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# Variable assignments can be extended recursively to

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Unformatted text preview: structure and be a variable assignment. An atom A in L is true under the semantic structure S with respect to the variable assignment , denoted S j= A, i S has an object, relationship, or a value with properties speci ed in (A). Formally: 7!d 7!s 7! 7! 7!d 7!s 7! O 7! For an is-a atom q :: p, S j= q :: p i (q) O (p). This says that object (q) is a subclass of object (p), equivalently an instance of (p), in the semantic structure S 11. For an is-a atom q : p, S j= q : p i (q) O (p). This says that object (q) is an immediate subclass of object (p), equivalently an immediate instance of (p), in the semantic structure S . For a functional d-atom of the form p m(a1 : : : an ) ! v], S j= p m(a1 : : : an ) ! v ] i Id (m)( (p) (a1) : : : (an )) = (v ). Similarly, for a set-valued d-atom of the form p m(a1 : : : an )! v1 : : : vn g], S j= p m(a1 : : : an )! !f ! fv1 : : : vng] i fv1 : : : vn g Id(m)( (p) (a1) : : : (an)). Recall the dual treatment of instances and subclasses. This is important for keeping the semantics rst-order. 11 49 s For an s-atom of the form p m(t1 : : : tn)7! ftype1 : : : typek g], S j= p m(t1 s : : : tn)7! ftype1 : : : typekg] i < (p) t1 : : : tn ftype1 : : : typek g > 2 Is(m). For a p-atom r t1 : : : tn, S j= r t1 : : : tn i r is a relation of type t1 : : : tn. For an l-atom of the form p mk ], S j= p mk ] i 7! 7! 7! (p) 2 L (m)(k). 7! 7! 7! For an i-atom of the form p [email protected] ], S j= p [email protected] ] i C (m)(k)( (p)) = (o). For a w-atom of the form p mk < o] (or o mk > p]), S j= p mk < o] (or S j= o mk > p]) i (o) 2 W (m)(k)( (p)). 7! 7! 7! 7! 7! Satisfaction of complex formulas can be de ned inductively in terms of atomic satisfaction. Let and be any formulae. Then, - S j= ^ i S j= and S j= . - S j= _ i S j= or S j= . - S j= : i S 6j= . - S j= i S 6j= or S j= - S j= (8X ) if for every that agrees with everywhere, except possibly on X , S j= . - S j= (9X ) if for some that agrees with everywhere, except possibly on X , S j= . Satisfaction of other formulae can be obviously derived from the above. If is a closed formula, its meaning is independent of a variable assignment, and its satisfaction in S can be simply written as S j= . Since our goal is to account for inheritance directly into the semantic structures of ORLog, we impose additional restrictions on our interpretation structures. It is essential that every ORLog structure assign unique inheritability of properties in the objects in the structure. We now de ne the concept of unique inheritability of properties in ORLog structures in the light of De nition 3.3. 50 following holds: For every o 2 O, method symbol m of arity k, the following condition is satis ed: (for all other r 2 O : (o]r ! (C (m)(k)(r) = r &r 62 L (m)(k)) _ C (m)(k)(r) = p _o 2 W (m)(k)(r))) then C (m)(k )(o) = p. In all other cases C (m)(k )(o) = o. 2 7! 7! 7! 7! 7! 7! De nition 3.5 An ORLog structure S satis es unique inheritability property if the if o 62 L (m)(k ) & 9q such that o]q & C (m)(k )(q ) = p & p 2 L (m)(k ) & 7! 7! 7! The above de nition asserts that for every object o and method denotation mk , o can inherit the code (the clasues that de ne the method m of arity k of the type 7!) from another object p i p has a local de nition for mk , o does not locally de ne mk , some object q (not necessarily distinct from p) which is a superclass of o inherits mk from p, and all other superclasses r of o (if exists) inherits mk either from p, or they do not inherit it from any object, or mk is withdrawn from r and o. Otherwise, o must use its own de nition for mk . The essence of this di nition is that there must exist a unique path from an object o to a superclass object p for o to be able to inherit a locally de ned method mk at p and no object q has a local de nition of the same method that is a superclass of o but a subclass of p. Furthermore, all other paths to any superclass r of o which has a local de nition for the same method mk are either withdrawn (blocked or inhibited) or are blocked due to inheritance con ict. The functions Id and Is in ORLog structures assign meaning to each symbol in the language independently of each other. Similarly the functions L C and W assign meaning to symbols in isolation. For example, the data concepts into which symbols are interpreted by Id should be consistent with the types associated with these symbols by Is. Hence we require that every ORLog structure satisfy some goodness property to be a candidate for a meaningful structure. The following goodness properties enforce this consistency12. We say that an ORLog structure S is admissible if it satis es the following goodness properties. 7! 7! 7! 7! 7! 7! 7! 7! 7! 7! 7! 7! Id and Is satisfy the following properties: 12Some of our goodness conditions are reminiscent of the well typing conditions of F-logic 47]. However, since our language and semantic structure are quite di erent from those of F-logic, there are important di erences between the two sets of well typing conditions. 51 { For each constant c in C , Id(c) 2 Id(Is(c)), where = Is(c) is the type of c...
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## This document was uploaded on 01/10/2011.

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