# Functors Today we will use the standard library's Map module, practice writing our own functors, and take a closer look at compilation units. ## The Map module The [Map module][map] in the OCaml standard library is an implementation of a dictionary data structure. Recall that dictionaries map *keys* to *values*. If a key \\(k\\) maps to a value \\(v\\), we say that \\(v\\) is *bound* to \\(k\\). [map]: http://caml.inria.fr/pub/docs/manual-ocaml/libref/Map.html ##### Exercise: make char map [&#10029;] To create a map, we first have to use the `Map.Make` functor to produce a module that is specialized for the type of keys we want. Type the following in utop: ``` # module CharMap = Map.Make(Char);; ``` The output tells you that a new module named `CharMap` has been defined, and it gives you a signature for it. Find the values `empty`, `add`, and `remove` in that signature. Explain their types in your own words. &square; ##### Exercise: char ordered [&#10029;] The `Map.Make` functor requires its input module to match the `Map.OrderedType` signature. Look at [that signature][ord] as well as the [signature for the `Char` module][char]. Explain in your own words why we are allowed to pass `Char` as an argument to `Map.Make`. [ord]: http://caml.inria.fr/pub/docs/manual-ocaml/libref/Map.OrderedType.html [char]: http://caml.inria.fr/pub/docs/manual-ocaml/libref/Char.html &square; ##### Exercise: use char map [&#10029;&#10029;] Using the `CharMap` you just made, create a map that contains the following bindings: * `'A'` maps to `"Alpha"` * `'E'` maps to `"Echo"` * `'S'` maps to `"Sierra"` * `'V'` maps to `"Victor"` Use `CharMap.find` to find the binding for `'E'`. Now remove the binding for `'A'`. Use `CharMap.mem` to find whether `'A'` is still bound. Use the function `CharMap.bindings` to convert your map into an association list. Are the correct three bindings active in it? &square; ##### Exercise: bindings [&#10029;&#10029;] Investigate the [documentation of the `Map.S`][map.s] signature to find the specification of `bindings`. Which of these expressions will return the same association list? 1. `CharMap.(empty |> add 'x' 0 |> add 'y' 1 |> bindings)` 2. `CharMap.(empty |> add 'y' 1 |> add 'x' 0 |> bindings)` 3. `CharMap.(empty |> add 'x' 2 |> add 'y' 1 |> remove 'x' |> add 'x' 0 |> bindings)` Check your answer in utop. *Note: make sure you understand the syntax being used in the expressions above. If not, ask your TA to review it.* &square; [map.s]: http://caml.inria.fr/pub/docs/manual-ocaml/libref/Map.S.html Here is a type for dates: ``` type date = { month:int; day:int } ``` For example, March 31st would be represented as `{month=3; day=31}`. Our goal in the next few exercises is to implement a map whose keys have type `date`. Obviously it's possible to represent invalid dates with type `date`&mdash;for example, `{ month=6; day=50 }` would be June 50th, which is [not a real date][parksandrec]. The behavior of your code in the exercises below is unspecified for invalid dates. [parksandrec]: http://nbcparksandrec.tumblr.com/post/46760908046/march-31st-is-a-day ##### Exercise: date order [&#10029;&#10029;] To create a map over dates, we need a module that we can pass as input to `Map.Make`. That module will need to match the `Map.OrderedType` signature. Create such a module. Here is some code to get you started: ``` module Date = struct type t = date let compare ... end ``` Recall the [specification of `compare`][ord] in `Map.OrderedType` as you write your `Date.compare` function. ##### Exercise: calendar [&#10029;&#10029;] Use the `Map.Make` functor with your `Date` module to create a `DateMap` module. Then define a `calendar` type as follows: ``` type calendar = string DateMap.t ``` The idea is that `calendar` maps a `date` to the name of an event occurring on that date. Using the functions in the `DateMap` module, create a calendar with a few entries in it. If you need inspiration, here are some good ideas for entries: * 10/12: CS 3110 Fall 2017 Prelim * 12/06: CS 3110 Fall 2017 Final Exam * 12/07: CS 3110 Fall 2017 Final Project Due &square; ##### Exercise: print calendar [&#10029;&#10029;] Write a function `print_calendar : calendar -> unit` that prints each entry in a calendar in a format similar the inspiring examples in the previous exercise. *Hint: use `DateMap.iter`, which is documented in the [`Map.S` signature][map.s].* &square; ## Writing functors Our goal in the next series of exercises is to write a functor that, given a module supporting a `to_string` function, returns a module supporting a `print` function that prints that string. ##### Exercise: ToString [&#10029;&#10029;] Write a module type `ToString` that specifies a signature with an abstract type `t` and a function `to_string : t -> string`. &square; ##### Exercise: Print [&#10029;&#10029;] Write a functor `Print` that takes as input a module named `M` of type `ToString`. The structure returned by your functor should have exactly one value in it, `print`, which is a function that takes a value of type `M.t` and prints a string representation of that value. &square; ##### Exercise: Print Int [&#10029;&#10029;] Create a module named `PrintInt` that is the result of applying the functor `Print` to a new module `Int`. You will need to write `Int` yourself. The type `Int.t` should be `int`. *Hint: do not seal `Int`.* Experiment with `PrintInt` in utop. Use it to print the value of an integer. &square; ##### Exercise: Print String [&#10029;&#10029;] Create a module named `PrintString` that is the result of applying the functor `Print` to a new module `MyString`. You will need to write `MyString` yourself. *Hint: do not seal `MyString`.* Experiment with `PrintString` in utop. Use it to print the value of a string. &square; ##### Exercise: Print reuse [&#10029;] Explain in your own words how `Print` has achieved code reuse, albeit a very small amount. &square; ## Compilation units *This section more properly belongs in the previous lab, but we moved it forward to here. The notes on compilation units are with the notes on modules from the previous lecture.* As the lecture notes on modules discuss, if you have a pair of files named `foo.ml` and `foo.mli` they together form a *compilation unit*. The `.ml` file is called the *implementation*, and the `.mli` file is called the *interface*. If, for example, `foo.mli` contains exactly the following: ``` val x : int val f : int -> int -> int ``` and `foo.ml` contains exactly the following: ``` let x = 0 let y = 12 let f x y = x + y ``` Then compiling `foo.ml` will have the same effect as defining the module `Foo` as follows: ``` module Foo : sig val x : int val f : int -> int -> int end = struct let x = 0 let y = 12 let f x y = x + y end ``` ##### Exercise: implementation without interface [&#10029;] Create a file named `date.ml`. In it put exactly the following code: ``` type date = { month:int; day:int } let make_date month day = {month; day} let get_month d = d.month let get_day d = d.day let to_string d = (string_of_int d.month) ^ "/" ^ (string_of_int d.day) ``` Compile that file to bytecode: ``` $ ocamlbuild date.cmo ``` Now start utop and type the following to use the module you've just created: ``` # #directory "_build";; # #load "date.cmo";; # let j1 = Date.make_date 1 1;; val j1 : Date.date = {Date.month = 1; day = 1} # j1.day;; - : int = 1 # Date.to_string j1;; - : string = "1/1" ``` &square; ##### Exercise: implementation with interface [&#10029;] After doing the previous exercise, also create a file named `date.mli`. In it put exactly the following code: ``` type date = { month:int; day:int; } val make_date : int -> int -> date val get_month : date -> int val get_day : date -> int val to_string : date -> string ``` Recompile `date.ml` to bytecode: ``` $ ocamlbuild date.cmo ``` Restart utop and re-issue the same phrases as before: ``` # #directory "_build";; # #load "date.cmo";; # let j1 = Date.make_date 1 1;; val j1 : Date.date = {Date.month = 1; day = 1} # j1.day;; - : int = 1 # Date.to_string j1;; - : string = "1/1" ``` &square; ##### Exercise: implementation with abstracted interface [&#10029;] After doing the previous two exercises, edit `date.mli` and change the first declaration in it to be exactly the following: ``` type date ``` The type `date` is now abstract. Recompile `date.ml` to bytecode: ``` $ ocamlbuild date.cmo ``` Restart utop and re-issue the same phrases as before. The responses to two of them will change. Explain in your own words those changes. ``` # #directory "_build";; # #load "date.cmo";; # let j1 = Date.make_date 1 1;; # j1.day;; # Date.to_string j1;; ``` &square; ## Additional exercises ##### Exercise: is for [&#10029;&#10029;&#10029;] Write a function `is_for : string CharMap.t -> string CharMap.t` that given an input map with bindings from \\(k_1\\) to \\(v_1\\), ..., \\(k_n\\) to \\(v_n\\), produces an output map with the same keys, but where each key \\(k_i\\) is now bound to the string "\\(k_i\\) is for \\(v_i\\)". For example, if `m` maps `'a'` to `"apple"`, then `is_for m` would map `'a'` to `"a is for apple"`. *Hint: there is a one-line solution that uses a function from the `Map.S` signature. To convert a character to a string, you could use `String.make`. An even fancier way would be to use `Printf.sprintf`.* <!--bigger hint: mapi --> &square; ##### Exercise: first after [&#10029;&#10029;&#10029;] Write a function `first_after : calendar -> Date.t -> string` that returns the name of the first event that occurs strictly after the given date. If there is no such event, the function should raise `Not_found`, which is an exception already defined in the standard library. *Hint: there is a one-line solution that uses two functions from the `Map.S` signature.* <!--bigger hint: split --> &square; ##### Exercise: sets [&#10029;&#10029;&#10029;] The standard library `Set` module is quite similar to the `Map` module. Use it to create a module that represents sets of *case-insensitive strings*. Strings that differ only in their case should be considered equal by the set. For example, the sets {"grr", "argh"} and {"aRgh", "GRR"} should be considered the same, and adding "gRr" to either set should not change the set. Assuming your module is named `CisSet`, here is some test code: ``` # CisSet.(equal (of_list ["grr"; "argh"]) (of_list ["GRR"; "aRgh"])) - : bool = true ``` &square; ##### Exercise: Print String reuse revisited [&#10029;&#10029;] The `PrintString` module you created above supports just one operation: `print`. It would be great to have a module that supports all the `String` module functions in addition to that `print` operation, and it would be super great to derive such a module without having to copy any code. Define a module `StringWithPrint`. It should have all the values of the built-in `String` module. It should also have the `print` operation, which should be derived from the `Print` functor rather than being copied code. *Hint: use two `include` statements.