378
M. Weber and E. A. Lee
information is knowable and could be expressed in an idealized (but hypothetical
3 )
ontology A
∗ , we may consider logical inference as a means to obtain information
available in A
∗ but not A.
We formalize this notion below, but first some definitions. Let ⊥ be the symbol
for “unknown”.
4
Definition 1 (Partial Order on Functions)
We define a (pointwise) partial order on n-valued function f: A
n
→ (A ∪ ⊥)
with f ≤ f
iff for x ∈ A, f (a 1 , a 2 , … a n ) = x → f
(a 1 , a 2 , … a n ) = x.
Observe this definition allows f (a 1 , a 2 , … a n ) = ⊥ with f
(a 1 , a 2 , … a n ) = x. In
other words, f agrees with f
everywhere where f is not unknown, but may disagree
where f is unknown.
Let open ontology A and its idealized A
∗ both be structures with the same signature
and the same domain. A may be missing some information available in A
∗ .
Definition 2 (Partial Order on Open Ontologies)
We define a pointwise partial order on open ontologies A and A
∗ with the same
signature and domain (A) by ordering relation . The relation indicates A
∗ has
more information than A when:
• A’s functions may have unknown value (⊥) over some elements of the domain
where A
∗ ’s functions are known. With f A as the interpretation of function symbol
f in A and f A ∗ as the interpretation of f in A
∗ , f A ≤ f A ∗ .
• A’s relations may be missing tuples which are available in the analogous relations
of A
∗ . For example, with r A and r A ∗ as interpretations of relation symbol r in
models A and A
∗ respectively, r A ⊆ r A ∗ .
• A’s interpretation of constant symbols may be less complete than the interpretation
of A
∗ . With k as the set of constant symbols in A’s signature and c: k → (A ∪
{⊥}), as the function mapping constant symbols to domain elements, c A ≤ c A ∗ .
Not only does the ordering relation defined by relate A to A
∗ , it also relates A to
a chain of non-idealized open ontologies A A
A
… A
∗ with progressively
more information than A. Applying a logical inference procedure to A, and filling in
an unknown function, relation, or constant with a concrete value can be interpreted
as finding an A
with A A
.
It may not be possible to definitively determine whether or not an open ontology
models a formula which depends on unknown functions, relations, and constants.
If the true/false value of a formula depends on evaluating a function where it is
unknown, an unknown constant, or the negation of a relation which is not explicitly
given in the model, the formula may not be evaluated with respect to the open model.
3 Of course we don’t actually know the contents of A ∗ because it contains the information we
currently don’t know in A. But it is nevertheless useful to define A ∗ as a model so we may make
explicit our assumptions about the missing information.
4 We do not always explicitly augment the domain of an open ontology to include ⊥, but this may
be assumed.
M. Weber and E. A. Lee
information is knowable and could be expressed in an idealized (but hypothetical
3 )
ontology A
∗ , we may consider logical inference as a means to obtain information
available in A
∗ but not A.
We formalize this notion below, but first some definitions. Let ⊥ be the symbol
for “unknown”.
4
Definition 1 (Partial Order on Functions)
We define a (pointwise) partial order on n-valued function f: A
n
→ (A ∪ ⊥)
with f ≤ f
iff for x ∈ A, f (a 1 , a 2 , … a n ) = x → f
(a 1 , a 2 , … a n ) = x.
Observe this definition allows f (a 1 , a 2 , … a n ) = ⊥ with f
(a 1 , a 2 , … a n ) = x. In
other words, f agrees with f
everywhere where f is not unknown, but may disagree
where f is unknown.
Let open ontology A and its idealized A
∗ both be structures with the same signature
and the same domain. A may be missing some information available in A
∗ .
Definition 2 (Partial Order on Open Ontologies)
We define a pointwise partial order on open ontologies A and A
∗ with the same
signature and domain (A) by ordering relation . The relation indicates A
∗ has
more information than A when:
• A’s functions may have unknown value (⊥) over some elements of the domain
where A
∗ ’s functions are known. With f A as the interpretation of function symbol
f in A and f A ∗ as the interpretation of f in A
∗ , f A ≤ f A ∗ .
• A’s relations may be missing tuples which are available in the analogous relations
of A
∗ . For example, with r A and r A ∗ as interpretations of relation symbol r in
models A and A
∗ respectively, r A ⊆ r A ∗ .
• A’s interpretation of constant symbols may be less complete than the interpretation
of A
∗ . With k as the set of constant symbols in A’s signature and c: k → (A ∪
{⊥}), as the function mapping constant symbols to domain elements, c A ≤ c A ∗ .
Not only does the ordering relation defined by relate A to A
∗ , it also relates A to
a chain of non-idealized open ontologies A A
A
… A
∗ with progressively
more information than A. Applying a logical inference procedure to A, and filling in
an unknown function, relation, or constant with a concrete value can be interpreted
as finding an A
with A A
.
It may not be possible to definitively determine whether or not an open ontology
models a formula which depends on unknown functions, relations, and constants.
If the true/false value of a formula depends on evaluating a function where it is
unknown, an unknown constant, or the negation of a relation which is not explicitly
given in the model, the formula may not be evaluated with respect to the open model.
3 Of course we don’t actually know the contents of A ∗ because it contains the information we
currently don’t know in A. But it is nevertheless useful to define A ∗ as a model so we may make
explicit our assumptions about the missing information.
4 We do not always explicitly augment the domain of an open ontology to include ⊥, but this may
be assumed.