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https://www.bilibili.com/video/BV1dL411j7L7

## 问题1

Check a box if and only if it is an accurate description of ML

• ML uses lexical scope for the semantics of looking up variables in the environment
• 正确
• ML has no language constructs for creating mutable data
• ML has a REPL as part of the definition of the language
• ML is statically typed
• 正确

## 问题2

Here is a particular list of pairs in ML:

[(4,19), (1,20), (74,75)]

For each pattern below, check the box if and only if this pattern matches the value above.

• x::y
• 正确
• x::(y::z)
• 正确
• (a,b,c)::d
• []
• (a,b)::(c,d)::(e,f)::[]
• 正确
• (a,b)::(c,d)::(e,f)::g
• 正确

## 问题3

For each of the statements below, check the box if and only if the statement is true regarding this ML code:

fun mystery f xs =
let
fun g xs =
case xs of
[] => NONE
| x::xs' => if f x then SOME x else g xs'
in
case xs of
[] => NONE
| x::xs' => if f x then g xs' else mystery f xs'
end
• mystery uses currying to take two arguments.
• 正确
• mystery uses tupling to take two arguments.
• If the second argument to mysteryis a zero-element list, then whenever mystery produces a result, the result is NONE.
• 正确
• If the second argument to mystery is a one-element list, then whenever mystery produces a result, the result is NONE.
• 正确
• If the second argument to mystery is a two-element list, then whenever mystery produces a result, the result is NONE.
• The argument type of f can be any type, but it must be the same type as the element type of xs.
• 正确
• The result type of f can be any type, but it must be the same type as the element type of xs.
• If you replace the first line of the code with fun mystery f = fn xs =>, then some callers of mystery might no longer type-check.
• If you replace the first line of the code with fun mystery f = fn xs =>, then some callers of mystery might get a different result.
• g is a tail-recursive function.
• 正确
• For the entire computation of a call like mystery someFun someList, the total number of times someFun is called is always the same as the length of someList (for any someFun and someList).
• For the entire computation of a call like mystery someFun someList, the total number of times someFun is called is sometimes the same as the length of someList (for any someFun and someList).
• 正确
• For the entire computation of a call like mystery someFun someList, the total number of times someFun is called is never the same as the length of someList (for any someFun and someList).

fun mystery f xs =
let
fun g xs =
case xs of
[] => NONE
| x::xs' => if f x then SOME x else g xs'
in
case xs of
[] => NONE
| x::xs' => if f x then g xs' else mystery f xs'
end

fun mystery f = fn xs =>
let
fun g xs =
case xs of
[] => NONE
| x::xs' => if f x then SOME x else g xs'
in
case xs of
[] => NONE
| x::xs' => if f x then g xs' else mystery f xs'
end

fun f x = x > 0;

val r1 = mystery f [];
val r2 = mystery f [~1];
val r3 = mystery f [1];
val r4 = mystery f [1, 2, 3];
val r5 = mystery f [1, 2];

val f = fn : int -> bool
val r1 = NONE : int option
val r2 = NONE : int option
val r3 = NONE : int option
val r4 = SOME 2 : int option
val r5 = SOME 2 : int option

## 问题4

The null function is predefined in ML’s standard library, but can be defined in many ways ourselves. For each suggested definition of null below, check the box if and only if the function would behave the same as the predefined null function whenever the function below is called.

Note: Consider only situations where calls to the functions below type-check.

• fun null xs = case xs of [] => true | _ => false
• 正确
• fun null xs = xs=[]
• 正确
• fun null xs = if null xs then true else false
• fun null xs = ((fn z => false) (hd xs)) handle List.Empty => true
• 正确

## 问题5

The next four questions, including this one, relate to this situation: Suppose somebody has written a library for a collection of strings (perhaps implemented as some sort of linked list of strings or tree of strings, but the details do not matter). The library includes higher-order functions map, filter, and fold that operate on these collections and have their conventional meanings. For each problem below, decide which of these library functions is the best to use for implementing the desired function.

(For those needing a precise definition of best: On this exam, the best function, given appropriate arguments, returns the final result you need, meaning you need no more computation after calling the function. If multiple functions can do this, choose the one that can be used by passing it the function argument that itself does the least amount of work.)

Desired function: Take a collection of strings and produce a new collection where each string in the output is like a string in the input except the string has any space characters removed.

• map
• 正确
• filter
• fold

## 问题6

Desired function: Take a collection of strings and return a string that is the concatenation of all the strings in the collection.

• map
• filter
• fold
• 正确

## 问题7

Desired function: Take a collection of strings and a number n and return how many strings in the collection have a length that is a multiple of n.

• map
• filter
• fold
• 正确

## 问题8

Desired function: Take a collection of strings and return a collection containing the strings in the input collection that start with a capital letter.

