Functional Programming

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Functional Programming

• Universitatea Politehnica Bucuresti2008-2009
• Adina Magda Florea
• http//aimas.cs.pub.ro/fp_09

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L1 Course content

• Introduction, programming paradigms, basic
  concepts
• Introduction to Scheme
• Expressions, types, and functions
• Lists
• Programming techniques
• Name binding, recursion, iterations, and
  continuations
• Introduction to Haskell
• Problem solving paradigms in FP

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Course materials

• Scheme resources
• Almost all you want to know and find about Scheme
• http//www.schemers.org/
• Development and execution environment
• http//www.drscheme.org/
• Books and research articles on Scheme
• http//library.readscheme.org/page2.html
• R. Kent DybvigThe Scheme Programming Language,
  Second Edition online book
• http//www.scheme.com/tspl2d/index.html
• Haskell
• http//www.haskell.org/
• Haskell language design
• http//haskell.readscheme.org/lang_sem.html

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Requirements

• Laboratory min 6
• Laboratory assignments
• Homeworks
• Final exam
• Grading
• Laboratory assignments and homewwork 50
• Final exam 50

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Lecture No. 1

• Introduction to FP
• Mathematical functions
• LISP
• Introduction to Scheme

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1. 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

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1.1 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
• lamda calculus provides a theoretical framework
  for describing functions and their evaluation
• it is a mathematical abstraction rather than a
  programming language

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1.2 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

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1.3 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

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1.4 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.

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1.5 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.

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Examples

• Describing a game as a function
• Text processing
• Text processing translation
• Compiler

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1.6 Why functional programming

• Functional programming languages are carefully
  designed to support problem solving.
• There are many features in these languages which
  help the user to design clear, concise, abstract,
  modular, correct and reusable solutions to
  problems.
• The functional Style of Programming allows the
  formulation of solutions to problems to be as
  easy, clear, and intuitive as possible.
• Since any functional program is typically built
  by combining well understood simpler functions,
  the functional style naturally enforces
  modularity.

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Why functional programming

• Programs are easy to write because the system
  relieves the user from dealing with many tedious
  implementation considerations such as memory
  management, variable declaration, etc .
• Programs are concise (typically about 1/10 of the
  size of a program in non-FPL)
• Programs are easy to understand because
  functional programs have nice mathematical
  properties (unlike imperative programs) .
• Functional programs are referentially transparent
  , that is, if a variable is set to be a certain
  value in a program this value cannot be changed
  again. That is, there is no assignment but only a
  true mathematical equality.

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Why functional programming

• Programs are easy to reason about because
  functional programs are just mathematical
  functions
• hence, we can prove or disprove claims about our
  programs using familiar mathematical methods and
  ordinary proof techniques (such as those
  encountered in high school Algebra).
• For example we can always replace the left hand
  side of a function definition by the
  corresponding right hand side.

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1.7 Examples of FP languages

• Lisp (1960, the first functional
  language.dinosaur, has no type system)
• Hope (1970s an equational fp language)
• ML (1970s introduced Polymorphic typing systems)
• Scheme (1975, static scoping)
• Miranda (1980s equational definitions,
  polymorphic typing
• Haskell (introduced in 1990, all the benefits of
  above facilities for programming in the large.)
• Erlang (1995 - a general-purpose concurrent
  programming language and runtime system,
  introduced by Ericsson)
• The sequential subset of Erlang is a functional
  language, with dynamic typing.

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2. Mathematical functions

• Def A mathematical function is a mapping of
  members of one set, called the domain set, to
  another set, called the range set
• A lambda expression specifies the parameter(s)
  and the mapping of a function in the following
  form
• ?(x) x x x
• for the function cube (x) x x x

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Mathematical functions

• Lambda expressions describe nameless functions
• Lambda expressions are applied to parameter(s) by
  placing the parameter(s) after the expression
• e.g. (?(x) x x x)(3)
• which evaluates to 27

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Mathematical functions

• Functional Forms
• Def A higher-order function, or functional form,
  is one that either takes functions as parameters
  or yields a function as its result, or both

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Functional forms

• 1. Function Composition
• A functional form that takes two functions as
  parameters and yields a function whose value is
  the first actual parameter function applied to
  the application of the second
• Form h ? f g
• which means h (x) ? f ( g ( x))
• For f (x) ? x x x and g (x) ? x
  3,
• h ? f g yields (x 3) (x 3) (x 3)

