Modules and Abstract Types (Due November 22)

Due Tuesday, November 22, 2022 at 11:59PM

Due Tuesday, November 22, 2022 at 11:59PM

(A printable version of this homework, with some loss of formatting, is available in PDF format.)


To build systems at scale, we break them into modules, and we put abstraction barriers between the modules. The most effective abstraction barrier we have is data abstraction. It’s a key element in the design of systems implemented in Ada, C, C++, Eiffel, Go, Haskell, Java, and related languages. Modules and abstract types are key mechanisms in Standard ML, which, not coincidentally, has one of the most expressive and powerful system-level abstraction mechanisms ever created.

In this assignment, you will

You’ll do this in two parts:

You may do this work on your own or with a partner.


The code in this handout is installed in /comp/105/lib. Your own code will be compiled using a special script, compile105-sml, which is available on the servers via use comp105. This script does not produce any executable binaries. Instead, it creates binary modules (“.uo files”) that you can load into Moscow ML, as in load "bignum";. The script can compile all files or just a single file:

compile105-sml bignum.sml

Loading binaries into Moscow ML requires an additional argument to mosmlc and mosml, as in

    mosml -I /comp/105/lib -P full

If anything surprises you, consult Appendix I below, which explains your options for compiling.

Dire warnings

Unless otherwise noted below, functions and constructs that were forbidden on earlier assignments remain forbidden.

In addition,


For this homework, you will need to understand the ML Modules section of “Learning Standard ML.” The book chapter on modules, chapter 9, is not included in your abridged edition—the language there is a little too different from Standard ML. Instead of the book chapter, we recommend the sixth lesson on program design: “Program design with abstract data types.”

Reading comprehension

Before starting the programming problems, answer the reading-comprehension questions for this module, which you will find online. It is also OK to alternate between reading-comprehension questions and related homework questions.

Arbitrary-precision signed integers

Standard ML’s primitive type int is limited to machine precision. If arithmetic on int results in a value that is too large or too small to fit in one word, the primitive functions raise Overflow. In this part of the homework, you will implement a true integer abstraction, called bigint, which never raises Overflow. (The worst it can do is make your program run out of memory.)

Large integers are described by this interface:

signature BIGNUM = sig
   type bigint
   exception BadDivision                (* contract violation for sdivmod *)

   val ofInt   : int -> bigint
   val negated : bigint -> bigint       (* "unary minus" *)
   val <+>     : bigint * bigint -> bigint
   val <->     : bigint * bigint -> bigint
   val <*>     : bigint * bigint -> bigint
   val sdivmod : bigint * int -> { quotient : bigint, remainder : int }

     (* Contract for "short division" sdivmod, which is defined only on
        *nonnegative* integers:

        Provided 0 < d <= K and n >= 0,
          sdivmod (n, d) returns { quotient = q, remainder = r }
        such that
          n == q /*/ ofInt d /+/ ofInt r
          0 <= r < d

        Given a negative n or d <= 0, sdivmod (n, d) raises BadDivision.

        Given d > K, the program halts with a checked run-time error.
        The constant K depends on the number of bits in a machine integer,
        so it is not specified, but it is known to be at least 10.
   val compare : bigint * bigint -> order

   val toString : bigint -> string

     (* toString n returns a string giving the natural 
        representation of n in the decimal system.  If n is
        negative, toString should use the conventional minus sign "-".

        And when toString returns a string containing two or more digits,
        the first digit must not be zero.

  val invariant : bigint -> bool  (* representation invariant---instructions below *)


You will not build your implementation from scratch. Instead, you will use ML’s functor mechanism to build on top of natural numbers. Natural numbers implement this interface:

signature NATURAL = sig
   type nat
   exception Negative     (* the result of an operation is negative *)
   exception BadDivisor   (* divisor zero or negative *)

   val ofInt   : int -> nat          (* could raise Negative *)
   val /+/     : nat * nat -> nat
   val /-/     : nat * nat -> nat    (* could raise Negative *)
   val /*/     : nat * nat -> nat
   val sdivmod : nat * int -> { quotient : nat, remainder : int }

     (* Contract for "Short division" sdivmod:

        Provided 0 < d <= K,
          sdivmod (n, d) returns { quotient = q, remainder = r }
        such that
          n == q /*/ ofInt d /+/ ofInt r
          0 <= r < d

        Given a d <= 0, sdivmod (n, d) raises BadDivisor.

