Chapter 9: Functional Programming in a Typed Language
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Transcript Chapter 9: Functional Programming in a Typed Language
Chapter 9: Functional
Programming in a Typed
Language
Essence
• What is the function that must be applied to
the initial machine state by accessing the
initial set of variables and combining them
in a specific ways in order to get an answer?
• Languages that emphasize this view are
called functional languages
Essence
• Program development proceeds by
developing functions from previously
developed function in order to build more
complex functions that manipulate the
initial set of data until the final function can
be used to compute and answer for the
initial data
Standard ML
• ML is the working language of this chapter
because it best describes the topic of
functional programming.
• The name of ML is an acronym for Meta
Language.
• ML was initially designed for computerassisted reasoning.
Functional Programming
• Precise definition is up to debate
• “pure functional programming” - do not
allow side effects
• Scheme (Lisp), ML are all impure, do allow
side effects
• There are some “pure”, Haskell, Miranda
• However when we compare programs written in the
pure and programs written in the pure part of
Scheme, Lisp and ML there is little difference
9.1 Exploring a List
• Lists are considered to be the original data
structure of functional programming.
• Most of the functions explore the structure
of the list.
– Examples of these functions include:
• APPEND FUNCTION
• REVERSE FUNCTION
Operations on Lists
• ML lists are written in between brackets and
separated by commas.
– Example:
[11, 23, 34] , [ ]
• A list has the form of a::y. Where a is the
head of the list and y represents the tail of
the list.
– Example: [7] => 7::[ ]
– Example: [11,23,34] => 11::34
Functions for Lists
• NULL (null)
• Test for emptiness.
• HEAD (hd)
• Return the first
element.
• Return all except the
first element.
• Infix list constructor
• TAIL (tl)
• CONS (::)
Linear Functions on Lists
• Most functions consider the elements of a list one
by one.
• In other words, they behave as follows:
– fun length(x) = if null (x) then 0
else 1 + length(tl(x))
(Recursive)
• An empty list has a length of 0.
• Length of a nonempty list x is 1 greater than the
length of the tail of x.
Definition of Append & Reverse
• APPEND:
fun append(x, z) = if null(x) then z
else hd (x) :: append(tl(x), z)
• REVERSE:
fun reverse(x, z) = if null(x) then z
else reverse(tl(x), hd(x) ::
z)
Append Function
• The append function uses the @ symbol
which combines two lists. For example:
– append ([1,2], [3, 4, 5]) => [1, 2] @ [3, 4, 5]
=> [1, 2, 3, 4, 5]
• Other examples:
– append ([ ], z) => z
– append (a::y, z) => a :: append (y, z)
Reverse Function
• The function reverse can be used to reverse
a list. Following examples:
– reverse([ ], z) => z
– reverse(a::y) => reverse(y, a :: y)
• The reverse function is related to the ML
function rev, which basically implements in
the same way. For example:
– rev(x) => reverse(x, [ ])
Reverse/Append Phases
• Linear functions like reverse and append
contain two different phases:
– 1. A winding in which the function examines
the tail of the list, and
– 2. an unwinding phase in which control
unwinds back to the beginning of the list.
Reverse/Append Phases
• Example of the reverse winding phase:
– reverse([2, 3, 4], [1]) => reverse([3, 4], [2, 1])
=> reverse([4], [3, 2, 1])
=> reverse([ ], [4, 3, 2, 1])
=> [4, 3, 2, 1]
• Example of the append winding phase:
– append([2, 3, 4], [1]) calls append([3, 4], [1])
append([3, 4], [1]) calls append([4], [1])
append([4], [1]) calls append([ ], [1])
Reverse/Append Functions
• Here is the unwinding of the previous
function.
– append([ ], [1]) => [1]
append([4], [1]) => [4, 1]
append([3, 4], [1]) => [3, 4, 1]
append([2, 3, 4], [1]) => [2, 3, 4, 1]
9.2 Function Declaration by
Cases
• The format of function declarations is:
fun <name> <formal-parameter> = <body>
• An example of this format is the successor
function:
fun succ n = n + 1
• The application of a function f to an
argument x can be written either with
parentheses f(x), or without f x.
Function Applications
• Function application has higher precedence than
the following operators:
<, <=, =, <>, >=, >
::, @
+, -, ^
*, /, div, mod
• Examples:
3 * succ 4; => 3 * 5 => 15
3 * succ 4 :: [ ]; => 3 * list [5] => list [15]
Patterns
• Functions with more than one argument can
be declared using the following syntax:
fun <name> <pattern> = <body>
• A <pattern> has the form of an expression
made up of variables, constants, pairs of
tuples, and list constructors.
• Examples:
(x,y), (a :: y), (x, _)
Patterns and Case Analysis
• Patterns and case analysis give ML a readable
notation. Cases are separated by a vertical bar.
fun length([ ]) = 0
|
length(a :: y) = 1 + length(y)
• The declaration of a function in ML can have the
following form.
fun f<pattern1> = <expression1>
|
f<pattern2> = <expression2>
…
Patterns and Case Analysis
• The ML interpreter complains if the cases in a
function declaration are not complete, or in other
words taking each case into consideration.
• Example:
fun head (a :: y) = a;
[WARNING]
(Not a case for empty lists!!)
• Other warnings come from misspellings or
repeated formals in patterns such as:
fun f(nul) = 0 …
null misspelled
strip(1, [1, 1, 2]) = … 1 is repeated in formal
9.3 Functions as First-Class
Values
• This section includes a small library with
useful functions such as map, remove_if,
and reduce.
