Overview
In this assignment, you will
 Learn about abstract data types
 Learn about arbitraryprecision arithmetic, which you will see again
 Use Standard ML modules to write client code for an interface that is not yet implemented
 Use Standard ML modules to put together a nontrivial program
 See how to reuse code that depends not just on other values, but also on other types
You’ll accomplish these ends by completing the four parts of the assignment:
In reading comprehension, you read about new language features (ML modules), new ideas (abstraction function and representation invariant), and new algorithms (arithmetic)
In the problem Q, you write client code that uses a priority queue—without an implementation. In effect, you implement heapsort despite not having a heap. This part involves a lot of reading and understanding what is being presented and asked. There is just a tiny amount of programming.
In problem N, you implement arbitraryprecision arithmetic on natural numbers. These socalled “bignums” are an important programminglanguage abstraction, and they are an ideal problem for exploring the freedom of choice offered to you by abstract types. Substantial thought and substantial programming are required.
In problem A, you implement an unsophisticated (yet unbeatable) computer opponent that can be used to play many different twoplayer games. To help you understand and test this very popular problem, you will also implement problem S: an unsophisticated game called “pick up the last stick.” The computer opponent requires careful thought but not a lot of code: in essence, it boils down to one carefully crafted recursive function. The simple game requires almost no thought, and not a lot of code—though it will seem like a lot, because it is all boilerplate.
As usual, you do readingcomprehension questions on your own. You may tackle the remaining problems either on your own or with a partner.
Setup
The code in this handout is installed for you in /comp/105/lib
, where you don’t have to look at it. You will compile your own code using a special script, compile105sml
, which is available on the servers through the usual command use comp105
. This script does not produce any executable binary. Instead, it creates binary modules (“.uo
files”) that you can load into Moscow ML, as in load "ags";
. You can use it to compile all files or just a single file:
compile105sml
compile105sml ags.sml
To be able to load the binaries that we provide, you must supply an additional argument to mosmlc
and mosml
, as in
mosml I /comp/105/lib P full
Reading comprehension
These problems will help guide you through the reading. We recommend that you complete them before starting the other problems below. You can download the questions.
Using one of the sources in the ML learning guide, read about structures, signatures, and matching. Then answer questions about the structure and signature below.
The following structure contains definitions that should be familiar from the ML homework and from code you may have seen in the course interpreters:
structure ExposedEnv = struct type name = string type 'a env = (name * 'a) list exception NotFound of name val emptyEnv = [] fun lookup (name, []) = raise NotFound name  lookup (name, (x, v) :: pairs) = if x = name then v else lookup (name, pairs) fun bindVar (name, value, env) = (name, value) :: env end
Here is a signature:
signature ENV = sig type name = string type 'a env val emptyEnv : 'a env val lookup : name * 'a env > 'a val bindVar : name * 'a * 'a env > 'a env end
Answer these questions:
Does the signature match the structure? That is, if we write
structure Env :> ENV = ExposedEnv
does the resulting code typecheck?
Does the signature expose enough information for us to write the following function? Explain why or why not.
fun extendEnv (names, vals, rho) = ListPair.foldrEq Env.bindVar rho (names, vals)
Does the signature expose enough information for us to write the following function? Explain why or why not.
fun isBound (name, rho) = (Env.lookup (name,rho) ; true) handle Env.NotFound _ => false
If in part (b) or part (c), it is not possible to write the function given, explain how you would change the code to make it possible.
Suppose you change the
ENV
signature to make thename
type abstract, so the signature readssignature ENV = sig type name type 'a env ... end
Is more abstraction better here? Or does the change have unfortunate consequences? Justify your answer.
You now have the basic ideas needed to understand what is being asked of you in this assignment, and you know enough to implement most of the “pick up the last stick” game (problem S).
An ML functor is a function that operates on the module level. Its formal parameters, if any, are specified by a sequence of declarations, and its actual parameters are given by a sequence of definitions. Its result is a structure. Read about functors in Harper, as recommended in the ML learning guide.
Here’s a typical application of functors. To keep track of the thousands of tests we run on students’ code, I need an efficient “test set” data structure. But not all tests have the same type. To reuse the data structure with tests of different types, I need a functor. Here is what my “test set” functor needs to know about a test:
 A string identifying the student who wrote it
 A comparison function that provides a total order on tests
 A function that converts a test to a string, for printing
Using this information, answer parts (a) and (b):
Write down the information needed for a test set in the form of formal parameters for the functor
TestSetFun
, keeping in mind that a functor’s formal parameters are written as a sequence of declarations:functor TestSetFun( ... fill in formal parameters here ... ) :> TEST_SET = struct ... end (* ignore this part *)
Now focus your attention on one particular test, the
checktype
test:The test is identified by a student’s user ID and a sequence number 1, 2, or 3.
The test contains an expression of type
exp
and a type of typety
.Comparison can be implemented using code like this:
case String.compare (uid1, uid2) of EQUAL => Int.compare (seqno1, seqno2)  diff => diff
A test can be converted to a string code like this:
concat ["(checktype ", expString e, " " , tyString tau, ")"]
Show how to create a set of
checktype
tests by filling in the actual parameters for theTestSetFun
functor:structure CheckTypeSet :> (TEST_SET where type test = check_type_test) = TestSetFun( ... fill in actual parameters here ... )
You now understand functors well enough to use them in problems 1 and 2.
Read about “signature refinement or specialization” in the ML learning guide. Now,
Explain what, in part (b) of the previous question, is going on with the
