Transcript function application - Computer Science & Engineering

Introduction to Functional
Programming
Notes for CSCE 190
Based on Sebesta, Hutton, Ullman
Marco Valtorta
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Language Families
• Imperative (or Procedural, or Assignment-Based)
• Functional (or Applicative)
• Logic (or Declarative)
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Imperative Languages
• Mostly influenced by the von Neumann computer
architecture
• Variables model memory cells, can be assigned to,
and act differently from mathematical variables
• Destructive assignment, which mimics the movement
of data from memory to CPU and back
• Iteration as a means of repetition is faster than the
more natural recursion, because instructions to be
repeated are stored in adjacent memory cells
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Functional Languages
• Model of computation is the lambda calculus (of
function application)
• No variables or write-once variables
• No destructive assignment
• Program computes by applying a functional form
to an argument
• Program are built by composing simple functions
into progressively more complicated ones
• Recursion is the preferred means of repetition
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Logic Languages
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Model of computation is the Post production system
Write-once variables
Rule-based programming
Related to Horn logic, a subset of first-order logic
AND and OR non-determinism can be exploited in
parallel execution
• Almost unbelievably simple semantics
• Prolog is a compromise language: not a pure logic
language
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What is a Functional Language?
Opinions differ, and it is difficult to give a precise
definition, but generally speaking:
• Functional programming is style of programming in
which the basic method of computation is the
application of functions to arguments;
• A functional language is one that supports and
encourages the functional style.
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Example
Summing the integers 1 to 10 in Java:
total = 0;
for (i = 1; i  10; ++i)
total = total+i;
The computation method is variable assignment.
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Example
Summing the integers 1 to 10 in Haskell:
sum [1..10]
The computation method is function application.
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Simple Function Definition
fact 0 = 1
fact n = n * fact (n -1)
fact1 = prod [1..n]
fib 0 = 1
fib 1 = 1
fib (n + 2) = fib (n + 1) + fib n
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Guards and otherwise
fact n
| n == 0 = 1
| n > 0 = n * fact (n – 1)
• fact will not loop forever for a negative argument!
sub n
| n < 10 = 0
| n > 100 = 2
| otherwise = 1
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Local scope: where
quadratic_root a b c =
[minus_b_over_2a – root_part_over_2a,
minus_b_over_2a + root_part_over_2a]
where
minus_b_over_2a = -b / (2.0 * a)
root_part_over_2a =
sqrt(b ^ 2 – 4.0 * a * c) / (2.0 * a)
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List comprehensions, generators,
and tests
• List comprehensions are used to define lists that
represent sets, using a notation similar to
traditional set notation: [body | qualifiers], e.g.,
[n * n * n | n <- [1..50]] defines a list of cubes of
the numbers from 1 to 50
• Qualifiers can be generators (as above) or tests
• The following function returns the list of factors of n
factors n = [i | i <- [1 .. n div 2], n mod i == 0]
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A Taste of Haskell
f []
= []
f (x:xs) = f ys ++ [x] ++ f zs
where
ys = [a | a  xs, a  x]
zs = [b | b  xs, b > x]
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Quicksort
The quicksort algorithm for sorting a list of integers
can be specified by the following two rules:
• The empty list is already sorted;
• Non-empty lists can be sorted by sorting the tail
values  the head, sorting the tail values  the head,
and then appending the resulting lists on either side
of the head value.
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Using recursion, this specification can be translated
directly into an implementation:
qsort
:: [Int]  [Int]
qsort []
= []
qsort (x:xs) =
qsort smaller ++ [x] ++ qsort larger
where
smaller = [a | a  xs, a  x]
larger = [b | b  xs, b  x]
Note:
• This is probably the simplest implementation of
quicksort in any programming language!
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For example (abbreviating qsort as q):
q [3,2,4,1,5]
q [2,1] ++ [3] ++ q [4,5]
q [1] ++ [2] ++ q []
[1]
[]
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q [] ++ [4] ++ q [5]
[]
[5]
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Lazy evaluation
• Lazy evaluation allows infinite data structures, e.g.:
positives = [0 ..]
squares = [n * n | n <- [0..]]
• member squares 16
• member [] b = False
member (a:x) b = (a == b) || member x b
• member2 (m:x) n
| m < n = member2 x n
| m == n true
otherwise = False
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