3 We're going to make gradual improvements to the calculator we developed in [week7](/reader_monad_for_variable_binding).
5 ##Original Calculator##
7 In a real programming application, one would usually start with a string that needs to be parsed and interpreted, such as:
9 let x = 1 in let y = x + 2 in x * y
11 The parsing phase converts this to an "abstract syntax tree" (AST), which in this case might be:
14 Let ('y', Addition (Variable 'x', Constant 2),
15 Multiplication (Variable 'x', Variable 'y')))
17 Then the interpreter (or "evaluator") would convert that AST into an "expressed value": in this case, to the integer 3. We're not concerning ourselves with the parsing phase here, so we're just thinking about how to interpret expressions that are already in AST form.
19 The language we had in week 7 looked like this:
21 type term = Constant of int
22 | Multiplication of (term * term)
23 | Addition of (term * term)
25 | Let of (char * term * term)
28 and the evaluation function looked like this:
30 let rec eval (t : term) (e: (char * int) list) = match t with
32 | Multiplication (t1, t2) -> (eval t1 e) * (eval t2 e)
33 | Addition (t1, t2) -> (eval t1 e) + (eval t2 e)
35 (* lookup the value of c in the current environment
36 This will fail if c isn't assigned anything by e *)
39 (* evaluate t2 in a new environment where c has been associated
40 with the result of evaluating t1 in the current environment *)
41 eval t2 ((c, eval t1 e) :: e)
45 ##Adding Booleans and Pairs##
47 Let's tweak this a bit.
49 First, let's abstract away from the assumption that our terms always evaluate to `int`s. Let's suppose they evaluate to a more general type, which might have an `int` payload, or might have, for example, a `bool` payload.
51 type expressed_value = Int of int | Bool of bool;;
53 We'll add one boolean predicate, `Iszero`, and an `If...` construction.
55 Similarly, we might allow for some terms to express pairs of other terms:
57 type expressed_value = Int of int | Bool of bool | Pair of expressed_value * expressed_value;;
59 We'd then want to add the ability to construct pairs, and extract their components.
61 We won't try here to catch any type errors, such as attempts to add a `bool` to an `int`, or attempts to check whether a `bool` iszero. Neither will we try here to monadize anything: these will be implementations of a calculator with all the plumbing exposed. What we will do is add more and more features to the calculator.
63 We'll switch over to using variable `g` for assignment functions, which is a convention many of you seem familiar with. As we mentioned a few times in week 9, for some purposes it's easier to implement environment or assignment functions as functions from `char`s to `int`s (or whatever variables are bound to), rather than as lists of pairs. However, we'll stick with this implementation for now. We will however abstract out the type that the variables are bound to. For now, we'll suppose that they're bound to the same types that terms can express.
65 type bound_value = expressed_value;;
66 type assignment = (char * bound_value) list;;
68 Here's where we should be now. We expand some of the clauses in the `eval` function for clarity, and we rename a few variables:
72 | Multiplication of (term * term)
73 | Addition of (term * term)
75 | Let of (char * term * term)
77 | If of (term * term * term)
78 | Makepair of (term * term)
82 let rec eval (t : term) (g : assignment) = match t with
83 Intconstant x -> Int x
84 | Multiplication (t1, t2) ->
85 (* we don't handle cases where the subterms don't evaluate to Ints *)
86 let Int i1 = eval t1 g
87 in let Int i2 = eval t2 g
88 (* Multiplication (t1, t2) should evaluate to an Int *)
90 | Addition (t1, t2) ->
91 let Int i1 = eval t1 g
92 in let Int i2 = eval t2 g
95 (* we don't handle cases where g doesn't bind var to any value *)
97 | Let (var_to_bind, t2, t3) ->
98 (* evaluate t3 under a new assignment where var_to_bind has been bound to
99 the result of evaluating t2 under the current assignment *)
100 let value2 = eval t2 g
101 in let g' = (var_to_bind, value2) :: g
104 (* we don't handle cases where t1 doesn't evaluate to an Int *)
105 let Int i1 = eval t1 g
106 (* Iszero t1 should evaluate to a Bool *)
109 (* we don't handle cases where t1 doesn't evaluate to a boolean *)
110 let Bool b1 = eval t1 g
111 in if b1 then eval t2 g
113 | Makepair (t1, t2) ->
114 let value1 = eval t1 g
115 in let value2 = eval t2 g
116 in Pair (value1, value2)
118 (* we don't handle cases where t1 doesn't evaluate to a Pair *)