where type
syntax.Explain what would go wrong if we wrote this code instead:
structure CheckTypeSet :> TEST_SET = TestSetFun(...)
You now understand functors well enough to know how to refine the result signature of your Abstract Game Solver in problem A.
Many useful abstractions can be implemented using the humble binary tree. Let’s consider how we might use a binarysearch tree to store a large data structure containing all of the software testing results from 105. I want to know for each test, for each student, did the student’s submission pass the test.
Read Ramsey, section 8.1, on abstractions, representations, and invariants. Focus on places where binary trees are mentioned. Also read the first three pages or section 8.8.4 on hash tables, which represent a similar abstraction. (Skip all the code parts.)
In informal English (“the world of ideas”), what is the abstraction represented by the binarysearch tree?
Suppose a binarysearch tree is represented as follows:
datatype tree = EMPTY  NODE of tree * (test * student * bool) * tree
Using some combination of ML code and informal English, write the abstraction function which maps this representation onto the abstraction that it stands for:
Using ML code, and assuming whatever functions you need on types
test
andstudent
, write the representation invariant of the binarysearch tree. (The representation invariant of a binarysearch tree is the order invariant which guarantees the correctness ofinsert
andlookup
.)
You now know enough to design an abstraction function and representation invariant for your chosen representation of natural numbers in problem 2.
In “Mastering Multiprecision Arithmetic”, read the sections on addition, subtraction, and multiplication.
Given that you may be adding two numbers of different lengths (e.g., 1777944 + 903), what sort of case analysis do you expect to need to implement addition?
Given that you may be subtracting two numbers of different lengths (e.g., 1777944  903), what sort of case analysis do you expect to need to implement subtraction?
Given that you may be multiplying two numbers of different lengths (e.g., 1777944 * 903), what sort of case analysis do you expect to need to implement multiplication?
You are now prepared to tackle the implementation part of problem 2.
Programming problem Q: The module warmup
Q. Abstract data type: priority queues. A priority queue is a collection of elements with an operation that quickly finds and removes a minimal element. (The priority queue assumes that a total order exists on the elements; a minimal element is an element of the priority queue that is at least as small as any other element. A priority queue may contain more than one minimal element, in which case it is not specified which such element is removed.)
Like most data structures, priority queues come in both immutable and mutable forms. Immutable data structures are easier to test, specify, and share. Mutable data structures usually have lower space costs and sometimes lower time costs. Here is an interface that describes an immutable priority queue:
signature PQUEUE = sig
type elem (* element (a totally ordered type) *)
val compare_elem : elem * elem > order (* observer: total order of elems *)
type pqueue (* Abstraction: a sorted list of 'elem' *)
val empty : pqueue (* the empty list *)
val insert : elem * pqueue > pqueue (* producer: insert (x, xs) == sort (x::xs) *)
val isEmpty : pqueue > bool (* observer: isEmpty xs == null xs *)
exception Empty
val deletemin : pqueue > elem * pqueue
(* observer: deletemin [] = raises the Empty exception
deletemin (x::xs) = (x, xs) *)
(* cost model: insert and deletemin take at most logarithmic time;
isEmpty takes constant time*)
end
The type pqueue
is abstract, so its representation is not specified, but the interface says what abstraction a pqueue
corresponds: a sorted list of values of type elem
.^{1} Each operation is described by giving the imagined effect on the abstraction. When specifying an interface that uses abstract data types, working with an abstraction in this way is a best practice—but not always easy or even possible.
Here is a mutable version of the same abstraction:
signature MPQUEUE = sig
type elem (* element (a totally ordered type) *)
val compare_elem : elem * elem > order (* observer: total order of elems *)
type pqueue (* Abstraction: mutable cell containing a sorted list of 'elem' *)
val new : unit > pqueue (* returns `ref []` *)
val insert : elem * pqueue > unit (* q := sort (x :: !q) *)
val isEmpty : pqueue > bool (* observer: isEmpty xs == null xs *)
exception Empty
val deletemin : pqueue > elem (* remove and return the first element *)
end
I can conclude that the abstraction is mutable just by inspecting the types: I see unit
types, and I observe that deletemin
doesn’t return a pqueue
.
Programming problem Q. Use a priorityqueue abstraction to implement a sort module matching this signature:
signature SORT = sig
type elem
val compare : elem * elem > order
val sort : elem list > elem list
end
You may use either of the abstractions defined above. Your choice of abstraction will determine the details of your solution:
If you choose the immutable abstraction, write a functor
PQSortFn
that takes a structure matching signaturePQUEUE
and produces a structure matchingSORT
, like this:functor PQSortFn(structure Q : PQUEUE) :> SORT where type elem = Q.elem = struct ... end
If you choose the mutable abstraction, write a functor
MPQSortFn
that takes a structure matching signatureMPQUEUE
and produces a structure matchingSORT
, like this:functor MPQSortFn(structure Q : MPQUEUE) :> SORT where type elem = Q.elem = struct ... end
Either way, place your solution in file sort.sml
. You can compile it by running
compile105sml sort.sml
You need not implement PQUEUE
or MPQUEUE
. This is the whole point! (Just keep in mind that without an implementation, you cannot run unit tests.)
Also, you do not need the source code that defines the signatures PQUEUE
and MPQUEUE
—the compile105sml
script finds those files in /comp/105/lib
.
How big is it? The body of the functor is under 10 lines of code.
Related reading:
Read the brief discussion of the priorityqueue abstraction on page 538 of Ramsey.
If you wish, you may model your sort on the heapsort presented in Section 8.8.2, which starts on page 594. Instead of CLU’s
for
loop, you can use a higherorder function.Read about functors in sources from the ML learning guide.
What’s the point? This problem illustrates a key advantage of the ML module system: you can program against an unimplemented interface. When you’re designing an interface, you often need to know if the interface meets the needs of its clients, and you want to get the design right before you spend the budget needed to create an implementation. Several languages provide mechanisms that will do the job (e.g., Java interfaces); in ML, those mechanisms are signatures and functors.
