Vocabulary of Minimal Scheme

Outline

• Functions, operators, and programs
• Arithmetic operators
• Predefined numeric constants
• The define keyword
• Boolean operators
• Conditionals
• Defining structures
• Lists

Functions, Operators, and Programs

I use the words "function", "operator", and "program" almost interchangeably. All three words mean "something that takes in some information, does some computation, and produces some information as a result".

The word "operator" tends to be used for built-in arithmetic functions like addition, subtraction, multiplication, division, square root, and comparison. The word "program" tends to be used for functions you or I have defined. The word "function" applies equally to both. Don't worry too much about this distinction.

For example, consider the familiar arithmetic operators "+", "-", etc. from grade school: in the algebraic expression 3 + 4, the operator "+" operates on the two pieces of information "3" and "4" and produces the result "7".

Arithmetic operators

• +, -, *, / all take two or more numbers and return a number
• sqrt takes one number and returns a number
• remainder and modulo (two names for the same operator) both take two integers and return an integer: they divide the first operand by the second, returning not the quotient but the remainder.

Examples:

• (sqrt (+ (* 3 (remainder 5 3)) (* 4 4)))
• (sqrt 3 4)
  Illegal because the sqrt function only accepts one parameter.

Predefined numeric constants

• pi has the value 3.14159....
• e has the value 2.71828182....

The define keyword

The define keyword is used in two common ways: defining variables and defining functions.

Boolean operators

• true and false are the two built-in boolean values.
• <, >, =, <=, >= all take two or more numbers and return a boolean
• eq? takes two objects (numbers, symbols, booleans, or anything else) and returns a boolean
• and, or each takes two or more booleans and returns a boolean
• not takes a boolean and returns a boolean
• zero?, positive?, negative? all take a number and return a boolean.
  (zero? x) is just another way of saying (= x 0).
  (positive? x) is just another way of saying (> x 0).
  (negative? x) is just another way of saying (< x 0).

Examples:

(or (<= price 10.00) (eq? my-symbol 'bluebird))

Legal. This determines whether price is less than or equal to 10.00; if so, it returns true. If not, it then tests whether my-symbol is the symbol 'bluebird; if so, it returns true, otherwise false.

(or 3 4)

Illegal because or works on booleans, not numbers

(< true false)

Illegal because < works on numbers, not booleans

(+ 3 (< 4 5))

The first step of simplification turns this into (+ 3 true), because the value of (< 4 5) is true. The second step fails because + only works on numbers, not on a number and a boolean.

Conditionals

Scheme contains several conditional constructs, but the only one we need in this course is cond.

Defining structures

The define-struct keyword allows you to add new aggregate data types to the language.

Lists

• empty is a variable, not a function; its value is the empty list.
• cons takes an object and a list, and returns a new list with the specified object as its first element, and the specified list as the rest. In Beginner mode, it signals an error if the second parameter is not a list.
• first takes a non-empty list and returns the first element. It signals an error message if it is given an empty list or anything other than a list.
• rest takes a non-empty list and returns a list of everything but the first element. It signals an error message if it is given an empty list or anything other than a list.
• empty? takes an object and returns a boolean indicating whether it is the empty list.
• cons? takes an object and returns a boolean indicating whether it is a non-empty list.

Examples:

(cons 'bluebird empty)

returns exactly the same thing

(define bird-list (cons 'bluebird (cons 'mallard (cons 'crow empty)))

returns nothing, but defines bird-list to be a list of three symbols.

bird-list

returns (cons 'bluebird (cons 'mallard (cons 'crow empty)))

(first bird-list)

returns 'bluebird

(rest bird-list)

returns (cons 'mallard (cons 'crow empty))

(first (rest bird-list))

returns 'mallard

(empty? bird-list)

returns false

(cons? bird-list)

returns true

(empty? (rest (rest (rest bird-list))))

returns true