Lab #1 Solution




Welcome to 15-150’s first lab!  In each lab we give you problems  to work on with  the assistance  of the  TAs.  This week we cover some of the  basics.  Think  of this  lab as a first date,  where you and SML are just getting  to know each other a little.

We are recording attendance each week. Please show up on time! Don’t worry if you run out of time.  You will still get credit for the lab, because credit is based on participation, not completeness.



1    Getting Started


1.1     Setting Your Path


If you are logged in to a cluster machine running Linux or OS X, or ssh’d to a UNIX machine, we provide a macro called smlnj that  you can use in




To be able to conveniently  use this without  typing the entire path  each time, you will need to modify your $PATH environment variable,  as follows.




Task 1.1 First,  determine  which shell you are running  with the command echo ${SHELL}.


  • If $SHELL is /bin/bash (or something ending in bash), add the line



PATH=”/afs/andrew/course/15/150/bin:${PATH}”; export PATH



to the file .bashrc in your home directory.


To actually  bring this  new definition for PATH into effect, you can either  start  a new shell or type



source ~/.bashrc


  • If $SHELL is /bin/csh (or something ending in csh), add the line




set path = ( $path /afs/andrew/course/15/150/bin/ .)



to the  file .cshrc in your home directory.   If you get an error  when you try  to run

smlnj after adding this line to your .cshrc, ask a TA to look at your .cshrc.


To actually  bring this  new definition for PATH into effect, you can either  start  a new shell or type



source ~/.cshrc



Now that  you have done this, you can run SML with the command smlnj.



1.1.1     Additional SML  implementations


If you  want  to  run  SML from your  own machine,  you  have  a  few options.   If you  have SML installed  locally, you just  type sml.  You may notice that  this  does not  allow you to conveniently navigate  the interface with the arrow keys to access other places on the line or your command history; if you want to do this, you have to run rlwrap sml, where rlwrap is part  of GNU Readline.  You should be able to install this with your package manager  on a linux distribution, or with MacPorts  on OS X. Also note that  smlnj is just  a macro we defined for rlwrap sml, so you may find it beneficial to do the  same thing  on your local machine.  There are installation instructions for SML on the course website.

Keep in mind that  you can access the unix machines from any machine with an internet connection  by sshing to; this  may be more convenient than  setting up SML to run locally.

When you run SML, you should get something that  looks like:


hqbovik@unix14 hqbovik % smlnj

Standard ML of New Jersey v110.69 [built: Wed Apr 29 12:25:34 2009]


This is the SML REPL (read-eval-print-loop): it reads  the programs you enter, evaluates

them,  prints  the result, and then waits for more input.

To get out of the REPL,  type Control-d.



1.2     Cloning The git Repository


git is a popular version control system.  We will be using a git repository to distribute all of the files for your homeworks and labs to you for this semester.  It will also include compiling versions of code from lecture for you to play with,  and various other  helpful documents  as the course progresses.


Task 1.2 Follow the instructions in Section 2 of




and check out the git repository  to get the files for this lab.

Once the git repository has been copied over, you should change into the code directory for this lab.  Assuming you named your copy of it 15150, that  means you should do


cd 15150/lab/01/code/


The  whole process  of checking  out  the  git repository  should  look like the  following transcript:


hqbovik@unix14 hqbovik % ls oldfiles  private public www hqbovik@unix14 hqbovik % cd private hqbovik@unix14 private % ls

hqbovik@unix14 private % git clone /afs/ 15150

Initialized empty Git repository in


hqbovik@unix14 private % ls


hqbovik@unix14 private % cd 15150/

hqbovik@unix14 15150 % ls docs lab lecture src

hqbovik@unix14 15150 % cd lab/01/code hqbovik@unix14 code % ls


hqbovik@unix14 code % smlnj

Standard ML of New Jersey v110.69 [built: Wed Apr 29 12:25:34 2009]




2    Expressions


From  here, we can type  in expressions for sml/nj  to evaluate.   For example,  if we want  to add 2 + 2, we write 2 + 2;


Task 2.1 Enter  the text


2 + 2;


into the REPL  and press Enter.  What  is SML’s output?



The output line, val it = 4 : int, indicates  the type and the result of evaluating  the expression.  it is used as a default name for the value if a name is not provided by you, 4 is the value or result,  and int is the type of the expression – in this case, meaning an integer. SML uses types to ensure at compile time that  programs cannot  go wrong in certain  ways; the phrase 4 : int can be read “4 has type int”.

Notice that  the  expression was terminated with a semicolon; if we do not  do this,  the

REPL  does not know to evaluate  the expression and expects more input.


Task 2.2 Enter  the text


2 + 2


(no semicolon) into the REPL  and press Enter.  What  is SML’s output? After doing that,  enter a semicolon. What  happens  now?

As you can see, it is possible to put  the  semicolon on the  next  line and still get the  same result.



2.1     Parentheses


In an arithmetic class long ago, you probably learned some standard rules of operator  prece- dence – for example,  multiply  before you add,  but  anything  grouped  in parentheses  gets evaluated  first.  SML follows the exact same rules of precedence.  You can, of course, insert parentheses  into expressions to force a particular order of evaluation.



