3B Symbolic Differentiation; Quotation⁠

# Part 1

# Recap

• Don’t solve a particular problem at each level: solve a class of problems in the neighborhood of the particular problem by building a language suited to them.
• We’ve seen the power of embedding languages.
• We’re going to confuse the distinction between procedures and data even more badly!

# Derivatives and integrals

• Instead of computing a numerical approximation of the derivative, we want to apply the rules of calculus to algebraic expressions.
• Producing derivatives are easy. Producing integrals, the inverse operation, is hard.
• Why is it easy to go one way but not the other? Because the rules for differentiation are reduction rules.

# Differentiation procedure

• We can write the rules of differentiation as a case analysis that uses recursion.
• But first, we need to be able to represent the expressions.
• Taking an expression as an argument is very different from taking a function.
• You can’t open up a function. You can only get answers from it.
• An expression is data representing the algebraic expression that defines a function.
• We can write the procedure deriv using lots of wishful thinking.

# Expression representation

• To represent sums, products, quotients, etc., why not use the same language we’re writing the program in? We can just use list structure.
• There are more built-in predicates we haven’t seen yet: atom? and eq?.

# Quotation

• We are also introducing quotation. The words of English are ambiguous, so we need punctuation: “say your name” is different from “say ‘your name’.”
• The notation in Lisp is to precede the expression with a single quotation mark.
• Quotation makes a language more complex because we can no longer substitute equals for equals (see Bertrand Russell’s “On Denoting”).
• The names car and cdr have survived (originally referring to the address register and the decrement register) because we can use shorthands like cadddr and cadar.

# Part 2

# Simplifying expressions

• The expressions we get from the deriv procedure are ugly.
• Nothing is simplified, and it’s hard to read.
• The solution is to change the representation to add a simplification step.

So the way we might solve this problem is … change the representation … it’s one of the pieces of artillery we have in our war against complexity. —Sussman

• We have an abstraction barrier between the rules of differentiation and the representation of algebraic expressions.

# Conclusion

• Quotation stops and says, “I’m talking about this expression itself.”
• We can write languages that are not only embedded in Lisp, but that are completely different, using quotation. Quotation gives us tremendous power.