Programming Languages and Compilers (CS 421)

04/20/23 1

Elsa L Gunter

2112 SC, UIUC

http://courses.engr.illinois.edu/cs421

Based in part on slides by Mattox Beckman, as updated by Vikram Adve and Gul Agha

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Mapping Functions Over Lists

# let rec map f list = match list with [] -> [] | (h::t) -> (f h) :: (map f t);;val map : ('a -> 'b) -> 'a list -> 'b list =

<fun># map plus_two fib5;;- : int list = [10; 7; 5; 4; 3; 3]# map (fun x -> x - 1) fib6;;: int list = [12; 7; 4; 2; 1; 0; 0]

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Mapping Recursion

One common form of structural recursion applies a function to each element in the structure

# let rec doubleList list = match list with [ ] -> [ ] | x::xs -> 2 * x :: doubleList xs;;val doubleList : int list -> int list = <fun># doubleList [2;3;4];;- : int list = [4; 6; 8]

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Mapping Recursion

Can use the higher-order recursive map function instead of direct recursion

# let doubleList list = List.map (fun x -> 2 * x) list;;val doubleList : int list -> int list = <fun># doubleList [2;3;4];;- : int list = [4; 6; 8]

Same function, but no explicit rec

Your turn now

Write a function

make_app : ((‘a -> ‘b) * ‘a) list -> ‘b list

that takes a list of function – input pairs and gives the result of applying each function to its argument. Use map, no explicit recursion.

let make_app l =

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Folding Recursion

Another common form “folds” an operation over the elements of the structure

# let rec multList list = match list with [ ] -> 1 | x::xs -> x * multList xs;;val multList : int list -> int = <fun># multList [2;4;6];;- : int = 48 Computes (2 * (4 * (6 * 1)))

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Folding Functions over Lists

How are the following functions similar?# let rec sumList list = match list with [ ] -> 0 | x::xs -> x + sumList xs;;val sumList : int list -> int = <fun># sumList [2;3;4];;- : int = 9# let rec multList list = match list with [ ] -> 1 | x::xs -> x * multList xs;;val multList : int list -> int = <fun># multList [2;3;4];;- : int = 24

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Base Case

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Recursive Call

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Head Element

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Combining Operation

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How are the following functions similar?# let rec sumList list = match list with [ ] -> 0 | x::xs -> x + sumList xs;;val sumList : int list -> int = <fun># sumList [2;3;4];;- : int = 9# let rec multList list = match list with [ ] -> 1 | x::xs -> x * multList xs;;val multList : int list -> int = <fun># multList [2;3;4];;- : int = 24R

Rec value

Combining Operation

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Recursing over lists

# let rec fold_right f list b = match list with [] -> b | (x :: xs) -> f x (fold_right f xs b);;val fold_right : ('a -> 'b -> 'b) -> 'a list -> 'b ->

'b = <fun># fold_right (fun s -> fun () -> print_string s) ["hi"; "there"] ();;therehi- : unit = ()

The Primitive Recursion Fairy

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Folding Recursion

multList folds to the right Same as:# let multList list = List.fold_right (fun x -> fun rv -> x * rv) list 1;;val multList : int list -> int = <fun># multList [2;4;6];;- : int = 48

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Encoding Recursion with Fold

# let rec append list1 list2 = match list1 with [ ] -> list2 | x::xs -> x :: append xs list2;;val append : 'a list -> 'a list -> 'a list = <fun>

Base Case Operation Recursive Call

# let append list1 list2 = fold_right (fun x rv -> x :: rv) list1 list2;;val append : 'a list -> 'a list -> 'a list = <fun># append [1;2;3] [4;5;6];; - : int list = [1; 2; 3; 4; 5; 6]

Question

let rec length l = match l with [] -> 0 | (a :: bs) -> 1 + length bsHow do you write length with fold_right, but no explicit recursion?

