Lecture 5: Functional Programming

TI1220 2012-2013
Concepts of Programming Languages

Eelco Visser / TU Delft

John McCarthy (September 4, 1927 –
October 24, 2011) was an American
computer scientist and cognitive scientist.
He coined the term "artificial
intelligence" (AI), developed the Lisp
programming language family, significantly
influenced the design of the ALGOL
programming language, popularized
timesharing, and was very influential in the
early development of AI.

Around 1959, he invented so-called "garbage collection" methods to
solve problems in Lisp. Based on the lambda calculus, Lisp soon
became the programming language of choice for AI applications after
its publication in 1960.

http://en.wikipedia.org/wiki/John_McCarthy_(computer_scientist)

Outline
From the lab:
Unit testing
Graded assignment 1
Functional objects in Scala
Pattern matching
Recursion and induction
Algebraic data types
Binary trees
Algebraic data types in C

Tests

• check that your code is correct
• regression testing: don’t make the same mistake twice
Coverage

• a test for each representative case
Test-driven development

• (1) define tests for representative cases
• (2) write code
• (3) test

Unit Testing in Scala

import org.scalatest.Suite

class <NameOfTestClass> extends Suite {
import <ClassUnderTest>._

def <testNameOfTest> {
expect(<expected result>) {
<computation>
}
}
}

/* import test framework .h */
#include "solution.c"
#include "CuTest.h"
#include "CuJoin.h"
Unit Testing in C
/* your imported libraries */
#include <string.h>

/* signatures of all functions being tested */
char* wc(char* data);

/* defined tests */
void test_1(CuTest *tc) {
char* wcout = wc("hellon world");
char* expected = "2 2 10 12";
CuAssertTrue(tc, !strcmp(wcout,expected));
}
/* hook all your tests into the harness */
void testHooker(CuSuite* intoSuite){
SUITE_ADD_TEST(intoSuite, test_1);
}

test("Changing properties", function() {
var obj = {x : 3};
expect(5);
ok(changeProp, "function exists");
equal(obj.x, 3);
equal(obj.y, undefined);
changeProp(obj);
equal(obj.x, 42);
equal(obj.y, 9);
});

Unit Testing in JavaScript

Graded Assignment 1
Algebraic datatypes in C
Dynamic dispatch in C

Important dates
Deadline: April 2, 2013 23:59
Extension: April 5, 2013 23:59

Submitting after extension date is not possible
Maximum penalty for submitting after deadline: 6 points
Minimum grade needed: 4
Grade: 70% unit tests, 30% check lists
Grade for GAs: average of four assignments

Algebraic Datatypes in C
abstract class XML
case class Text(t: String) extends XML
case class Elem(tag: String, elems: List[XML]) extends XML

object Solution {

def text(elems1: List[XML]): List[XML] =
elems1.flatMap(_ match {
case t@Text(_) => List[XML](t)
case Elem(_, elems2) => text(elems2)
})

}

translate this Scala program to an equivalent C program

// values

abstract class Value {
def value: Int Dynamic Dispatch in C
def isFailure: Boolean
def +(that: Value): Value
def *(that: Value): Value
}

object LookupFailure extends Value {
def value: Int = 0
translate this Scala program
def isFailure: Boolean = true to an equivalent C program
def +(that: Value) = LookupFailure
def *(that: Value) = LookupFailure
}

class IntValue(v : Int) extends Value {
val value = v
def isFailure: Boolean = false
def +(that: Value) = that match {
case v: IntValue => new IntValue(value + v.value)
case _ => LookupFailure
}
def *(that: Value) = that match {
case v: IntValue => new IntValue(value * v.value)
case _ => LookupFailure
}
}

// environments

abstract class Env { Dynamic Dispatch in C
def lookup(x: String): Value
}
class MtEnv extends Env {
def lookup(x: String): Value = LookupFailure
}
class Bind(key: String, value: Int, env: Env) extends Env {
def lookup(x: String): Value =
if(x == key) new IntValue(value) else env.lookup(x);
}

// expressions

abstract class Exp {
def eval(env: Env): Value;
Dynamic Dispatch in C
override def toString: String
}

class IntExp(value: Int) extends Exp {
def eval(env: Env) = new IntValue(value)
override def toString = value.toString
}

class VarExp(name: String) extends Exp {
def eval(env: Env) = env.lookup(name)
override def toString = name
}

class PlusExp(left: Exp, right: Exp) extends Exp {
def eval(env: Env) = left.eval(env) + right.eval(env)
override def toString = "(" + left.toString + " + " + right.toString + ")"
}

class MulExp(left: Exp, right: Exp) extends Exp {
def eval(env: Env) = left.eval(env) * right.eval(env)
override def toString = "(" + left.toString + " * " + right.toString + ")"
}

