Defining relationships between URIs and resources using predicate calculus and schemata

Jonathan Borden The Open Healthcare Group

[1] Valid(a,A) := a document x is valid with respect to a schema A

The predicate valid(a,A) is true for all documents a which are valid with respect to a schema A.

[2] Instances(A) := the set of all a such that Valid(a,A)

The set of all documents valid with respect to a schema A.

Theorem: instance equivalence

[3] equivalent(a,b) := exists(t) and exists(t') such that b = t(a) and a = t'(b)

A document a is equivalent to another document b if there exists a transform t which produces b from a, and there exists a transform t' which produces a from b.

Theorem: schema equivalence

[4] Equivalent(A,B) := Instances(A) = Instances(B)

The schemata A and B are equivalent if the set of documents valid with respect to A is equal to the set of documents valid with respect to B.

Corrollary for instance equivalence

[4a] equivalent(a,b) := exists schema A and exists schema B, such that a in Instances(A) and b in Instances(B) and Equivalent(A,B)

Theorem: instance restriction

[5] restriction(a,b) := exists t such that a = t(b) but not exists t' such that b = t'(a)

a is a restriction of b if a transformation exists that maps b to a but not exists a transform t' that maps a into b.