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Question

let rec length l = match l with [] -> 0 | (a :: bs) -> 1 + length bsHow do you write length with fold_right, but no explicit recursion?let length list = List.fold_right (fun x -> fun n -> n + 1) list 0

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Map from Fold

# let map f list = fold_right (fun x -> fun y -> f x :: y)

list [ ];;val map : ('a -> 'b) -> 'a list -> 'b list =

<fun># map ((+)1) [1;2;3];;- : int list = [2; 3; 4] Can you write fold_right (or fold_left)

with just map? How, or why not?

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Iterating over lists

# let rec fold_left f a list = match list with [] -> a | (x :: xs) -> fold_left f (f a x) xs;;val fold_left : ('a -> 'b -> 'a) -> 'a -> 'b list ->

'a = <fun># fold_left (fun () -> print_string) () ["hi"; "there"];;hithere- : unit = ()

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Encoding Tail Recursion with fold_left

# let prod list = let rec prod_aux l acc = match l with [] -> acc | (y :: rest) -> prod_aux rest (acc * y) in prod_aux list 1;;val prod : int list -> int = <fun>

Init Acc Value Recursive Call Operation

# let prod list = List.fold_left (fun acc y -> acc * y) 1 list;;val prod: int list -> int = <fun># prod [4;5;6];; - : int =120

Question

let length l = let rec length_aux list n = match list with [] -> n | (a :: bs) -> length_aux bs (n + 1)in length_aux l 0How do you write length with fold_left, but no explicit recursion?

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Question

let length l = let rec length_aux list n = match list with [] -> n | (a :: bs) -> length_aux bs (n + 1)in length_aux l 0How do you write length with fold_left, but no explicit recursion?let length list = List.fold_left (fun n -> fun x -> n + 1) 0 list

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Folding

# let rec fold_left f a list = match list with [] -> a | (x :: xs) -> fold_left f (f a x) xs;;val fold_left : ('a -> 'b -> 'a) -> 'a -> 'b list -> 'a

= <fun>fold_left f a [x1; x2;…;xn] = f(…(f (f a x1) x2)…)xn

# let rec fold_right f list b = match list with [ ] -> b | (x :: xs) -> f x (fold_right f xs b);;val fold_right : ('a -> 'b -> 'b) -> 'a list -> 'b ->

'b = <fun>fold_right f [x1; x2;…;xn] b = f x1(f x2 (…(f xn b)…))

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Recall

# let rec poor_rev list = match list with [] -> [] | (x::xs) -> poor_rev xs @ [x];;val poor_rev : 'a list -> 'a list = <fun>

What is its running time?

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Tail Recursion - Example

# let rec rev_aux list revlist = match list with [ ] -> revlist | x :: xs -> rev_aux xs (x::revlist);;val rev_aux : 'a list -> 'a list -> 'a list =

# let rev list = rev_aux list [ ];;val rev : 'a list -> 'a list = <fun>

What is its running time?

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Comparison

poor_rev [1,2,3] = (poor_rev [2,3]) @ [1] = ((poor_rev [3]) @ [2]) @ [1] = (((poor_rev [ ]) @ [3]) @ [2]) @ [1] = (([ ] @ [3]) @ [2]) @ [1]) = ([3] @ [2]) @ [1] = (3:: ([ ] @ [2])) @ [1] = [3,2] @ [1] = 3 :: ([2] @ [1]) = 3 :: (2:: ([ ] @ [1])) = [3, 2, 1]

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Comparison

rev [1,2,3] = rev_aux [1,2,3] [ ] = rev_aux [2,3] [1] = rev_aux [3] [2,1] = rev_aux [ ] [3,2,1] = [3,2,1]

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Folding - Tail Recursion

- # let rev list =- fold_left- (fun l -> fun x -> x :: l) //comb

op [] //accumulator cell list

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Folding

Can replace recursion by fold_right in any forward primitive recursive definition Primitive recursive means it only recurses

on immediate subcomponents of recursive data structure

Can replace recursion by fold_left in any tail primitive recursive definition

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Continuations

A programming technique for all forms of “non-local” control flow: non-local jumps exceptions general conversion of non-tail calls to

tail calls Essentially it’s a higher-order

function version of GOTO

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Continuations

Idea: Use functions to represent the control flow of a program

Method: Each procedure takes a function as an extra argument to which to pass its result; outer procedure “returns” no result

Function receiving the result called a continuation

Continuation acts as “accumulator” for work still to be done

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Continuation Passing Style

Writing procedures such that all procedure calls take a continuation to which to give (pass) the result, and return no result, is called continuation passing style (CPS)

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Continuation Passing Style

A compilation technique to implement non-local control flow, especially useful in interpreters.

A formalization of non-local control flow in denotational semantics

Possible intermediate state in compiling functional code

Why CPS?

Makes order of evaluation explicitly clear Allocates variables (to become registers) for

each step of computation Essentially converts functional programs into

imperative ones Major step for compiling to assembly or byte

code Tail recursion easily identified Strict forward recursion converted to tail

recursion At the expense of building large closures in heap

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Example

Simple reporting continuation:# let report x = (print_int x; print_newline( ) );;val report : int -> unit = <fun>

Simple function using a continuation:# let addk a b k = k (a + b);;val addk : int -> int -> (int -> ’a) -> ’a = <fun># addk 22 20 report;;2- : unit = ()

Simple Functions Taking Continuations

Given a primitive operation, can convert it to pass its result forward to a continuation

Examples:# let subk x y k = k(x + y);;val subk : int -> int -> (int -> 'a) -> 'a = <fun># let eqk x y k = k(x = y);;val eqk : 'a -> 'a -> (bool -> 'b) -> 'b = <fun># let timesk x y k = k(x * y);;val timesk : int -> int -> (int -> 'a) -> 'a = <fun>

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Nesting Continuations

# let add_three x y z = (x + y) + z;;val add_three : int -> int -> int -> int = <fun># let add_three x y z= let p = x + y in p + z;;val add_three : int -> int -> int -> int = <fun># let add_three_k x y z k = addk x y (fun p -> addk p z k );;val add_three_k : int -> int -> int -> (int -> 'a)

-> 'a = <fun>

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add_three: a different order

# let add_three x y z = x + (y + z);; How do we write add_three_k to use a

different order?

let add_three_k x y z k =

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Recursive Functions

Recall:# let rec factorial n = if n = 0 then 1 else n * factorial (n -

1);; val factorial : int -> int = <fun># factorial 5;;- : int = 120

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Recursive Functions

# let rec factorial n = let b = (n = 0) in (* First computation *) if b then 1 (* Returned value *) else let s = n – 1 in (* Second computation

*) let r = factorial s in (* Third

computation *) n * r in (* Returned value *) ;;val factorial : int -> int = <fun># factorial 5;;- : int = 120

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Recursive Functions

# let rec factorialk n k = eqk n 0 (fun b -> (* First computation *) if b then k 1 (* Passed value *) else subk n 1 (* Second computation *) (fun s -> factorialk s (* Third computation *) (fun r -> timesk n r k))) (* Passed value *)val factorialk : int -> int = <fun># factorialk 5 report;;120- : unit = ()

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Recursive Functions

To make recursive call, must build intermediate continuation to take recursive value: r build it to final result: n * r And pass it to final continuation: times n r k = k (n * r)

Example: CPS for length

let rec length list = match list with [] -> 0 | (a :: bs) -> 1 + length bsWhat is the let-expanded version of this?

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let rec length list = match list with [] -> 0 | (a :: bs) -> 1 + length bsWhat is the let-expanded version of this?let rec length list = match list with [] -> 0 | (a :: bs) -> let r1 = length bs in 1 + r1

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#let rec length list = match list with [] -> 0 | (a :: bs) -> let r1 = length bs in 1 + r1What is the CSP version of this?