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Functional forms

• 2. Construction
• A functional form that takes a list of functions
  as parameters and yields a list of the results of
  applying each of its parameter functions to a
  given parameter
• Form f, g
• For f (x) ? x x x and g (x) ? x 3,
• f, g (4) yields (64, 7)

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Functional forms

• 3. Apply-to-all
• A functional form that takes a single function as
  a parameter and yields a list of values obtained
  by applying the given function to each element of
  a list of parameters
• Form ?
• For h (x) ? x x x
• ?( h, (3, 2, 4)) yields (27, 8, 64)

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3. LISP

• Lambda notation is used to specify functions and
  function definitions. Function applications and
  data have the same form.
• e.g., If the list (A B C) is interpreted as
  data it is
• a simple list of three atoms, A, B,
  and C
• If it is interpreted as a function
  application,
• it means that the function named A is
  
• applied to the two parameters, B and
  C
• The first LISP interpreter appeared only as a
  demonstration of the universality of the
  computational capabilities of the notation

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4. Introduction to Scheme

• A mid-1970s dialect of LISP, designed to be a
  cleaner, more modern, and simpler version than
  the contemporary dialects of LISP
• Invented by Guy Lewis Steele Jr. and Gerald Jay
  Sussman
• Emerged from MIT
• Originally called Schemer
• Shortened to Scheme because of a 6 character
  limitation on file names

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Introduction to Scheme

• Designed to have very few regular constructs
  which compose well to support a variety of
  programming styles
• Functional, object-oriented, and imperative
• All data type are equal
• What one can do to one data type, one can do to
  all data types

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Introduction to Scheme

• Uses only static scoping
• Functions are first-class entities
• They can be the values of expressions and
  elements of lists
• They can be assigned to variables and passed as
  parameters

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Introduction to Scheme

• Primitive Functions
• 1. Arithmetic , -, , /, ABS, SQRT, REMAINDER,
  MIN, MAX
• e.g., ( 5 2) yields 7

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Introduction to Scheme

• 2. QUOTE -takes one parameter returns the
  parameter without evaluation
• QUOTE is required because the Scheme interpreter,
  named EVAL, always evaluates parameters to
  function applications before applying the
  function. QUOTE is used to avoid parameter
  evaluation when it is not appropriate
• QUOTE can be abbreviated with the apostrophe
  prefix operator
• e.g., '(A B) is equivalent to (QUOTE (A B))

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Introduction to Scheme

• 3. CAR takes a list parameter returns the first
  element of that list
• e.g., (CAR '(A B C)) yields A
• (CAR '((A B) C D)) yields (A B)
• 4. CDR takes a list parameter returns the list
  after removing its first element
• e.g., (CDR '(A B C)) yields (B C)
• (CDR '((A B) C D)) yields (C D)

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Introduction to Scheme

• 5. CONS takes two parameters, the first of which
  can be either an atom or a list and the second of
  which is a list returns a new list that
  includes the first parameter as its first element
  and the second parameter as the remainder of its
  result
• e.g., (CONS 'A '(B C)) returns (A B C)

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Introduction to Scheme

• 6. LIST - takes any number of parameters returns
  a list with the parameters as elements

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Introduction to Scheme

• Lambda Expressions
• Form is based on ? notation
• e.g., (LAMBDA (L) (CAR (CAR L)))
• L is called a bound variable
• Lambda expressions can be applied
• e.g.,
• ((LAMBDA (L) (CAR (CAR L))) '((A B) C D))

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Introduction to Scheme

• A Function for Constructing Functions
• DEFINE - Two forms
• 1. To bind a symbol to an expression
• e.g.,
• (DEFINE pi 3.141593)
• (DEFINE two_pi ( 2 pi))

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Introduction to Scheme

• 2. To bind names to lambda expressions
• e.g.,
• (DEFINE (cube x) ( x x x))
• Example use
• (cube 4)

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Introduction to Scheme

• Evaluation process (for normal functions)
• 1. Parameters are evaluated, in no particular
  order
• 2. The values of the parameters are substituted
  into the function body
• 3. The function body is evaluated
• 4. The value of the last expression in the body
  is the value of the function
• (Special forms use a different evaluation process)

