51 for each constant c in c idc 2 idisc where

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Unformatted text preview: and Id( ) = D is the domain of . That is, each constant is mapped to an element of a domain of its intended type. { For a method m, an object o 2 O, and elements p1 : : : pk 2 D, if Id(m)(o p1 : : : pk ) is de ned, then (i) there is a type tuple of the form < o 1 : : : k ft1 : : : tlg >, in Is(m), where o is the type of the context o and i is the type of pi , i = 1 : : : k, and (ii) if m is a single valued method, then the value Id(m)(o p1 : : : pk ) belongs to all types in ft1 : : : tng, and if m is a set-valued method, then all the values in Id(m)(o p1 : : : pk ) belong to all types in ft1 : : : tng. { For each r in P , if the type of r is Is(r) =< 1 : : : n >, then Id(r) D 1 : : : D . n If a structure S satis es a clause of the form p m(a1 : : : ak ) 7! a] r1]q1 : : : r n ]q n G , where G is \is-a free", i.e., S j= p m(a1 : : : ak ) 7! a] r 1 ]q 1 : : : r n ]q n G then we require that S j= p mk ] r 1 ]q 1 : : : rn ]q n also. 7! For every object o 2 O and method denotation mk , C de nes inheritability of mk in o and satis es the unique inheritability property as de ned in De nition 3.5. 7! 7! 7! For every o m k and p, o 2 W (m)(k)(p) implies o 7! O p. ORLog structures also satisfy clauses due to structural and behavioral inheritance in a non-trivial and unique way. To this end, we require that every admissible ORLog structure also satisfy the following closure condition: the following holds. De nition 3.6 An admissible ORLog structure is inheritance closed, or i-closed, if For any clause p m(a1 : : : ak ) 7! a] G, if S j= p m(a1 : : : ak ) 7! a] G and S j= o p@mk ], then S j= (p m(a1 : : : ak ) 7! a] G ) p= o], where p= o] is a clause= = wise context substitution that replaces every occurrence of p by o. 2 7! 52 3.5 Model Theoretic Semantics We now develop an Herbrand semantics for our language and introduce the notions of satisfaction and models. Given an ORLog language L, the Herbrand universe U of L is the set of all ground id-terms { in our case just the individual constants. The Herbrand base H of L is the set of all ground atoms that can be built from U and the vocabulary of L. Let H denote the set of id-, r-, l-, is-a and w-atoms, H denote the set of i-atoms, and nally H denote the set of pred-, r- and p-atoms (i.e., d-atoms and s-atoms) in H such that H = H H H . An Herbrand structure H of L is a triple hH H H i, where H H , H H , and H H . 3.5.1 Canonical Models Ground instances of formulas are de ned as in the classical case. We de ne satisfaction in a manner identical to the classical case. De nition 3.7 (Herbrand Structures) Let H be an Herbrand structure. Then a ground atom, A, is true in H, denoted H j= A, i A 2 H13 a ground negative literal, :A, is true in H, denoted H j= :A, i A 62 H a ground conjunction of literals is true in H, denoted H j= B1 ^ : : : ^ Bm , i H j= Bi i = 1 : : : m a ground clause cl is true in H, denoted H j= A and G , i H j= G =) H j= A 2 a clause cl is true in H i all ground instances of cl are true in H. To account for the notion of inheritance outlined in the Example 3.2, we require that every Herbrand structure be \proper" in the sense of De nition 3.8 below. De nition 3.8 (Proper Structures) An Herbrand structure H of ORLog is called proper i it satis es the following properties, where o p q r tis are arbitrary ground id-terms, m is any method symbol. 13An atom A is in H i A is either in H , H or H . 53 1. o mk ] 2 H =) o :: o 2 H . Also p : o 2 H =) o :: o 2 H , and o : p 2 H =) o :: o 2 H . 7! 2. o : q 2 H and q :: p 2 H =) o :: p 2 H . 3. Whenever H satis es a ground p-clause o m(t1 : : : tk ) 7! t] r1]q1 : : : rn]qn G , where G is \is-a free", we require that H also satisfy the ground l-clause o mk ] r1]q1 : : : rn]qn. 7! 4. o mk > p] 2 H =) o mk > p] 2 H . Similarly, o mk < p] 2 H =) o mk < p] 2 H . 7! s 7!d 7! s 7!d 5. Nothing else is in H . 6. Whenever (i) p mk ] 2 H , (ii) for some o, o :: p 2 H , and (iii) r(H mk o) = p, we have o p@mk ] 2 H . 7! 7! 7! 7. Whenever H satis es a ground p-clause cl = o m(t1 : : : tk) 7! t] G and p o@mk ] 2 H , we require that H also satisfy the ground p-clause cl = (o m(t1 : : : tk) 7! t] G) o= ], which denotes the clause obtained from =p cl by replacing every occurrence of o by p. (See also De nition 3.4.) 2 7! 0 The purpose of a proper structure is intuitive and is a model-theoretic counterpart of the informal semantics of inheritance we discussed earlier. Conditions (1)-(2) establish \::" as the re exive transitive closure of \:". Condition (3) says whenever H satis es a ground p-clause, it also asserts the locality of the p-clause. (4) and (5) say signature withdrawal implies withdrawal of the corresponding method. (6) says the inheritability as de ned by the operator r (see De nition 3.3) must be respected. (7) correctly enforces what it means for a structure to respect behavioral inheritance. This is accomplished by the context switch o= ]. =p In practice, as in the case of F-logic, we only want those m...
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