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#let rec length list = match list with [] -> 0 | (a :: bs) -> let r1 = length bs in 1 + r1What is the CSP version of this?#let rec lengthk list k = match list with [ ] -> k 0 | x :: xs -> lengthk xs (fun r -> addk r 1 k);;val lengthk : 'a list -> (int -> 'b) -> 'b = <fun># lengthk [2;4;6;8] report;;4- : unit = ()

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CPS for Higher Order Functions

In CPS, every procedure / function takes a continuation to receive its result

Procedures passed as arguments take continuations

Procedures returned as results take continuations

CPS version of higher-order functions must expect input procedures to take continuations

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Example: all

#let rec all p l = match l with [] -> true | (x :: xs) -> let b = p x in if b then all p xs else falseval all : ('a -> bool) -> 'a list -> bool = <fun>What is the CPS version of this?

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Example: all

#let rec all p l = match l with [] -> true

| (x :: xs) -> let b = p x in if b then all p xs else falseval all : ('a -> bool) -> 'a list -> bool = <fun>What is the CPS version of this?#let rec allk pk l k =

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Example: all

| (x :: xs) -> let b = p x in if b then all p xs else falseval all : ('a -> bool) -> 'a list -> bool = <fun>What is the CPS version of this?#let rec allk pk l k = match l with [] -> true

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Example: all

| (x :: xs) -> let b = p x in if b then all p xs else falseval all : ('a -> bool) -> 'a list -> bool = <fun>What is the CPS version of this?#let rec allk pk l k = match l with [] -> k true

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Example: all

#let rec all p l = match l with [] -> true | (x :: xs) -> let b = p x in if b then all p xs else falseval all : ('a -> bool) -> 'a list -> bool = <fun>What is the CPS version of this?#let rec allk pk l k = match l with [] -> k true | (x :: xs) -> pk x (fun b -> if b then allk pk xs k else k false)val allk : ('a -> (bool -> 'b) -> 'b) -> 'a list -> (bool -> 'b) -> 'b = <fun>04/20/23 57

Example: all

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Terminology

Tail Position: A subexpression s of expressions e, such that if evaluated, will be taken as the value of e if (x>3) then x + 2 else x - 4 let x = 5 in x + 4

Tail Call: A function call that occurs in tail position if (h x) then f x else (x + g x)

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Terminology

Available: A function call that can be executed by the current expression

The fastest way to be unavailable is to be guarded by an abstraction (anonymous function, lambda lifted). if (h x) then f x else (x + g x) if (h x) then (fun x -> f x) else (g (x + x))

Not available

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CPS Transformation

Step 1: Add continuation argument to any function definition: let f arg = e let f arg k = e Idea: Every function takes an extra

parameter saying where the result goes Step 2: A simple expression in tail position

should be passed to a continuation instead of returned: return a k a Assuming a is a constant or variable. “Simple” = “No available function calls.”

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CPS Transformation

Step 3: Pass the current continuation to every function call in tail position return f arg f arg k The function “isn’t going to return,” so we

need to tell it where to put the result.

CPS Transformation

Step 4: Each function call not in tail position needs to be converted to take a new continuation (containing the old continuation as appropriate) return op (f arg) f arg (fun r -> k(op r)) op represents a primitive operation

return f(g arg) g arg (fun r-> f r k)

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Example

Before:let rec add_list lst =match lst with [ ] -> 0| 0 :: xs -> add_list

xs| x :: xs -> (+) x

(add_list xs);;

After:let rec add_listk lst k = (* rule 1 *)match lst with| [ ] -> k 0 (* rule 2 *)| 0 :: xs -> add_listk xs k (* rule 3 *)| x :: xs -> add_listk xs (fun r -> k ((+) x r));; (* rule 4 *)

Programming Languages and Compilers (CS 421)

int list int list

list b list

match list

rec multlist list

rec sumlist list

rec doublelist list

xs x sumlist xsval sumlist

multlist xsval multlist

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