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Introduction to Scheme

• Example
• (DEFINE (square x) ( x x))
• (DEFINE (hypotenuse side1 side1)
• (SQRT ( (square side1)
• (square side2)))
• )

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Introduction to Scheme

• Example
• (define
• (list-sum lst)
• (cond
• ((null? lst) 0)
• ((pair? (car lst))
• ((list-sum (car lst)) (list-sum (cdr lst))))
• (else
• ( (car lst) (list-sum (cdr lst))))))

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Some conventions

• Naming Conventions
• A predicate is a procedure that always returns a
  boolean value (t or f). By convention,
  predicates usually have names that end in ?'.
• A mutation procedure is a procedure that alters a
  data structure. By convention, mutation
  procedures usually have names that end in !'.

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Cases

• Uppercase and Lowercase
• Scheme doesn't distinguish uppercase and
  lowercase forms of a letter except within
  character and string constants in other words,
  Scheme is case-insensitive.
• For example, Foo is the same identifier as FOO,
  but 'a' and 'A' are different characters.

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Predicate functions

• Predicate Functions (t is true and f or ()is
  false)
• 1. EQ? takes two symbolic parameters it returns
  T if both parameters are atoms and the two are
  the same
• e.g., (EQ? 'A 'A) yields t
• (EQ? 'A '(A B)) yields ()
• Note that if EQ? is called with list parameters,
  the result is not reliable
• Also, EQ? does not work for numeric atoms

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Predicate functions

• Predicate Functions
• 2. LIST? takes one parameter it returns T if
  the parameter is a list otherwise()
• 3. NULL? takes one parameter it returns T if
  the parameter is the empty list otherwise()
• Note that NULL? returns T if the parameter
  is()
• 4. Numeric Predicate Functions
• , ltgt, gt, lt, gt, lt, EVEN?, ODD?, ZERO?,
  NEGATIVE?

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Control flow

• Control Flow
• 1. Selection- the special form, IF
• (IF predicate then_exp else_exp)
• e.g.,
• (IF (ltgt count 0)
• (/ sum count)
• 0
• )

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Control flow

• Control Flow
• 2. Multiple Selection - the special form, COND
• General form
• (COND
• (predicate_1 expr expr)
• (predicate_1 expr expr)
• ...
• (predicate_1 expr expr)
• (ELSE expr expr)
• )
• Returns the value of the last expr in the
  first
• pair whose predicate evaluates to true

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Example

• (DEFINE (compare x y)
• (COND
• ((gt x y) (DISPLAY x is greater than y))
• ((lt x y) (DISPLAY y is greater than x))
• (ELSE (DISPLAY x and y are equal))
• )
• )

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Example

• 1. member - takes an atom and a simple list
  returns T if the atom is in the list ()
  otherwise
• (DEFINE (member atm lis)
• (COND
• ((NULL? lis) '())
• ((EQ? atm (CAR lis)) T)
• ((ELSE (member atm (CDR lis)))
• ))

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Example

• 2. equalsimp - takes two simple lists as
  parameters returns T if the two simple lists
  are equal () otherwise
• (DEFINE (equalsimp lis1 lis2)
• (COND
• ((NULL? lis1) (NULL? lis2))
• ((NULL? lis2) '())
• ((EQ? (CAR lis1) (CAR lis2))
• (equalsimp(CDR lis1)(CDR lis2)))
• (ELSE '())
• ))

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Example

• 3. equal - takes two general lists as parameters
  returns T if the two lists are equal
  ()otherwise
• (DEFINE (equal lis1 lis2)
• (COND
• ((NOT (LIST? lis1))(EQ? lis1 lis2))
• ((NOT (LIST? lis2)) '())
• ((NULL? lis1) (NULL? lis2))
• ((NULL? lis2) '())
• ((equal (CAR lis1) (CAR lis2))
• (equal (CDR lis1) (CDR lis2)))
• (ELSE '())
• ))

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Example

• 4. append - takes two lists as parameters
  returns the first parameter list with the
  elements of the second parameter list appended at
  the end
• (DEFINE (append lis1 lis2)
• (COND
• ((NULL? lis1) lis2)
• (ELSE (CONS (CAR lis1)
• (append (CDR lis1) lis2)))
• ))