• The tools may use functions as arguments.
• A function is called “higher order” if either
its arguments or its results are themselves
functions.
Mapping Functions
• A “filter” is a function that copies a list, and
makes useful changes to the elements as
they are copied.
• The idea behind the function map is for
each element a of a list, do something with
a and return a list of the results.
• For example:
map square [1, 2, 3, 4] => [1, 4, 9, 16]
The Utility of Map
• The beauty of functional programming lies in
the ability to combine functions in interesting
ways.
• Examples use the following functions:
– Square Multiply an integer argument by itself
– First Return the first element of a pair
– Second Return the second element of a pair.
• Before defining new functions, we will consider a
short example involving the map.
The Utility of Map
• Example:
hd [ [11, 12, 13, 14],
[21, 22, 23, 24],
[31, 32, 33, 34] ]; => [11, 12, 13, 14]
map hd [ [11, 12, 13, 14],
[21, 22, 23, 24],
[31, 32, 33, 34] ]; => [11, 21, 31]
Anonymous Functions
• In ML, an anonymous function, a function
without a name, has the form
fn <formal-parameter> => <body>
• Example:
fn x => x * 2
• These functions are helpful for adapting
existing functions so they can be used
together with tools like map.
• map (fn x => x*2) [1,2,3,4,5];
– [2,4,6,8,10]
Selective Copying
• The “remove_if” function is another higher
order function that removes elements from
lists if some condition holds.
• It is used in the same fashion as map.
• Example:
fun odd = (x mod 2) = 1
remove_if odd [0, 1, 2, 3, 4, 5];
=> [0, 2, 4]
Accumulate a Result
• The “reduce” function accumulates a result
from a list.
• Most of the time this function is used with a
few simple functions such as sum_all, add,
multiply, etc.
• For example:
reduce add [1, 2, 3, 4, 5] 0;
=> 15
9.4 ML: Implicit Types
• Even when ML checks its types at compile
time, ML expressions are surprising free of
type declarations.
• This section will consider two aspects of
types in ML.
– Inference
– Polymorphism
Type Inference
• ML refers types when the user has not specified
the type. For example:
3 * 4;
=> val it = 12 : int
(Since the 3 and 4 are integers the product yields an
integer)
• The type of an expression can be specified by
writing
<expression> : <type>
• Overloading yields an error. For example:
fun add (x, y) = x + y; => Error
Parametric Polymorphism
• A polymorphic function can be applied to
arguments of more than one type.
• We concentrate on parametric polymorphism,
a special kind of polymorphism, in which
type expressions are parameterized.
– Example: alpha -> alpha
(with parameter alpha)
Parametric Polymorphism
• Example:
fun length (nil) = 0
|
length (a :: y) = 1 + length (y);
=> fn : (alpha-> int)
• Example:
length ([“hello”, “world”]);
=> 2 : int (remember the # in the list)
(Holds the strings as ints.)
9.5 Data Types
• Datatype declarations in ML are useful for
defining types that correspond to data
structures.
• Examples of data structures include binary
trees, arithmetic expressions, etc.
Value Constructors
• A datatype in ML introduces a basic type as
a set of values. Here is an example of a
datatype direction.
datatype direction = north| south| east| west
=>
{ north, south, east, west }
• These values become atomic; they are
constants with no components.
Value Constructors with
Parameters
•
A datatype declaration involves two parts.
1. A type name.
2. A set of value constructors for creating values of that
type.
•
•
Value Constructors can have parameters, as in
the following declaration of datatype bitree.
datatype bitree = leaf| nonleaf of bitree*bitree
In words, a value of type bitree is either the
constant leaf or it is constructed by applying
nonleaf to a pair of values of type bitree.
Binary Trees
Leaf
Nonleaf (leaf, leaf)
Nonleaf (nonleaf (leaf, leaf), leaf)
Nonleaf (leaf, nonleaf (leaf, leaf))
…
Operations on Constructed
Values
• Patterns can be used to examine the
structure of a constructed value.
• Example:
nonleaf ( leaf, nonleaf (leaf, leaf))
fun leafcount (leaf) = 1
|
leafcount (nonleaf (s,t)) = leafcount (s) +
leafcount (t);
=> 3 leaves
Differentiation: A Traditional
Example
• Symbolic differentiation of expressions like
x*(x+y) is a standard example.
• An expression is either a constant, variable,
sum, or a product. For example:
datatype expr = constant of int
| variable of string
| sum of expr *expr
| product of expr * expr;
val zero constant (0); (Declaration)
=> val zero = constant 0 : expr
Differentiation: A Traditional
Example
• Example where d => derivative:
fun d x (constant_) = zero
(This statement reads the derivative of any
constant is zero.)
Polymorphic Datatypes
• Lists are polymorphic. In other words, there can be
lists of integers, lists of strings, lists of type alpha, for
any type alpha.
• Example of a datatype declaration would be:
datatype alpha List = Nil | Cons of alpha * (alpha List)
• Example:
Nil : alpha List
(This statement reads that the value of Nil must
denote an empty list, where List is of alpha datatype.)
Functional Programming
• Exception handling will be in a separate lecture
• look at how little quilt is implemented in ML starting in
section 9.7
• Do exercise 9.1 page 380. Remember use existing ML
functions to create these