Task 2.3 Enter  the text


1 + 2 * 3 + 4;


into the REPL.  What  would you expect the result to be? What  is the actual  result? Now, enter

(1 + 2) * (3 + 4);


into the REPL.  Is the result the same?  Why?




3    Evaluation


As you were determining  how your expressions evaluated,  you may have gone step-by-step, evaluating each of the arithmetic operations one at a time.  As it turns out, this is how we can determine  the runtime  of an expression.  For example, (1 + 1) + 1 steps to 2 + 1, which steps to 3, which is a value, at which point evaluation  stops.  In this case, then,  evaluation takes  two  steps  to complete.   For  our purposes,  we assume  that  all arithmetic operations take  exactly  one step  of computation and  numbers  take  zero steps.   Also remember  that arithmetic operators  like + and * evaluate  from left to right.  We introduce  the  notation  e

=⇒ e’ to mean that  evaluation  of e steps to e’ in a single step, and e =⇒* e’ for a finite

number  of steps.  So for example,


(1+1)+1 =⇒ 2+1 =⇒ 3 (1+1)+1 =⇒* 3


Task 3.1  Figure out how many steps it takes to evaluate  (1 + 2) * (3 + 4) all the way, writing out each intermediate step.




3.1     Computation Trees


An important property  of additions  is that  the final result of a bunch of additions  does not depend  on the order in which the adds were performed.  Suppose we have some expression (…) + (…), where each parenthesized  sub-expression is built from additions and numer- als. Instead  of arbitrarily choosing a side to evaluate  first, in a parallel setting  it is possible to evaluate  both  sides at  once, and the  result  will always come out  the  same as if we had evaluated  sequentially.

It  is useful to draw an expression as a   computation  tree  (later,  we will see that  these expression trees are a special case of what is called a cost graph ).  For example, the tree for (1 + 1) + (2 + 2) looks like:



/ \

+  +

/ \ / \

1 1 2 2




The leaves represent values that  do not need any computation to evaluate.  In this case, the only other nodes in the tree are arithmetic operations,  which we have defined to have a cost of 1.




Task 3.2 Draw the tree for (1 + 2) * (3 + (4 * 5)).




3.2     Work and Span


Given an expression,  we can determine  its work, which is the  total  number  of operations needed to evaluate  the  expression, and  its span,  which is the  length  of the  longest critical path  in the  computation tree; a critical  path  is a sequence of operations  that  each depend on the  results  of the  previous  one.  The  work is the  number  of steps  it  takes  to  evaluate the  expression sequentially,  on a machine  with  only one processor.   The  span  is the  best possible number  of steps  for parallel  evaluation,  assuming  enough processors—if there  are not enough processors, the  cost would be somewhere between the  work and the  span.  We will do a bunch of work and span analysis this semester.

Once we have written  the expression as a computation tree, the work is the size (number of non-leaf nodes) of the tree, and the span is the depth (length  of the longest path)  of the tree.

For example, the expression (1 + (2 + (3 + 4))) , with computation tree



/ \

1  +

/ \

2  +

/ \

3  4


has work 3, because there are three additions  that  need to be made.  It also has a span of 3, since the result of the outermost  add cannot  be done until  the result of the other  two adds has been found, and the result of the middle add needs the result of the innermost  add.

On the other hand the expression (1 + 2) + (3 + 4) has work 3 but  span 2.


Task 3.3 What  is the work associated with the tree for (1 + 2) * (3 + (4 * 5))?






Task 3.4 What  is the span associated with the tree for (1 + 2) * (3 + (4 * 5))?




4    Types


There are more types than  just int in SML. For example, there is a type string for strings of text.


Task 4.1 Enter  the text




into the REPL.  What  is the result?



Instead  of seeing a number  as the  output, you see a string  here.  It is possible to con- catenate  two  strings,  using the  infix ^ operator.   This  can be used just  like + is used on integers.


Task 4.2 Enter  the text


“foo” ^ ” bar”;


into the REPL.  What  is the result?



We can write a program  that  is not well-typed to see what SML does in that  situation. For example, you can only concatenate  two strings.


Task 4.3 What  happens  when you enter the expression


3 ^ 7;


into the REPL?


This  is an  example  of one of SML’s error  messages – you  should  start   to  familiarize yourselves with them, as you will be seeing them quite a lot this semester, at least until you get used to types!




5    Variables


Above,  we mentioned  that  the  results  of computations are  bound  to  the  variable  it by default.  This means that  once we have done one computation, we can refer to its result  in the next computation:


Task 5.1 Enter


2 + 2;


into the REPL.  Then,  enter


it * 2;


into the REPL.  What  is the result?



As you see, before the second evaluation  the value bound to it was 4 (the value of 2+2), and now it is bound to 8, the result of the most recent expression evaluation  (the  value of it * 2 with it bound to 4).


Of course, you shouldn’t get into the habit of using it like this! The SML runtime system only uses it (i.e.  it) as a convenient  default  and  a way to help you debug code.  Usually you will want to choose a more mnemonic name by which to refer to a value, and the SML syntax  for declarations  allows you to do this.

