## Such as for instance reps arise definitely whenever details are 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 beneficial->b]`. For instance, `o[?A beneficial->?B ?C->?D]` becomes `o[a->b an effective->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 a good->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_{specifics}`?`(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 connective and quantifier. For aggregates, I_{connective} maps them to functions D > D and additional restrictions are imposed on the mapping I defined below.

## Particularly, in the event the

I in addition to establish the second name-interpreting mapping to your well-formed conditions, hence i signify utilizing the same symbol I that is used to the semantic design alone. Which overloading is simpler and will not bring about ambiguity.

Here ` denotes an empty listing 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 bag 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 good->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 puts 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} * (`?X _{1}`), . I

_{V}* (

`?X`)) | for all semantic structures I * such that I * (

_{n}`?`) = 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,x_{1}, . 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.