Functional Programming
(in Scala)

Ingredients of functional programming

Immutable objects

Pattern matching

Inductive (algebraic) data types

Recursive functions

First-class functions (next week)

functional object: the ﬁelds
of an object are immutable

Example: Rational Numbers

• Rational = Int x Int

• Notation: numerator/denominator

• Addition

• example: 1/2 + 2/3 = 3/6 + 4/6 = (3 + 4)/6 = 7/6
• general: n1/d1 + n2/d2 = (n1*d2 + n2*d1) /
(d1*d2)

• Multiplication

• n1/d1 + n2/d2 = (n1 * n2) / (d1 * d2)
• Division

• n1/d1 / n2/d2 = n1/d2 * d2/n2

Constructing a Rational

class parameters

class Rational(n: Int, d: Int) {
println("Created " + n + "/" + d)
}

scala> new Rational(1, 2)
Created 1/2
res0: Rational = Rational@2d83e895

Immutable object trade-offs

Advantages

• easier reasoning
• pass around freely (no risk of undesired mutations)
• cannot be changed concurrently in two threads
• immutable object make safe hash table keys
Disadvantages

• copying large object graphs vs in-place update

Overriding Methods

override def toString = n + "/" + d
}

scala> val half = new
Rational(1, 2)
half: Rational = 1/2

Checking Pre-conditions

require(d != 0)
}

scala> val half = new Rational(1, 0)
java.lang.IllegalArgumentException: requirement failed

require(d != 0)
def add(that: Rational): Rational =
new Rational(n * that.d + that.n * d, d * that.d)
}

Adding Rationals Functionally

Visibility of Class Parameters

require(d != 0)
}

$ fsc Rational.scala
Rational.scala:5: error: value d is not a member of Rational
^
Rational.scala:5: error: value d is not a member of Rational
^
two errors found

Functional Fields

require(d != 0)
val numer: Int = n
val denom: Int = d
override def toString = numer + "/" + denom
new Rational(
numer * that.denom + that.numer * denom,
denom * that.denom)
}

scala> new Rational(1,2) add new Rational(2,3)
res0: Rational = 7/6

Non-functional Objects

class ImperativeRational(n: Int, d: Int) {
require(d != 0)
var numer: Int = n
var denom: Int = d
def add(that: ImperativeRational) {
numer = numer * that.denom + that.numer * denom;
denom = denom * that.denom;
}
}

scala> val half = new ImperativeRational(1, 2)
Destructive Update half: ImperativeRational = 1/2

scala> val twothirds = new ImperativeRational(2,3)
twothirds: ImperativeRational = 2/3

scala> half.add(twothirds)

scala> half
res1: ImperativeRational = 7/6

Self References

def lessThan(that: Rational) =
this.numer * that.denom < that.numer * this.denom

def max(that: Rational) =
if (this.lessThan(that)) that else this

this: optional when referring to ﬁelds

Auxiliary Constructors

require(d != 0)
val numer = n
val denom = d
def this(n: Int) = this(n, 1)
// auxiliary constructor
...
}

scala> new Rational(6)

Private Fields and Methods

require(d != 0)
private val g = gcd(n.abs, d.abs)
val numer = n / g
val denom = d / g
...
private def gcd(a: Int, b: Int): Int =
if (b == 0) a else gcd(b, a % b)
}

scala> new Rational(6,42)

Using Functions as Inﬁx Operators

new Rational(numer * that.denom + that.numer * denom,
denom * that.denom)

scala> new Rational(1,2).add(new Rational(2,3))
scala> new Rational(1,2) add new Rational(2,3)

def +(that: Rational): Rational =
new Rational(numer * that.denom + that.numer * denom,
denom * that.denom)

def *(that: Rational): Rational =
new Rational(numer * that.numer, denom * that.denom)

scala> val d = a + b * c
d: Rational = 11/14
Operator Identiﬁers
scala> val d = a.+(b.*(c))
d: Rational = 11/14
scala> val d = a * b + c
d: Rational = 16/21
scala> val d = (a.*(b)).+(c)
d: Rational = 16/21

Invoking Operators

How many Rational objects are created while executing:

require(d != 0)
val numer: Int = n
val denom: Int = d
def +(that: Rational): Rational =
new Rational(
numer * that.denom + that.numer * denom,
denom * that.denom)
}
var half = new Rational(1,2)
half = half + half + half

(a) 1
(b) 2
(c) 3
(d) 4 Quiz

Alphanumeric identifier

• identifier: [$A-Za-z_][$A-Za-z_0-9]* ($ reserved for Scala compiler)

• camel-case convention: toString, HashSet

Operator identifier

• Unicode set of mathematical symbols(Sm) or other symbols(So), or
to the 7-bit ASCII characters that are not letters, digits,
parentheses, square brackets, curly braces, single or double quote,
or an underscore, period,semi-colon, comma, or back tick
character.