Programming problem N: Arbitraryprecision natural numbers
Standard ML 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 implementation raises Overflow
. In this problem, you implement natural numbers—or if you prefer, unsigned integers—of arbitrary precision. Arithmetic will never overflow; the worst that can happen is to run out of memory.
You will implement this interface:
signature NATURAL = sig
type nat
exception Negative (* raised if the result of any operation would be negative *)
exception BadDivisor
val of_int : int > nat (* could raise Negative *)
val /+/ : nat * nat > nat
val // : nat * nat > nat (* could raise Negative *)
val /*/ : nat * nat > nat
val sdiv : nat * int > { quotient : nat, remainder : int }
(* Contract for "Short division" sdiv:
Provided 0 < d <= base,
sdiv (n, d) returns { quotient = q, remainder = r }
such that
n == q /*/ of_int d /+/ of_int r
0 <= r < d
Given a d out of range, sdiv (n, d) raises BadDivisor
We know that base >= 10, but otherwise its value is unspecified
*)
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 (of_int 123) = [1, 2, 3]
decimal 0 = [0]
It must never return an empty list.
*)
end
N. Abstract data type: natural numbers. You will design and build your implementation in two parts.
Abstraction function and representation invariant.
A natural number should be represented by a sequence of digits. But this still leaves you with interesting design choices:
Do you prefer an array or a list?
If you prefer an array, do you want a mutable array (type
'a array
) or an immutable array (type'a vector
)?If you prefer a list, do you prefer to store the digits of a number in bigendian order (most significant digit first) or littleendian order (least significant digit first)?
(If you prefer an array, the only order that makes sense is to store the least significant digit in slot 0, and in general, to store the digit that is multiplied by b^{n} in slot n.)
Some implementations of arithmetic operations, especially multiplication, can leave leading zeroes in a representation. The number of leading zeroes must be controlled so that it doesn’t grow without bound.
Do you prefer a representation invariant the eliminates leading zeroes?
If not, how will you track the number of leading zeroes in each representation? Will you store it in the representation itself, or will you compute it dynamically?
The beauty of abstract data types is that the interface isolates these decisions, so you can easily change them.
In part (a) of this problem, you pick a representation and an invariant. You document them by writing an abstraction function and representation invariant. Do so by completing this template:
functor NaturalRep (structure N : NATURAL) :> sig datatype digit = D of int type rep = ... (* your choice of representation *) val invariant : rep > bool val abstraction : rep > N.nat end = struct datatype digit = D of int type rep = ... exception LeftAsExercise of string val base = raise LeftAsExercise of string fun invariant r = raise LeftAsExercise "invariant" fun abstraction r = raise LeftAsExercise "abstraction" end
Place your solution in file
naturalrep.sml
. Compile bycompile105sml naturalrep.sml
Put a note in your README saying which representation you picked and why.
Hints:
No one representation is equally good for all operations. You can pick one representation and stick with it, or you can convert temporarily as needed to facilitate each operation. You can even define a representation as a set of alternatives, as in
datatype nat = ARRAY of digit array  LITTLE_ENDIAN of digit list  BIG_ENDIAN of digit list
A little freedom can be dangerous: if you choose a representation like this one, you risk having to handle at least nine cases per binary operator. But the choice is yours.
As you complete part (b), you may want to revisit your decisions. That’s part of the point—representations can change without affecting any external code.
Related reading:
Read about “Choice of representation” on the first page of the handout “Mastering Multiprecision Arithmetic”
Read the algebraic formulation of arithmetic in Section 8.9.2, which starts on page 604 of Ramsey, and look at the examples in that section.
Read about representation invariants and abstraction functions on page 584 of Ramsey. If you want more depth, the first part of Section 8.7, which starts on page 582, may be useful. Don’t bother with sections 8.7.1, 8.7.2, or 8.7.3.
By this point, I expect you’re solid on ML signatures, structures, and functors. If not, there’s the ML learning guide.
Implementation of all the operations.
In file
natural.sml
, write a structureNatural
that implements signatureNATURAL
. The right way to do it is with code that looks like this:(* inconvenient code *) structure Natural :> NATURAL = struct ... type nat = ... (* abstraction: ... *) (* invariants: ... *) ... end
But there’s a small problem here: once the module is sealed, you will find it almost impossible to debug. Here’s a trick of the ML masters:
(* convenient code *) structure ExposedNatural = struct (* Look! No ascription *) ... type nat = ... (* abstraction: ... *) (* invariants: ... *) ... end structure Natural :> NATURAL = ExposedNatural (* the module is sealed here *)
You can debug
ExposedNatural
, while your client code uses the sealedNatural
.Whichever way you choose to write it, be sure you seal structure
Natural
with the:>
operator, so that no client code can see the representation and violate its invariants. Sealing is especially important if you choose a mutable representation of this immutable abstraction.Be sure also to document your abstraction function and representation invariant here also. A couple of informal sentences will do nicely.
How big is it? I wrote two implementations: one arraybased and one listbased. Together they total about 270 lines of code, of which 50 of those lines are blank. The two implementations are roughly the same size.
Related reading:
The short handout “Mastering Multiprecision Arithmetic” recommends exactly how to implement natural numbers, including what helper functions you will find most useful and what operations to implement first.
You will find many hints in Exercise 39 on page 636.
For addition and subtraction, read Ramsey, page 605; and Hanson, pages 305 to 307.
For multiplication, read Ramsey, page 606; and Hanson, pages 307 and 308.
For short division, read Ramsey, pages 606 and 607; and Hanson, pages 311 and 312 (the
XP_quotient
function). What Ramsey calls a “remainder” r_{i} is calledcarry