• map
• filter
• 正确
• fold

## 问题9

This datatype binding and type synonym are useful for representing certain equations from algebra:

datatype algebra_exp = Variable of string
| Integer of int
| Decimal of real
| Addition of algebra_exp * algebra_exp
| Multiplication of algebra_exp * algebra_exp
| Exponent of algebra_exp * int
type equation = algebra_exp * algebra_exp

Which of the mathematical equations below could not be elegantly represented by a value of type equation?

• $x+y=z$
• $(x+4)+z = 7\cdot y(x+4)+z=7⋅y$
• $x^3 . y^2 =z^0$
• $14.2+3=17.2$
• $x^y = z$
• 正确

## 问题10

Here is a particular polymorphic type in ML:

'a * 'b -> 'b * 'a * 'a

For each type below, check the box if and only if the type is an instantiation of the type above, which means the type above is more general.

• string * int -> string * int * int
• int * string -> string * int * int
• 正确
• int * int -> int * int * int
• 正确
• {foo : int, bar : string} -> {a : string, b : int, c : int}
• 'a * 'a -> 'a * 'a * 'a
• 正确

## 问题11

The next 5 questions, including this one, are similar. Each question uses a slightly different definition of an ML signature COUNTER with this same structure definition:

structure NoNegativeCounter :> COUNTER =
struct

exception InvariantViolated

type t = int

fun newCounter i = if i <= 0 then 1 else i

fun increment i = i + 1

fun first_larger (i1,i2) =
if i1 <= 0 orelse i2 <= 0
then raise InvariantViolated
else (i1 - i2) > 0

end

In each problem, the definition of COUNTER matches the structure definition NoNegativeCounter, but different signatures allow clients to use the structure in different ways. You will answer the same question for each COUNTER definition by choosing the best description of what it allows clients to do.In this question, the definition of COUNTER is:

signature COUNTER =
sig
type t = int
val newCounter : int -> t
val increment : t -> t
val first_larger : t * t -> bool
end
• This signature allows (some) clients to cause the NoNegativeCounter.InvariantViolated exception to be raised.
• 正确，方式为调用NoNegativeCounter.first_larger(~1, ~3)
• This signature makes it impossible for any client to call NoNegativeCounter.first_larger at all (in a way that causes any part of the body of NoNegativeCounter.first_larger to be evaluated).
• This signature makes it possible for clients to call NoNegativeCounter.first_larger, but never in a way that leads to the NoNegativeCounter.InvariantViolated exception being raised.

## 问题12

In this question, the definition of COUNTER is:

signature COUNTER =
sig
type t = int
val newCounter : int -> t
val first_larger : t * t -> bool
end
• This signature allows (some) clients to cause the NoNegativeCounter.InvariantViolated exception to be raised.
• 正确，方式为调用NoNegativeCounter.first_larger(~1, ~3)
• This signature makes it impossible for any client to call NoNegativeCounter.first_larger at all (in a way that causes any part of the body of NoNegativeCounter.first_larger to be evaluated).
• This signature makes it possible for clients to call NoNegativeCounter.first_larger, but never in a way that leads to the NoNegativeCounter.InvariantViolated exception being raised.

## 问题13

In this question, the definition of COUNTER is:

signature COUNTER =
sig
type t
val newCounter : int -> int
val increment : t -> t
val first_larger : t * t -> bool
end
• This signature allows (some) clients to cause the NoNegativeCounter.InvariantViolated exception to be raised.
• This signature makes it impossible for any client to call NoNegativeCounter.first_larger at all (in a way that causes any part of the body of NoNegativeCounter.first_larger to be evaluated).
• 正确，因为无法实例化t类型的变量
• This signature makes it possible for clients to call NoNegativeCounter.first_larger, but never in a way that leads to the NoNegativeCounter.InvariantViolated exception being raised.

## 问题14

In this question, the definition of COUNTER is:

signature COUNTER =
sig
type t
val newCounter : int -> t
val increment : t -> t
val first_larger : t * t -> bool
end
• This signature allows (some) clients to cause the NoNegativeCounter.InvariantViolated exception to be raised.
• This signature makes it impossible for any client to call NoNegativeCounter.first_larger at all (in a way that causes any part of the body of NoNegativeCounter.first_larger to be evaluated).
• This signature makes it possible for clients to call NoNegativeCounter.first_larger, but never in a way that leads to the NoNegativeCounter.InvariantViolated exception being raised.
• 正确，因为newCounter不会产生负数，first_larger也无法直接调用负数

## 问题15

In this question, the definition of COUNTER is:

signature COUNTER =
sig
type t = int
val newCounter : int -> t
val increment : t -> t
end
• This signature allows (some) clients to cause the NoNegativeCounter.InvariantViolated exception to be raised.
• This signature makes it impossible for any client to call NoNegativeCounter.first_larger at all (in a way that causes any part of the body of NoNegativeCounter.first_larger to be evaluated).
• 正确，因为签名中没有first_large
• This signature makes it possible for clients to call NoNegativeCounter.first_larger, but never in a way that leads to the NoNegativeCounter.InvariantViolated exception being raised.