Literal Identifier
Identiﬁer Syntax
• arbitrary string enclosed in back ticks (` . . . `).

Method Overloading


def *(i: Int): Rational =
new Rational(numer * i, denom)

scala> val c = new Rational(3,7)
c: Rational = 3/7
In a method call, the
scala> c * 2
compiler picks the version
of an overloaded method
that correctly matches the
types of the arguments.

Method Overloading
does not apply to this



scala> 2 * c
<console>:7: error: overloaded method value * with alternatives:
(Double)Double <and> (Float)Float <and> (Long)Long <and> (Int)Int <and>
(Char)Int <and> (Short)Int <and> (Byte)Int cannot be applied to (Rational)
2 * c
^

Implicit Conversions



implicit def intToRational(x: Int) = new Rational(x)

scala> 2 * c

Functional Objects - Summary

Immutable objects

• class parameters
• immutable fields (val)
• methods don’t change object, but return value
Natural, concise notation

• methods as infix operators, operator identifiers
• method overloading
• implicit conversion

e0 match {
case 1 => e1;
case 2 => e2; Scala’s Match
case _ => e3
}

Match is similar to Java switch. Differences:

• Match is expression: returns a value
• Alternatives never fall through
• MatchError when no pattern matches

val firstArg = if (args.length > 0) args(0) else ""
firstArg match {
case "salt" => println("pepper")
case "chips" => println("salsa")
case "eggs" => println("bacon")
Choosing between Actions
case _ => println("huh?")
}

val firstArg = if (!args.isEmpty) args(0) else ""
val friend =
firstArg match {
case "salt" => "pepper"
case "chips" => "salsa"
case "eggs" => "bacon"
case _ => "huh?" Choosing between Values
}
println(friend)

def describe(x: Any) = x match {
case 5 => "five"
case true => "truth"
case "hello" => "hi!" Constant Patterns
case Nil => "the empty list"
case _ => "something else"
}

expr match {
case 0 => "zero"
case somethingElse => "not zero: " + somethingElse
}

Pattern Variables

Inductive Deﬁnitions

Natural number

• 0 is a number
• if n is a number then n + 1 is a number
• nothing else is a natural number

def property(n: Int): Boolean =
n match {
case 0 => // base case
case m => ... property (m - 1) ...
// recursive case for n + 1
}

def f(n: Int): Int =
n match {
case 0 => // base case
case m => ... f(m - 1) ...
// recursive case for n + 1
}

Induction Principle

def isEven(n: Int): Boolean =
n match {
case 0 => ?
case m => ?
}

def isOdd(n: Int): Boolean =
n match {
case 0 => ?
case m => ?
}

def isEven(n: Int): Boolean =
n match {
case 0 => true
case m => isOdd(m - 1)
}

def isOdd(n: Int): Boolean =
n match {
case 0 => false
case m => isEven(m - 1)
}

def power(n: Int, exp: Int): Int =
exp match {
case 0 => ?
case m =>
?

}

def power(n: Int, exp: Int): Int =
exp match {
case 0 => 1
case m =>
if(exp % 2 == 1)
n * power(n, m - 1)
else
power(n * n, m / 2)
}

Inductive Data Structures

Natural number

• 0 is a number
• if n is a number then n + 1 is a number
List

• Empty list is a list
• If L is a list then adding an element in
front of L produces a list

abstract class IntList

case class Nil() extends IntList

// Nil() is a list (the empty list)

case class Cons(hd: Int, tail: IntList) extends IntList

// if hd is an Int and tail is an IntList
// then Cons(hd, tl) is an IntList

Nil()
Cons(1, Nil()) Scala: Case Classes
Cons(2, Cons(1, Nil()))
Cons(1, Cons(2, Cons(3, Nil())))
...

abstract class IntList
case class Nil() extends IntList
case class Cons(hd: Int, tail: IntList) extends IntList

def f(xs: IntList): T =
xs match {
case Nil() => // base case
case Cons(x, ys) => ... f(ys) ...
// recursive case
}

