452 CSE Due:
Homework #3 Thursday September 25
Your answers to these questions are to be submitted as a single ML program using handin. Use ML comments to clearly delineate where the answer to each problem starts and ends. Embed your answers to non-coding questions as comments in this same le. 1. (a) Write the function zip to compute the product of two lists of arbitrary length that is, given two lists l1 and l2, the application zip l1 l2 should return the list of pairs obtained by pairing up corresponding elements of l1 and l2; if the lists do not have the same lengths, then it should pair up elements until the shorter of the two lists is exhausted. You should use pattern matching to dene this function. Here are some examples to illustrate the behavior of zip: - zip [1,3,5,7] ["a","b","c","de", "f"]; val it = [(1,"a"),(3,"b"),(5,"c"),(7,"de")]: - zip [] [1, 5, 6]; val it = nil (int * string) list
Now write the inverse function, unzip, which behaves as follows: - unzip [(1,"a"),(3,"b"),(5,"c"),(7,"de")]; val it = ([1,3,5,7], ["a","b","c","de"]): int list * string list Write zip3, to zip three lists. - zip3 [1,3,5,7] ["a","b","c","de", "f"] [1,2,3,4,5,6]; val it = [(1,"a",1),(3,"b",2),(5,"c",3),(7,"de",4)]: (int * string * int) list (b) In principal, you could dene a function zip1000 that zips together 1000 lists in the same fashion that you dened zip and zip3, although actually writing out the denition of this function would be onerous. (You could write a program that would generate the function denition.) However, you cannot write a function zipAny that can zip together any arbitrary number of liststhat is, a function that takes a list of lists ll and returns the list obtained by zipping together all the lists in ll. Explain why it is impossible to dene such a list in ML. 2. Certain programming languages (and HP calculators) evaluate expressions using a stack. You are going to implement a simple evaluator for such a language. Computation is expressed as a sequence of operations, which are drawn from the following data type: datatype OpCode = | | | | PUSH of real ADD MULT SUB DIV 1
| SWAP ; The operations have the following eect on the operand stack. (The top of the stack is shown on the left.) OpCode PUSH r ADD SUB DIV SWAP Initial Stack ... a b... a b... a b... a b... Resulting Stack r... (a+b)... (a-b)... (a/b)... b a...
The stack may be represented using a list for this example, although you could also dene a stack data type for it. type Stack = real list; Write a recursive evaluation function with the signature eval: OpCode list * Stack -> real
This function should take a list of operations and a stack. It should perform each operation in order and return what is left on the top of the stack when no operations are left to be performed. For example, eval([PUSH 2.0, PUSH 1.0, SUB],[]) returns 1.0. The eval function will have the following basic form: fun eval (nil,a::st) = (* ... *) | eval (PUSH(n)::ops,st) = (* ... | (* ... *) | eval ( , ) = 0.0 ; *)
You need to ll in the blanks and add cases for the other opcodes. last The rule handles illegal cases by matching any operation list and stack not handled by the cases you write. These illegal cases include ending with an empty stack, performing addition when fewer than two elements are on the stack, and so on. You may ignore divide-byzero errors for now (or look at exception handling in ML; we will cover that topic in a few weeks). 3. This problem is based on problem 5.4 of Mitchell.
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(a) Answer part (a) of problem 5.4 in Mitchell. In addition, show how to use the tree datatype by declaring identiers tree1 and tree2 that represent the trees shown in the diagram below.
tree1 tree2
"waste" 3 2 ~6 1 0 "not" "want"
"not"
"!"
Next, show how to use your maptree function in declaring identiers tree3 and tree4 representing new trees as described below. Do this without declaring any additional new identiers. (Recall that functions can be anonymous.) tree3: the tree produced from tree1 by negating the values at the leaves of tree1. tree4: the tree produced from tree2 by concatenating a single blank space to the front of each string value in the leaves of tree2. (b) Read part (b) of problem 5.4 in Mitchell. I am not sure why he refers to the type expression given in the problem as the expected type I would not have expected it to have that type. So, rather than answer the question asked in this part of the problem, show an ML expression that applies maptree to a function whose domain and range are of dierent types and to one of the trees created in part (a) of this problem. Use this example to explain what is meant by the statement that The ML type inference algorithm computes the most general type possible for an ML expression. 4. This problem is based on problem 5.5 in Mitchell. (a) Write a function reduce, as described in part (a) of problem 5.5 in Mitchell. (You do not need to explain your denition, as requested in Mitchell.) (b) Write an ML expression that uses the function reduce to compute the product of the values at the leaves of tree1. (c) Write an ML expression that uses the function reduce to compute the string " waste not want not !" from tree2. Do not declare any additional identiers in doing so. (Again, you will need to use an anonymous function.) 3
5. This problem is based on problem 5.6 in Mitchell. (a) Write functions Curry and UnCurry as requested in part (a) of problem 5.5 in Mitchell. As some additional explanation, your version of Curry should transforms a function f that takes a pair of arguments into a function fcurry of a single argument x such that the result of applying fcurry (x) to an argument y is f (x, y). For example, Curry op+ 2 3 should compute 5, because op+(2,3) computes 5. UnCurry is the inverse of Curry. It converts a function g that returns another function into a function gUnCurry on a pair of arguments. For example, UnCurry map (op~,[1, ~1]) should compute [~1, 1], because map op~ [1, ~1] computes [~1, 1]. (b) Use your Curry function to declare a function insertFront: a -> a list -> a list so that the function insertFront x has the eect of inserting x at the front of each list in a list of lists. For example, - insertFront 5 [[2, 1], [0], []]; val it = [[5,2,1],[5,0],[5]] : int list list (c) Answer part (b) of problem 5.6 in Mitchell.
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