119 let Pair (value1, value2) = eval t1 g
123 The complete code is available [here](/code/calculator/calc1.ml).
125 ##Adding Function Values##
127 Now we want to add function values to our language, so that we can interpret (the abstract syntax trees of) expressions like this:
129 let x = 1 in let f = lambda y -> y + x in apply f 2
131 What changes do we need to handle this?
133 We can begin with our language:
137 | Multiplication of (term * term)
138 | Addition of (term * term)
140 | Let of (char * term * term)
142 | If of (term * term * term)
143 | Makepair of (term * term)
145 | Lambda of (char * term)
146 | Apply of (term * term)
149 Next, we need to expand our stock of `expressed_value`s to include function values as well. How should we think of these? We've several times mentioned the issue of how to handle free variables in a function's body, like the `x` in `lambda y -> y + x`. We'll follow the usual functional programming standard for these (known as "lexical scoping"), which keeps track of what value `x` has in the function declaration's lexical environment. That shouldn't get shadowed by any different value `x` may have when the function value is later applied. So:
151 let x = 1 in let f = lambda y -> y + x in let x = 2 in apply f 2
153 should evaluate to `3` not to `4`. To handle this, the function values we construct need to keep track of the present values of all free variables in the function's body. The combination of the function's body and the values of its free variables is called a "function closure." We'll implement these closures in a straightforward though inefficient way: we'll just stash away a copy of the assignment in effect when the function value is being constructed. Our function values also need to keep track of which of their variables are to be bound to the arguments they get applied to. All together, then, we need three pieces of information: which variables are to be bound to arguments, what the function's body is, and something that keeps track of the right values for the free variables in the function body. We'll pack this all together into an additional variant for our `expressed_value` type:
155 type expressed_value = Int of int | Bool of bool | Pair of expressed_value * expressed_value | Closure of char * term * assignment;;
157 We'd like to define `bound_value`s and `assignment`s just as before:
159 type bound_value = expressed_value;;
160 type assignment = (char * bound_value) list;;
162 However, note that we have a recursive relation between these types: `expressed_value` is defined partly in terms of `assignment`, which is defined partly in terms of `bound_value`, which is equivalent to `expressed_value`. In OCaml one has to define such types using the following form:
164 type bound_value = expressed_value
165 and assignment = (char * bound_value) list
166 and expressed_value = Int of int | Bool of bool | Pair of expressed_value * expressed_value | Closure of char * term * assignment;;
168 Now our evaluation function needs two further clauses to interpret the two new expression forms `Lambda (...)` and `Apply (...)`:
170 let rec eval (t : term) (g : assignment) = match t with
172 | Lambda (arg_var, t2) -> Closure (arg_var, t2, g)
174 (* we don't handle cases where t1 doesn't evaluate to a function value *)
175 let Closure (arg_var, body, savedg) = eval t1 g
176 in let value2 = eval t2 g
177 (* evaluate body under savedg, except with arg_var bound to value2 *)
178 in let savedg' = (arg_var, value2) :: savedg
182 The complete code is available [here](/code/calculator/calc2.ml).
184 ##Adding Recursive Functions##
186 There are different ways to include recursion in our calculator. First, let's imagine our language expanded like this:
188 let x = 1 in letrec f = lambda y -> if iszero y then x else y * apply f (y - 1) in apply f 3
190 where the AST would be:
192 Let ('x', Intconstant 1,
195 If (Iszero (Variable 'y'),
197 Multiplication (Variable 'y',
199 Addition (Variable 'y', Intconstant (-1)))))),
200 Apply (Variable 'f', Intconstant 3)))
202 Here is the expanded definition for our language type:
206 | Multiplication of (term * term)
207 | Addition of (term * term)
209 | Let of (char * term * term)
211 | If of (term * term * term)
212 | Makepair of (term * term)
214 | Lambda of (char * term)
215 | Apply of (term * term)
216 | Letrec of (char * term * term)
219 Now consider what we'll need to do when evaluating a term like `Letrec ('f', Lambda (...), t2)`. The subterm `Lambda (...)` will evaluate to something of the form `Closure ('y', body, savedg)`, where `Variable 'f'` may occur free in `body`. What we'll want to do is to ensure that when `body` is applied, it's applied using not the assignment `savedg` but a modified assignment `savedg'` which binds `'f'` to this very function value. That is, we want to bind `'f'` not to:
221 Closure ('y', body, savedg)
225 let orig_closure = Closure ('y', body, savedg)
226 in let savedg' = ('f', orig_closure) :: savedg
227 in let new_closure = Closure ('y', body, savedg')
230 Except, this isn't quite right. It's almost what we want, but not exactly. Can you see the flaw?
232 The flaw is this: inside `new_closure`, what is `'f'` bound to? It's bound by `savedg'` to `orig_closure`, which in turn leaves `'f'` free (or bound to whatever existing value it had according to `savedg`). This isn't what we want. It'll break if we need to make applications of `Variable 'f'` which recurse more than once.
234 What we really want is for `'f'` to be bound to `new_closure`, something like this:
236 let rec new_closure = Closure ('y', body, ('f', new_closure) :: savedg)
239 And as a matter of fact, OCaml *does* permit us to recursively define cyclical lists in this way. So a minimal change to our evaluation function would suffice:
241 let rec eval (t : term) (g : assignment) = match t with
243 | Letrec (var_to_bind, t2, t3) ->
244 (* we don't handle cases where t2 doesn't evaluate to a function value *)
245 let Closure (arg_var, body, savedg) = eval t2 g
246 in let rec new_closure = Closure (arg_var, body, (var_to_bind, new_closure) :: savedg)
247 in let g' = (var_to_bind, new_closure) :: g
251 However, this is a somewhat exotic ability in a programming language, so it would be good to work out how to interpret `Letrec (...)` forms without relying on it.
253 If we implemented assignments as functions rather than as lists of pairs, the corresponding move would be less exotic. In that case, our `Let (...)` and `Letrec (...)` clauses would look something like this:
256 | Let (var_to_bind, t2, t3) ->
257 let value2 = eval t2 g
258 in let g' = fun var -> if var = var_to_bind then value2 else g var
261 | Letrec (var_to_bind, t2, t3) ->
262 let Closure (arg_var, body, savedg) = eval t2 g
263 in let rec savedg' = fun var -> if var = var_to_bind then Closure (arg_var, body, savedg') else savedg var
264 in let g' = fun var -> if var = var_to_bind then Closure (arg_var, body, savedg') else g var
268 and this is just a run-of-the-mill use of recursive functions. However, for this exercise we'll continue using lists of pairs, and work out how to interpret `Letrec (...)` forms using them.
270 The way we'll do this is that, when we bind a variable to a value, we'll keep track of whether the term was bound via `Let` or `Letrec`. We'll rely on that to interpret pairs of terms like these differently:
274 Let ('f', Lambda ('y', Variable 'f')),
279 Letrec ('f', Lambda ('y', Variable 'f')),
282 In the first case, an application of `Variable 'f'` to any argument should evaluate to `Int 1`; in the second case, it should evaluate to the same function closure that `Variable 'f'` evaluates to. We'll keep track of which way a variable was bound by expanding our `bound_value` type:
284 type bound_value = Nonrecursive of expressed_value |
285 Recursive_Closure of char * char * term * assignment
286 and assignment = (char * bound_value) list
287 and expressed_value = Int of int | Bool of bool | Pair of expressed_value * expressed_value | Closure of char * term * assignment;;
290 Since we're not permitting ourselves OCaml's ability to recursively define cyclical lists, we're not going to be able to update the saved assignment in a closure when that closure is recursively bound to a variable. Instead, we'll just make a note that variable `'f'` is supposed to be the recursively bound one---by binding it not to `Nonrecursive (Closure (arg_var, body, savedg))` but rather to `Recursive_Closure ('f', arg_var, body, savedg)`. We'll do the work to make the saved assignment recursive in the right way *later*, when we *evaluate* `Variable 'f'`. The result will look like this:
292 let rec eval (t : term) (g : assignment) = match t with
294 | Variable (var) -> (
295 (* we don't handle cases where g doesn't bind var to any value *)
296 match List.assoc var g with
297 | Nonrecursive value -> value
298 | Recursive_Closure (self_var, arg_var, body, savedg) as rec_closure ->
299 (* we update savedg to bind self_var to rec_closure here *)
300 let savedg' = (self_var, rec_closure) :: savedg
301 in Closure (arg_var, body, savedg')
303 | Let (var_to_bind, t2, t3) ->
304 (* evaluate t3 under a new assignment where var_to_bind has been bound to
305 the result of evaluating t2 under the current assignment *)
306 let value2 = eval t2 g
307 (* we have to wrap value2 in Nonrecursive *)
308 in let g' = (var_to_bind, Nonrecursive value2) :: g
311 | Lambda (arg_var, t2) -> Closure (arg_var, t2, g)
313 (* we don't handle cases where t1 doesn't evaluate to a function value *)
314 let Closure (arg_var, body, savedg) = eval t1 g
315 in let value2 = eval t2 g
316 (* evaluate body under savedg, except with arg_var bound to Nonrecursive value2 *)
317 in let savedg' = (arg_var, Nonrecursive value2) :: savedg
319 | Letrec (var_to_bind, t2, t3) ->
320 (* we don't handle cases where t2 doesn't evaluate to a function value *)
321 let Closure (arg_var, body, savedg) = eval t2 g
322 (* evaluate t3 under a new assignment where var_to_bind has been recursively bound to that function value *)
323 in let g' = (var_to_bind, Recursive_Closure (var_to_bind, arg_var, body, savedg)) :: g
327 The complete code is available [here](/code/calculator/calc3.ml).
329 ##Adding Mutable Cells##
331 Next, we'll add mutable cells (explicit-style mutation) to our calculator, as we did in [[week9]].
333 We'll add a few more syntactic forms to the language:
337 | Multiplication of (term * term)
338 | Addition of (term * term)
340 | Let of (char * term * term)
342 | If of (term * term * term)
343 | Makepair of (term * term)
345 | Lambda of (char * term)
346 | Apply of (term * term)
347 | Letrec of (char * term * term)
350 | Setref of (term * term)
353 And we now have to allow for `Mutcell`s as an additional kind of `expressed_value`. These are implemented as wrappers around an index into a `store`:
357 type bound_value = Nonrecursive of expressed_value |
358 Recursive_Closure of char * char * term * assignment
359 and assignment = (char * bound_value) list
360 and expressed_value = Int of int | Bool of bool | Pair of expressed_value * expressed_value | Closure of char * term * assignment | Mutcell of index;;
362 type store = expressed_value list;;
364 Our evaluation function will now expect a `store` argument as well as an `assignment`, and will return an `expressed_value * store` pair:
366 let rec eval (t : term) (g : assignment) (s : store) = match t with
367 Intconstant x -> (Int x, s)
369 | Variable (var) -> (
370 (* we don't handle cases where g doesn't bind var to any value *)
371 match List.assoc var g with
372 | Nonrecursive value -> value
373 | Recursive_Closure (self_var, arg_var, body, savedg) as rec_closure ->
374 (* we update savedg to bind self_var to rec_closure here *)
375 let savedg' = (self_var, rec_closure) :: savedg
376 in Closure (arg_var, body, savedg')
379 | Lambda (arg_var, t2) -> (Closure (arg_var, t2, g), s)
382 also, we'll need to be sure to thread the store argument through the evaluation of any subterms, as here:
385 | Multiplication (t1, t2) ->
386 (* we don't handle cases where the subterms don't evaluate to Ints *)
387 let (Int i1, s') = eval t1 g s
388 in let (Int i2, s'') = eval t2 g s'
389 (* Multiplication (t1, t2) should evaluate to an Int *)
390 in (Int (i1 * i2), s'')
391 | Addition (t1, t2) ->
392 let (Int i1, s') = eval t1 g s
393 in let (Int i2, s'') = eval t2 g s'
394 in (Int (i1 + i2), s'')
396 | Let (var_to_bind, t2, t3) ->
397 (* evaluate t3 under a new assignment where var_to_bind has been bound to
398 the result of evaluating t2 under the current assignment *)
399 let (value2, s') = eval t2 g s
400 (* we have to wrap value2 in Nonrecursive *)
401 in let g' = (var_to_bind, Nonrecursive value2) :: g
404 (* we don't handle cases where t1 doesn't evaluate to an Int *)
405 let (Int i1, s') = eval t1 g s
406 (* Iszero t1 should evaluate to a Bool *)
407 in (Bool (i1 = 0), s')
409 | Makepair (t1, t2) ->
410 let (value1, s') = eval t1 g s
411 in let (value2, s'') = eval t2 g s'
412 in (Pair (value1, value2), s'')
414 (* we don't handle cases where t1 doesn't evaluate to a Pair *)
415 let (Pair (value1, value2), s') = eval t1 g s
419 (* we don't handle cases where t1 doesn't evaluate to a function value *)
420 let (Closure (arg_var, body, savedg), s') = eval t1 g s
421 in let (value2, s'') = eval t2 g s'
422 (* evaluate body under savedg, except with arg_var bound to Nonrecursive value2 *)
423 in let savedg' = (arg_var, Nonrecursive value2) :: savedg
424 in eval body savedg' s''
425 | Letrec (var_to_bind, t2, t3) ->
426 (* we don't handle cases where t2 doesn't evaluate to a function value *)
427 let (Closure (arg_var, body, savedg), s') = eval t2 g s
428 (* evaluate t3 under a new assignment where var_to_bind has been recursively bound to that function value *)
429 in let g' = (var_to_bind, Recursive_Closure (var_to_bind, arg_var, body, savedg)) :: g
433 The clause for `If (...)` is notable:
437 (* we don't handle cases where t1 doesn't evaluate to a boolean *)
438 let (Bool b1, s') = eval t1 g s
439 (* note we thread s' through only one of the then/else clauses *)
440 in if b1 then eval t2 g s'
444 Now we need to formulate the clauses for evaluating the new forms `Newref (...)`, `Deref (...)`, and `Setref (...)`.
448 let (starting_val, s') = eval t1 g s
449 (* note that s' may be different from s, if t1 itself contained any mutation operations *)
450 (* now we want to retrieve the next free index in s' *)
451 in let new_index = List.length s'
452 (* now we want to insert starting_val there; the following is an easy but inefficient way to do it *)
453 in let s'' = List.append s' [starting_val]
454 (* now we return a pair of a wrapped new_index, and the new store *)
455 in (Mutcell new_index, s'')
457 (* we don't handle cases where t1 doesn't evaluate to a Mutcell *)
458 let (Mutcell index1, s') = eval t1 g s
459 (* note that s' may be different from s, if t1 itself contained any mutation operations *)
460 in (List.nth s' index1, s')
462 (* we don't handle cases where t1 doesn't evaluate to a Mutcell *)
463 let (Mutcell index1, s') = eval t1 g s
464 (* note that s' may be different from s, if t1 itself contained any mutation operations *)
465 in let (new_value, s'') = eval t2 g s'
466 (* now we create a list which is just like s'' except it has new_value in index1 *)
467 in let rec replace_nth lst m =
469 | [] -> failwith "list too short"
470 | x::xs when m = 0 -> new_value :: xs
471 | x::xs -> x :: replace_nth xs (m - 1)
472 in let s''' = replace_nth s'' index1
473 (* we'll arbitrarily return Int 42 as the expressed_value of a Setref operation *)
477 The complete code is available [here](/code/calculator/calc4.ml).
479 ##Adding Mutable Pairs##
481 Suppose we wanted to work with pairs where we could mutate either component of the pair. Well, we've already given ourselves pairs, and mutable cells, so we could just work here with pairs of mutable cells. But it might sometimes be more wieldy to work with a structure that fused these two structures together, to give us a mutable pair. With the mutable pair, we wouldn't ask for the first element, and then apply `Deref` to it to get the value it then temporarily contains. Instead, asking for the first element would *constitute* asking for the value the mutable pair then temporarily contains in its first position.
483 This means a mutable pair is an interesting hybrid between explicit-style and implicit-style mutation. Looked at one way, it's just a generalization of an explicit mutable cell: it's just that where the mutable cells we implemented before were boxes with only one position, now we have boxes with two positions. Looked at another way, though, mutable pairs are similar to implicit-style mutation: for we don't have separate ways of referring to the first position of the mutable pair, and its dereferenced value. Peeking at the first position *just will be* peeking at its current dereferenced value.
485 To keep our codebase smaller, we'll implement mutable pairs instead of, not in addition to, the mutable cells from the previous section. Also, we'll leave out the immutable pairs we've been working with up to this point; in this implementation, all pairs will be mutable.
487 This implementation will largely parallel the previous one. Here are the differences. First, we remove the `Newref`, `Deref`, and `Setref` forms from the language. Our existing form `Makepair` will serve to create mutable pairs, and so will take over a role analogous to `Newref`. Our existing form `First` will take over a role analogous to `Deref`. We'll introduce one new form `Setfirst` that will take over a role analogous to `Setref`:
491 | Multiplication of (term * term)
492 | Addition of (term * term)
494 | Let of (char * term * term)
496 | If of (term * term * term)
497 | Makepair of (term * term)
499 | Lambda of (char * term)
500 | Apply of (term * term)
501 | Letrec of (char * term * term)
502 | Setfirst of (term * term)
505 Our `expressed_value` type changes in two ways: first, we eliminate the `Mutcell` variant added in the previous implementation. Instead, we now have our `Pair` variant wrap `index`es into the `store`:
509 type bound_value = Nonrecursive of expressed_value |
510 Recursive_Closure of char * char * term * assignment
511 and assignment = (char * bound_value) list
512 and expressed_value = Int of int | Bool of bool | Pair of index * index | Closure of char * term * assignment;;
514 type store = expressed_value list;;
516 Finally, here are the changed or added clauses to the evaluation function:
518 let rec eval (t : term) (g : assignment) (s : store) = match t with
520 | Makepair (t1, t2) ->
521 let (value1, s') = eval t1 g s
522 in let (value2, s'') = eval t2 g s'
523 (* now we want to retrieve the next free index in s'' *)
524 in let new_index = List.length s''
525 (* now we want to insert value1 and value2 there; the following is an easy but inefficient way to do it *)
526 in let s''' = List.append s'' [value1; value2]
527 in (Pair (new_index, new_index + 1), s''')
529 (* we don't handle cases where t1 doesn't evaluate to a Pair *)
530 let (Pair (index1, index2), s') = eval t1 g s
531 (* note that s' may be different from s, if t1 itself contained any mutation operations *)
532 in (List.nth s' index1, s')
534 | Setfirst (t1, t2) ->
535 (* we don't handle cases where t1 doesn't evaluate to a Pair *)
536 let (Pair (index1, index2), s') = eval t1 g s
537 (* note that s' may be different from s, if t1 itself contained any mutation operations *)
538 in let (new_value, s'') = eval t2 g s'
539 (* now we create a list which is just like s'' except it has new_value in index1 *)
540 in let rec replace_nth lst m =
542 | [] -> failwith "list too short"
543 | x::xs when m = 0 -> new_value :: xs
544 | x::xs -> x :: replace_nth xs (m - 1)
545 in let s''' = replace_nth s'' index1
549 Compare these to the clauses for `Newref`, `Deref`, and `Setref` in the previous implementation.
551 The complete code is available [here](/code/calculator/calc5.ml).
553 ##Adding Implicit Mutation##
555 Next we implement implicit-style mutation, as we did in [[week9]]. Here we don't have any explicit reference cells or mutable pairs; we'll return pairs back to their original immutable form. Instead, all variables will have mutable bindings. New reference cells will be implicitly introduced by the `Let` form. They'll also be implicitly introduced by the `Apply` form---we didn't have function values on the table during the [[week9]] discussion, so this didn't come up then. The reason we introduce new reference cells when `Apply`ing a function value to arguments is that we don't want mutation of those arguments inside the body of the function to propagate out and affect the reference cell that may have supplied the argument. When we call functions in this implementation, we just want to supply them with *values*, not with the reference cells we may be drawing those values from. Below, after we discuss *aliases*, we'll consider another strategy, where function bodies are given the ability to mutate the reference cells implicitly associated with the arguments they're supplied.
557 Our language for the present implementation will be the language for the calculator with recursive functions, with one added syntactic form, `Change (...)`:
561 | Multiplication of (term * term)
562 | Addition of (term * term)
564 | Let of (char * term * term)
566 | If of (term * term * term)
567 | Makepair of (term * term)
569 | Lambda of (char * term)
570 | Apply of (term * term)
571 | Letrec of (char * term * term)
572 | Change of (char * term * term)
575 In the present implementation, we separate the roles of the `bound_value` and `expressed_value` types. As we discussed in [[week9]], our assignment will bind all variables to indexes in the store, and the latter will contain the `expressed_value`s that the variables evaluate to. A consequence of this is that our definitions of the `bound_value` and `expressed_value` types no longer need to be mutually recursive:
579 type bound_value = index;;
580 type assignment = (char * bound_value) list;;
581 type expressed_value = Int of int | Bool of bool | Pair of expressed_value * expressed_value | Closure of char * term * assignment;;
583 type store = expressed_value list;;
585 Our evaluation function still interacts with a `store` argument in much the same way it did with explicit-style mutation. The clause for `Variable ...` works differently, because all `expressed_value`s now need to be retrieved from the `store`:
587 let rec eval (t : term) (g : assignment) (s : store) = match t with
590 (* we don't handle cases where g doesn't bind var to any value *)
591 let index = List.assoc var g
592 (* get value stored at location index in s *)
593 in let value = List.nth s index
597 So this clause takes over the roles that were separately played by `Variable` and `Deref` in the calculator with mutable cells. The role played by `Newref` is absorbed into `Let`, `Letrec`, and `Apply`:
600 | Let (var_to_bind, t2, t3) ->
601 let (value2, s') = eval t2 g s
602 (* note that s' may be different from s, if t2 itself contained any mutation operations *)
603 (* get next free index in s' *)
604 in let new_index = List.length s'
605 (* now we want to insert value2 there; the following is an easy but inefficient way to do it *)
606 in let s'' = List.append s' [value2]
607 (* bind var_to_bind to location new_index in the store *)
