The purpose of this assignment is to help you get acclimated to programming in ML:
You will be more effective if you are aware of the
Standard ML Basis Library.
Jeff Ullman's text (Chapter 9) describes the 1997 basis,
but today's compilers use the 2004 basis, which is a standard.
You will find a few differences in I/O, arrays, and elsewhere;
the most salient difference is in TextIO.inputLine.
The most convenient guide to the basis is the Moscow ML help system; type
- help "";at the mosml interactive prompt. The help file displays an incorrect date in its banner, but as far as I know, it is up to date.
length function.
The entire assignment can and should be solved without using
length.
Solutions that use length will earn No Credit.
null, hd, and tl;
use patterns instead.
Code using hd or tl will earn No Credit.
local
or let.
Problems defining auxiliary functions at top level will earn No Credit.
open;
if needed, use one-letter abbreviations for common structures.
Code using open may earn No Credit for your entire assignment.
<patterns>= [D->] [] [x] [x, y] [a, b, c]
and also these patterns, which match lists of at least 0, 1, 2, or 3 elements:
<patterns>+= [<-D] xs x::xs x1::x2::xs a::b::c::xs
When using these patterns, remember that function application has higher precedence than any infix operator! This is as true in patterns as it is anywhere else.
Use the standard basis extensively.
Moscow ML's help "lib"; will tell you all about the library.
And if you use
ledit mosml -P fullas your interactive top-level loop, it will automatically load almost everything you might want from the standard basis.
All the sample code we show you is gathered in one place online.
As you learn ML, this table may help you transfer your knowledge of μScheme:
μScheme ML val val define fun lambda fn
Put all your solutions in one file: warmup.sml. (If separate files are easier, combine them with cat.) At the start of each problem, please label it with a short comment, like
(***** Problem A *****)To receive credit, your warmup.sml file must compile and execute in the Moscow ML system. For example, we must be able to compile your code without warnings or errors:
% /usr/sup/bin/mosmlc -c warmup.sml %Please remember to put your name and the time you spent in the warmup.sml file.
fold, but it works on binary operators.
val compound : ('a * 'a -> 'a) -> int -> 'a -> 'a
that ``compounds'' a binary operator rator so that
compound rator n x is x if n=0, x rator x if n = 1,
and in general x rator (x rator (... rator x)) where rator is applied
exactly n times.
compound rator need not behave well when applied to negative integers.
compound function to define a
Curried
function for integer
exponentiation
val exp : int -> int -> intso that, for example,
exp 3 2 evaluates to 9.
Hint: take note of the description of op in Ullman S5.4.4, page 165.
Don't get confused by infix vs prefix operators. Remember this:
My estimate of difficulty: Mostly easy. You might be troubled by an off-by-one error.
firstVowel
that takes a list of lower-case letters and
returns true if the first character is a vowel (aeiou) and false if
the first character is not a vowel or if the list is empty.
Use the wildcard symbol _
whenever possible, and avoid if.
Remember that the ML character syntax is #"x", as decribed
in Ullman, page 13.
null, which when applied to a list
tells whether the list is empty. Avoid if, and make sure the
function takes constant time. Make sure your function has the same type as the
null in the Standard Basis.
foldl and foldr are predefined with type
('a * 'b -> 'b) -> 'b -> 'a list -> 'b
They are like the μScheme versions except the ML versions are Curried.
rev (the function known in
μScheme as "reverse") using foldl or foldr.
minlist, which returns the smallest
element of a non-empty list of integers.
Your solution should work regardless
of the representation of integers (e.g., it should not matter how many bits are used to
represent integers).
Your solution can fail (e.g., by raise Match) if given an empty
list of integers.
Use foldl or foldr.
My estimate of difficulty: Medium
foldl and foldr using recursion. Do not
create unnecessary cons cells. Do not use if.
My estimate of difficulty: Easy
val pairfoldr : ('a * 'b * 'c -> 'c) -> 'c -> 'a list * 'b list -> 'c
that applies a three-argument function to a pair of lists of equal
length, using the same order as foldr.
val flatten : 'a list list -> 'a listwhich takes a list of lists and produces a single list containing all the elements in the correct order. For example,
<sample>+= [<-D->] - flatten [[1], [2, 3, 4], [], [5, 6]]; > val it = [1, 2, 3, 4, 5, 6] : int list
To get full credit for this problem, your function should use no unnecessary cons cells.
My estimate of difficulty: Easy
val nth : int -> 'a list -> 'ato return the nth element of a list. (Number elements from 0.) If
nth is given arguments on which it is not defined,
raise a suitable exception.
You may
define one or more suitable exceptions or you may choose
to use an appropriate one from the initial basis.
(If you have doubts about what's appropriate, play it safe and define
an exception of your own.)
I expect you to implement nth yourself
and not simply call List.nth.
My estimate of difficulty: Easy
'a env and functions
<sample>+= [<-D->] type 'a env = (* you fill in this part *) exception NotFound of string val emptyEnv : 'a env = (* ... *) val bindVar : string * 'a * 'a env -> 'a env = (* ... *) val lookup : string * 'a env -> 'a = (* ... *)
DefinesbindVar,emptyEnv,env,lookup,NotFound(links are to index).
such that you can use 'a env for a type environment or a value
environment.
On an attempt to look up an
identifier that doesn't exist,
raise the exception NotFound.
Don't worry about efficiency.
type 'a env = string -> 'a, and let
<sample>+= [<-D->] fun lookup (name, rho) = rho name
Defineslookup(links are to index).
val isBound : string * 'a env -> boolthat works with both representations of environments. That is, write a single function that works regardless of whether environments are implemented as lists or as functions. You will need imperative features, like sequencing (the semicolon). Don't use
if.
val extendEnv : string list * 'a list * 'a env -> 'a envthat takes a list of variables and a list of values and adds the corresponding bindings to an environment. It should work with both representations. Do not use recursion. Hint: you can do it in two lines using the higher-order list functions defined above.
You shoud put these functions in the same file; don't worry about the later ones shadowing the earlier ones.
My estimate of difficulty: Medium, because of those pesky functions as values again
<sample>+= [<-D->] datatype 'a tree = NODE of 'a tree * 'a * 'a tree | LEAF
Definestree(links are to index).
To make a search tree, we need to compare values at nodes.
The standard idiom for comparison is to define a function that returns a value of type
order.
As discussed in Ullman, page 325,
order is predefined by
<sample>+= [<-D->] datatype order = LESS | EQUAL | GREATER (* do not include me in your code *)
Definesorder(links are to index).
Because order is predefined, if you include it in your program, you
will hide the predefined version (which is in the initial
basis) and other things may break mysteriously.
So don't include it.
We can use the order type to define a higher-order insertion function by, e.g.,
<sample>+= [<-D->] fun insert cmp = let fun ins (x, LEAF) = NODE (LEAF, x, LEAF) | ins (x, NODE (left, y, right)) = (case cmp (x, y) of LESS => NODE (ins (x, left), y, right) | GREATER => NODE (left, y, ins (x, right)) | EQUAL => NODE (left, x, right)) in ins end
Definesinsert(links are to index).
This higher-order insertion function accepts a comparison function as argument,
then returns an insertion function.
(The parentheses around case aren't actually necessary here, but
I've included them because if you leave them out when they
are needed, you will be very confused by the
resulting error messages.)
We can use this idea to implement polymorphic sets in which we store the comparison function in the set itself. For example,
<sample>+= [<-D->] datatype 'a set = SET of ('a * 'a -> order) * 'a tree fun nullset cmp = SET (cmp, LEAF)
Definesnullset,set(links are to index).
val addelt : 'a * 'a set -> 'a setthat adds an element to a set.
val treeFoldr : ('a * 'b -> 'b) -> 'b -> 'a tree -> 'b
that folds a function over every element of a tree, rightmost element first.
Calling treeFoldr op :: [] t should return the elements of t in order.
Write a similar function
val setFold : ('a * 'b -> 'b) -> 'b -> 'a set -> 'b
The function setFold should visit every element of the set exactly
once, in an unspecified order.
My estimate of difficulty: Easy to Medium
This is a challenge problem. The other problems on the homework all involve old wine in new bottles. To solve this problem, you have to think of something new.
'a flist.
(Before you can define a representation, you will want to study
the rest of the parts of this problem, plus the test cases.)
Document your representation by saying, in a short comment,
what sequence is
meant by any value of type 'a flist.
val singletonOf : 'a -> 'a flistwhich returns a sequence containing a single value, whose finger points at that value.
val atFinger : 'a flist -> 'awhich returns the value that the finger points at.
val fingerLeft : 'a flist -> 'a flist val fingerRight : 'a flist -> 'a flistCalling
fingerLeft xs creates a new list that is like xs,
except the finger is moved one position to the left.
If the finger belonging to xs already points to the leftmost position,
then fingerLeft xs should
raise the same exception that the Basis Library raises for array access out of bounds.
Function fingerRight is similar.
Both functions must run in constant time and space.
Please think of these functions as "moving the finger", but remember no mutation is involved. Instead of changing an existing list, each function creates a new list.
val deleteLeft : 'a flist -> 'a flist val deleteRight : 'a flist -> 'a flistCalling
deleteLeft xs creates a new list that is like xs,
except the value x to the left of the finger has been removed.
If the finger points to the leftmost position, then deleteLeft should
raise the same exception that the Basis Library raises for array access out of bounds.
Function deleteRight is similar.
Both functions must run in constant time and space.
As before, no mutation is involved.
val insertLeft : 'a * 'a flist -> 'a flist val insertRight : 'a * 'a flist -> 'a flistCalling
insertLeft (x, xs) creates a new list that is like xs,
except the value x is inserted to the left of the finger.
Function insertRight is similar.
Both functions must run in constant time and space.
As before, no mutation is involved.
(These functions are related to "cons".)
val ffoldl : ('a * 'b -> 'b) -> 'b -> 'a flist -> 'b
val ffoldr : ('a * 'b -> 'b) -> 'b -> 'a flist -> 'b
which do the same thing as foldl and foldr, but
ignore the position of the finger.
Here is a simple test case, which should produce a list containing the numbers
1 through 5 in order.
You can use ffoldr to confirm.
val test = singletonOf 3 val test = insertLeft (1, test) val test = insertLeft (2, test) val test = insertRight (4, test) val test = fingerRight test val test = insertRight (5, test)You'll want to test the delete functions as well.
Hints: The key is to come up with a good representation for "list with finger." Once you have a good representation, the code is easy: over half the functions can be implemented in one line each, and no function requires more than two lines of code.
My estimate of difficulty: Hard
This problem asks you to explore the fact that a compiler doesn't need to store an entire environment in a closure; it only needs to store the free variables of its lambda expression. For the details you will want the handout on free variables, which is also available as Section 5.9 in your textbook.
You'll solve the problem in a prelude and four parts:
mlscheme-improved.sml.
You will edit mlscheme-improved.sml.
val freeIn : exp -> name -> bool.
This predicate tells when a variable appears free in an expression,
and it implements the proof rules on page 224 of the
handout.
During this part I recommend that you compile early and often using
/usr/sup/bin/mosmlc -c mlscheme-improved.sml
LAMBDA body and an environment,
and returns a better pair containing the same LAMBDA body
paired with an environment that contains only the free variables of
the LAMBDA.
(In the handout, on page 225, this environment is explained as the
restriction of the environment to the free variables.)
