Deductive Database

Consider a deductive database with the following rules:

Notice that PARENT(X, Y) means that X and Y are human beings and Y is the (biological) parent of X; ANCESTOR(X, Y) means that Y is the ancestor of X.

Consider the following fact base:

PARENT(john, steve), PARENT(john, olivia), PARENT(olivia, emma), PARENT(olivia, william).

• Construct either a model-theoretic interpretation or a proof-theoretic interpretation of the above rules using the given facts.

Consider that a database contains the following relations PARENT(X, Y), and a relation BIRTH(X, B), where B is the birth date of a person X.

The following information is used for questions (b), (c), and (d).
Assume that we have the following family tree:

• State a new rule named SIBLING(X, Y) and construct a proof-theoretic interpretation of this rule to find all siblings.
• State a new rule named as DESCENDANT(X, Y) and construct a proof-theoretic interpretation of this rule to find all descendants of Martin. Note DESCENDANT(X, Y) means Y is a descendant of X.
• Given the following rules: