# Software Foundations 558

## meta stuff

• Programming in Haskell can read in one sitting, intro to prog
• Introduction to Functional Programming part math, part prog more in depth, prove correctness etc…
• Practical Foundations for Programming Languages online notes, can serve as alternative to textbook
• Programming Language Pragmatics undergrad PL and compiler textbook
• Real World Haskell Orielly, in case we actually wanted to write real programs in Haskell

### homework hand in

Turn In

• There is a turn-in mechanisms through cssupport.
• electronic through CS machines (need cs account)

### exams

• 2 midterms
• 1 half way through content
• 1 cumulative final

#### DONE 1st midterm 2009-10-01 Thu

covers programming and reasoning in Haskell

• present this data structure or algorithm in Haskell
• may be whole period may just be parts of the period

preparation

• work through examples

non-cumulative

## class notes

### 2009-08-25 Tue [2/2]

• OO
• imperative
• most features of current programming languages
• possibly more
• [X] get text book

### 2009-08-27 Thu

#### sum integers

C: creates the location sum in memory, then modifies it 11 times

sum = 0;
for (i=1, i<11, 1++)
{
sum = sum + 1;
}


Haskell: no store, initialization, variables, etc…

sum [1..10]


sum over all integers in Haskell

sum [0..]


#### QuickSort

let f [] = []
let f (x:xs) = f ys ++ [x] ++ f zs
where
ys = [a | a <- xs, a <= x]
zs = [b | b <- xs, b > x]


this is much clearer in functional language, but would be difficult to specify an in-place QuickSort. see "persistent data structures v.s. ephemeral data structures" for more on this

#### expressions

Prelude: contains basic mathematical expressions (sqrt, +, etc…), and is located in Prelude.hs

common functions in the prelude

Prelude> head [2,3,5,1]
2
Prelude> tail [2,3,5,1]
[3,5,1]
Prelude> [2,3,5,1] !! 2
5
Prelude> take 3 [2,3,5,1]
[2,3,5]
Prelude> drop 3 [2,3,5,1]
[1]
Prelude> length [2,3,5,1]
4
Prelude> sum [2,3,5,1]
11
Prelude> product [2,3,5,1]
30
Prelude> [1,2,3] ++ [2,3,5,1]
[1,2,3,2,3,5,1]
Prelude> reverse [2,3,5,1]
[1,5,3,2]


#### function application

• in Java f(x)
• in math f(x)

f x + b -> f(x) + b

style in Haskell is to minimize parenthesis

Curried functions

functions always take a single argument, so in take 2 [2,5,3] the portion take 2 returns a function which is then applied to the list.

f x y -> (f(x))(y)

#### program

let double x = x + x
let quadruble x = double (double x)


in the interactive session you'd load a file with

Prelude> :l example.hs


#### pattern matching

pattern matching happens from top to bottom

let fac 0 = 1
let fac n = n * fac(n - 1)


so in the above example 0 matches first.

also guards (same as such that) can be used to be more explicit

let fac n
| n > 0 = n * fac(n - 1)
| n = 0 = 1


### 2009-09-01 Tue

#### technical questions

undefined is a valid token in Haskell code.

Prelude> undefined
*** Exception: Prelude.undefined


if your guards for a function don't cover some argument, then the result for that argument is undefined

fact n
| n >= 1 = n * fact(n - 1)
| n == 0 = 1
-- is equal to
fact n
| n >= 1 = n * fact(n - 1)
| n == 0 = 1
| n < 0  = undefined


write all function definitions in files

#### more Haskell features / syntax

• when you want to use a curried function infix, then you must surround it in back quotes
average xs = sum xs div length xs
-- or
average xs = div (sum xs) (length xs)


funny ns = sum ns + length ns
-- or
funny ns = (+) (sum ns) (length ns)


• indentation matters, code is 2-D (like python), however there are 1-D options for all expression, and in fact these 1-D options are the basic or core of Haskell which the complex 2-D layout stuff is translated to.
• the !! notation is used to index into a list
• single line starts with --
• multi line functions use {- -} notation

#### types

• can be explicit about types
• :: means has the type
• all types are capitalized
False :: Bool
'5' :: Char
5 :: Int
"a" :: String
"a" :: [Char]
[False, True] :: [Bool]



we will use type declarations for every function which we will write

types

Int
machine size integers
Integer
integers of arbitrary size
Float
numbers with decimals
Bool
boolean
Char
characters
String
is the same as a list of characters [Char]
uppcase :: String -> String
uppcase str = map f s
where
f :: Char -> Char
f 'a' = 'A'
f 'b' = 'B'
...
f c = c


### 2009-09-03 Thu

#### question: executable with main

module Main where
fac 0 = 1
fac n = n * fac(n - 1)
main =
putStrLn "hello"


#### types

[False, False] :: [Bool]
[1..] :: [Integer]

tuple
is like a record, can contain multiple values of different types. tuples can be any size but it's size if fixed.
(False, True) :: (Bool, Bool)
(False, 'a', 1) :: (Bool, Char, Int)


div 5 9
• div 5 returns a function which divides 5 by it's argument. the type of div is div :: Integer -> (Integer -> Integer)
->
we just say that the -> constructor associates to the right, so we don't have to write the parenthesis as above.