        Given d > K, the program halts with a checked run-time error.
        The constant K depends on the number of bits in a machine integer,
        so it is not specified, but it is known to be at least 10.
   val compare : nat * nat -> order

   val decimal : nat -> int list

     (* decimal n returns a list giving the natural decimal
        representation of n, most significant digit first.
        For example,  decimal (ofInt 123) = [1, 2, 3]
                      decimal (ofInt 0)   = [0]
        It must never return an empty list.
        And when it returns a list of two or more digits,
        the first digit must not be zero.

  val invariant : nat -> bool  (* representation invariant---instructions below *)


In this homework, you implement BIGNUM as a functor that takes an implementation of NATURAL as a parameter (Exercise I).

Exercise I: Integers from natural numbers (30%)

Abstract data type: large integers. In file bignum.sml, define a functor BignumFn that takes as its argument a structure N matching signature NATURAL, and returns as its result a structure matching signature BIGNUM. Your functor should look like this:

functor BignumFn(structure N : NATURAL) :> BIGNUM
  ... sequence of definitions here ...

Within the body of the functor, you refer to the representation of natural numbers as type N.nat, and you call operations using the fully qualified names of the functions as in N./+/ (n, m).

ML syntax hint: The introduction form for the quotient/remainder record is

{ quotient = e1, remainder = e2 }

for any expressions e1 and e2.

The elimination form is

let val { quotient = q, remainder = r } = …

for any patterns q and r. (In typical usage, q and r will both be names.) You can see some examples in the section on implementing short division in “Mastering Multiprecision Arithmetic”.

How big is it? My implementation is about 70 lines, of which about 15 are blank.

Representation, abstraction function, and invariant

What you need to know about an integer is how big it is and whether it is negative: its magnitude and sign. Within that constraint, you have your choice of representations. Several representations will work; here are three good ones:

  • Represent the magnitude and sign independently.

  • Encode the sign in a value constructor, and apply the value constructor to the magnitude, as in NEGATIVE mag.

  • Define three value constructors: one each for positive numbers, negative numbers, and zero. A value constructor for a positive or negative number is applied to a magnitude. The value constructor for zero is an integer all by itself.

Each of these representations has its advantages, and they all work. Pick what you think will make your job easy.

Document your representation by writing down the abstraction function and invariant:

  • Define the abstraction function in a comment. We recommend defining the abstraction function using algebraic laws. The right-hand sides should use standard arithmetic notation for operations on the mathematical integers.

  • Write the invariant in an ML function named invariant of type bigint -> bool, located inside the module. Because it is located inside the module, the invariant function has complete access to the representation of bigint.

    The invariant function must typecheck. You may unit-test it if you wish, but you don’t have to.

    It is OK if the invariant function for bigint is not interesting—depending on your chosen representation, it might even be a function that returns true on every input.

Put the abstraction function and invariant inside your BignumFn functor, right after the definition of type bigint.

Two mild warnings

This abstraction has just one pitfall: the integer zero is likely to have two or even three different representations. You must make it impossible for client code to distinguish them. That is, it must be impossible for anyone to write a program that uses the BIGNUM interface and can tell one representation of zero from another. The risks are greatest in exported functions compare and toString.

There is also a minor pitfall: if an adversary hands you the most negative integer (see Int.minInt), its magnitude cannot be represented as a machine integer. For example, if a machine integer had only four bits, it would represent integers in the range from  − 8 to 7, and if someone were to ask you to convert  − 8, you wouldn’t be able to represent positive 8 as a machine integer.

To compute the magnitude of a negative integer safely, use the following steps:

  1. Add one to the negative integer (from  − 8, this would give  − 7)
  2. Negate the sum (this would give 7, no worries!)
  3. Convert the result to nat
  4. Add one to the nat (no worries; nat can be arbitrarily large!)

You would be wise to write a test case involving Int.minInt.

Recommendation: infix operators

Inside your BignumFn functor, I recommend you change the “fixity” of the major operators so you can write both calls and definitions using infix notation. You can change the fixity using these declarations:

infix 6 <+> <->
infix 7 <*> sdivmod

If you also want to use the operations from the NATURAL interface as infix operators, try something like this:

val /+/ = N./+/
val /-/ = N./-/
val /*/ = N./*/

infix 6 /+/ /-/
infix 7 /*/ 

Following these declarations, you can write both definitions and calls using infix notation:

fun thing1 <+> thing2 = ...

... mag1 /+/ mag2 ...

Guidance: algebraic laws

Arithmetic on signed integers requires algorithms you may not have thought about since elementary school. The heavy lifting is done in the implementation of natural numbers (below); the integer abstraction mostly has to get the signs right. To help you get signs right, we provide some algebraic laws. In the laws, magnitudes appear as variables N and M; signs appear as symbols + and -, and the BIGNUM and NATURAL operations are written using infix notation.

  • Multiplication:

    +N <*> +M  ==  +(N /*/ M)
    +N <*> -M  ==  -(N /*/ M)
    -N <*> +M  ==  -(N /*/ M)
    -N <*> -M  ==  +(N /*/ M)
  • Addition:

    +N <+> +M  ==  +(N /+/ M)
    +N <+> -M  ==  +(N /-/ M)  ==  -(M /-/ N)
    -N <+> +M  ==  -(N /-/ M)  ==  +(M /-/ N)
    -N <+> -M  ==  -(N /+/ M)

    Warning: As shown in the laws, adding integers of opposite signs requires subtracting magnitudes. Unless the magnitudes are equal, only one of the subtractions will work—the other will raise exception N.Negative.

  • Subtraction can be implemented by changing the sign of the subtrahend and adding the result to the minuend:2

    +N <-> +M  == +N <+> -M
    ... and so on ...
  • Short division is defined only on nonnegative integers, and it just delegates to N.sdivmod.

  • When “signed zeroes” are involved, comparison becomes tricky. A call to compare mustn’t be able to distinguish a “plus zero” from a “minus zero.” That is, the following algebraic law must hold:

    compare (+0, -0)  ==  EQUAL

    Here are some more general laws:

    compare (+N, +M) = (N, M)
    compare (-N, -M) = (M, N)   (* order is swapped *)
    compare (-N, +M) = LESS,    provided N is not 0 or M is not 0
    compare (+N, -M) = GREATER, provided N is not 0 or M is not 0

Guidance: short division

If it gets a divisor that’s too big, function sdivmod is allowed to fail with a checked run-time error. How big is too big? Anything that makes division on natural numbers fail. In other words, your BignumFn only has to check the sign of the dividend and divisor; it doesn’t have to worry about the magnitude of the divisor.

Related reading: Using the information in the ML learning guide, read about ML signatures, structures, and functors.

Testing your arithmetic

Signed integers are much less tricky than natural numbers, and they don’t require extensive testing. Make sure every line of code you write is exercised by some test, and you’ll be good to go.

Testing is optional, but if you’re keen to do it, I provide some computer-generated tests for signed integers. These tests can be compiled using compile105-sml, but they’re useful only if you also have an implementation of natural numbers. Such an implementation can be thrown together based on the model solutions to the ML homework—as long as you can implement decimal, you can just skip sdivmod. If you create an implementation of NATURAL that uses the representation from the ML homework, you are welcome to share it with other students.

Once you have the modules you need, you can load them into Moscow ML and run the tests:

- load "natural";
- load "bignum";
- load "bignum-tests";
- structure B  = BignumFn(structure N = Natural);
- structure BT = UnitTestsFun(structure Bignum = B);

Avoid a common mistake

It’s a surprisingly common mistake to write a BIGNUM implementation of ofInt that doesn’t work with negative inputs.

It’s a common mistake to think you need to write code in BignumFn to deal with K. The K in the division contract is dealt with entirely in the implementation of natural numbers. All you need to do is catch the exception from NATURAL and raise the exception from BIGNUM.

Two-player games

Data abstraction supports reuse. In this part of the assignment, you’ll reuse code for an algorithm that can play a game. Because the game is abstract, your algorithm will be capable of playing any finite, two-player game of complete information. It will be tested on several!

An abstraction of two-player games

The data in a two-player game are as follows:

Every game is a state machine: the game starts in some initial state, and the game is played by transitioning from state to state. Each transition is triggered by a move. When the game reaches a final state, the game is over. The GAME interface enables software not only to drive the state transitions, but also to ask such questions as “what moves are legal in this state?”, “is the game over?”, and “whose turn is it?”

Manifest (exposed) types: Players and outcomes

The representations of players and outcomes are exposed. (In ML, exposed types are called “manifest”. In other languages, they are “public.”) The types are exposed by the signature PLAYER:

signature PLAYER = sig
  datatype player  = X | O    (* 2 players called X and O *)
  datatype outcome = WINS of player | TIE

  val otherplayer : player -> player
  val unparse     : player -> string 
  val outcomeString : outcome -> string

The signature PLAYER also includes some functions that compute with players and outcomes. Signature PLAYER is implemented by a structure called Player:

structure Player :> PLAYER = struct
  datatype player  = X | O
  datatype outcome = WINS of player | TIE

  fun otherplayer X = O
    | otherplayer O = X

  fun unparse X = "X"
    | unparse O = "O"

  fun outcomeString TIE = "a tie"
    | outcomeString (WINS p) = unparse p ^ " wins"

To refer to Player types, constructors, and functions, you will use the “fully qualified” ML module syntax, as in the examples Player.otherplayer p, Player.X, Player.O, and Player.WINS p. The last three expressions can also be used as patterns.

First abstraction and abstract type: Moves

The move abstraction helps communicate gameplay to a human player. Functions visualize, prompt, and parse can respectively show, request, and understand moves.

signature MOVE = sig
  type move               (* A move---perhaps an (x,y) location *)
  exception Move          (* Raised for invalid moves *)

  (* CREATOR *)
  val parse : string -> move
        (* Converts the given string to a move; If the string 
           doesn't correspond to a valid move, parse raises Move *)

  val prompt : Player.player -> string
        (* A request for a move from the given player *)
        (* Example: "What square does player O choose?" *)

  val visualize : Player.player -> move -> string
        (* A short string describing a move. 
           Example: "Player X picks up ...".
           The string may not contain a newline. *)

Second abstraction: a game (with abstract states and moves)

A game implements signature GAME. This signature (and its contract) subsumes the contracts for the abstract types move and state, as well as all the exported functions. The game state includes complete information about a game in progress, even “invisible” state like whose turn it is. Each operation’s contract is expressed in terms of how it affects states.

The state abstraction is immutable—if a mutable representation is chosen, it must be impossible for a client to tell if a mutation has taken place.

signature GAME = sig
  structure Move : MOVE
  type state 

  (* CREATORS *)

  val initial : Player.player -> state
    (* Initial state for a game in which 
       the given player moves first. *)


  val makemove: state -> Move.move -> state
    (* Returns the state obtained by making the 
       given move in the given state.  Raises Move.Move
       if the state is final or if the given move is not
       legal in the given state. *)


  val visualize : state -> string
    (* A string representing the given state.  
       The string must show whose turn it is, 
       and it must end in a newline. *)

  val whoseturn  : state -> Player.player
    (* Given a _non-final_ state, returns the player
       whose turn it is to move.    When given a
       final state, behavior is unspecified. *)

  val isOver : state -> bool
    (* Tells if the given state is final. *)

  val outcome : state -> Player.outcome option
    (* When given a final state, returns SOME applied to the outcome.
       When given a non-final state, returns NONE. *)

  val legalmoves : state -> Move.move list
    (* Lists all moves that are valid inputs to `makemove` in the
       given state.  The list is empty if and only if the given
       state is final. *)


This interface is not minimal. For example, there are three different ways to tell if a game is over. They must all agree!

Third abstraction: a game solver

The GAME interface supports a computer opponent based on exhaustive search: an abstract game solver (AGS). An AGS searches possible moves until it finds a move that leads to a win. If it can’t find such a move, it tries to force a draw.

The AGS knows nothing about the rules of any game, or even whose turn it is—it knows only which moves can be made in which states, which states are final, and what outcomes are good. It gets all that information from the GAME interface. Provided there aren’t too many states, the AGS makes a worthy opponent.

The AGS interface exports just one function: all a client can do is present a state and ask the AGS what the current player (whose turn it is) should do in that state. If there are any legal moves, the AGS recommends one. And it always says what final outcome is expected, assuming that its recommendation (if any) is followed and that no player ever makes a suboptimal move.

In addition to the advice function, the AGS contains a complete copy of the game itself! In effect, the AGS extends the Game with new functionality. In ML, this idiom is common.

signature AGS = sig
  structure Game : GAME
  type advice = { recommendation : Game.Move.move option
                , expectedOutcome : Player.outcome
  val advice : Game.state -> advice
    (* Given a non-final state, returns a recommended move,
       plus the expected outcome if the recommendation is followed.
       Given a final state, returns no recommendation,
       plus the outcome that has already been achieved. *)

The cost model of an AGS is that it tries all possible combinations of moves. AGS functions can be slow. Wait patiently.

Exercise G: A game (30%)

Implementing a game will solidify your understanding of the GAME interface, which in turn will help you build a good solver. You will choose from these three games:

Each game is more interesting to play (and more time-consuming to implement) than the game before it.

G. Implement a two-player game. Choose a game and implement one of the following:

  • Functor SticksFun in file sticks.sml
  • Structure Coins in file coins.sml
  • Structure TTT in file ttt.sml

Detailed instructions, rules for play, and hints for implementation can be found on the games page. The instructions include the strings that must be recognized by Move.parse. You need these instructions!

All three games can be played on the departmental servers: run coins, sticks, or ttt.5 In the first two games, if you play perfectly, you can beat my AGS. In the third, the best you can do is tie.

What to watch out for. We’ll test your code to see if your observers provide a consistent picture of a game’s state. For example, if your implementation says that player X won, it had better also say that the game is over.

How big are they? Not counting embedded unit tests, my implementations look like this:

  • My pick up the last stick is 51 lines of Standard ML, but 13 of those lines are blank.

  • My take the last coin is 107 lines of code and comments, but 23 of those lines are blank. Most of the length is in visualize (11 lines) and in makemove (12 lines). Depending on what representations you choose, your hot spots will be different.

  • My tic-tac-toe is 89 lines of Standard ML, of which 5 are blank. Even though it’s shorter than my coins game, it was more tiresome to write. (A less clever implementation written by a student at CMU is 144 lines.)

Related reading: The contract of the GAME signature, the instructions for the game of your choice on the games page, and the section on ML modules in the ML learning guide.

Test your game with my AGS

Once you’re satisfied with your game, you can test it with my AGS. You’ll load a functor I’ve written that instantiates my AGS with your game, then runs the “play manager” with that AGS. Using Moscow ML interactively, follow these steps:

  1. Start mosml with the options -I /comp/105/lib -P full.
  2. Load .uo files with the load command.
  3. Use functor application to create a player.
  4. Play interactively.

The steps are illustrated by this the annotated transcript below, which uses take the last coin as an example:6

nr@homedog /tmp> mosml -I /comp/105/lib -P full
Moscow ML version 2.10-4 (Tufts University, April 2017)
Enter `quit();' to quit.
- load "playvsnr";   <---- play manager with my AGS
> val it = () : unit
- load "coins";      <---- your game
> val it = () : unit
- structure P = PersonVsAgs(structure Game = Coins);
> structure P : {val go : unit -> outcome}
- P.go ();
Player X sees 4 quarters, 10 dimes, and 7 nickels.
What coins does player X take?

If you want to watch the computer play both sides, try this:

nr@homedog /tmp> mosml -I /comp/105/lib -P full
Moscow ML version 2.10-4 (Tufts University, April 2017)
Enter `quit();' to quit.
- load "playvsnr";   <---- play manager with my AGS
> val it = () : unit
- load "coins";      <---- your game
> val it = () : unit
- structure C = AgsVsAgs(structure Game = Coins);
> structure C : {val go : unit -> outcome}
- C.go ();
Player X sees 4 quarters, 10 dimes, and 7 nickels.
Player X takes [redacted]

With this experience in hand, you’re ready for the final module of the assignment: the AGS itself.

Exercise A: Abstract Game Solver (40%)

Given a state, an AGS should advise us on the best move for the player whose turn it is.

The AGS considers legal moves one by one—but to compute the consequences of a move, the AGS calls itself recursively. So the search can be exponential in the length of the game. If you’ve ever studied AI or search algorithms, you may be aware that there are lots of fancy tricks you can use to cut down the size of a search like this. Ignore all of them. Implement the advice function in the simplest way possible.

Because the state type is abstract, the AGS can’t break down the state by forms of data. What the AGS can and should do is break down observations. I recommend breaking down these observations:

Implement an Abstract Game Solver. Choose one of the implementation tactics and write an AGS in file ags.sml. Use the following template:

functor AgsFun (structure Game : GAME) :> 
     where type Game.Move.move = Game.Move.move
     and   type Game.state     = Game.state
= struct
  structure Game = Game

  ... helper functions, if any ...

  fun advice state = ...

The where type declarations are annoying: they look tautological, but they’re not. Complain to Dave MacQueen and Bob Harper.


  • ML syntax: The introduction form for the advice record is

    { recommendation = e1, expectedOutcome = e2 }

    for any expressions e1 and e2.

    The elimination form is

    let val { recommendation = x1, expectedOutcome = x2 } = …

    for any names x1 and x2.9

  • Do not assume that players take turns, that the last player to move always wins, that there are no ties, or any other property of the game you have implemented. Use whoseturn and outcome instead. We will test your AGS on games that are quite different, including Connect 3 and others. Even a solitaire!

  • Unit tests are useful, but they are hard to write inside an AGS. If you want unit tests, write them for a particular game. Start with a known configuration and check the two fields returned by advice. For example, if a computer player sees a table with only one denomination of coin, its best move is to take all the coins and win. Unit tests like these are game-specific and will have to go into another module. Put them in file ags-tests.sml.

To test your AGS, restart Moscow ML and load "ags";. As long as there is an ags.uo in the current working directory, Moscow ML will prefer it to the one we provide in /comp/105/lib. You’ll be able to run your unit tests. You can also run game-playing integration tests designed for your AGS. You can do this with your own game or one of ours.

How big is it? My CPS version takes about 65 lines of Standard ML. My direct recursion takes under 40 lines. My version using argAtCeilingOtherwiseMax takes only 25 lines, but a module exporting atLeastNOrBestOffer takes another 20 lines.

Related reading:

A common mistake to avoid when debugging your AGS

If you build a simple AGS that fits in 40 lines of code, it is not going to try to fool you: if the AGS cannot force a win, it will pick a move more or less arbitrarily. A simple AGS has no notion of “better” or “worse” moves; it knows only whether it can force a win.

Here’s the common mistake: you’re playing against the AGS, and it makes a terrible move. You think it’s broken. For example, suppose you are playing Tic-Tac-Toe, with you as X, the AGS as O, and play starting in this position:

        |   | O |   |
        |   | X |   |
        |   |   |   |

You place your X in the upper left corner. The AGS does not move lower right to block you. Is it broken? No—the AGS recognizes that you can force a win, and it just gives up.

If you want an AGS that won’t give up, for extra credit you can implement an aggressive version that will delay the inevitable as long as possible. An aggressive AGS searches more states so that it can (a) win as quickly as possible and (b) hold on in a lost position as long as possible.

How big is it? My aggressive AGS is under 60 lines of Standard ML code.

Test your AGS with my game

Testing your AGS with one of my games requires a slightly different play manager. As an example, I test an AGS with the binary implementation of Tic-Tac-Toe from file /comp/105/lib/ttt.uo:

nr@homedog /tmp> mosml -I /comp/105/lib -P full
Moscow ML version 2.10-4 (Tufts University, April 2017)
Enter `quit();' to quit.
- load "ags";     <--- your AGS
> val it = () : unit
- load "ttt";     <--- my game
> val it = () : unit
- load "playvsags";  <--- play manager with no AGS
> val it = () : unit
- structure TTTAgs = AgsFun(structure Game = TTT);
> structure TTTAgs : ...
- structure P = PersonVsMyAgs(structure Ags = TTTAgs);
> structure P : {val go : unit -> outcome}
- P.go();
|   |   |   |
|   |   |   |
|   |   |   |
Player X is to move

Square for player X?

If you want to test your own AGS with our Tic-Tac-Toe, our parser recognizes squares upper left, upper middle, upper right, middle left, middle, middle right, lower left, lower middle, and lower right, as well as abbreviations ul, um, ur, ml, m, mr, ll, lm, and lr.

Extra Credit

75c option. Implement a game that is like take the last coin, but that has an additional rule: if you take exactly 75 cents’ worth of coins, you have the option to move again immediately, before your opponent takes a turn. (As implied by the word “option,” the extra move is a choice—a player is never required to exercise the option.) If you choose this extra credit, document your code with an explanation of what you chose to change and why.

Game theory. Professor Ramsey challenges you to a friendly game of “pick up the last stick,” with one thousand sticks. The stakes are a drink at the Tower Cafe. As the person challenged, you get to go first. Should you accept the challenge, or should you insist, out of deference to the professor’s age and erudition, that the professor go first? Justify your answer.

More. Implement a second game from the approved list.

Even more. Implement all three games on the approved list.

Four. Implement Connect 4.

Aggression. If it can’t win, a standard AGS will “give up”—if every move leads to a loss, all moves are equally bad, and it apparently moves at random. That’s because a standard AGS assumes that its opponent is perfect. In the dual situation, when the standard AGS knows it can win no matter what, it picks a winning move at random instead of winning as quickly as possible. This behavior may lead you to suspect bugs in your AGS. Don’t be fooled.

Change your AGS so that it delays losing as long as possible, and when it can win, it wins as quickly as possible. (For example, it should prune the search only if it finds a win on the next move.) One technique is to assign each move a floating-point “benefit” (of type real) and to choose the move with the highest benefit.

What and how to submit

Do not copy or submit the signatures provided with the assignment. Submit just the following files:

The ML files that you submit should contain all structure and function definitions that you write for this assignment (including any helper functions that may be necessary), in the order they should be compiled. Don’t include copies of files we provide; we already have them, and duplicates can confuse the compiler.

The files you submit must compile with Moscow ML, using the compile105-sml script we give you. We will reject files with syntax errors or type errors. Your files must compile without warning messages.

As soon as you have the files listed above, run submit105-sml-pair to submit a preliminary version of your work. Keep submitting until your work is complete; we grade only the last submission.


Appendix I: Two ways to compile Standard ML modules

The Definition of Standard ML does not specify where or how a compiler should look for modules in a filesystem. And each compiler looks in its own idiosyncratic way. This issue should be handled by compile105-sml, but if something goes wrong, this appendix explains not only what is going on but also how to compile with MLton. (When the time comes to test your game, MLton is going to be important.)

Compiling Standard ML modules using Moscow ML

To compile an individual module using Moscow ML, you type

mosmlc -I /comp/105/lib -c -toplevel filename.sml

This command puts compiler-interface information into filename.ui and implementation information into filename.uo. Perhaps surprisingly, either a signature or a structure will produce both .ui and .uo files. This behavior is an artifact of the way Moscow ML works; don’t let it alarm you.

