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CIS425-Homework 3 Define SML Functions Solved
You will define several SML functions. Many will be very short because they will use other higher-order functions. You may use functions in ML’s library; the problems point you toward the useful functions and often require that you use them. The sample solution is about 120 lines, including the provided code. Note that problems with 1-line answers can still be challenging, perhaps because the answers are intended to be so short.
1. Write a function only_lowercase that takes a string list and returns a string list that has only the strings in the argument that start with an lowercase letter. Assume all strings have at least 1 character. Use List.filter, Char.isLower, and String.sub to make a 1-2 line solution.
2. Write a function longest_string1 that takes a string list and returns the longest string in the list. If the list is empty, return "". In the case of a tie, return the string closest to the beginning of the list. Use foldl, String.size, and no recursion (other than the implementation of foldl is recursive).
3. Write a function longest_string2 that is exactly like longest_string1 except in the case of ties it returns the string closest to the end of the list. Your solution should be almost an exact copy of longest_string1. Still use foldl and String.size.
4. Write functions longest_string_helper, longest_string3, and longest_string4 such that:
• longest_string3 has the same behavior as longest_string1 and longest_string4 has the same behavior as longest_string2.
• longest_string_helper has type (int * int - bool) - string list - string
(notice the currying). This function will look a lot like longest_string1 and longest_string2 but is more general because it takes a function as an argument.
• If longest_string_helper is passed a function that behaves like (so it returns true exactly when its first argument is stricly greater than its second), then the function returned has the same behavior as longest_string1.
• longest_string3 and longest_string4 are defined with val-bindings and partial applications of longest_string_helper.
5. Write a function longest_lowercase that takes a string list and returns the longest string in the list that begins with an lowercase letter, or "" if there are no such strings. Assume all strings have at least 1 character. Use a val-binding and the ML library’s o operator for composing functions. Resolve ties like in problem 2.
6. Write a function caps_no_X_string that takes a string and returns the string that is like the input except every letter is capitalized and every “x” or “X” is removed (e.g., “aBxXXxDdx” becomes “ABDD”). Use a val binding, ML’s o operator, 2–3 library functions in the String module, 1 in the Char module, and 1–2 in the List module. Browse the module documentation to find the most useful functions. Note ML has strange syntax for character literals, e.g., #"A".
The next two problems involve writing functions over lists that will be useful in later problems.
7. Write a function first_answer of type (’a - ’b option) - ’a list - ’b (notice the 2 arguments are curried). The first argument should be applied to elements of the second argument in order until the first time it returns SOME v for some v and then v is the result of the call to first_answer. If the first argument returns NONE for all list elements, then first_answer should raise the exception NoAnswer. Hints: Sample solution is 5 lines and does nothing fancy.
8. Write a function all_answers of type (’a - ’b list option) - ’a list - ’b list option (notice the 2 arguments are curried). The first argument should be applied to elements of the second argument. If it returns NONE for any element, then the result for all_answers is NONE. Else the calls to the first argument will have produced SOME lst1, SOME lst2, ... SOME lstn and the result of all_answers is SOME lst where lst is lst1, lst2, ..., lstn appended together (order doesn’t matter). Hints: The sample solution is 8 lines. It uses a helper function with an accumulator and uses @. Note all_answers f [] should evaluate to SOME [].
The remaining problems use these type definitions, which are inspired by the type definitions an ML implementation would use to implement pattern matching:
datatype pattern = WildcardP | VariableP of string | UnitP | ConstantP of int
| ConstructorP of string * pattern | TupleP of pattern list
datatype valu = Constant of int | Unit | Constructor of string * valu | Tuple of valu list
Given valuv and patternp, either pmatches v or not. If it does, the match produces a list of string * valu pairs; order in the list does not matter. The rules for matching should be unsurprising:
• WildcardP matches everything and produces the empty list of bindings.
• VariableP s matches any value v and produces the one-element list holding (s,v).
• UnitP matches only Unit and produces the empty list of bindings.
• ConstantP 17 matches only Constant 17 and produces the empty list of bindings (and similarly for other integers).
• ConstructorP(s1,p) matches Constructor(s2,v) if s1 and s2 are the same string (you can compare them with =) and p matches v. The list of bindings produced is the list from the nested pattern match. We call the strings s1 and s2 the constructor name.
• TupleP ps matches a value of the form Tuple vs if ps and vs have the same length and for all i, the ith element of ps matches the ith element of vs. The list of bindings produced is all the lists from the nested pattern matches appended together.
• Nothing else matches.
9. (This problem uses the pattern datatype but is not really about pattern-matching.)
fun g f1 f2 p = let
val r = g f1 f2
in
case p of
WildcardP = f1 ()
| VariableP x = f2 x
| ConstructorP(_,p) = r p
| TupleP ps = List.foldl (fn (p,i) = (r p) + i) 0 ps
| _ = 0
end
(a) In an ML comment, describe in a few English sentences the arguments that g takes and what g computes (not how g computes it, though you will have to understand that to determine what g computes). Note you write no code for this subproblem.
(b) Use g to define a function count_wildcards that takes a pattern and returns how many WildcardP patterns it contains.