by having every function take exactly one argument, Haskell and ML avoid the complexities of needing to have having apply operators like in lisp

when passing complex arguments to functions you can use pattern matching to automatically deconstruct the complex argument

f (b,n) = b + n

polymorphic functions
can take multiple types of arguments
length :: [Integer] -> Integer
-- or
length :: [Char] -> Integer


type variables
can be used to represent multiple types
length :: [a] -> Integer
-- or
take :: Integer -> [a] -> [a]
fst :: (a,b) -> a
id :: a -> a


when you want multiple types, but not all types you basically are overloading the function (defining it multiple times for multiple argument types)
type classes
Haskell manages overloading by defining classes of multiple types. see the prelude for type classes. type class constraints can be placed in front of type class constructions
product :: Num a => [a] -> a
+ :: Num a => a -> a -> a


Num
numerical types
Eq
equality types
Ord
ordered types

### 2009-09-08 Tue

#### pattern matching

abs :: Int -> Int
abs n
| n < 0 = -n
| otherwise = n

not :: Bool -> Bool
not False = True
not True = False


wildcard _

(and) :: Bool -> Bool -> Bool
True and True = True
_ and _ = False


note that the above and clause is not short-circuiting, while the next one is

(and) :: Bool -> Bool -> Bool
True and b = b
False and _ = False


#### lazy evaluation

allows the use of infinite data structures

[1..] is the infinite list

lists use the : cons operator

[1, 2, 3]

#### anonymous functions lambda (written \)

(\ x -> x + 2) 4 -- is equal to 6


useful when combined with map

firstodss n = map (\ x -> x + 1) [0..n-1]


#### sections (partial infix operators)

shorthand for lambda extractions

-- the following are all equivalent
1 + 2
(+) 1 2
(1+) 2
(+2) 1


can be handy

map (*2) [1, 2, 3, 4]


#### list comprehension

[x^2 | x <- [1..5]]
-- is equal to
map (^2) [1..5]


more interesting list comprehension

-- multiple generators
[(x,y) | x < [1, 2, 3], y <- [4,5]]
-- triangular
[(x,y) | x <- [1, 2, 3], y <- [x..3]]

concat
flattening function for lists
concat xss = [[x] | xs <- xss, x <- xs]



guards

[[x] | x <- [1..10], even x]


#### finding primes

factors n = [[x] | x <- [1..n], n mod x == 0]
prime n = factors n == [1,2]
primes n = [[x] | x <- [2..n], prime x]


### 2009-09-10 Thu

continue where we left off (more examples)

#### zip

combines lists

zip : [a] -> [b] -> [(a, b)]
zip [] _  = []
zip _ []  = []
zip [a] [b] = [(a,b)]
zip (a:as) (b:bs) = (a,b) : (zip as bs)


turn a list into a list of pairs

pairs [a] -> [(a, a)]
pairs xs = zip xs (tail xs)

sorted :: [Ord] -> Bool
sorted [x] = True
sorted xs = foldl (and) True (map (\ x y -> if (x < y) True else False) (pairs xs))
-- more concisely
sorted xs = [ x <= y | (x,y) <- pairs xs]


#### count lower-case letters in string

numlower :: String -> Int
numlower str = length [x | x < str, isLower x]


#### recursive list functions

product Num a => [a] -> a
product [] = 1
product (x : xs) = x * (product xs)

length :: [a] -> Int
length [] = 0
length (_:xs) = 1 + (length xs)

reverse [a] -> [a]
reverse [] = []
reverse (x:xs) = (reverse xs) ++ [x]


how fast is reverse… I'd think its O(n)

reverse [1, 2, 3]
(reverse [2,3]) ++ [1]
((reverse [3]) ++ [2]) ++ [1]
(((reverse []) ++ [3]) ++ [2]) ++ [1]
(([] ++ [3]) ++ [2]) ++ [1]
([3] ++ [2]) ++ 1
[3, 2] ++ [1]
[3, 2, 1]


++ must traverse the list, so it doesn't run in unit time, so we run in n2 time

maybe

reverse [a] -> [a]
reverse xs = foldl (++) [] xs

concat :: [[a]] -> [a]
concat xss = [x | xs <- xxs, x <- xs]
-- or recursively
concat [] = []
concat (x:xs) = x ++ (concat xs)

replicate :: Int a -> [a]
replicate 0 _ = []
replicate n x = x : (replicate (n - 1) x)

(!!) :: [a] -> Int -> a
(x:xs) !! 0 = x
(x:xs) !! n = xs !! (n - 1)

elem :: Eq a => a -> [a] -> Bool
elem _ [] = False
elem x (y:ys) = (x == y) or (elem x ys)


What types would not be comparable? Functions

map :: (a -> b) -> [a] -> [b]
map f xs = [f x | x <- xs]
-- now recursively
map f [] = []
map f (x:xs) = f x : map f xs

filter :: (a -> Bool) -> [a] -> [a]
filter f xs = [x | x <- xs, fx]
-- recursively
filter f [] = []
filter f (x:xs)
| p x = x : filter xs
| otherwise = filter xs


### 2009-09-17 Thu

#### our own types, and their classes

How to define types and make them instances of type classes

In standard Haskell it is not possible to have a type which is dependent on a value.

Type Synonym
do not introduce new types
type Point = (Float, Float)


New Type
does introduce a new type
data Point2D = Point2D (Float, Float)



this actually does introduce a new type, which we can use

pointA, pointB :: Point2D
pointA = Point2D(2.0, 3.0)
pointB = Point2D(3.0, 4.0)
-- we can then define functions on these types
distance (Point2D(x1, y1)) (Point2D(x2, y2)) = sqrt((x1 - x2)^2 + (y1-y2^2))



the new Point2D syntactic element is a tag which has the type of a function from a pair of Floats to the type Point2D. It is not necessary for the tag and the type to have the same name.

types w/variants
a type which has multiple constructors
data Point2D = Rect (Float, Float)  -- (x,y)
| Polar (Float, Float) -- (theta, r)
-- we can then use these types
pointC :: Point2D
pointC = Polar(2.0, 1.5)
-- and define functions
distance(Rect(x1, y1))(Rect(x2, y2))   = sqrt((x1 - x2)^2 + (y1-y2^2))
distance(Polar(x1, y1))(Rect(x2, y2))  =
distance(Rect(x1, y1))(Rect(x2, y2))   = distance(Polar(x1, y1))(Rect(x2, y2))
distance(Polar(x1, y1))(Polar(x2, y2)) =


recursive data types
for real fun, we need to recurse
data BnIntTree = Leaf Int
-- normally in haskell curried functions are prefered to tuples
| Node BnIntTree BnIntTree



and to use said structure…

tree1 :: BinIntTree
tree1 = Leaf 1
tree123 :: BinIntTree
tree123 = Node (Node (Leaf 1) (Leaf 2)) (Leaf 3)



for more stuff…

-- lets count our leaves
countLeaves :: BinIntTree -> Int
countLeaves (Leaf n) = 1
countLeaves (Node t1 t2) = (countLeaves t1) + (countLeaves t2)
-- lets add our leaf values
-- left to right traversal
traverseTreeLeft :: BinIntTree -> [Int]
traverseTreeLeft (Leaf n) = [n]
traverseTreeLeft (Node t1 t2) = (traverseTreeLeft t1) ++ (traverseTreeLeft t2)
-- right to left traversal
traverseTreeRight :: BinIntTree -> [Int]
traverseTreeRight (Leaf n) = [n]
traverseTreeRight (Node t1 t2) = (traverseTreeRight t2) ++ (traverseTreeRight t1)
-- map some function across a tree
mapBinIntTree :: (Int -> Int) -> BinIntTree -> BinIntTree
mapBinIntTree f (Leaf n) = Leaf (f n)
mapBinIntTree f (Node t1 t2) = Node (mapBinIntTree f t1) (mapBinIntTree f t2)
-- height
heightBinIntTree :: BinIntTree -> Int
heightBinIntTree (Leaf _) = 0
heightBinIntTree (Node t1 t2) = 1 + (heightBinIntTree t1 max heightBinIntTree t2)


new tree data type
abstract arithmetic syntax tree
data ArExp = Mul ArExp ArExp
| Number Num



now to evaluate said expressions

evaluate :: ArExp -> Int
evaluate (Mul e1 e2) = evaluate e1 * evaluate e2
evaluate (Add e1 e2) = evaluate e1 + evaluate e2
evaluate Number n = n



pretty print

-- just do the pattern above only construct/concat strings


### 2009-09-22 Tue

parameters to type constructors

another way to define Point2D

type Pair a = (a,a)
type Point2D = Pair Float


can also do this with data (remember data actually defines a new structure while type defines something more like an alias)

data BinIntTree = Leaf Int
| Node BinIntTree BinIntTree
-- or can be written as
data BinTree a = Leaf a
| Node (BinTree a) (BinTree a)
type BinIntTree = BinTree Int


if we wanted to define lists

data MyList a = Empty
| Nonempty a (MyList a)
-- to construct values of this type
list1 :: MyList Char
list1 = Nonempty 'r' (Nonempty 'e' (Nonempty '1' Empty))


lists in the prelude (same functionality, different syntax)

-- this isn't really legal haskell
data [] a = []
| a : ([] a)
-- to construct values of this type
list2 :: [Char]
list2 = 'r' : ('e' : ('1' : []))


#### foldr abstract the structure or recursive list traversal

foldr is a higher order function to hold this common form

f [] = v
f (x:xs) = x (+) xs


so

sum = foldr (+) 0
product = foldr (*) 1
and = foldr (&&) True


type of foldr

foldr :: (a -> b -> b) -> b -> [a] -> b


when list is thought of as a series of cons'd lists, then foldr can be thought of as

• replacing the cons with it's function
• replacing the empty list with it's base value
• length using foldr
length = foldr (\ _ a = 1 + a) 0
-- or more satisfyingly
length = foldr (const (1+)) 0


• reverse using foldr
reverse = foldr (\ x r -> r ++ [x]) []



#### function composition

even :: Int -> Bool
not :: Bool -> Bool
-- lets use the above to define the below
odd :: Int -> Bool
odd n = not (even n)
-- even better (using function composition)
odd = not . even


this last example is an instance of the point free style of definition

to define .

(f . g) a = f (g a)
(f . g) = (\ a -> f (g a))


### 2009-09-24 Thu

#### some more higher order functions…

map
applies function to list
filter
filters list on some function

express list comprehension using map and filter

[f x | x <- xs, p x]
-- in terms of map and filter
(map f . filter p) xs -- could be point free


map :: a -> b -> [a] -> [b]
map f = foldr accum (\ x accum -> (f x):accum) []
-- or more point free
map f = foldr (\ x -> ((fx):)) []
-- can the above be simplified to eliminate all lambdas

-- my first guess
filter p = foldr (\ x -> ((if (p x) then x else []) ++)) []
-- better
filter p = foldr (\ x -> (if (p x) then (x :) else id)) []
-- if we hate if-then-else
cond p f g = if (p x) then (f x) else (g x)
filter p = foldr (cond p (:) (\ _ -> id)) []
filter p = foldr (cond p (:) (const id)) [] -- lookup const


algebra on functions like the above is an example of "eta-expansion" or "eta-reduction"

back to map

map = foldr (\ x -> ((f x):)) []
map = foldr ((:) . f) []


#### equational reasoning (of the map redefinition above)

for algebraic program transformation

1. first we calculate y
 y: by definition (:) y by two eta expansions (\a -> \b -> a:b) by function application (beta-reduction) \b -> y:b
2. in particular \r -> fx:r = (fx:) where y is fx
 (\x -> (fx:)) a b by function application (fa:) b by def of sections (:) (fa) b make parenthesis explicit ((:) (fa)) b by def. of function composition (((:) . f) a) b dropping parens (: . f) a b

### 2009-09-29 Tue

a Haskell is a set of defining equations

#### lazy evaluation

(for now) use instead of meaning

infinite list
(e.g. [6..] is list of 6 to infinity). to practically use an infinite list we must only deal with some finite portion of the list, this can be done using the take prefix operator, or the !! index operator.
-- create an infinite list of factorials
factorials = map fact [0..]
-- first 5 factorial values
take f factorials
-- factorial of 9
factorials !! 9


w/list comprehension
can create infinite lists with list comprehension
-- squares
squares = [x^2 | x <- [0..]]
-- powers of an integer
powers :: Integer -> [Integer]
powers n = [2^x | x <- [0..]]
-- using a nice build in function iterate'
powers n = iterate (p*) 1


iterate
how to write iterate
iterate :: (a -> b) -> a -> [b]
iterate f x = x : iterate f (f x)


primes
primes :: [Integer]
primers = [p | p <- [2..], prime p]
where
prime :: Integer -> Bool
prime n = null (nonTrivialFactors n)
where
nonTrivialFactors n = filter (\ x -> n mod x == 0) [2..n-1]


so now what is this
it is an infinite list of infinite lists.
map powers primes


• how do we access elements of this list w/o evaluating the entirety of it's predecessors?
example
example working with infinite lists
ints = 1 : map (1+) ints



#### odds and ends

• literate programming
a program is a work of literature

-- comment
{- literate comment -}


.lhs files can be ingested by all Haskell compilers. These files can be constructed in the following files

Bird-style
all text is considered to be a comment, to write code first input an empty line following by code lines prefixed with "> ", the code is then ended with another empty line
LaTeX-style
main body is Latex code, to write code wrap it in a \begin{code}...\end{code} LaTeX environment
lhs2TeX
this is a preprocessor for the LaTeX-style of .lhs files, that can be used to make the haskell code look more like math if desired (i.e. converting -> to arrows, Greek letters to Greek, etc…).

From now on we will be required to submit our code in a literate programming style.

• fold for generic data types
Arithmetic Data Type
data Expr = Num Int
| Mul Expre Expr
eval :: Expr => Int
eval (Num n) = n
eval (Mul e1 e2) = eval e1 * eval e2

-- now to define fold for the Expr type
foldExpr (Int -> a) -> (a -> a -> a) -> Expr -> a
foldExpr f g (Num n) = f n
foldExpr f g (Mul e1 e2) = foldExpr e1 g foldExpr e2
-- or using case
foldExpr fNum fMul e =
case e of
Num n -> fNum n
Mul e1 e2 -> foldExpr e1 fMul foldExpr e2

-- evaluating with our combining function
eval = foldExpr (id) (*)
eval = foldExpr (\ (Num n) -> n) (\ (Mul e1 e2) -> eval e1 * eval e2)

-- printing with the combining function
to_string = foldExpr show (\ s1 s2 -> "("++s1++" * "++s2++")")


note that this can be done automatically in generic Haskell (feel free to investigate extensions that provide this, but don't use them as they're not part of the standard)

using data genericity certain functions are defined automatically whenever a data structure is defined.

### 2009-10-06 Tue

homework

 median 100 average 94

midterm

 median 40 average 42

#### notes/topics from homework/midterm

• homework problem 2
data Expr = Num Integer
| Let {var :: String, value :: Expr, body :: Expr}

-- same as
data Expr = Num Integer
| Let String, Expr, Expr

• midterm problem 1
jj
where
j x = j x

• type is a
• value diverges
f (2, [1, 2, 3])
where
f (n, x:xs) = f (n-1, xs)
f (0, xs) = xs

• type is [Int]
• value is fails, because eventually the list is empty and that case isn't matched (because when the list is empty n doesn't equal 0)
• midterm problem 3

my solution which was wrong

inits [] = [[]]
inits (x:xs) = (\ rest -> rest ++ (head (reverse rest) : [x])) (inits xs)


in general on tests, you should evaluate your function by hand on a small input to ensure that the mechanics work

• midterm problem 4
keep around the depth information in the same way that you would have in C?

### 2009-10-08 Thu

#### homework questions

question 2.2.4 (the assembler)
turn a large string (including newlines) into a program, the large string will look like the example in 2.2.5 (i.e. the output of "the disassembler" from question 2.2.3)

the homework introduces some new concepts

IO
we need to learn a little bit about IO, but don't need to understand monads
PRINT
the PRINT statements will return a type IO() all of which must be accumulated during the course of executing the program

#### reasoning about functions & data structures

• pairs (a,b)
-- take the first of a pair
fst :: (a, b) -> a
fst (x, _) = x
-- take the second of a pair
snd :: (a, b) -> b
fst(_, y) = y

pair
no special function is needed to create a pair, rather the function pair will be used to do something more subtle (apply a pair of functions to a value).
pair :: (a -> b, a -> c) -> a -> (b, c)
pair (f, g) x = (f x, g x)


cross
like pair but applies a pair of functions to a pair of values
cross :: (a -> c, b -> d) -> (a, b) -> (c, d)
cross (f, g) (x, y) = (f x, g y)



can we define cross in terms of pair? or course we can.

cross (f, g) = pair (f . fst, g . snd)


now lets prove some properties of pair and cross

prove fst . pair (f, g) = f
in the calculational proof style
-- being painfully verbose
(fst . pair (f, g)) x
== {- definition of function composition -}
fst (pair (f, g) x)
== {- definition of pair -}
fst (f x, g x)
== {- definition of fst -}
f x



we have established, by calculation, that for any arbitrary x

fst . pair (f, g) x == f x



By extensionality

fst . pair (f, g) == f


• pair exercises
prove that snd . pair (f, g) = g
we know that ∀ x
-- snd . pair (f, g) = g
pair (f, g) x == (f x, g x)  -- by the definition of pair
snd (f x, g x) == g x -- by the definition of snd
-- so for any x
(snd . pair (f, g)) x == g x


prove that pair (f, g) . h = pair (f . h, g . h)
x
-- pair (f, g) . h = pair (f . h, g . h)
(pair (f, g) . h) x == pair (f, g) (h x)    -- by the definition of (.)
pair (f, g) h x == (f (h x), g (h x))       -- by the definition of pair
(f (h x), g (h x)) == (f . h x, g . h x)    -- by the definition of (.)
(f . h x, g . h x) == pair (f . h, g . h) x -- by the definition of pair


prove that cross (f, g) . pair (h, k) = pair (f . h, g . k)

x

-- cross (f, g) . pair (h, k) = pair (f . h, g . k)
cross (f, g) . pair (h, k) x
-- definition of pair
== cross (f, g) . (h x, k x)
--


???
prove that cross (f, g) . cross (h, k) = cross (f . h, g . k)
???
what are the functions pair and cross good for?
• laws of map
Laws of map
1. map id = id, we must be careful here as the second id is an id over lists
2. map f . g = map f . map g so for example
map (square . succ) [1, 2] == [4, 9]



so the left hand side of this rule is one list traversal of a complex function (square . succ) and the right hand side is two list traversals applying two simpler functions succ and square. Note that the compiler may prefer the left hand side to the right hand side because the right hand side requires 2 list traversals (time) and an intermediate data structure (space).

3. map f . tail = tail . map f so map f and tail commute
4. f . head = head . map f
5. map f . reverse = reverse . map f
6. map f (xs ++ ys) = map f xs ++ map f ys
7. map f . concat = concat map (map f)
• filter
1. filter p . concat = concat . map (filter p)
2. filter p . filter q = filter (\ x -> p x && q x)
• typing exercise
what are these types?
map (map square) -- [[1, 2], [3, 4]] -> [[1, 4], [9, 16]]
-- what is the type of this equation?
-- we need to know the type of square
map (map square) :: [[Int]] -> [[Int]]
-- and
map square :: [Int]

map map :: [a -> b] -> [[a] -> [b]]


to explain the above just cram the entire type of map into the (a -> b) portion of the type of map

### 2009-10-13 Tue

Depak is lecturer

#### unification

Unification (or equation solving or constraint solving) this topic comes from Heibrand's PhD thesis – proving first order predicate calculus is complete – then "rediscovered" by Pravitz and Robinson (resolution principle, initially exponential algorithm, eventually quadratic, by others down to n log(n) or n α(n)).

α(n)
inverse of the Ackerman function

Universe

• constants
• variables
• function symbols (making no assumptions of properties)
• terms/expressions (combination of the above)

given finite set of simultaneous equations (composed of terms) we will

1. determine if solvable (or unifiable), if ∃ a substitution σ (values for variables) s.t. when the variables are applied they become equal (unifier)
2. if so find the most general solution (most general unifier)

possibilities

1. not solvable
2. solvable with single solution
3. solvable with ∞ solutions

depending on the nondeterministic choices (which equation/variable to solve first) it is possible (in the case of ∞ solutions) to arrive at syntactically different solutions which will still be equivalent.

operations on the system of equations must be both

sound
complete
no operation removes possible solutions

basically you should be operating equally on either side of the equality relation in the equations

#### solvability when dealing with terms and expressions

No Solutions

1. two functions which are not equal
2. two functions which are not equal
3. can't be equal as x is on both sides

Solutions

1. solvable iff is solvable

#### unification algorithm

at every point the state of the system will include

 set of equations partial solution

at each step there are two possibilities

1. no solution if "function clash" (2 above) or "occurs check" (3 above)
2. can progress by removing equations from system and possibly adding terms to the partial solution
• can remove equation of the form
• can remove equation of the form and add to partial solution
• you have like 2 in the solutions above and add to the partial solution
• you have case 4 above in the solutions section

### 2009-10-20 Tue

when checking whether a program has a type we will be implementing unification-type algorithms

thus begins the second part of the course

• from textbook
• more interactive
t ::=
true
false
if t then t else t
0
succ t
pred t
iszero t

