Such as for example reps happen however whenever details is actually instantiated that have constants

Bags are employed here because the order of the attribute/value pairs in a frame is immaterial and the pairs may repeat. For instance, o[a->b a->b]. For instance, o[?A great->?B ?C->?D] becomes o[a->b an excellent->b] if variables ?A and ?C are instantiated with the symbol a and ?B, ?D with b. (We shall see later that o[a->b an excellent->b] is equivalent to o[a->b].)

The operator ## is required to be transitive, i.e., cstep one ## c2 and c2 ## c3 must imply c1 ## c3. This is ensured by a restriction in Section Interpretation of Formulas.

# and ## are required to have the usual property that all members of a subclass are also members of the superclass, i.e., o # cl and cl ## scl must imply o # scl. This is ensured by a restriction in Section Interpretation of Formulas.

For every outside schema, ?, associated with the language, I_external(?) is assumed to be specified externally in some document (hence the name external schema). ? is a schema of a RIF built-in predicate or function, I_external(?) is specified in [RIF-DTB] so that:

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• If ? is a schema of a built-in function then I_external(?) must be the function defined in the aforesaid document.
• If ? is a schema of a built-in predicate then I_knowledge? (I_external(?)) (the composition of I_truth and I_external(?), a truth-valued function) must be as specified in [RIF-DTB].

Further restrictions on the interaction of this function with I_truth will be imposed in order to ensure the intended semantics for each conjunctive and quantifier. For aggregates, I_connective maps them to functions D > D and additional restrictions are imposed on the mapping I defined below.

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Particularly, in the event the

I and define another identity-interpreting mapping on the well-shaped conditions, and that we signify using the same symbol I which is used for the semantic build alone. That it overloading are much easier and won’t end up in ambiguity.

Here denotes an empty number of elements of D. (Note that the domain of I_list is D * , so D 0 is an empty list of elements of D.)

Here denotes a purse of attribute/value pairs. Jumping ahead, we note that duplicate elements in such a bag do not affect the meaning of a frame formula. So, for instance, o[a->b a great->b] and o[a->b] always have the same truth value.

Note that, by definition, External(t loc) is well-formed only if it is an instantiation of an external schema. Furthermore, by the definition of coherent lays of external schemas, it can be an instantiation of at most one such schema, so I(External(t loc)) is well-defined.

S = <(I_V * (?X),I_V * (?X₁), . I_V * (?X_n)) | for all semantic structures I * such that I * (?) = t and I * is exactly like I except that I_V * (?X) can be different from I_V(?X)>.

In addition, let S _set denote the set of all elements x such that (x,x₁, . x_n) ? S and S _bag denote the bag of all such elements x (i.e., S _bag can have repeated occurrences of the same element).

where L is a sorted list of the elements in S _set. Since sorting requires an ordering, the above is well-defined only for semantic structures with totally ordered domains. If L is infinite then the value of the aggregate in I is indeterminate (i.e., it can be any element of the domain D).

The requirement that the list L must be sorted comes from the fact that there can be many ways to represent S _set as a list, while I(setof
1 . ?X_n] | ?>) must be defined as one concrete element of the domain D. Sorting a set is a standard way of providing the requisite unique representation.