Functional Programming

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Functional Programming Standard ML

• Hossein Hojjat et al.

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By doing it mathematically, you provide a firm
foundation which will enable people to go
further. Robin Milner, Turing Award
Lecturer(1991) and the ML designer
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Outline

• Introduction
• Why functional programming?
• Some History
• Standard ML
• ML Syntax
• Programming in Standard ML
• Some Good References

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Introduction
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Introduction

• We use a zillion different programming languages
• general purpose programming  Fortran, Lisp,
  Basic, C, Pascal, C, Java, etc.
• scripting  Visual Basic, awk, sed, perl, tcl,
  sh, csh, bash, REXX, Scheme, etc.
• search regular expressions, browser queries,
  SQL, etc.

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Introduction

• display and rendering  PostScript, HTML, XML,
  VRML, etc.
• hardware  SystemC, VHDL, Esterelle
• theorem proving and mathematics  Mathematica,
  Maple, Matlab, NuPRL, Coq
• others?

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Introduction

• ML is very different from what most of us have
  seen it is functional
• Before considering ML, we will summarize the
  imperative languages properties

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Introduction
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Introduction

• Design of imperative languages is based directly
  on the von Neumann architecture

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Introduction

• Programs in imperative languages rely heavily on
  modifying the values of a collection of
  variables, called the state
• Before execution, the state has some initial
  value s
• During execution, each command changes the state

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Introduction

• Example
• In a sorting program, the state initially
  includes an array of values
• When the program has finished, the state has been
  modified in such a way that these values are
  sorted
• Intermediate states represent progress towards
  this goal

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Introduction

• The state is typically modified by assignment
  commands
• By using control structures, one can execute
  these commands conditionally, or repeatedly,
  depending on other properties of the current state

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Introduction

• But

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Introduction

• Functional programs dont use variables - there
  is no state
• Therefore they cannot use assignments there is
  nothing to assign to

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Introduction

• The idea of executing commands in sequence is
  meaningless
• The first command can make no difference to the
  second there are not any state between them
• They are based on Mathematical functions

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Introduction

• Functions can be treated in exactly the same way
  as simpler objects like integers
• They can be passed to other functions as
  arguments and returned as results
• Most traditional languages provide poor
  facilities in these areas

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Introduction

• Instead of sequencing and looping, functional
  languages use recursive functions

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Introduction

• Example
• f(n)
• 1 if n1
• f(5n-1) if n is odd, n?1
• f(n/4-3) if n is even
• Question You said there are not any variables,
  but what about n?

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Introduction

• n is an identifier
• In a Functional Language, the identifier bind to
  values
• Variable is something that can be assigned a
  value
• Functions have no side effect
• They do not update any variables
• It is easy to define the semantics

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Introduction

• Haskell is an example of a pure functional
  language.
• Haskell is, as of 2002, the functional language
  on which the most research is being performed.
• ML is not a pure functional language in that it
  is possible to write procedural programs (with
  assignments and side-effects)

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Why Functional Programming?
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Why Functional Programming?
The von Neumann bottleneck

• Backus' famous paper encouraged much interest in
  functional languages as a means of breaking the
  von-Neumann bottleneck

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Why Functional Programming?

• Von Neumann bottleneck pumping single words back
  and forth between CPU and store
• Task of a program change store in some major
  way.
• It has kept us tied to word-at-a-time thinking
  instead of of encouraging us to think in terms of
  the larger conceptual units of the task at hand.

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Why Functional Programming?

• The assignment statement is the von Neumann
  bottleneck of programming languages
• Pure functional programming languages remove
  state and assignments
• Concurrency possible order of evaluation doesnt
  matter

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Why Functional Programming?
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Why Functional Programming?

• System is referentially transparent if, in a
  fixed context, the meaning of the whole can be
  determined solely by the meaning of its parts.
• Independent of the surrounding expression.

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Why Functional Programming?

• Do we have such property in imperative languages?
• If the function has side-effects (updating a
  global variable, doing input or output), then
  f(3) f(3) may not be the same as 2 f(3).
• The second f(3) has a different meaning than the
  rst

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Why Functional Programming?

• Purely declarative languages guarantee
  referential transparency
• It makes it easier to understand how a program
  works

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Why Functional Programming?

• Many features of imperative languages have arisen
  by a process of abstraction from typical computer
  hardware
• Perhaps the right approach is not to start from
  the hardware and work upwards

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Why Functional Programming?