A simple form of declaration  has the syntax


val <varname> = <exp>


This declaration  binds the value of <exp> to <varname>.  If we want to mention  the desired type of this variable explicitly we can say write


val <varname> : <type> = <exp>


We can use a val declaration  at the prompt  in the SML runtime  system, or as part  of a

let-expression with syntax


let val <varname>=<exp1> in <exp2> end


The value of this let-expression is obtained  by evaluating  <exp1>, binding its result to the variable  named  <varname>, and using this  binding while evaluating  <exp2>. This binding is not available  for use outside  of <exp2>, and we say that  the  scope of the  binding is this expression.


The  SML declaration  syntax  is much  more general  than  we have  indicated  here,  but this gives us enough to work with for now and also introduces  the key notions of scope and binding.




Task 5.2 Enter  the declaration


val x : int = 2 + 2;


into the REPL.  What  is the result?  How does it differ from just typing 2 + 2;?



As you can see, that  declaration  binds the value of 2 + 2 to the variable x. We can now use the variable.


Task 5.3 Enter




into the REPL;  what is the result?




Task 5.4 Now enter


val y : int = x * x;


into the REPL.  What  is the result?




Task 5.5 How about


val y : int list = x * x;


What  happens?  Why?




Task 5.6 After that,  enter


z * z;


into the REPL.  What  happens?  Why?




Task 5.7  Variables in SML refer to values, but  are not assignable like variables  in impera- tive programming  languages.  Each time a variable is declared, SML creates a fresh variable and binds it to a value.  This binding is available,  unchanged,  throughout the  scope of the declaration  that  introduced  it.  If the  name was already  taken,  the  new definition shadows the previous definition:  the old definition is still around,  but uses of the variable refer to the new definition.  We’ll talk about  this more in lecture and subsequently.


Type the following in the REPL:


val x : int = 3;

val x : int = 10;

val x : string = “hello, world”;


What  are the value and type of x after each line?




6    Using Files


Now that  we have written  some basic SML expressions, we can take a look at something  a little more interesting:  getting input from files. We have provided the file lab01.sml for you in the git repo you cloned in task 1.1. under lab/01/code/



Task 6.1 In the REPL,  type use “lab01.sml”;.  The output from SML should look like


– use “lab01.sml”; [opening lab01.sml]

val it = () : unit


Now that  you have done this, you have access to everything that  was defined in lab01.sml, as if you had copied and pasted  the contents  of the file into the REPL.




7    Functions


7.1     Applying functions


In this file, notice that  there are functions defined. For example, there is


val intToString : int -> string


In this  case, the  function  can be invoked by writing  intToString(37).   However, the parentheses  around  the  argument are actually  unnecessary.   It  doesn’t matter whether  we write intToString 37 or (((intToString) (37))) – both are evaluated  exactly the same.


Task 7.1 Enter


(intToString 37) ^ ” ” ^ (intToString 42);


into the REPL.  What  is the result?





7.2     Defining functions


We  will generally  require  you  to  use a  simple standard format  for commenting  function definitions:  the  SML code for a function  definition  should be preceded by comments  that specify the function’s type, a “requires”-condition, and an “ensures”-condition that  describes how the function behaves when applied to arguments  that  satisfy the pre-condition.  We may waive this  requirement when the  function  is part  of the  SML basis,  or has been specified earlier.  You may omit the requires-condition  when it there is none (i.e. it requires true).

For example, here is a function from class, treated  in this style.


(* sum : int list -> int  *) (* REQUIRES: true        *)

(* ENSURES:  sum(L) evaluates to the sum of the integers in L. *)


fun sum [ ] = 0

| sum (x::L) = x + (sum L);


We read this specification as saying that:  For all values L : int list, sum(L) evaluates to the sum of the integers  in L.




Task 7.2  Complete  the  following template  for an SML function  mult that  multiplies  the integers in a list.  Choose a sensible base case for the empty list.


(* mult : int list -> int   *) (* REQUIRES: true         *)

(* ENSURES:  mult(L) evaluates to the product of the integers in L. *)


fun mult [ ] =   (* FILL IN *)

| mult (x::L) = (* FILL IN *);






Task 7.3 Complete the specification for the following function:


(* mult’ : int list * int -> int *) (* REQUIRES: true            *)

(* ENSURES:  mult’(L, a) . . . (* FILL IN *)    *)


fun mult’ ([ ], a) = a

| mult’ (x::L, a) = mult’ (L, x*a);






Task 7.4 Using mult, define an SML function


Mult : int list list -> int


such that  for all lists of integer lists R Mult(R)

evaluates  to the product  of all the integers in the lists of R.

You can do this by filling in the following template:


fun Mult [ ] =


| Mult (r::R) =




Task 7.5 Using mult’, define an SML function


Mult’ : int list list * int -> int


such that  for all lists of integer lists R and all integers a,


Mult’ (R, a)


evaluates  to the product  of a with the product  of all the integers in the lists of R.

You can do this by filling in the following template:


fun Mult’ ([ ], a) =


| Mult’ (r::R, a) =