Induction Principle for IntList

length of list xs

def length(xs: IntList): Int =
xs match {
case Nil() => ?
case Cons(x, ys) => ?
}

length of list xs

def length(xs: IntList): Int =
xs match {
case Nil() => 0
case Cons(x, ys) => 1 + length(ys)
}

sum of the integers in xs

def sum(xs: IntList): Int =
xs match {
case Nil() => ?
}

sum of the integers in xs

def sum(xs: IntList): Int =
xs match {
case Nil() => 0
case Cons(x, ys) => x + sum(ys)
}

def product(xs: IntList): Int =
xs match {
case Nil() => ?
}

product of the integers in xs

def product(xs: IntList): Int =
xs match {
case Nil() => 1
case Cons(x, ys) => x * product(ys)
}

product of the integers in xs

def append(xs: IntList, ys: IntList): IntList =
xs match {
case Nil() => ?
case Cons(x, zs) => ?
}

append elements of ys to xs

def append(xs: IntList, ys: IntList): IntList =
xs match {
case Nil() => ys
case Cons(x, zs) => Cons(x, append(zs, ys))
}

append elements of ys to xs

elements of xs in reverse

def reverse(xs: IntList): IntList =
xs match {
case Nil() => ?
case Cons(y, ys)
=> ?
}

elements of xs in reverse

xs match {
case Nil() => Nil()
case Cons(y, ys)
=> append(reverse(ys), Cons(y, Nil()))
}

xs match {
case Nil() => Nil()
case Cons(y, ys)
=> append(reverse(ys), Cons(y, Nil()))
}

reverseAcc(xs, Nil())

def reverseAcc(xs: IntList, rest: IntList): IntList =
xs match {
case Nil() => rest
case Cons(y, ys)
=> reverseAcc(ys, Cons(y, rest))
}

reverse in linear time using accumulator

the list of elements of xs that are even

def filterEven(xs: IntList): IntList =
xs match {
case Nil() => Nil()
case Cons(y, ys)
=> if(y % 2 == 0) Cons(y, filterEven(ys))
else filterEven(ys)
}

def insert(x: Int, xs: IntList): IntList =
xs match {
case Nil() => Cons(x, Nil())
case Cons(y, ys) => if(x < y) Cons(x, Cons(y, ys))
else Cons(y, insert(x, ys))
}

def isort(xs: IntList): IntList =
xs match {
case Nil() => Nil()
case Cons(y, ys) => insert(y, isort(ys))
}

sort elements in ascending order

def msort(xs: IntList): IntList = { Merge Sort
val n = lenght(xs) / 2
n match {
case 0 => xs
case _ => val (as, bs) = splitAt(xs, n)
merge(msort(as), msort(bs))
}
}

def splitAt(xs: IntList, n: Int): (IntList, IntList) =
xs match {
case Nil() => (Nil(), Nil())
case Cons(y, ys) => n match {
case 0 => (Nil(), xs)
case _ =>
val (as, bs) = splitAt(ys, n - 1)
(Cons(y, as), bs)
}
}

Merge Sort

define merge(xs: IntList, ys: IntList): IntList =
xs match {
case Nil() => ys
case Cons(a, as) =>
ys match {
case Nil() => xs
case Cons(b, bs) =>
if(a < b) Cons(a, merge(as, ys))
else Cons(b, merge(xs, bs))
}
}

abstract class BinTree

case class Empty() extends BinTree

case class Node(left: BinTree,
value: Int,
right: BinTree) extends BinTree

Empty()

Node(Empty(), 42, Empty())

Node(Empty(), 5,
Node(Empty(), 42, Empty()))
def f(t: BinTree): A =
t match { Node(Node(Empty(), 2, Empty()),
5, Node(Empty(), 42, Empty()))
case Empty() => ...
case Node(t1, i, t2) =>
... f(t1) ... f(t2) ...
}

def replace(x: Int, y: Int, t: BinTree): BinTree =
t match {
case Empty() => ?
case Node(l, n, r) =>
?

}

replace occurrence of x by y

def replace(x: Int, y: Int, t: BinTree): BinTree =
t match {
case Empty() => Empty()
Node(
replace(x, y, l),
if(n == x) y else n,
replace(x, y, r)
)
}

replace occurrence of x by y

def toList(t: BinTree): IntList = t match {
case Empty() => ?
?
}

transform binary tree to list

def toList(t: BinTree): IntList = t match {
case Empty() => List()
append(toList(l), Cons(n, toList(r)))
}

transform binary tree to list

Binary Search Trees

Invariant:

Node(t1, n, t2)
- all numbers in t1 are smaller than n
- all numbers in t2 are larger than n

Functions:
def insert(x: Int, t: BinTree): BinTree
def lookup(x: Int, t: BinTree): Boolean

Correctness
lookup(x, insert(x, t)) == true

Lookup in Binary
Search Tree

def lookup(x: Int, t: BinTree): Boolean = {
t match {
case Empty() => ?
case Node(left, value, right) =>
?

