**demodulator**in terminology) is an equation with a specified direction. For example, the rewrite rule $x+0 \rightarrow x$ suggests replacing any expression that matches $x+0$ with the expression $x$. Rewrite rules are a key component of equational reasoning systems. Use the predicate rewrite(X,Y) to represent rewrite rules. For example, the earlier rewrite rule is written as rewrite(X+0,X). Some terms are

*primitive*and cannot be further simplified; thus, we write primitive(0) to say that 0 is a primitive term.

1. Write a definition of a predicate simplify(X,Y), that is true when Y is a simplified version of X—that is, when no further rewrite rules apply to any subexpression of Y.

2. Write a collection of rules for the simplification of expressions involving arithmetic operators, and apply your simplification algorithm to some sample expressions.

3. Write a collection of rewrite rules for symbolic differentiation, and use them along with your simplification rules to differentiate and simplify expressions involving arithmetic expressions, including exponentiation.

This exercise looks at the recursive
application of rewrite rules, using logic programming. A rewrite rule
(or **demodulator** in terminology) is an
equation with a specified direction. For example, the rewrite rule
$x+0 \rightarrow x$ suggests replacing any expression that matches $x+0$
with the expression $x$. Rewrite rules are a key component of equational
reasoning systems. Use the predicate rewrite(X,Y) to
represent rewrite rules. For example, the earlier rewrite rule is
written as rewrite(X+0,X). Some terms are
*primitive* and cannot be further simplified; thus, we
write primitive(0) to say that 0 is a primitive term.

1. Write a definition of a predicate simplify(X,Y), that
is true when Y is a simplified version of
X—that is, when no further rewrite rules apply to any
subexpression of Y.

2. Write a collection of rules for the simplification of expressions
involving arithmetic operators, and apply your simplification
algorithm to some sample expressions.

3. Write a collection of rewrite rules for symbolic differentiation,
and use them along with your simplification rules to differentiate
and simplify expressions involving arithmetic expressions,
including exponentiation.