If your module depends on another module, you will have to mention the .ui file on the command line as you compile. For example, a BignumFn functor depends on both NATURAL and BIGNUM signatures. If BignumFn is defined in bignum.sml, NATURAL is defined in natural-sig.sml, and BIGNUM is defined in bignum-sig.sml, then to compile BignumFn you run

mosmlc -I /comp/105/lib -toplevel -c natural-sig.ui bignum-sig.ui bignum.sml

The script compile105-sml knows about the files that are assigned for the homework, and in most situations it inserts the .ui references for you.

Calling mosmlc produces a bignum.uo file. The .uo files are good for two things:

Compiling Standard ML to native machine code using MLton

If your games are running too slow, compile them with MLton. MLton is a whole-program compiler that produces optimized native code. Because of its superior error messages, MLton can be worth using on other codes, too.

To use MLton, you list all your modules in an MLB file, and MLton compiles them at one go. If you want to try MLton on your own codes, download file, then compile with

mlton -verbose 1

You can then run any Unit tests with ./sml-homework.10

More information about MLton is available on the man page and at

If you want to try this with the coins game, download files and runcoins.sml, and then compile with

mlton -output coins -verbose 1

(You can also download and compile, runsticks.sml,, and runttt.sml.)

Because MLton requires source code, you will be able to use it only after you will have finished your own AGS.

Appendix II: How your work will be evaluated

Program structure

We’ll be looking for you to seal all your modules. We’ll also be looking for the usual hallmarks of good ML structure.

Exemplary Satisfactory Must Improve

• All modules are sealed using the opaque sealing operator :>

• Code uses basis functions effectively, especially higher-order functions on list and vector types.

• Code has no redundant case analysis

• Code is no larger than is necessary to solve the problem.

• Most modules are sealed using the opaque sealing operator :>

• Code uses the familiar functions, but misses opportunities to use unfamiliar functions like Vector.tabulate.

• Code has one redundant case analysis

• Code is somewhat larger then necessary to solve the problem.

• Only some or no modules are sealed using the opaque sealing operator :>

• A module is defined without ascribing any signature to it (unsealed)

• Code misses opportunities to use map, fold, or other familiar HOFs.

• Code has more than one redundant case analysis

• Code is almost twice as large as necessary to solve the problem.

Correctness and performance

We’ll look to be sure your code meets specifications, and that the performance of your AGS is as good as reasonably possible.

Exemplary Satisfactory Must improve

• Game code fulfills the contracts specified in the GAME and MOVE signatures. In particular, every observer and producer presents a consistent view of whether the game is over.

• AGS code makes no additional assumptions about the implementations of Player, Move, or Game.

• Game code fulfills the contracts specified in the GAME and MOVE signatures, except that it permits illegal moves.

• Game code fulfills the contracts specified in the GAME and MOVE signatures, except it sometimes raises the wrong exception.

• Game code violates one of its contracts.

• AGS code assumes that players take turns.


• The AGS implements its advice function using a single, pruned search that stops once the best move or outcome is known.

Or, the AGS implements advice by making just one search through the state space of possible game configurations.

• Function advice may search the state space of possible configurations more than once.

• Function advice may search the state space of possible configurations more than twice.

  1. If you copy a signature, sml-lint will complain. Don’t copy signatures.

  2. Words like “subtrahend” and “minuend” are useful, but I always have to look them up.

  3. The abstraction is based on a design by George Necula.

  4. If you want to tackle Tic-Tac-Toe, a good representation of the grid can be created using Standard ML’s Vector module.

  5. All binaries can be found in /comp/105/bin.

  6. If you choose the sticks game, you’ll have to instantiate your functor SticksFun with val N = 14.

  7. To make the pattern matching exhaustive, the compiler also requires you to handle the empty list. Since an empty list is impermissible according to the contract for legalmoves, I recommend that you handle this case with an expression like “let exception ContractViolation in raise ContractViolation end.”

  8. Function argAtCeilingOtherwiseMax is gnarly, and I can’t see a way around it.

  9. Please choose good names!

  10. The mlton-with-unit script we’ve been using won’t work here, because it’s limited to one source file at a time.