• in the above t is a place holder for any term
• the above is shorthand for an inductive definition of a set of terms
• context free grammars define sets of strings
• the above defines terms (tree-like structures) rather than strings, these kinds of grammars may be called abstract grammars, as opposed to the context free grammars which are concerned with the string representations of the programs as written by the programmer
• for now we won't worry about the parsing issues, but only with the abstract trees

type systems will help to divide terms into those that are definitely meaningless and those that might have meaning

the standard notation for these inductive set definitions of the form if then is

a more concrete method of constructing the set of terms

then applying these rules

• S0 = empty-set
• S1 = {true, false, 0}
• S2 = …

#### from syntax to semantics

evaluation is moving from sets of terms to sets of values

-- terms
t ::=
true
false
if t then t else t

-- values
v ::=
true
false


evaluation rules

• if true then t1 else t2 t1
• if false then t1 else t2 t2

### 2009-10-22 Thu

#### homework2 notes

• definitely use LaTeX, lhs, etc…
• include test code in the resulting pdf
• include discussion in the resulting pdf
• could even include the test log into the resulting pdf file
• need to be able to print a single pdf file
• minimum font is 11pt
• use the following LaTeX font \usepackage{mathpazo}
• don't use ellipses in a proof
• don't use a narrative style in a proof (as much as possible the proof should be a system of equations)
• these calculational proofs are intended to be mechanically checkable (in style at least if not in practice)

here is an example .lhs file excerpt-CircuitDesigner.lhs

#### current homework notes

• we can use our own types if we prefer
• fail with cycle is just an occurs check failure

#### operational semantics

• if t1 then t2 else t3 is strict in t1 (meaning it must be known/evaluated) but is lazy in t2 and t3 in that they may not be evaluated
• rules for if/then/else
• (E-IFTRUE)
• (E-IFFALSE)
• (E-IF)
• repeat evaluation ->* is the transitive, reflexive closure of the single-step evaluation relation
• the normal form of a term is what it ultimately evaluates to

actually stepping-through/evaluating a program

#### now adding numbers to our simple boolean language

sadly we now have normal forms that we don't like, for example iszero false. Adding a type system to our system will allow us to find these stuck situations before evaluating.

well typed
evaluation will not result in a stuck term

### 2009-10-27 Tue

homework2 back today average score is ~90

recall the language of arithmetic expressions

t ::= true
false
if t then t else t
0
succ t
pred t
iszero t


inductive function definition examples

size
returns the "size" of a term
size true  = 1
size false = 1
size 0     = 1
size (succ t)   = 1 + size t
size (pred t)   = 1 + size t
size (iszero t) = 1 + size t
size (if t1 then t2 else t3) = (size t1) + (size t2) + (size t3)


consts
returns the set of constants present in a term
consts 0 = [0]
consts true = [true]
consts 0 = [false]
consts (succ t) = consts t
consts (pred t) = consts t
consts (iszero t) = consts t
consts (if t1 then t2 else t3) =
consts t1 union consts t2 union consts t3



#### Theorem

• by induction on the structure of t
• base cases are :
• inductive size
• :

#### Operational Semantics

what if we want to describe a language on booleans where both cases of the conditional are evaluated and in addition we'd like to fix the order of evaluation to t2 then t3 then t1?

 E-IF-THEN E-IF-ELSE T E-IF-ELSE F E-IF-GUARD TT and three more … E-TRUE TT and eight more …

we should really have a metavariable v over values

 E-IF-THEN E-IF-ELSE E-IF-GUARD E-TRUE E-FALSE

#### Lets type our language of arithmetic

the goal here being to eliminate normal terms which are not values (e.g. succ(false)).

we'd prefer types to be a relation rather than a partition, so that terms can have multiple types (e.g. 1 is a natural number and an int.)

: will be our type assignment operator t:T mean t has type T

T := Nat
Bool


rules for values

now rules for compound terms

### 2009-10-29 Thu

lambda calculus – there are only functions and function definitions.

what special things can you do in a language where you know that all terms will terminate

#### lambda calculus

 t ::= x
\x . t
t t


Reduction rule – -reduction rule

• will not get more than one normal form
• is possible to end up in an infinite re-write loop
• normal order strategy – always work with the outermost redex
• call by name strategy – don't dive into a lambda
• call by value – only values can be substituted into a lambda (abstractions are considered to be values)

note that the two previous strategies result in different normal forms and define different lambda calculi. call by name can vaguely be thought of as a strategy in which parts of the program are set aside which can be compiled into machine code (because they won't serve as data at any point). call by name is the flavor of lambda calculus which lives in the core of Haskell – with the addition of laziness which makes it call by need.

call by need
call by value
java, C, ML, etc…

from here on out we will restrict ourselves to the call by value form of lambda calculus

#### untyped, call-by-value, -calculus

terms

t ::= x
\x . t
t t


values

v ::= \x . t


evaluation relation

E-APPABS

staging rules

E-APP1
E-APP2

tricky spot (what does mean): which only means free occurrences of should be replaced. free terms mean those which are not inside of a term.

### 2009-11-03 Tue

the next homework will be out soon…

#### lambda calculus (Church constants)

• – true
• – false

the above are like conditionals

• true A B –> A
• false A B –> B

Church could not discover an encoding of the predecessor function, it was discovered by Fellini while he was having a tooth extracted.

– Legend

• – and

#### y combinator

om = (\x -> x x)(\x -> x x)


results in

Occurs check: cannot construct the infinite type: t = t -> t1
Probable cause: x' is applied to too many arguments
In the expression: x x
In the expression: (\ x -> x x) (\ x -> x x)


never converges, it has no normal form

### 2009-11-05 Thu

#### formal definition of our lambda calculus / notation

very formal definition of λ-terms

• infinite sequence of expressions called variables:
• finite, infinite or empty sequence of expressions called atomic constants which will be different from variables. (if there are none of these we have a pure λ calculus)

given the variables and atomic constants we will define λ terms

• all variables and atomic constants are λ terms and we will call these atoms
• if M and N are λ-terms, then (M N) is a λ term, this term will be called an application
• if M is a λ-term and x is a variable, then (λ x . M) is a λ term, this term will be called an abstraction

Notation

• Capital letters will denote arbitrary items
• Letters will denote variables

Parenthesis will be omitted as follows

• MNPQ denotes (((MN)P)Q)
• λ x . P Q denotes (λ x (P Q))
• λ x_1 x_2 ... x_n . M denotes (λ x_1 .(λ x_2 . (... (λ x . M) ...)))