(c) Use g to define a function count_wild_and_variable_lengths that takes a pattern and returns the number of Wildcard patterns it contains plus the sum of the string lengths of all the variables in the variable patterns it contains. (Use String.size. We care only about variable names; the constructor names are not relevant.)
(d) Use g to define a function count_a_var that takes a string and a pattern (as a pair) and returns the number of times the string appears as a variable in the pattern. We care only about variable names; the constructor names are not relevant.
10. Write a function check_pat that takes a pattern and returns true if and only if all the variables appearing in the pattern are distinct from each other (i.e., use different strings). The constructor names are not relevant. Hints: The sample solution uses two helper functions. The first takes a pattern and returns a list of all the strings it uses for variables. Using foldl with a function that uses @ is useful in one case. The second takes a list of strings and decides if it has repeats. List.exists may be useful. Sample solution is 15 lines. These are hints: We are not requiring foldl and List.exists here, but they make it easier.
11. Write a function match that takes a valu * pattern and returns a (string * valu) list option, namely NONE if the pattern does not match and SOME lst where lst is the list of bindings if it does. Note that if the value matches but the pattern has no patterns of the form VariableP s, then the result is SOME []. Hints: Sample solution has one case expression with 7 branches. The branch for tuples uses all_answers and ListPair.zip. Sample solution is 13 lines. Remember to look above for the rules for what patterns match what values, and what bindings they produce. These are hints: We are not requiring all_answers and ListPair.zip here, but they make it easier.
12. Write a function first_match that takes a value and a list of patterns and returns a
(string * valu) list option, namely NONE if no pattern in the list matches or SOME lst where lst is the list of bindings for the first pattern in the list that matches. Use first_answer and a handle-expression. Hints: Sample solution is 3 lines.
Type Summary: Evaluating a correct homework solution should generate these bindings, in addition to the bindings for datatype and exception definitions:
val only_lowercase = fn : string list - string list val longest_string1 = fn : string list - string val longest_string2 = fn : string list - string val longest_string_helper = fn : (int * int - bool) - string list - string val longest_string3 = fn : string list - string val longest_string4 = fn : string list - string val longest_lowercase = fn : string list - string val caps_no_X_string = fn : string - string val first_answer = fn : (’a - ’b option) - ’a list - ’b val all_answers = fn : (’a - ’b list option) - ’a list - ’b list option val count_wildcards = fn : pattern - int val count_wild_and_variable_lengths = fn : pattern - int val count_a_var = fn : string * pattern - int val check_pat = fn : pattern - bool val match = fn : valu * pattern - (string * valu) list option val first_match = fn : valu - pattern list - (string * valu) list option
Of course, generating these bindings does not guarantee that your solutions are correct.
1. Write a function only_lowercase that takes a string list and returns a string list that has only the strings in the argument that start with an lowercase letter. Assume all strings have at least 1 character. Use List.filter, Char.isLower, and String.sub to make a 1-2 line solution.
2. Write a function longest_string1 that takes a string list and returns the longest string in the list. If the list is empty, return "". In the case of a tie, return the string closest to the beginning of the list. Use foldl, String.size, and no recursion (other than the implementation of foldl is recursive).
3. Write a function longest_string2 that is exactly like longest_string1 except in the case of ties it returns the string closest to the end of the list. Your solution should be almost an exact copy of longest_string1. Still use foldl and String.size.
4. Write functions longest_string_helper, longest_string3, and longest_string4 such that:
• longest_string3 has the same behavior as longest_string1 and longest_string4 has the same behavior as longest_string2.
• longest_string_helper has type (int * int - bool) - string list - string
(notice the currying). This function will look a lot like longest_string1 and longest_string2 but is more general because it takes a function as an argument.
• If longest_string_helper is passed a function that behaves like (so it returns true exactly when its first argument is stricly greater than its second), then the function returned has the same behavior as longest_string1.
• longest_string3 and longest_string4 are defined with val-bindings and partial applications of longest_string_helper.
5. Write a function longest_lowercase that takes a string list and returns the longest string in the list that begins with an lowercase letter, or "" if there are no such strings. Assume all strings have at least 1 character. Use a val-binding and the ML library’s o operator for composing functions. Resolve ties like in problem 2.
6. Write a function caps_no_X_string that takes a string and returns the string that is like the input except every letter is capitalized and every “x” or “X” is removed (e.g., “aBxXXxDdx” becomes “ABDD”). Use a val binding, ML’s o operator, 2–3 library functions in the String module, 1 in the Char module, and 1–2 in the List module. Browse the module documentation to find the most useful functions. Note ML has strange syntax for character literals, e.g., #"A".
The next two problems involve writing functions over lists that will be useful in later problems.
7. Write a function first_answer of type (’a - ’b option) - ’a list - ’b (notice the 2 arguments are curried). The first argument should be applied to elements of the second argument in order until the first time it returns SOME v for some v and then v is the result of the call to first_answer. If the first argument returns NONE for all list elements, then first_answer should raise the exception NoAnswer. Hints: Sample solution is 5 lines and does nothing fancy.
