Vocabulary of Minimal Scheme

Outline

• Functions, operators, and programs
• Arithmetic operators
• Predefined numeric constants
• The define keyword
• Boolean operators
• Conditionals
• Defining structures
• Graphics
• 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
• symbol=? takes two symbols and returns a boolean
<--
• eq? takes two objects (numbers, symbols, booleans, or anything else) and returns a boolean
-->
• and and or each takes two or more booleans and returns a boolean
• not takes a boolean and returns a boolean
• zero? takes a number and returns a boolean.
  (zero? x) is just another way of saying (= x 0).
• even? and odd? each takes an integer and returns a boolean.
  (even? x) is just another way of saying (zero? (remainder x 2)).
  (odd? x) is just another way of saying (not (even? x)).
  Note that these functions produce an error message if used on non-integers.
• number?, integer?, symbol?, string?, boolean?, and struct? all take in an object (of any type) and return a boolean.

Examples:

(or (<= price 10.00) (symbol=? 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.

Graphics

Note: to use any of these functions, you need to load the image.ss teachpack (or the world.ss teachpack, which includes all the same stuff and more). See also Dr. Felleisen's exercises on animation worlds.

See updated documentation for the image teachpack. (I'll add some of my own if I think it necessary.)

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.
• list takes zero or more objects and returns a list of them all. We'll start with cons, which is more unwieldy but conceptually simpler, and add the list function later in the workshop.

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

(rest (first bird-list))

produces an error message: (first bird-list) returns the symbol 'bluebird, but rest can only be applied to a non-empty list, not a symbol.
Note: A one-element list is not the same thing as that one element, e.g. 'bluebird is a symbol, whereas (cons 'bluebird empty) is a list whose one element is a symbol.

(empty? bird-list)

returns false

(cons? bird-list)

returns true

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

returns true

(list 'bluebird 'mallard 'crow)

returns the exact same list as (cons 'bluebird (cons 'mallard (cons 'crow empty))).

(list 'bluebird (list 'mallard (list 'crow empty)))

is a perfectly legal list, but it's probably not what you meant; in particular, it is not the same list as in the previous example. In fact, this is a list of two (not three) elements, the second of which is itself a list of two elements, the second of which is a list of two elements, the second of which is an empty list.