part I $(12$ points$)$

In Part I, you are going to write a Prolog knowledge base and use it to answer some queries.

Let the relation $$parent$($A$,$ B$).$ means A is a parent of B $($or B is a child of A$)$; and let the relation $$sibling$($A$,$ B$).$

means A and B are siblings.

Encoding the Knowledge Base

Encode in your knowledge base these $($and only these$)$ six facts:

$$ Amy is the daughter of Bob.

$$ Bob is the father of Christine.

$$ David is the son of Bob.

$$ Emily is the mother of Frank.

$$ Gilbert is the son of Emily.

$$ Heidi is the daughter of Emily.

Define a Rule for Siblings

Also encode in your knowledge base this $($and only this$)$ rule:

A and B are siblings only if A and B are children of the same person. However, a person cannot be a sibling of

himself$/$herself$.($Note that in Prolog, we use $$A $\backslash =$ B$$ to mean that A and B are not the same object.$)$

Querying the Knowledge Base

Put the facts and the rule in a prolog file. Load the file into Prolog; and then perform the following queries:

$$ Is Bob a parent of Amy?

$$ Is Bob a parent of Heidi?

$$ Who are the children of Emily?

$$ Are Christine and David siblings?

$$ Are Amy and Gilbert siblings?

$$ Who are Frank$$s siblings?

Draw the Search Tree

Draw the search tree for the last question: Who are Frank$$s siblings?

Part II $(8$ points$)$

In Chapter $3$ of the prolog book, we have discussed how to implement the add predicate using the definition of numeral

with the successor $($complex$)$ term. Now we would like to extend the system to detect if a numeral is a power of $2.$

Detecting Power of Two

Convention:

$$ powerOf$2($X$).$ means X is a power of $2.$

We are going to define this predicate recursively. The followings are the fact and rule for detecting whether a numeral

is a power of $2$ or not:

$$ The base clause is a fact: $1$ is a power of $2($because $1=20)$;

$$ The recursive clause is a rule: X is a power of $2$ only if there is a Y such that when adding Y to Y the result is

X$,$ and Y is a power of $2.$

For example, the following shows how the queries should be performed:

$|?-$ powerOf$2($succ$($succ$($succ$($succ$(0))))).$

true $?$

yes

$|?-$ powerOf$2($succ$($succ$($succ$(0)))).$

no

The first query shows that $4$ is a power of $2$; while the second shows that $3$ is not.

Querying the Knowledge Base

Put the relations and the rule in a prolog file. Load the file into Prolog; and then perform the following queries:

$$ Is $8$ a power of $2?$

$$ Is $10$ a power of $2?$