jamilthesis

# 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 &lt; o 1 : : : k ft1 : : : tlg &gt;, 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) =&lt; 1 : : : n &gt;, 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&quot;, 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&quot; 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&quot;, we require that H also satisfy the ground l-clause o mk ] r1]q1 : : : rn]qn. 7! 4. o mk &gt; p] 2 H =) o mk &gt; p] 2 H . Similarly, o mk &lt; p] 2 H =) o mk &lt; 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 \::&quot; as the re exive transitive closure of \:&quot;. 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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