* <!-- bigger hint: include String include Print(MyString) --> &square; ##### Exercise: printer for date [&#10029;&#10029;&#10029;, recommended] After finishing **implementation with abstracted interface**, and after reading the notes in the previous lecture about `#install_printer`, add a declaration to `date.mli`: ``` val format : Format.formatter -> date -> unit ``` And add a definition of `format` to `date.ml`. *Hint: use `Format.fprintf` and `Date.to_string`.* Now recompile, load utop, and install the printer by issuing the directive ``` #install_printer Date.format;; ``` after loading `date.cmo`. Reissue the other phrases to utop as you did in the exercises above. The response from one phrase will change in a helpful way. Explain why. &square; ## Challenge exercise: Algebra Download this file: [algebra.ml](algebra.ml). It contains two signatures and four structures: * `Ring` is signature that describes the algebraic structure called a *[ring]*, which is an abstraction of the addition and multiplication operators. * `Field` is a signature that describes the algebraic structure called a *[field]*, which is like a ring but also has an abstraction of the division operation. * `IntRing` and `FloatRing` are structures that implement rings in terms of `int` and `float`. * `IntField` and `FloatField` are structures that implement fields in terms of `int` and `float`. * `IntRational` and `FloatRational` are structures that implement fields in terms of ratios (aka fractions)&mdash;that is, pairs of `int` and pairs of `float`. *(For afficionados of abstract algebra: of course these representations don't necessarily obey all the axioms of rings and fields because of the limitations of machine arithmetic. Also, the division operation in `IntField` is ill-defined on zero. Try not to worry about that.)* Using this code, you can write expressions like the following: ``` # FloatField.(of_int 9 + of_int 3 / of_int 4 |> to_string);; - : string = "9.75" # IntRational.( let half = one / (one+one) in let quarter = half*half in let three = one+one+one in let nine = three*three in to_string (nine + (three*quarter)) );; - : string = "39/4" ``` [ring]: https://en.wikipedia.org/wiki/Ring_(mathematics) [field]: https://en.wikipedia.org/wiki/Field_(mathematics) ##### Exercise: refactor arith [&#10029;&#10029;&#10029;&#10029;] The file [algebra.ml](algebra.ml) contains a great deal of duplicated code. Refactor the code to improve the amount of code reuse it exhibits. To do that, use `include`, functors, and introduce additional structures and signatures as needed. There isn't necessarily a right answer here, but it is possible to eliminate all the duplicated code. Here's some advice to guide you toward a good solution: * No name should be *directly declared* in more than one signature. For example, `(+)` should not be directly declared in `Field`; it should be reused from an earlier signature. By "directly declared" we mean a declaration of the form `val name : ...`. An indirect declaration would be one that results from an `include`. * You need only three *direct definitions* of the algebraic operations and numbers (plus, minus, times, divide, zero, one): once for `int`, once for `float`, and once for ratios. For example, `IntField.(+)` should not be directly defined as `Pervasives.(+)`; rather, it should be reused from elsewhere. By "directly defined" we mean a definition of the form `let name = ...`. An indirect definition would be one that results from an `include` or a functor application. * The rational structures can both be produced by a single functor that is applied once to `IntField` and once to `FloatField`. * It's possible to eliminate all duplication of `of_int`, such that it is directly defined exactly once, and all structures reuse that definition; and such that it is directly declared in only one signature. This will require the use of functors. It will also require inventing an algorithm that can convert an integer to an arbitrary `Ring` representation, regardless of what the representation type of that `Ring` is. [dsub]: http://caml.inria.fr/pub/docs/manual-ocaml/extn.html#sec234 When you're done, the types of all the modules should remain unchanged. You can easily see those types by running `ocamlc -i algebra.ml`, which will originally output the following: ``` module type Ring = sig type t val zero : t val one : t val ( + ) : t -> t -> t val ( ~- ) : t -> t val ( * ) : t -> t -> t val to_string : t -> string val of_int : int -> t end module type Field = sig type t val zero : t val one : t val ( + ) : t -> t -> t val ( ~- ) : t -> t val ( * ) : t -> t -> t val ( / ) : t -> t -> t val to_string : t -> string val of_int : int -> t end module IntRing : Ring module IntField : Field module FloatRing : Ring module FloatField : Field module IntRational : Field module FloatRational : Field ``` The final output of that command on your solution might define additional types, but the ones above should remain literally identical. &square;