to object-oriented concepts. These are the so called translation based approaches. (ii) The proponents of the other class of languages advocate a more direct semantics that does not require a translation. 71 These are the languages that have their own syntax and underlying semantics. Within each of these categories, they can be classi ed again into two classes. (a) Languages that are given a partial logical semantics and rely on a non-logical, meta-logical, or procedural semantics at varying degrees for some of their features. (b) The second class gives a complete logical interpretation to whatever features they embody in their model. Figure 8 shows some of the languages and their classi cation according to this scheme. Direct F-logic Orlog Ordered Theories SelfLog Gulog Translational LLO ConceptBase ROCK & ROLL L&O Monteiro and Porto CORAL++ Bertino and Montesi OOLP+ Approach Partial Declarativeness Full Figure 10: Classi cation of Languages According to their Semantics and Approach. It is our thesis that a deductive language should be given a direct semantics which captures the actual logical meaning of most, if not all, of its features. In the translational approach, on the other hand, the insight is lost, and there is no control on how the translated program would behave. This results into a less than intuitive, and at times a complicated semantics that is hard to grasp. A direct semantics is \readily comprehensible" if the underlying model on which the language is based is so. This approach also preserves the declarativeness of the language and eliminates possible impedance mis-match, and has a greater appeal from a theoretical standpoint. Our contention is supported by a recent work by Kifer 46] where he argues that we can take whatever approach we would like to develop and implement a logic based language only after we fully understand the language from a logical standpoi...
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## This document was uploaded on 01/10/2011.

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