Functional Programming

1
Concepts from Functional Programming Languages

• Peter Gorm Larsen

2
Agenda

• Introduction to the Functional Programming
  Paradigm
• The notion of higher order functions
• Polymorphic examples of standard higher order
  functions

3
Introduction to FP

• The design of the imperative languages is based
  directly on the von Neumann architecture
• Efficiency is the primary concern, rather than
  the suitability of the language for software
  development
• The design of the functional languages is based
  on mathematical functions
• A solid theoretical basis that is also closer to
  the user, but relatively unconcerned with the
  architecture of the machines on which programs
  will run

4
Principles of FP

• Treats computation as evaluation of mathematical
  functions (and avoids state)
• Data and programs are represented in the same way
• Functions as first-class values
• Higher-order functions functions that operate
  on, or create, other functions
• Functions as components of data structures
• Lambda calculus provides a theoretical framework
  for describing functions and their evaluation
• It is a mathematical abstraction rather than an
  imperative programming language

5
History

• lambda calculus (Church, 1932)
• simply typed lambda calculus (Church, 1940)
• lambda calculus as prog. lang. (McCarthy(?),
  1960, Landin 1965)
• polymorphic types (Girard, Reynolds, early 70s)
• algebraic types ( Burstall Landin, 1969)
• type inference (Hindley, 1969, Milner, mid 70s)
• lazy evaluation (Wadsworth, early 70s)
• Equational definitions Miranda 80s
• Type classes Haskell 1990s
• Microsoft F etc 2000s

6
Varieties of FP languages

• Typed (ML, Haskell) vs untyped (Scheme, Erlang)
• Pure vs Impure
• impure have state and imperative features
• pure have no side effects, referential
  transparency
• Strict vs Lazy evaluation

7
Declarative style of programming

• Declarative Style of programming - emphasis is
  placed on describing what a program should do
  rather than prescribing how it should do it.
• Functional programming - good illustration of the
  declarative style of programming.
• A program is viewed as a function from input to
  output.
• Logic programming another paradigm
• A program is viewed as a collection of logical
  rules and facts (a knowledge-based system). Using
  logical reasoning, the computer system can derive
  new facts from existing ones.

8
Functional style of programming

• A computing system is viewed as a function which
  takes input and delivers output.
• The function transforms the input into output .
• Functions are the basic building blocks from
  which programs are constructed.
• The definition of each function specifies what
  the function does.
• It describes the relationship between the input
  and the output of the function.

9
Agenda

• Introduction to the Functional Programming
  Paradigm
• The notion of higher order functions
• Polymorphic examples of standard higher order
  functions

10
First-Class Functions

• Data values are first-class if they can
• be assigned to local variables
• be components of data structures
• be passed as arguments to functions
• be returned from functions
• be created at run-time

11
Higher-order Functions

• Every function has an order
• A function that does not take any functions as
  parameters, and does not return a function value,
  has order 1
• A function that takes a function as a parameter
  or returns a function value has order n1, where
  n is the order of its highest-order parameter or
  returned value
• A small example
• Twice (int -gt int) int -gt int
• Twice(f,x)
• f( f (x))
• Or
• TwiceCur (int -gt int)-gt int -gt int
• TwiceCur(f)(x)
• f( f (x))

12
Functions in Programming Languages

• How functions are treated by programming
  languages?

Language passed as arguments returned from functions nested scope
Java No No No
C Yes Yes No
C Yes Yes No
Pascal Yes No Yes
Modula-3 Yes No Yes
Scheme Yes Yes Yes
ML Yes Yes Yes
13
Nested Functions and Closures

• Return a function from function call
• function f(x)
• var y x
• return function (z)y z return y
• var h f(5)
• h(3)
• In order to handle this one needs to introduce
  closures
• A closure is a function that captures the
  bindings of free variables in its lexical context.

14
Agenda

• Introduction to the Functional Programming
  Paradigm
• The notion of higher order functions
• Polymorphic examples of standard higher order
  functions

15
Predefined Higher-Order Functions in Functional
Languages

• We will use three important predefined
  higher-order functions
• map
• filter
• foldr
• foldl
• Actually, foldr and foldl are very similar, as
  you might guess from the names

16
The Map Function

• Map applies a function to every element of a
  list
• Map_at_A,_at_B (_at_A -gt _at_B) -gt seq of _at_A -gt seq of _at_B
• Map(f)(list)
• f(list(i)) i in set inds list

17
The Filter Function

• Filter selects every element that satisfies a
  predicate
• Filter_at_A (_at_A -gt bool) -gt seq of _at_A -gt seq of
  _at_A
• Filter(pred)(list)
• list(i) i in set inds list pred(list(i))

18
The FoldR Function

• Folds all elements in a list from the right into
  one value (a simple pattern of recursion)
• FoldR_at_A,_at_B (_at_A _at_B -gt _at_B) -gt _at_B -gt seq of _at_A
  -gt _at_B
• FoldR(f)(neutral)(list)
• if list
• then neutral
• else f(hd list,FoldR(f)(neutral)(tl list))
• Example usage
• Sum FoldR()(0)
• Product FoldR()(1)
• Or FoldR(or)(false)
• And FoldR(and)(true)

19
Summary

• What have I presented today?
• Introduction to the Functional Programming
  Paradigm
• The notion of higher order functions
• Polymorphic examples of standard higher order
  functions
• What do you need to do now?
• Complete your distributed real time model for
  your project

20
Quote of the day

• Program designers have a tendency to think of the
  users as idiots who need to be controlled. They
  should rather think of their program as a
  servant, whose master, the user, should be able
  to control it. If designers and programmers think
  about the apparent mental qualities that their
  programs will have, they'll create programs that
  are easier and pleasanter more humane to deal
  with.
• John McCarthy