by Hanson.By this point, I expect you’re solid on ML signatures, structures, and functors. If not, there’s the ML learning guide.
Mistake to avoid: Don’t try to multiply by repeated addition—multiplication of large numbers won’t terminate in your lifetime. You must multiply each pair of digits (one each from the multiplicand and the multiplier) and add up the partial products, appropriately shifted.
What’s the point? The key point here is that abstract data types put a firewall between an interface and its implementation, so that you can easily change the base of natural numbers, or even the representation, and no program can tell the difference. In Standard ML, this degree of abstraction is achieved through opaque signature ascription, known to a select group of 105 students as “the beak.”
Testing arbitraryprecision arithmetic
To be able to test anything at all, you need of_int
and decimal
, which probably also means sdiv
. If you can’t get sdiv
working and you’re desperate to write tests, try using base 10 and writing a specialpurpose implementation of decimal
.
With these in place, compute some 20digit numbers by taking a list of digits and folding with multiply and add. Since you only have to multiply by 10, you can test addition without multiplication. Here’s a function that multiplies by 10:
fun tenTimes n =
let infix 6 /+/
val p = n
val p = n /+/ n (* p == 2n *)
val p = p /+/ p (* p == 4n *)
val p = p /+/ n (* p == 5n *)
val p = p /+/ p (* p == 10n *)
in p
end
Once you are confident that addition works, you can test subtraction of natural numbers by generating a long random sequence, then subtracting the same sequence in which all digits except the most significant are replaced by zero.
You can create more ambitious tests of subtraction by generating random natural numbers and using the algebraic law (m + n)−m = n. You can also check to see that unless n is zero, m − (m + n) raises the Negative
exception.
It is harder to test multiplication, but you can at least use repeated addition to test multiplication by small values.
You can also easily test multiplication by large powers of 10.
Beyond these simple kinds of tests, civilized people don’t write unit tests by hand—we write computer programs that generate unit tests. At any point, you can just generate random expressions that compute with large numbers, then compare your results with a working implementation of bignums. Both MLton and Standard ML of New Jersey (command sml
) provide a structure IntInf
that implements arbitraryprecision signed integers.
I have provided some computergenerated tests.
Understanding and representing adversary games
In problems S and A below, you will implement and use a system for playing simple adversary games. The program will show game configurations, accept moves from the user, and choose the best move.
The system is based on an abstract game solver (AGS) which, given a description of the rules of the game, will be able to select the best move in a particular configuration. An AGS is obtained by abstracting (separating) the details of a particular game from the details of the solving procedure. The solving procedure uses exhaustive search: it tries all possible moves and picks the best. Such a search can solve games of complete information, provided the configuration space is small enough. And the search is general enough that we can abstract away details of many games, separating the implementation of the solver from the implementation of the game itself.
Separating game from solver in such a way that a single solver can be used with many games requires a carefully designed interface. In this problem, we give you such an interface, which is specified using the SML signature GAME
. (The signature was designed by George Necula and modified by Norman Ramsey.)
The GAME
signature declares all the types and functions that an Abstract Game Solver must know about a game. The signature is general enough to cover a variety of games. Even details like “the players take turns” are considered to be part of the rules of the game—such rules are hidden behind the GAME
interface, and the AGS operates correctly no matter what order players move in. (I have even implemented a solitaire as a “twoplayer” game in which the second player never gets a turn!)
You will use twoplayer games in the last two parts of this assignment: implement a particular game and implement an AGS of your own.
The idea behind the Abstract Game Solver (AGS)
As players move, the state of a game moves from one configuration to another. Think of a configuration as a markedup tictactoe board, or even just a number of sticks sitting on a table. In any given configuration, our solver considers all possible moves. After each move, it examines the resulting configuration and tries all possible moves from that configuration, and so on. In each configuration, the solver assumes that the player plays perfectly, that is, whenever possible the player will choose a move that forces a win.
This method (“exhaustive search”) is suitable only for very small games. Nobody would use it for a game like chess, for example. Nevertheless, variations of this idea are used successfully even for chess; the idea is to stop or “prune” the search before it goes too far. Many advanced pruning techniques have been developed for solving games, and if you wish, you are welcome to try them, but for this assignment, you don’t have to—exhaustive search works really well.
Basic data in the problem: Players and outcomes
Representation is the essence of programming. We start by describing basic representations for the essential facts we assume about each game:
There are two players.
A game ends in an outcome: either one of the players has won, or the outcome is a tie.
The representations of these central concepts are exposed, not abstract. They are given by the signature PLAYER
:
signature PLAYER = sig
datatype player = X  O (* 2 players called X and O *)
datatype outcome = WINS of player  TIE
(* Returns the other player *)
val otherplayer : player > player
val toString : player > string
val outcomeToString : outcome > string
end
The signature PLAYER
also includes some functions that compute with players and outcomes. Here’s the implementation of signature PLAYER
in 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 toString X = "X"
 toString O = "O"
fun outcomeToString TIE = "a tie"
 outcomeToString (WINS p) = toString p ^ " wins"
end
Although it might seem overly pedantic, we prefer to isolate details like the player names and how to convert them to a printable representation.
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.
Specification of an abstract game: the GAME
signature
The AGS can play any game that meets the specification given in signature GAME
. This signature gives a contract for an entire module, which subsumes the contracts for all its exported functions.
signature GAME = sig
structure Move : sig (* information related to moves *)
eqtype move (* A move (perhaps a set of coordinates) *)
exception Move (* Raised (by makemove & fromString) for invalid moves *)
val fromString : string > move
(* converts a string to a move; If the string does not
correspond to a valid move, fromString raises Move *)
val prompt : Player.player > string
(* Given a player, return a request for a move
for that player *)
val toString : Player.player > move > string
(* Returns a short message describing a
move. Example: "Player X moves to ...".
The message may not contain a newline. *)
end
type config (* A representation for a game configuration. It
must include a full description of the state
of a game at a particular moment, including
keeping track of whose turn it is to move.
Configurations must appear immutable.
If a mutable representation is used, it must
be impossible for a client to tell that a
mutation has taken place. *)
val toString : config > string
(* Returns an ASCII representation of the
configuration. The string must show whose turn it is. *)
val initial : Player.player > config
(* Initial configuration for a game when
"player" is the one to start. We need the
parameter because the configuration includes
the player to move. *)
val whoseturn : config > Player.player
(* Extracts the player whose turn is to move
from a configuration. We need this function because
the solver may need to know whose
turn it is, and the solver does not have
access to the representation of a configuration.
*)
val makemove: config > Move.move > config
(* Changes the configuration by making a move.
The player making the move is encoded in the
configuration. Be sure that the new
configuration knows who is to move. *)
val outcome : config > Player.outcome option
(* If the configuration represents a finished game,
return SOME applied to the outcome.
If the game isn't over, return NONE. *)
val finished : config > bool
(* True if the configuration is final. This
might be because one player has won,
or it might be that nobody can move
(which would be considered a tie). *)
val possmoves : config > Move.move list
(* A list of possible moves in a given
configuration. ONLY final configurations
might return nil. This means that a
configuration which is not final MUST have
some possible moves. In other words,
part of the contract is that if 'finished cfg'
is false, 'possmoves cfg' must return nonnil. *)
end
This is a broad interface. For example, there are three different ways to tell if a game is over!
Specification of a game solver: the AGS
signature
A solver for a GAME
exports only two new functions:
Function
bestmove
returns the best available move in any configuration. “Best” is always from the point of view of the player whose turn it is. If no moves are available—that is, if the configuration is final—bestmove
returnsNONE
. The computer player uses the result frombestmove
.Function
forecast
looks at a configuration and predicts what the outcome will be if both players make perfect moves. It is useful for testing.
In addition to these functions, 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
val bestmove : Game.config > Game.Move.move option
(* Given a configuration, returns the
* most beneficial move for the player
* to move *)
val forecast : Game.config > Player.outcome
(* Given a configuration, returns the
* best possible outcome for the player
* whose turn it is, assuming opponent
* plays optimally *)
end
The cost model of the AGS is that it tries all possible combinations of moves. For some games, the AGS functions are slow. Be patient.
Programming problem S: Implement “take the last stick”
The main foci of this assignment are the natural numbers and the Abstract Game Solver. But to understand how the solver works, it helps you to implement a simple twoplayer game. You’ll implement a very boring easy game called “pick up the last stick.” Here are the rules:
The game starts with
N
sticks on a table. We require N = 14.Players take turns.
When it’s your turn, you must pick up 1, 2, or 3 sticks.
The player who picks up the last stick wins.
Put the code for your game into a file called sticks.sml
, which be organized according to the following template:
functor SticksFun(val N : int) :> GAME = struct
...
type config = ... (* or possibly datatype config = ... *)
...
structure Move = struct
datatype move ... (* pick one of these two ways to define type `move` *)
type move = ...
...
end
...
end
Complete the implementation by following these stepbystep instructions for implementing twoplayer games:
Choose how you will represent the state of the game (i.e., define
config
). This step is crucial because it determines how complex your implementation will be. You want a representation that will make it easy to implementmakemove
,possmoves
, andoutcome
.In “pick up the last stick,” there’s an obvious choice: a pair containing the player who’s turn it is and the number of sticks on the table. For a more ambitious game, like Hexapawn or tictactoe, the representation is less obvious. I myself have implemented tictactoe in two different ways, one of which outperforms the other by a factor of four.
You might be tempted to use mutable data to represent a game state. Don’t! The contract of the
GAME
interface requires that any value of typeconfig
be available to the AGS indefinitely. Mutating a configuration is not safe.Choose a representation for moves. That is, define type
move
. Everything said for configurations applies here also, but this choice seems less critical.For “pick up the last stick”, there are two good choices: you could pick type
int
, in which case you’d have to enforce a representation invariant, or you could define an enumeration type such asdatatype move = ONE  TWO  THREE
The choice is yours.
Define the exception
Move
.Write function
initial
.Write function
whoseturn
.Write
makemove
. The contract requires it to be Curried.Write
outcome
. If the configuration is not final and nobody has won, returnNONE
.Hints for “pick up the last stick”:
 There are no ties.
 The game is over only if there are no sticks left on the table.
 If there are no sticks left, the player whose turn it is loses.
Write
finished
. This function should return true if somebody has won or if no move is possible (everybody is stuck).Write
possmoves
. This function must return a list of the possible moves (in no particular order). It is in everybody’s interest that the list have no duplicates. If the game is over, no further moves are possible, andpossmoves
must returnnil
. (In this case, according to contract,finished
must returntrue
.)

Write
Move.toString
. This function must return a string like “Player X picks up 1 stick” or “Player O moves to the middle square.” The string must not contain a newline. You can build your strings using concatenation (^
) and exported functions from other modules (e.g.Player.toString
). To convert integer values to strings you can use the functionInt.toString
. Write
toString
. You must return a simple ASCII representation of the state of the game configuration. The value should end in a newline. Don’t forget to include the player whose turn it is to move.Move.toString
andtoString
don’t affect the AGS; they are used by the interactive player to show you what’s happening.Write
Move.prompt
. It takes the player whose turn it is to move, and it returns a prompt message (without newline) asking the specified player to give a move.Write
Move.fromString
. This function should take a string (which is probably the reply given after a call toMove.prompt
, and it should return the move corresponding to that string. If there is no such move, it should raise an exception.So that we can test your code, please ensure that
Move.fromString
accepts either the strings"1"
,"2"
, and"3"
or the strings"one"
,"two"
, and"three"
.
The “take the last stick” game is so simple that it doesn’t need a lot of unit testing, but it is probably worth writing a couple of tests. When people implement twoplayer games, their most common mistake to permit players to continue to move even when the game is over—if I had any concerns about correctness, I would focus on tests to ensure that possmoves
, finished
, and outcome
are all consistent. Use the Unit signature from /comp/105/lib
; the compile
script should import it for you.
How big is it? Not counting embedded unit tests, my implementation is 43 lines of Standard ML. But ten of those lines are blank. The median function is 2 lines long, and no single function takes more than 6 lines of code.
Related reading: The lengthy description of the GAME
signature, and the section on ML modules in the ML learning guide.
What’s the point? Parametric polymorphism on a large scale requires something like modules. The main point of implementing the stick game is to enable you to understand games well enough to write an AGS.
Integration testing, part I: your game with my AGS
Once you’re satisfied with your game, you can test to see how it works when combined with my AGS as a computer player. You will create an instance of your game with N = 14, use the instances to create a gamespecific AGS, then use that instance with the computer player. All this will be done interactively using Moscow ML. The key steps are as follows:
 Start
mosml
with the optionsI /comp/105/lib P full
.  Load
.uo
files with theload
command.  Use functor applications to create the components you need.
 Play interactively.
Here is an annotated transcript:
: homework> mosml I /comp/105/lib P full
Moscow ML version 2.103 (Tufts University, April 2012)
Enter `quit();' to quit.
 load "sticks"; < your game
> val it = () : unit
 load "ags"; < my AGS
> val it = () : unit
 structure Sticks14 = SticksFun(struct val N = 14 end); < create Game structure
> structure Sticks14 : ...
 structure S14Ags = AgsFun(structure Game = Sticks14); < create Ags structure
> structure S14Ags : ...
Once you have your gamespecific AGS, you create an interactive player by applying functor PlayFun
to your AGS. To get PlayFun
, load file play.uo
:
 load "play";
> val it = () : unit
 structure P = PlayFun(structure Ags = S14Ags); < create the player
> structure P : ...
Functor PlayFun
returns a structure that implements the following signature:
signature PLAY = sig
structure Game : GAME
exception Quit
val getamove : Player.player list > Game.config > Game.Move.move
(* raises Quit if human player refuses to provide a move *)
val play : (Game.config > Game.Move.move) > Game.config > Player.outcome
end
The function getamove
expects a list of players for which the computer is supposed to play (the computer might play for X, for O, for both or for none). The return value is a function which the interactive player will use to request a move given a configuration. The idea is that the function returned will ask the AGS for a move if the computer is playing for the player to move, or will prompt the user and convert the user’s response into a move.
The function play
expects an input function (one built by getamove
) and a starting configuration. This function then starts an interactive loop printing the intermediate configurations and prompting the users for moves (or asking the AGS where appropriate). Here are some suitable definitions.
val computerxo = P.getamove [Player.X, Player.O]
(*Computer plays for both X and O *)
val computero = P.getamove [Player.O]
(*Computer plays only O *)
val cnfi = Sticks14.initial Player.X
(* Empty configuration with X to start *)
val contest = P.play computero
(* We play against the computer *)
With these definitions in place, you can start a game:
 P.play computero cnfi;
Player X sees              
How many sticks does player X pick up?
If you want to watch the computer play both sides, try this:
 P.play computerxo cnfi;
Player X sees              
Player X takes [redacted] sticks
...
With this experience in hand, you’re ready for the final problem of the assignment: the AGS itself.
Programming problem A: Build an Abstract Game Solver
A. Implement an Abstract Game Solver. Given a configuration, an AGS should pick the best move:
If the AGS finds a move that enables the current player to force a win, it’s done: it picks that move. It doesn’t even have to consider other moves.
If AGS can’t find a winning move, the next best move is one that forces a tie. (In the stick game, ties are impossible, but they are a common feature of tictactoe and other games.)
If the AGS can’t win or tie, then all moves lead to losses, and to the AGS, they are all equally bad.
The AGS looks at moves by simple linear search—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. Instead, build your AGS around this helper function:
val bestresult : Game.config > Game.move option * result
where result
is a representation you choose. The idea is that bestresult conf = (bestmove, whathappens)
where
If the player can’t move,
bestmove
isNONE
.If the player can move,
bestmove
isSOME m
, wherem
is the best possibleGame.move
for the player in this configuration.Value
whathappens
explains what the AGS predicts is the outcome of the game if both players play perfectly. It suffices to use a result of typePlayer.outcome
, but you can play around with this one some—for example, you might want to return an outcome like “Player X wins in 3 moves.” This would help you build an aggressive AGS.You might be tempted to use a “relative” outcome like “Win, Lose, or Tie.” This can be made to work, but it is harder to get right, especially in games where players don’t always take turns.
The
whathappens
value is computed inductively. In the base case,conf
is a finished configuration, andwhathappens
is determined by the return value fromGame.outcome
. In the inductive step, the AGS chooseswhathappens
by recursively evaluating the best possible result from each move. It then picks the result from the move that is best for the current player.
To compare outcomes, I recommend creating a helper function that calculates, for any given result
, the benefit that the result provides to the current player. A win confers maximum benefit; a tie confers medium benefit; and a loss confers minimum benefit. There are more sophisticated ways to view benefits; for example, we could assign larger benefits to winning quickly, and so on. But distinguishing wins, losses, and ties is good enough.
Write an AGS using the following template:
functor AgsFun (structure Game : GAME) :> AGS
where type Game.Move.move = Game.Move.move
and type Game.config = Game.config
= struct
structure Game = Game
fun bestresult conf = ...
fun bestmove conf = ...
fun forecast conf = ...
end
Note how annoying the where type
declarations are: they look tautological, but they’re not. Complain to Dave MacQueen and Bob Harper.
Hints:
You can implement the inductive step by using
map
with the result ofGame.possmoves
. But if you use this code, your AGS will always search every possible move, even if it finds a winning strategy on the very first move. This code will make your AGS slow and no fun to play. Instead, write a simple recursive function so that if the AGS finds a move that confers maximum benefit, it can halt the search right away and just return the good move. It will take just a few lines of code, and you will have a lot more fun.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 “pick up the last stick.” Use
whoseturn
andoutcome
instead. We will test your AGS on games that are quite different from “pick up the last stick”, including TicTacToe, Connect 3, and others. Probably even a solitaire!It is hard to write unit tests inside an AGS. If you want unit testing, write unit tests for a particular game. Start with a known configuration and check
forecast
andbestmove
. For example, if a computer player sees a table with two sticks, its best move is to pick up both sticks and win. Unit tests like these are gamespecific and will have to go into another module. Put them in fileagstests.sml
.
To test your AGS, all you need to do is restart Moscow ML and once again 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, as well as the same kind of gameplaying integration tests you used wih your stick game. If you want to test your AGS with a more sophisticated game, try our tictactoe game, as described below.
How big is it? My AGS takes about 40 lines of Standard ML.
Related reading: the section on ML modules in the ML learning guide.
What’s the point? The AGS requires one short but subtle recursive function, and it presents a simple, narrow interface. But look at the parameters! To try to implement an AGS using parametric polymorphism, you would need at least two type parameters (configuration and move), and you would need at least four function parameters (whoseturn
, makemove
, outcome
, and possmoves
). Writing the code would become very challenging—polymorphism and higherorder programming at the value level is the wrong tool for the job. The point of this exercise is to use a functor to take an entire Game
structure at one go: both types and values. Moreover, the same Game
structure also drives the computer player and the interactive player—nothing in the Game
module has to change. Programming at scale requires some sort of tool that bundles types and code into one unit.
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 TicTacToe, with you as X, the AGS as O, and play starting in this position:

  O  

  X  

   

You move 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.
Integration testing, part II: your AGS with my game
We supply a binary implementation of TicTacToe in file /comp/105/lib/ttt.uo
. You can use it as follows:
homework> mosml I /comp/105/lib P full
Moscow ML version 2.103 (Tufts University, April 2012)
Enter `quit();' to quit.
 load "ags";
> val it = () : unit
 load "ttt";
> val it = () : unit
 structure TTTAgs = AgsFun(structure Game = TTT);
> structure TTTAgs : ...
 structure PT = PlayFun(structure Ags = TTTAgs);
> structure PT : ...
 PT.play (PT.getamove [Player.O]) (SlickTTT.initial Player.X);

   

   

   

Player X is to move
Square for player X?
Descriptions of twoplayer games
Here are descriptions of 4 games: “Pick up the last stick,” “TicTacToe,” “Nim,” and “Connect 4”. Do not worry if you haven’t seen these games before—you can learn by playing against a perfect or nearperfect player. (The Connect 4 player would be perfect if it were faster.) For the purpose of this assignment you do not have to know any tricks of the games but only to understand their rules.
Pick up the last stick
The game starts with N
sticks on a table. Players take turns. When it’s your turn, you must pick up 1, 2, or 3 sticks. The player who picks up the last stick wins.
TicTacToe
This is an adversary game played by two persons using a 3x3 square board. The players (traditionally called X and O) take turns in placing X’s or O’s in the empty squares on the board (player X places only X’s and O only O’s). In the initial configuration, the board is empty.
The first player who managed to obtain a full line, column or diagonal marked with his name is the winner. The game can also end in a tie. In the picture below the first configuration is a win for O, the next two are wins for X and the last one is a tie.
   
 X   X     X   X  O    O  O  X 
   
  X    O  X  O   X  O    X  X  O 
   
 O  O  O   X   O   X   O   O  X  O 
   
In Britain, this game is called “noughts and crosses.” No matter what you call it, a player who plays perfectly cannot lose. All your base are belong to the AGS. You can play /comp/105/bin/ttt
.
Nim
This is an adversary game played by two persons. The game is played with number of sticks arranged in 3 rows. In the initial state the rows usually contain 3, 5 and 7 sticks respectively. The players take turns in removing sticks: each player can remove 1, 2 or 3 adjacent sticks from one row. The one that removes the last stick is the loser. Or, stated differently the first player who has no sticks to remove is the winner. Below are two configurations. The first one is the initial configuration (for the 3, 5 and 7) case and the other one is the configuration obtained after a few moves. A possible sequence of moves that might lead to this configuration is:
 X removes sticks 0, 1 and 2 from row 1
 O removes stick 1 from row 0
 X removes stick 6 from row 2
 O removes sticks 3 and 4 from row 2
Row 0:     _ 
Row 1:      _ _ _  
Row 2:           _ _  _
We have represented a stick using a 
and a missing stick using a _
. It might be wise to play with a smaller configuration (2, 3 and 4 for example) because otherwise the AGS will take too long to produce its answers.
For this game the first player can always win no matter what the other does. If you let the AGS start you have no chance. If you play first you can beat the AGS, but you have to play well. You can play /comp/105/bin/nim
.
Connect 4
This is an adversary game played by two persons using 6 rods and 36 balls. Imagine the rods standing vertically, and each ball has a hole in it, so you can drop a ball onto a rod. The balls are divided in two equal groups marked X and O. The players take turns in making moves. A move for a player consists in sliding one of its own balls down a rod which is not full (the capacity of a rod is 6). The purpose is to obtain 4 balls of the same type adjacent on a horizontal, vertical or diagonal line. The game ends in a tie when all the rods are full and no player has won. We represent below the initial configuration of the game and a final state where X has won.
           
