Laziness
Complete this assignment with the same team you worked with for Substitution (Written). You and your partner must each understand the answers to all the problems, so don't just split up the work.
For this assignment, we will use the Haskell programming language. The Glasgow Haskell Compiler (GHC) is freely available for most platforms and is installed on the CS department network. To start the interactive evaluator, simply run ghci.
Important Note: Yes, we realize that you don't know any Haskell. We expect that you'll spend much of your time on this assignment learning Haskell. Learning a new language shouldn't be that big of a deal. You are taking a programming languages course, after all.
Haskell has wonderful standard libraries which probably contain the
function you're looking for. To see treestyle browsable docs, check
out the Haskell
Hierarchical Libraries . You should particularly take advantage
of the
Prelude
, which is the set of functions that are
"built in" to Haskell. You're welcome to use other functions; if you
want to do so, you must put an import
statement at the top
of your code (there's an example in the template). Hoogle is a useful search engine for the Haskell API.
The following are library functions that you may find particularly
helpful, and which are all in the Haskell Prelude
:
Haskell Function  Scheme Analog 

==  eq? 
<, >, <=, >=  <, >, <=, >= 
mod  modulo 
div  quotient 
not  not 
!!  listref 
filter  filter 
all  andmap 
any  ormap 
take  (returns a prefix of a list) 
takeWhile  (returns the prefix of a list satisfying a predicate) 
Any function whose name begins with a symbolic character is an infix
operator. Other binary functions can be used as infix operators by
enclosing in backquotes (e.g., x `mod` y
). Also, infix
operators can be used as ordinary functions by enclosing them in
parentheses (e.g., (!!) [1, 2, 3] 2
).
Please submit a README file explaining how you tested your code. Actual test cases are preferable, but a clear summary of what you did will be sufficient as long as your code actually works as you claim it does.
Problem 1: Prime Numbers
 Write
isPrime :: Integer > Bool
, which determines whether a given integer is prime.  Define
primes :: [Integer]
, the list of all primes.  Revise
isPrime
so that it only tests divisibility by prime factors. Just turn in the revised version.
Problem 2: Longest Common Subsequence
 Write
buildList :: Int > (Int > a) > [a]
, where((buildList n f) !! i) == (f i)
(for all i in [0 .. n1]).  Write
buildTable :: Int > Int > (Int > Int > a) > [[a]]
, where(((buildTable n m f) !! i) !! j) == (f i j)
(for all i in [0 .. n1], j in [0 .. m1]).  Write
lcsLength :: String > String > Int
, which quickly computes the length of the longest common subsequence of two strings s1 and s2. Hint: you can easily computelcsLength (take i s1) (take j s2)
fromlcsLength (take (i1) s1) (take j s2), lcsLength (take i s1) (take (j1) s2), and lcsLength (take (i1) s1) (take (j1) s2).
and the knowledge of whether
(s1 !! i) == (s2 !! j)
. Note that you will lose credit if you use a slow, "brute force" method; make use of your table!
Problem 3: Minimax Search
In this exercise, you will implement a strategy to play a simple game. The game is called Mancala, but you won't need to worry about the specific rules, since we have implemented that part for you. Your job is to build a tree of possible move sequences and choose the move that appears best.
The support code below provides the following set of data types and functions:

Player
: values of this type represent the players of the game (PlayerA
orPlayerB
). 
State
: values of this type represent game configurations. 
GameValue
: values of this type represent scores used to evaluate the game state. SinceGameValue
is an instance ofOrd
, GameValues can be compared using standard comparison operators. 
initialState :: Player > State
represents the initial configuration of the game board (the given player goes first). 
getPlayer: State > Player
: given a configuration, returns the player who makes the next move. 
gameValue :: State > GameValue
returns Player A's score minus Player B's score. (Player A wants a big value and Player B wants a small value.) 
nextStates :: State > [State]
gives the possible configurations after the next move. If the returned list is empty, then the game is over. 
simulateGame :: IO ()
uses an IO loop to allow you to play Mancala. The valid moves are the numbers 1 through 6 and they indicate the player playing the nth cup on her side. Player A goes first; her cups are on the top row and are numbered from right to left. Player B is the bottom row numbered from left to right. 
simulateAIGame :: (Show a, Ord a) => (State > a) > IO ()
consumes a scoring function and plays the game where Player A is the AI using that function. We give you the helper function,level :: Int > IO ()
in the stub file, which consumes a difficulty and runs simulateAIGame using your functions. This may be helpful for testing.
 Define a datatype
GameTree
to represent the game state after any sequence of moves. Each node should have its current configuration and a list of trees, where each tree corresponds to the game states obtainable after making any one legal move.  Define
mkTree :: State > GameTree
, which consumes a state and returns the tree of all legal board configurations (those obtainable by repeated application ofnextStates
to the given state). Use this to definemancala :: GameTree
, the entire game of Mancala as a GameTree (remember that Player A goes first).  Define
prune :: Int > GameTree > GameTree
, which prunes a tree to a given depth.  Define
minimax :: GameTree > GameValue
, which consumes a (pruned) tree and evaluates the configuration by looking ahead and applying the minimax algorithm. If a node has no children, its value is thegameValue
of itsState
. Remember, Player A, Maxi, is always trying to get the maximum value, and Player B, Mini, is always trying to get the minimum value.
Support Code
Note that you must download both files (even for problems 1 and 2), and
you must put them both in the same directory. To test your code interactively,
first cd
into the directory with your code, and then do
ghci Laziness.hs
at which point you should either see a parse
error, type error, or this:
___ ___ _ / _ \ /\ /\/ __(_) / /_\// /_/ / /   GHC Interactive, version 6.6, for Haskell 98. / /_\\/ __ / /___  http://www.haskell.org/ghc/ \____/\/ /_/\____/_ Type :? for help. Loading package base ... linking ... done. [1 of 2] Compiling Game ( Game.hs, interpreted ) [2 of 2] Compiling Laziness ( Laziness.hs, interpreted ) Ok, modules loaded: Game, Laziness. *Laziness>
ghci
supports tab completion, which is a very handy feature.
A Haskell stub file for the problem set: Laziness.hs.
The support code for Problem 3: Game.hs. Remember, you need neither read nor understand this code.
Handin
Turn in one file, Laziness.hs
.