I recommend that you call this function improve, and that you give
it the type
val improve : (name list * exp) * 'a env -> (name list * exp) * 'a env
improve in the evaluation
case for LAMBDA, which appears in the book on page 228.
You simply apply improve to the pair that is already there, so
your improved interpreter looks like this:
(* more alternatives for ev ((mlscheme)) 228d *)
| ev (LAMBDA l) = CLOSURE (improve (l, rho))
mlton -verbose 1 -output mlscheme mlscheme.sml mlton -verbose 1 -output mlscheme-improved mlscheme-improved.sml(If plain
mlton doesn't work, try /usr/sup/bin/mlton.)
I have provided a script that you can use to measure the improvement. I also recommend that you compare the performance of the ML code with the performance of the C code in the course directory.
time run-exponential-arg-max 22 ./mlscheme
time run-exponential-arg-max 22 ./mlscheme-improved
time run-exponential-arg-max 22 /comp/105/bin/uscheme
Hints:
freeIn. This is the only recursive function
and the only function that requires case analysis on expressions.
And it is the only function that requires you to understand the
concept of free variables.
You will be using all of these concepts on future
assignments.
In Standard ML, the μScheme function exists? is called List.exists.
You'll have lots of opportunities to use it.
If you don't use it, you're making extra work for yourself.
In addition to List.exists, you may have a use for map,
foldr, foldl, or List.filter.
You might also have a use for these functions:
fun fst (x, y) = x
fun snd (x, y) = y
fun member (y, []) = false
| member (y, z::zs) = y = z orelse member (y, zs)
LETSTAR is gnarly, and writing it adds little to the
experience.
Here are two algebraic laws which may help:
freeIn (LETX (LETSTAR, [], e)) y = freeIn e y freeIn (LETX (LETSTAR, b::bs, e)) y = freeIn (LETX (LET, [b], LETX (LETSTAR, bs, e))) y
freeIn if you use nested functions.
Mostly the variable y doesn't change, so you needn't pass it
everywhere.
You'll see the same technique used in the eval and ev
functions in the chapter.
improve
on one line, without using any explicit recursion.
freeIn is 21 lines of ML.
... (three dots) special significance when it appears
as the last formal parameter in a lambda.
For example:
-> (val f (lambda (x y ...)) (+ x (+ x (foldl + 0 ...)))
-> (f 1 2 3 4 5) ; inside f, rho = { x |-> 1, y |->, ... |-> '(3 4 5) }
15
In this example, it is an error for f to get fewer than two arguments.
If f gets at least two arguments, any additional arguments are placed into
an ordinary list, and the list is used to initialize the location of the
formal parameteter associated with ....
lambda on
page 225 to
and lambda = name list * { varargs : bool } * exp
The type system will tell you what other code you have to change.
For the parser, you may find the following function useful:
fun newLambda (formals, body) =
case rev formals
of "..." :: fs' => LAMBDA (rev fs', {varargs=true}, body)
| _ => LAMBDA (formals, {varargs=false}, body)
The type of this function is
name list * exp -> name list * {varargs : bool} * exp;
thus it is designed exactly for you to adapt old syntax to new syntax;
you just drop it into the parser wherever LAMBDA was used.
call
primitive such that
(call f '(1 2 3))
is equivalent to
(f 1 2 3)
Sadly, you won't be able to use PRIMITIVE for this;
you'll have to invent a new kind of thing that has access to the
internal eval.
cons-logger that
counts cons calls in a private variable. It should
operate as follows:
-> (val cl (cons-logger))
-> (val log-cons (car cl))
-> (val conses-logged (cdr cl))
-> (conses-logged)
0
-> (log-cons f e1 e2 ... en) ; returns (f e1 e2 ... en), incrementing
; private counter whenever cons is called
-> (conses-logged)
99 ; or whatever else is the number of times cons is called
; during the call to log-cons
<sample>+= [<-D]
(* first declaration for sanity check *)
val compound : ('a * 'a -> 'a) -> int -> 'a -> 'a = compound
val exp : int -> int -> int = exp
val fib : int -> int = fib
val firstVowel : char list -> bool = firstVowel
val null : 'a list -> bool = null
val rev : 'a list -> 'a list = rev
val minlist : int list -> int = minlist
val foldl : ('a * 'b -> 'b) -> 'b -> 'a list -> 'b = foldl
val foldr : ('a * 'b -> 'b) -> 'b -> 'a list -> 'b = foldr
val zip : 'a list * 'b list -> ('a * 'b) list = zip
val pairfoldr : ('a * 'b * 'c -> 'c) -> 'c -> 'a list * 'b list -> 'c = pairfoldr
val unzip : ('a * 'b) list -> 'a list * 'b list = unzip
val flatten : 'a list list -> 'a list = flatten
val nth : int -> 'a list -> 'a = nth
val emptyEnv : 'a env = emptyEnv
val bindVar : string * 'a * 'a env -> 'a env = bindVar
val lookup : string * 'a env -> 'a = lookup
val isBound : string * 'a env -> bool = isBound
val extendEnv : string list * 'a list * 'a env -> 'a env = extendEnv
val addelt : 'a * 'a set -> 'a set = addelt
val treeFoldr : ('a * 'b -> 'b) -> 'b -> 'a tree -> 'b = treeFoldr
val setFold : ('a * 'b -> 'b) -> 'b -> 'a set -> 'b = setFold
val scons : 'a * 'a seq -> 'a seq = scons
val ssnoc : 'a * 'a seq -> 'a seq = ssnoc
val sappend : 'a seq * 'a seq -> 'a seq = sappend
val listOfSeq : 'a seq -> 'a list = listOfSeq
val seqOfList : 'a list -> 'a seq = seqOfList
val singletonOf : 'a -> 'a flist = singletonOf
val atFinger : 'a flist -> 'a = atFinger
val fingerLeft : 'a flist -> 'a flist = fingerLeft
val fingerRight : 'a flist -> 'a flist = fingerRight
val deleteLeft : 'a flist -> 'a flist = deleteLeft
val deleteRight : 'a flist -> 'a flist = deleteRight
val insertLeft : 'a * 'a flist -> 'a flist = insertLeft
val insertRight : 'a * 'a flist -> 'a flist = insertRight
val ffoldl : ('a * 'b -> 'b) -> 'b -> 'a flist -> 'b = ffoldl
val ffoldr : ('a * 'b -> 'b) -> 'b -> 'a flist -> 'b = ffoldr
(* last declaration for sanity check *)
Definesaddelt,atFinger,bindVar,compound,deleteLeft,deleteRight,emptyEnv,exp,extendEnv,ffoldl,ffoldr,fib,fingerLeft,fingerRight,firstVowel,flatten,foldl,foldr,insertLeft,insertRight,isBound,listOfSeq,lookup,minlist,nth,null,pairfoldr,rev,sappend,scons,seqOfList,setFold,singletonOf,ssnoc,treeFoldr,unzip,zip(links are to index).
I don't promise to have all the functions and their types here. Making sure that every function has the right type is your job, not mine.
length, hd, and tl.
Instant No Credit.
fun f x = ... stuff that is broken ... fun g (y, z) = ... stuff that uses 'f' ... fun f x = ... new, correct version of 'f' ...You now have a situation where
g is broken, and the
resulting error is very hard to detect.
Stay out of this situation;
instead, load fresh definitions from a file
using the use function.
op.
Ullman covers op in Section 5.4.4, page 165.
warmup.sml, and optionally varargs.sml,
using the script
submit105-ml-solo.
In comments at the top of your warmup.sml file, please include your name, the
names of any collaborators, and the
number of hours you spent on the assignment.
mlscheme-improved.sml,
using the script
submit105-ml-pair.