8. Write a function all_answers of type (’a - ’b list option) - ’a list - ’b list option (notice the 2 arguments are curried). The first argument should be applied to elements of the second argument. If it returns NONE for any element, then the result for all_answers is NONE. Else the calls to the first argument will have produced SOME lst1, SOME lst2, ... SOME lstn and the result of all_answers is SOME lst where lst is lst1, lst2, ..., lstn appended together (order doesn’t matter). Hints: The sample solution is 8 lines. It uses a helper function with an accumulator and uses @. Note all_answers f [] should evaluate to SOME [].
The remaining problems use these type definitions, which are inspired by the type definitions an ML implementation would use to implement pattern matching:
datatype pattern = WildcardP | VariableP of string | UnitP | ConstantP of int
| ConstructorP of string * pattern | TupleP of pattern list
datatype valu = Constant of int | Unit | Constructor of string * valu | Tuple of valu list
Given valuv and patternp, either pmatches v or not. If it does, the match produces a list of string * valu pairs; order in the list does not matter. The rules for matching should be unsurprising:
• WildcardP matches everything and produces the empty list of bindings.
• VariableP s matches any value v and produces the one-element list holding (s,v).
• UnitP matches only Unit and produces the empty list of bindings.
• ConstantP 17 matches only Constant 17 and produces the empty list of bindings (and similarly for other integers).
• ConstructorP(s1,p) matches Constructor(s2,v) if s1 and s2 are the same string (you can compare them with =) and p matches v. The list of bindings produced is the list from the nested pattern match. We call the strings s1 and s2 the constructor name.
• TupleP ps matches a value of the form Tuple vs if ps and vs have the same length and for all i, the ith element of ps matches the ith element of vs. The list of bindings produced is all the lists from the nested pattern matches appended together.
• Nothing else matches.
9. (This problem uses the pattern datatype but is not really about pattern-matching.)
fun g f1 f2 p = let
val r = g f1 f2
in
case p of
WildcardP = f1 ()
| VariableP x = f2 x
| ConstructorP(_,p) = r p
| TupleP ps = List.foldl (fn (p,i) = (r p) + i) 0 ps
| _ = 0
end
(a) In an ML comment, describe in a few English sentences the arguments that g takes and what g computes (not how g computes it, though you will have to understand that to determine what g computes). Note you write no code for this subproblem.
(b) Use g to define a function count_wildcards that takes a pattern and returns how many WildcardP patterns it contains.
(c) Use g to define a function count_wild_and_variable_lengths that takes a pattern and returns the number of Wildcard patterns it contains plus the sum of the string lengths of all the variables in the variable patterns it contains. (Use String.size. We care only about variable names; the constructor names are not relevant.)
(d) Use g to define a function count_a_var that takes a string and a pattern (as a pair) and returns the number of times the string appears as a variable in the pattern. We care only about variable names; the constructor names are not relevant.
10. Write a function check_pat that takes a pattern and returns true if and only if all the variables appearing in the pattern are distinct from each other (i.e., use different strings). The constructor names are not relevant. Hints: The sample solution uses two helper functions. The first takes a pattern and returns a list of all the strings it uses for variables. Using foldl with a function that uses @ is useful in one case. The second takes a list of strings and decides if it has repeats. List.exists may be useful. Sample solution is 15 lines. These are hints: We are not requiring foldl and List.exists here, but they make it easier.
11. Write a function match that takes a valu * pattern and returns a (string * valu) list option, namely NONE if the pattern does not match and SOME lst where lst is the list of bindings if it does. Note that if the value matches but the pattern has no patterns of the form VariableP s, then the result is SOME []. Hints: Sample solution has one case expression with 7 branches. The branch for tuples uses all_answers and ListPair.zip. Sample solution is 13 lines. Remember to look above for the rules for what patterns match what values, and what bindings they produce. These are hints: We are not requiring all_answers and ListPair.zip here, but they make it easier.
12. Write a function first_match that takes a value and a list of patterns and returns a
(string * valu) list option, namely NONE if no pattern in the list matches or SOME lst where lst is the list of bindings for the first pattern in the list that matches. Use first_answer and a handle-expression. Hints: Sample solution is 3 lines.
Type Summary: Evaluating a correct homework solution should generate these bindings, in addition to the bindings for datatype and exception definitions:
val only_lowercase = fn : string list - string list val longest_string1 = fn : string list - string val longest_string2 = fn : string list - string val longest_string_helper = fn : (int * int - bool) - string list - string val longest_string3 = fn : string list - string val longest_string4 = fn : string list - string val longest_lowercase = fn : string list - string val caps_no_X_string = fn : string - string val first_answer = fn : (’a - ’b option) - ’a list - ’b val all_answers = fn : (’a - ’b list option) - ’a list - ’b list option val count_wildcards = fn : pattern - int val count_wild_and_variable_lengths = fn : pattern - int val count_a_var = fn : string * pattern - int val check_pat = fn : pattern - bool val match = fn : valu * pattern - (string * valu) list option val first_match = fn : valu - pattern list - (string * valu) list option
Of course, generating these bindings does not guarantee that your solutions are correct.
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