• Start with programming languages as an abstract
  notation for specifying algorithms and then work
  down to the hardware. (Dijkstra 1976)

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Why Functional Programming?

• Makes programming into an engineering discipline
  rather than a trial-and-error process
• The Formalism Principle Correctness should be
  confirmed by reasoning and not by experiment-
  Marjan Sirjani

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Why Functional Programming?

• As a matter of fact, it is unlikely that
  programmers will have the patience to perform
  such proofs the proofs are usually long and
  boring

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• Beware of bugs in the above code I have only
  proved it correct, not tried it. ,Donald Knuth

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Why Functional Programming?

• Functional Programming is an area of current
  research
• ACM Conference on LISP and Functional Programming

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Some History
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Some History

• 1940s
• Alonzo Church and Haskell Curry developed the
  lambda calculus, a simple but powerful
  mathematical theory of functions.

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Some History

• Alonzo Church is a famous computer scientist
• He had many doctoral students , such as Stephen
  C. Kleene or Alan Turing

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Some History

• 1960s
• John McCarthy developed Lisp, the first
  functional language. Some influences from the
  lambda calculus, but still retained variable
  assignments.

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Some History

• 1978
• John Backus publishes award winning article on
  FP, a functional language that emphasizes
  higher-order functions and calculating with
  programs.

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Some History

• Mid 1970s
• Robin Milner develops ML, the first of the modern
  functional languages, which introduced type
  inference and polymorphic types.

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Some History

• Late 1970s - 1980s
• David Turner develops a number of lazy functional
  languages leading up to Miranda, a commercial
  product.
• 1988
• A committee of prominent researchers publishes
  the first definition of Haskell, a standard lazy
  functional language.

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Some History

• 1999
• The committee publishes the definition of Haskell
  98, providing a long-awaited stable version of
  the language.

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Standard ML
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Standard ML

• Historically, ML stands for metalanguage
• General-purpose functional programming language
• Developed by Robin Milner and others in the late
  1970s at Edinburgh University

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Standard ML

• In 1969 Dana Scott introduced LCF, his Logic for
  Computable Functions
• It was a core higher-order call-by-name
  functional programming language with arithmetic,
  booleans and recursion at all types
• That lead to Milner et al's LCF system and then
  the programming language ML

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ML Syntax

• A program in ML, like any other language, is made
  up of various kinds of expressions.

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ML Syntax
syntactic class syntactic variable(s) and grammar rule(s) examples
identifiers x, y a, x, y, x_y, ...
constants c ...2, 1, 0, 1, 2 (integers)  1.0, 0.001, 3.141 (reals) true, false (booleans) "hello", "", "!" (strings) "A", " " (characters)
unary operator u , not, size, ...
binary operators b , , -, gt, lt, gt, lt, , ...
expressions (terms) e x    c    u e    e1 b e2   if e then e else e     let d1...dn in e end    e (e1, ..., en) foo, 0.001, not b, 2 2, 
declarations d val x e    fun y (x1t1, ..., xntn) t  e val one 1 fun square(x int) int
types t int    real    bool    string    char    t1...tn-gtt int, string, int-gtint, boolint-gtbool
Adapted from Cornell lectures
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Programming in Standard ML

• Example A simple function declaration that
  computes the absolute value of a real number

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Programming in Standard ML

• The SML prompt lets you type either a term or a
  declaration that binds a variable to a term
• Running an ML program is just evaluating a term
• The ML evaluator takes the left-most expression
  that is not a value and reduces it to some
  simpler expression. Eventually the whole
  expression is a value and then evaluation stops
  the program is done

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Programming in Standard ML

• For example consider evaluating abs(2.01.0)
• abs(2.01.0) ?
• abs(3.0) ?
• if 3.0 lt 0.0 then 3.0 else 3.0 ?
• if false then 3.0 else 3.0 ?
• 3.0

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Programming in Standard ML

• The let expression works by
• Evaluating all of its bindings.
• Those bindings are substituted into the body of
  the let  expression (the expression in between
  in...end)
• Example
• let val x 14 in x3 ?
• let val x 5 in x3 ?
• 53 ? 15

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Programming in Standard ML

• Rather big example

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Programming in Standard ML
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Some Good References

• Functional Programming Using Standard ML Ake
  Wikstrom
• Introduction to Functional Programming John
  Harrison
• Introduction to Standard ML,Robert Harper
• Cornell CS312 lectures
• And lots more

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Any Questions ?!