#### definitions

P,Q λ-terms

P occurs in Q, or P is a subterm of Q, or Q contains P

inductive definition:

• P occurs in P
• if P occurs in M or in N, then P occurs in (MN)
• if P occurs in M or P==x, then P occurs in (λx.M)

Exercise:

• ((xy)(λx.(xy)))
• three occurrences of x
• two occurrences of (xy)
• (λxy.xy)
• only one occurrence of (xy)

scope

Def
for a particular occurrence of λx.M in a term P, the occurrence of M is called the scope of the occurrence of λx.
Example
in P==(λy.yx (λx.y(λy.z)x))vw
• the scope of the leftmost λy is yx (λx.y(λy.z)x
• the scope of λx is y(λy.z)x

An occurrence of a variable x in a term P is:

• bound if it is in the scope of a λx in P
• bound and binding if it is the x in λx
• free otherwise

bound and free in a term

• x is a bound variable of a term P if x has at least one binding occurrence in P
• x is a free variable of a term P if x has at least one free occurrence in P
• FV(P) is the set of free variables of the term P
• the term P is closed if it has no free variables

### 2009-11-10 Tue

#### homework notes

• is there something more elegant than string comparison for the highlighting? yes – there is an easier way if the function understands what the highlighting means
• when asking questions we should employ the phrasing "why is _ _"
• scanning should happen in two phases
1. all identifiers taken to be identifiers
2. see if these presumed identifiers are identifiers or if they are keywords
• although we may not alter/remove the given function, we may create our own additional functions
• note that the parenthesis are part of the grammar and must be included (the same is true of else and term for the if construct)
• the only place we really have leeway is in our acceptable identifiers
• we don't want to return any identifiers to the parser

#### formal definitions

substitution
for any M,N,x we define [N/x]M to be the result of substituting N for every free occurrence of x in M, and changing bound variables to avoid clashes. other notations include:
• [x N]M
• [x/N]M
• [N/x]M
induction on M
a number of rules
• [N/x]x == N - [N/x]a == a \forall atoms a != x
• [N/x](PQ) == ([N/x]P [N/x]Q) - [N/x](λx.P) == (λx.P)
• [N/x](λy.P) == (λy.P) if x $\not\in$ FV(P) - [N/x](λy.P) == (λy.[N/x]P) if x [[file:ltxpng/cs558_4ba518990fcebbf250a11f22405cb5524f2873a3.png]] FV(P) and y [[file:ltxpng/cs558_ecd413eb6f6a9015a9e6d75f9cc2e3455fd2eb4b.png]] FV(N)
• [N/x](λy.P) == (λz.[N/x][z/y]P) if x FV(P) and y FV(N) where z "is fresh"

### 2009-11-12 Thu

#### homework5

simple extension of homework4, λ calculus with reductions etc…

#### language of Booleans and Numbers

• avoiding non-meaningful(untyped) normal terms
• terms which do have types/meaning will evaluate and will do so to their type

we would like to find types w/o evaluating the term, so what do we do?

• type checking should be guaranteed to terminate
• should be quick (not so true in ML or Haskell)

new syntactic type

T ::= Bool
Nat


type of our syntactic elements

• true : Bool
• false : Bool
• 0 : Nat

so the typing system will exclude many terms which would evaluate w/o problem

goals

• well typed term will not get stuck, meaning it can be further evaluated or it is a value
• Preservation is the property that a well typed term will not evaluate in a single step to an untyped term
• Canonical forms lemma: if B has type Bool then it is either True or False

### 2009-12-01 Tue

#### homework

• many people's fresh variables weren't generating truly fresh variables
• overuse of isValue function
• average grade is around 70

Chapter 12 introduces a fixed point combinator which will allow recursion and make the language much more usable. this is done by removing the limitation of the inability to recur in the typed λ-calculus because combinators have no type (may not terminate).

#### lecture

would be nice to have…

let bindings
which are equivalent to function applications
pairs, tuples, records
which can also be nicely typed

### 2009-12-03 Thu

#### getting back the midterms

• typing went well
• some evaluation uncertainties
• class average 216
• look up the normal order reduction solution in the book
• in cases where a rule has no predicates like E-APPABS we can write the derivation tree sideways for clarity
• initial expression : (λx. (λz. λx. x z) x) (λx.xx)
• applying E-APPABS : [x |-> λx. x x] ((λz. λx. x z) x)
• resolving application : (λz. λx. x z) (λx. x x)
• applying E-APPABS : [z |-> λx. x x] (λz. λx. x z)
• resolving application : λx. x λx. x x
• I'm not going to latex this one out…
• Nat -> Nat -> Nat != (Nat -> Nat) -> Nat, because the convention is that -> associates to the right

#### λ-calculus

(bridging the gap between core λ-calculus and what we need in a language)

so far we have talked about

• pairs
• tuples
• records

Today we will talk about variant types

when several types are all injected into a single type with labels -- these are like the data types in Haskell

T ::= T+T

t ::= inject_left t
inject_right t
case t of { inject_left x => t | inject_right x => t }


data T = A T1 | B T2

x1 :: T1
x1 = ...
x2 :: T2
x2 = ...
t1 = A x1
t2 = B x2

-- using these types
case t3 of
A x1 -> ... x1 ...
B x2 -> ... x2 ...


typing rules

and also a much longer one for the case rule

in our Haskell implementation of this concept we will have no idea what T2 will be for an expression like inject_left x where x has type T1, so we may have to force the programmer to type annotate these expressions s.t.