}
}

Lookup in Binary
Search Tree

def lookup(x: Int, t: BinTree): Boolean = {
t match {
case Empty() => false
if(x < value)
lookup(x, left)
else if (x > value)
lookup(x, right)
else true
}
}

Insert in Binary
Search Tree

def insert(x: Int, t: BinTree): BinTree = {
t match {
case Empty() => ?
?

}
}

Insert in Binary
Search Tree

def insert(x: Int, t: BinTree): BinTree = {
t match {
case Empty() => Node(Empty(), x, Empty())
if(x < value)
Node(insert(x, left), value, right)
else if(x > value)
Node(left, value, insert(x, right))
else t
}
}

Insert in Binary
Search Tree

def toBinTree(xs: IntList): BinTree = xs match {
case Nil() => Empty()
case Cons(y, ys) => insert(y, toBinTree(ys))
}

def listSort(xs: IntList): IntList =
toList(toBinTree(xs))

Algebraic Datatypes in C

Based on: Simple algebraic data types for C by Pieter
Hartel and Henk Muller. Version 8, 2nd September, 2010

Algebraic Data Types in Scala

abstract class Tree
case class Leaf(v: Int)
case class Branch(v: Int, left: Tree, right: Tree)

def sum(t: Tree): Int = t match {
case Leaf(v) => v
case Branch(v, left, right) => v + sum(left) + sum(right)
}

ADT K&R Style

typedef struct tree_struct {
int val;
struct tree_struct *left;
struct tree_struct *right;
} tree;

tree *mkBRANCH(int val, tree *left, tree *right) {
tree *result = calloc(1, sizeof(struct tree_struct));
if (result == NULL) {
printf("panicn");
}
result->val = val;
result->left = left;
result->right = right;
return result;
}

int krsum1(tree *cur) { ADT K&R Style
if (cur == NULL) {
return 0;
} else {
return cur->val + krsum1(cur->left) + krsum1(cur->right);
}
}
int krsum2(tree_cc *cur) {
/*assert cur->left==NULL <==> cur->right==NULL*/
if (cur->left == NULL) {
return cur->val;
} else {
return cur->val + krsum1(cur->left) + krsum2(cur->right);
}
}
void test() {
tree *r = mkBRANCH(30, NULL, NULL);
tree *l = mkBRANCH(20, NULL, NULL);
tree *t = mkBRANCH(10, l, r);
printf("%dn", krsum1(t));
}

ADT K&R Style

No explicit cases for Leaf and Branch

• distinction by use of NULL values for branches

• Does not cater for more alternatives

Confusion about NULL

• uninitialized value

• end-of-list, leaf-of-tree

• => errors

Explicit allocation of tree nodes

Algebraic Data Type with Union

typedef enum {
LEAF = 0, BRANCH = 1
} tree_tag;

typedef struct tree_struct {
tree_tag tag;
char *filename;
union {
struct {
int _val;
} _LEAF;
struct {
int _val;
struct tree_struct *_left;
struct tree_struct *_right;
} _BRANCH;
} data;
} tree;

Algebraic Data Type with Union

tree *mkLEAF(int _val) {
printf("panicn");
}
result->tag = LEAF;
result->data._LEAF._val = val;
return result;
}

tree *mkBRANCH(int val, tree *left, tree *right) {
printf("panicn");
}
result->tag = BRANCH;
result->data._BRANCH._val = val;
result->data._BRANCH._left = left;
result->data._BRANCH._right = right;
return result;
}

Recursive Functions on ADT

int sum(tree *t) {
if (t == NULL) {
return 0;
} else if (t->tag == LEAF) {
return t->data._LEAF._val;
} else { // t->tag == BRANCH
return t->data._BRANCH._val
+ sum(t->data._BRANCH._left)
+ sum(t->data._BRANCH._right);
}
}

Reading & Programming in Week 5

Reading

Sebesta Chapter 6:

Scala Chapter 6: Functional Objects

Scala Chapter 15: Case Classes and Pattern Matching

WebLab:
C, JavaScript, Scala tutorials
Graded Assignment 1: Dynamic Dispatch in C
(deadline 2 April 2013, 23:59)

Week 6: First-Class Functions

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