           
           
      O     
      O  O   
      O X X X X 
 
Our version uses 5 rods and connects 3, because otherwise the AGS takes too long. You can play /comp/105/bin/four
.
Extra Credit
Proof. Prove that any of these simple games is always in one of these three states:
 The player whose turn it is can force a win.
 Either player can force a tie.
 The player whose turn it is can be forced to lose.
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.
Tictactoe. Implement tictactoe.
Four. Implement Connect 4.
Aggression. With the simple benefits outlined above, the AGS will “give up’’ if it can’t beat a perfect player—all moves are equally bad, and it apparently moves at random. What this scheme doesn’t account for is that the other player might not be perfect, so there is a reason to prefer the most distant loss. In the dual situation, when the AGS knows it can win no matter what, it will pick 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 benefits so that the AGS prefers the closest win and the most distant loss. (This means you can only prune the search if you find a win in one move.) If you are clever, you can encode all this information in one value of type real
.
What and how to submit
As soon as you have tackled the reading comprehension, run submit105smlsolo
to submit your answers to the CQ’s.
For the programming problems, submit the following files:
 A
README
file containing The names of the people with whom you collaborated
 A list of the problems that you solved (including any extra credit)
 A short statement explaining which representation you picked for natural numbers, and why
 Answers to Proof and Game Theory extra credits.
 File
sort.sml
, implementing your solution to Problem Q  Files
naturalrep.sml
andnatural.sml
, implementing your solutions to problem N  File
sticks.sml
, implementing your solution to Problem S  File
ags.sml
, implementing your solution to Problem A  File
agstests.sml
, containing any unit tests you may have written for your AGS  For problems S and A, your
compile
andsticks.mlb
files. (See Appendix 1 for information on the these files.)  For problems S and A, any other files you need in order to compile
sticks.sml
andags.sml
.
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. The files you submit must compile with Moscow ML, using the compile
script we give you. We will reject files with syntax or type errors. Your files must compile without warning messages. If you must, you can include multiple structures in your files, but please don’t make copies of the structures and signatures above; we already have them.
As soon as you have the files listed above, run submit105smlpair
to submit a preliminary version of your work. Keep submitting until your work is complete; we grade only the last submission.
Acknowledgments
The AGS assignment is derived from one graciously provided by Bob Harper. George Necula, who was his teaching assistant at the time (and is now a professor at Berkeley and is world famous as the inventor of proofcarrying code), did the bulk of the work.
Appendices
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. You should be able to get away with using compile105sml
, but if something goes wrong, this appendix explains not only what is going on but also how to compile with MLton.
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 puts compilerinterface 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 DictFn
functor depends on both DICT
and KEY
signatures. If DictFn
is defined in dict.sml
, KEY
is defined in keysig.sml
, and DICT
is defined in dictsig.sml
, then to compile DictFn
you run
mosmlc I /comp/105/lib toplevel c dictsig.ui keysig.ui dict.sml
The script compile105sml
knows about the files that are assigned for the homework, and in most situations it inserts the .ui
references for you.
To talk about what happens after you compile, I’ll use another example:
mosmlc I /comp/105/lib c toplevel /comp/105/lib/gamesig.ui /comp/105/lib/player.ui sticks.sml
This compilation produces two files:
sticks.ui
, which can be used on the command line when compiling other units that depend onSticksFun
sticks.uo
, which contains the compiled binary
You can do two things with the .uo
files:
When you are debugging, it is tremendously useful to get compiled modules into the interactive system. Load them directly using
load
, e.g.,: homework: mosml I /comp/105/lib P full Moscow ML version 2.103 (Tufts University, April 2012) Enter `quit();' to quit.  load "sticks"; > val it = () : unit  structure Sticks12 = SticksFun(struct val N = 12 end); > ...
Once you load a module, you cannot recompile it and reload it later. Loading it again has no effect, even if the code has changed; you have to start Moscow ML over again.
You can use
mosmlc
to link a bunch of.uo
files together to form an executable binary. To do anything interesting, one of the source files should have a toplevel call toplay
,forecast
, or some other interesting function.Here is an example of a command line I use on my system to build an interactive game player:
mosmlc I /comp/105/lib toplevel o games \ playersig.uo player.uo gamesig.uo \ agssig.uo playsig.uo slickttt.uo \ ags.uo aggress.uo nim.uo four.uo peg.uo mrun.uo
Order matters; for example, I have to put
player.uo
afterplayersig.uo
because thePlayer
structure defined inplayer.sml
uses thePLAYER
signature defined inplayersig.sml
.
Compiling Standard ML to native machine code using MLton
If your games are running too slow, compile them with MLton. MLton is a wholeprogram compiler that produces optimized native code. To use MLton, you list all your modules in an MLB file, and MLton compiles them at one go. If you want to try this, download files sticks.mlb
and runsticks.sml
, and then compile with compile with
mlton output sticks verbose 1 sticks.mlb
Because MLton requires source code, you will be able to use it only once you have your own AGS. More information about MLton is available on the man page and at mlton.org
.
Note: At press time, Unit
module wasn’t working with MLton.
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  

Structure 
• All modules are sealed using the opaque sealing operator • Code uses basis functions effectively, especially higherorder 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 • 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 • Code has more than one redundant case analysis • Code is almost twice as large as necessary to solve the problem. • Or, code contains nearduplicate functions (most likely in AGS) 
Performance and correctness
Finally, 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  

Correctness 
• AGS code makes no additional assumptions about the implementations of 
• AGS code assumes that players take turns. 

Performance 
• The AGS implements its
• Or, the AGS implements 
• Function 
• Function 