• inject_left x becomes inject_left x as Nat x T2
• inject_right x becomes inject_right x as Nat T1 x

and the rules above must be changed similarly

the data keyword in Haskell introduces both the records and variant types that we have discussed thus far as well as recursive data types which we will not have time to address this year.

#### recursive evaluation (in 4 minutes)

fact n in the λ-calculus

we can't say

fact n = λ n. if n == 0 then 1
else n * fact (n - 1)


but in λ-calculus the fact above doesn't refer to the fact being described

λf. λn. if n == 0 then 1
else n * fact (n - 1)


so we end up saying…

fact = Y (λf. λn. if n == 0 then 1
else n * f (n - 1))


where Y is some Y-combinator, however we can't write this in our type system, so we must add the new syntactic form fix s.t.

t ::= fix t


with typing rule

and the evaluation rules

so to perform the factorial of 6 we could do…

   (fix  (λf. λn. if n == 0 then 1
else n * f (n - 1))) 6


as these expand the fix(...) is inserted in place of fact in such a way that the exposed λ-expressions are applied to their arguments (initially 6) on every other step, and every other-other step the fix substitution will take place. basically it alternates between expanding and applying.

### 2009-12-10 Thu

#### type reconstruction algorithm

typing rules are similar to the evaluation rules, but in addition to requiring assumptions about the evaluation of subterms, there may also be type constraints generated during the subterm evaluation.

first, constraints C1, C2, and C3 are generated while evaluating the portions of the if statement, and two new constraints are added which are required by the if statement.

for a simpler typing constraint

while running a type constraint algorithm we will occasionally need to generate a fresh type variable which has no constraints (e.g. when calculating the type constraints on an application).

more formally the above fresh type variable creation requires a slightly more complicated stating that the new variable (say X) is not an element of any of the existing sets of variables used in our subexpressions.

one valid output of our type reconstruction is polymorphic functions. one possible problem of polymorphic functions is the potential for them to be applied to different values.

for example

let id = λn.n in
(id (λk.k)) (id 5)


id is used here with different types.

the work around for this issue – discovered during the implementation of ml – is called let polymorphism and involves calculation of polymorphism during the evaluation of a let-bound. so while the above would work the following previously equivalent expression would be rejected

(λid.((id(λk.k))(id 5)))(λn.n)


#### what we haven't covered

meta-theory
analysis/proofs of the properties and relationships of/between our types and terms
lanuage extensions
we skipped many possible extensions
references
clean way of adding mutable state to a core λ calculus
exceptions
chapter 14, models for raising and handling said
subtyping
type classes – complex interactions with type reconstruction, fundamental to Object Oriented programming. Type classes in Haskell have nothing to do with subtyping
recursive types
this is not hard, but we didn't cover it. here's a quick introduction to recursive types in Haskell.
data Tree = Leaf | Fork Tree Tree



If we think of types as sets of possible tokens, then we can recursively build the set(type) of type Tree. Chapters 20 and 21 discuss how we can handle recursing into types.

universal polymorphism
system F, module system, existential polymorphism
some really interesting stuff
quantification over terms and types
• dependent types
• types over types

## questions / topics

### BFS

Note the solution shown earlier today actually correct, the following does work however

data Tree a = Leaf a
| Fork [Tree a]

tree = (Fork [(Fork [(Leaf 1), (Fork [(Leaf 2), (Leaf 3)])]), (Leaf 4)])

dfs :: Tree a -> [a]
dfs (Leaf l) = [l]
dfs (Fork ts) = foldr (\ t rest -> dfs t ++ rest) [] ts

-- fixed version -- Thanks to Sunny
bfs :: Tree a -> [a]
bfs (Leaf l) = [l]
bfs (Fork []) = []
bfs (Fork xs) = (concat (map bfs ([y | y <- xs, isLeaf y])))++(bfs (Fork (collapseForks [y | y <- xs, not (isLeaf y)])))
where
isLeaf :: Tree a -> Bool
isLeaf (Leaf l) = True
isLeaf _ = False
collapseForks :: [Tree a] -> [Tree a]
collapseForks [] = []
collapseForks ((Fork a):xs) = a++(collapseForks xs)

tree = (Fork [(Fork [(Leaf 1), (Fork [(Leaf 2), (Leaf 3)])]), (Leaf 4)])

-- from George
bfs (Fork xs) = concatMap bfs ([y | y <- xs, isLeaf y] ++ [Fork (collapseForks [y | y <- xs, not (isLeaf y)])])
bfs (Fork xs) = [l | Leaf l <- xs] ++ bfs (Fork (concat [ts | Fork ts <- xs]))

-- broken version -- don't use
bfs :: Tree a -> [a]
bfs (Leaf l) = [l]
bfs (Fork ts) = foldr ((++).bfs) [] ([y | y <- ts, isLeaf y] ++ [y | y <- ts, not (isLeaf y)])
where
isLeaf :: Tree a -> Bool
isLeaf (Leaf l) = True
isLeaf (Fork ts) = False


a list of IO actions

todoList :: [IO ()]

todoList = [putChar 'a',
do putChar 'b'
putChar 'c',
do c <- getChar
putChar c]


processes the IO actions with

sequence_ todoList


### partial lists

Haskell has three types of lists, finite, infinite, and partial

a partial list has an undefined final element

a = [1, 2, 3, undefined]


### null operator

take a look at null

### Array

create and access an array

squares = array (1,10) [(i, i*i) | i <- [1..10]]
-- then to get at a value
squares!2