Function ARITY

• Defined in theory: Frame-ontology
• Source code: frame-ontology.lisp

Slots on this function:

Documentation:

Arity is the number of arguments that a relation can take. If a relation can take an arbitrary number of arguments, its arity is undefined. For example, a function such as `+' is of undefined arity; its last formal argument is specified with a sequence variable. The arity of a function is one more than the number of arguments it can take, in keeping with the unified treatment of functions and relations. The arity of the empty relation (i.e., with no tuples) is undefined.

Instance-Of: Function

Arity: 2

Domain: Relation

Range: Integer

Equivalence Axioms:

(<=> (Arity ?Relation)
     (And (Relation ?Relation)
          (Not (Empty ?Relation))
          (Integer ?N)
          (Forall (?Tuple)
                  (=> (Member ?Tuple ?Relation)
                      (= (Length ?Tuple) ?N)))))

Other Related Axioms:

(<= (Arity $X 1) (Class $X))

(<=> (Class ?Class) (And (Relation ?Class) (= (Arity ?Class) 1)))

(Arity Instance-Of 2)

(Arity Has-Instance 2)

(Arity All-Instances 2)

(Instance-Of (Arity One-Of) Undefined)

(Undefined (Arity One-Of))

(Arity Subclass-Of 2)

(Arity Superclass-Of 2)

(Arity Subrelation-Of 2)

(=> (Subrelation-Of ?Child-Relation ?Parent-Relation)
    (=> (Defined (Arity ?Parent-Relation))
        (= (Arity ?Child-Relation) (Arity ?Parent-Relation))))

(Arity Direct-Instance-Of 2)

(Arity Direct-Subclass-Of 2)

(=> (Arity $X $Y) (Integer $Y))

(=> (Arity $X $Y) (Relation $X))

(=> (= (Arity ?Relation) ?N)
    (Forall (?Tuple)
            (=> (Member ?Tuple ?Relation) (= (Length ?Tuple) ?N))))

(=> (= (Arity ?Relation) ?N) (Integer ?N))

(=> (= (Arity ?Relation) ?N) (Not (Empty ?Relation)))

(=> (= (Arity ?Relation) ?N) (Relation ?Relation))

(Arity Exact-Domain 2)

(Arity Exact-Range 2)

(Arity Total-On 2)

(Arity Onto 2)

(Arity Projection 3)

(=> (= (Projection ?Relation ?Column) ?Projection-Relation)
    (=< ?Column (Arity ?Relation)))

(=> (= (Projection ?Relation ?Column) ?Projection-Relation)
    (Defined (Arity ?Relation)))

(Arity Composition-Of 2)

(Instance-Of (Arity Compose) Undefined)

(Undefined (Arity Compose))

(Arity Alias 2)

(Arity Domain 2)

(Arity Domain-Of 2)

(Arity Range 2)

(Arity Range-Of 2)

(Arity Nth-Domain 3)

(<=> (Nth-Domain ?Relation ?N ?Type)
     (And (Defined (Arity ?Relation))
          (Positive-Integer ?N)
          (Class ?Type)
          (Forall (?Tuple)
                  (=> (Member ?Tuple ?Relation)
                      (And (>= (Length ?Tuple) ?N)
                           (Instance-Of (Nth ?Tuple ?N) ?Type))))))

(Arity Relation-Universe 2)

(Arity Has-Value 3)

(Arity All-Values 3)

(Arity Value-Type 3)

(Arity Same-Values 3)

(Arity Value-Cardinality 3)

(Arity Minimum-Value-Cardinality 3)

(Arity Maximum-Value-Cardinality 3)

(Arity Slot-Value-Type 3)

(Arity Slot-Cardinality 3)

(Arity Minimum-Slot-Cardinality 3)

(Arity Maximum-Slot-Cardinality 3)

(Arity Single-Valued-Slot 2)

(Arity Inherited-Slot-Value 3)

(Arity All-Inherited-Slot-Values 3)

(Arity Same-Slot-Values 3)

(Arity Inherited-Facet-Value 4)

(Arity Subclass-Partition 2)

(Arity Exhaustive-Subclass-Partition 2)

(Arity Documentation 2)

(Arity Related-Axioms 2)

Notes:

• KIF 2.2 doesn't require one to declare the arity of a relation, nor does it require one to use a relation with a consistent number of arguments. However, relations defined with Ontolingua are always of fixed arity, which Ontolingua asserts as part of the translation. This is to facilitate sharing over implemented frame systems, most of which do not support variable-arity relations.
• Asserting that the arity is undefined is not the same as saying that the arity is unconstrained. The arity can only exist if the relation is of fixed arity. Asserting (undefined (arity ?relation)) means that one knows that the relation has variable arity.