608 in let g' = ((var_to_bind, new_index) :: g)
612 (* we don't handle cases where t1 doesn't evaluate to a function value *)
613 let (Closure (arg_var, body, savedg), s') = eval t1 g s
614 in let (value2, s'') = eval t2 g s'
615 (* evaluate body under savedg, except with arg_var bound to a new location containing value2 *)
616 in let new_index = List.length s''
617 in let s''' = List.append s'' [value2]
618 in let savedg' = (arg_var, new_index) :: savedg
619 in eval body savedg' s'''
622 `Letrec` requires some reworking from what we had before. Earlier, we resorted to a `Recursive_Closure` variant on `bound_value`s because it gave us a non-exotic way to update the `savedg` component of a `Closure` to refer to a `new_closure` that contained that very updated `savedg`. Now that we we've got a mutation-supporting infrastructure in place, we can do this directly, without needing the unwieldy `Recursive_Closure` wrapper:
625 | Letrec (var_to_bind, t2, t3) ->
626 (* we don't handle cases where t2 doesn't evaluate to a function value *)
627 let (Closure (arg_var, body, savedg), s') = eval t2 g s
628 in let new_index = List.length s'
629 in let savedg' = (var_to_bind, new_index) :: savedg
630 in let new_closure = Closure (arg_var, body, savedg')
631 in let s'' = List.append s' [new_closure]
632 in let g' = (var_to_bind, new_index) :: g
636 Finally, here is the clause for `Change (...)`, which takes over the role earlier played by `Setref`:
638 | Change (var, t2, t3) ->
639 (* we don't handle cases where g doesn't bind var to any value *)
640 let index = List.assoc var g
641 in let (value2, s') = eval t2 g s
642 (* note that s' may be different from s, if t2 itself contained any mutation operations *)
643 (* now we create a list which is just like s' except it has value2 at index *)
644 in let rec replace_nth lst m =
646 | [] -> failwith "list too short"
647 | x::xs when m = 0 -> value2 :: xs
648 | x::xs -> x :: replace_nth xs (m - 1)
649 in let s'' = replace_nth s' index
650 (* evaluate t3 using original assignment function and new store *)
654 Note that because the `savedg` component of a `Closure` keeps track of which `index`es in the store free variables were bound to, the values at those `index`es can later be changed, and later applications of the `Closure` will use the changed values.
656 The complete code is available [here](/code/calculator/calc6.ml).
658 ##Adding Aliasing and Passing by Reference##
662 | Multiplication of (term * term)
663 | Addition of (term * term)
665 | Let of (char * term * term)
667 | If of (term * term * term)
668 | Makepair of (term * term)
670 | Lambda of (char * term)
671 | Apply of (term * term)
672 | Letrec of (char * term * term)
673 | Change of (char * term * term)
674 | Alias of (char * char * term)
675 | Applyalias of (term * char)
678 let rec eval (t : term) (g : assignment) (s : store) = match t with
680 | Alias (var_to_bind, orig_var, t3) ->
681 (* we don't handle cases where g doesn't bind orig_var to any value *)
682 let index = List.assoc orig_var g
683 (* bind var_to_bind to the same index in the store *)
684 in let g' = ((var_to_bind, index) :: g)
686 | Applyalias (t1, var) ->
687 (* we don't handle cases where t1 doesn't evaluate to a function value *)
688 let (Closure (arg_var, body, savedg), s') = eval t1 g s
689 (* we don't handle cases where g doesn't bind var to any value *)
690 in let index = List.assoc var g
691 (* evaluate body under savedg, except with arg_var bound to existing index *)
692 in let savedg' = (arg_var, index) :: savedg
693 in eval body savedg' s'
696 The complete code is available [here](/code/calculator/calc7.ml).