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:
23 | Multiplication of (term * term)
24 | Addition of (term * term)
26 | Let of (char * term * term)
29 and the evaluation function looked like this:
31 let rec eval (t : term) (e: (char * int) list) = match t with
33 | Multiplication (t1, t2) -> (eval t1 e) * (eval t2 e)
34 | Addition (t1, t2) -> (eval t1 e) + (eval t2 e)
36 (* lookup the value of c in the current environment
37 This will fail if c isn't assigned anything by e *)
40 (* evaluate t2 in a new environment where c has been associated
41 with the result of evaluating t1 in the current environment *)
42 eval t2 ((c, eval t1 e) :: e)
46 ##Adding Booleans and Pairs##
48 Let's tweak this a bit.
50 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.
52 type expressed_value = Int of int | Bool of bool;;
54 We'll add one boolean predicate, `Iszero`, and an `If...` construction.
56 Similarly, we might allow for some terms to express pairs of other terms:
58 type expressed_value = Int of int | Bool of bool | Pair of expressed_value * expressed_value;;
60 We'd then want to add the ability to construct pairs, and extract their components.
62 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.
64 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 [[week9]], 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.
66 type bound_value = expressed_value;;
67 type assignment = (char * bound_value) list;;
69 Here's where we should be now. We'll work with the language:
73 | Multiplication of (term * term)
74 | Addition of (term * term)
76 | Let of (char * term * term)
78 | If of (term * term * term)
79 | Makepair of (term * term)
83 Here is our evaluation function. We expand some of the clauses and rename a few variables for clarity. Our implementation should make it clear how to add additional constants or native predicates, such as a `Second` predicate for extracting the second element of a pair.
85 let rec eval (t : term) (g : assignment) = match t with
86 Intconstant x -> Int x
87 | Multiplication (t1, t2) ->
88 (* we don't handle cases where the subterms don't evaluate to Ints *)
89 let Int i1 = eval t1 g
90 in let Int i2 = eval t2 g
91 (* Multiplication (t1, t2) should evaluate to an Int *)
93 | Addition (t1, t2) ->
94 let Int i1 = eval t1 g
95 in let Int i2 = eval t2 g
98 (* we don't handle cases where g doesn't bind var to any value *)
100 | Let (var_to_bind, t2, t3) ->
101 (* evaluate t3 under a new assignment where var_to_bind has been bound to
102 the result of evaluating t2 under the current assignment *)
103 let value2 = eval t2 g
104 in let g' = (var_to_bind, value2) :: g
107 (* we don't handle cases where t1 doesn't evaluate to an Int *)
108 let Int i1 = eval t1 g
109 (* Iszero t1 should evaluate to a Bool *)
112 (* we don't handle cases where t1 doesn't evaluate to a boolean *)
113 let Bool b1 = eval t1 g
114 in if b1 then eval t2 g
116 | Makepair (t1, t2) ->
117 let value1 = eval t1 g
118 in let value2 = eval t2 g
119 in Pair (value1, value2)
121 (* we don't handle cases where t1 doesn't evaluate to a Pair *)
122 let Pair (value1, value2) = eval t1 g
126 The complete code is available [here](/code/calculator/calc1.ml).
128 ##Adding Function Values##
130 Now we want to add function values to our language, so that we can interpret (the abstract syntax trees of) expressions like this:
132 let x = 1 in let f = lambda y -> y + x in apply f 2
134 What changes do we need to handle this?
136 We can begin with our language:
140 | Multiplication of (term * term)
141 | Addition of (term * term)
143 | Let of (char * term * term)
145 | If of (term * term * term)
146 | Makepair of (term * term)
148 | Lambda of (char * term)
149 | Apply of (term * term)
152 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:
154 let x = 1 in let f = lambda y -> y + x in let x = 2 in apply f 2
156 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:
158 type expressed_value = Int of int | Bool of bool | Pair of expressed_value * expressed_value | Closure of char * term * assignment;;
160 We'd like to define `bound_value`s and `assignment`s just as before:
162 type bound_value = expressed_value;;
163 type assignment = (char * bound_value) list;;
165 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:
167 type bound_value = expressed_value
168 and assignment = (char * bound_value) list
169 and expressed_value = Int of int | Bool of bool | Pair of expressed_value * expressed_value | Closure of char * term * assignment;;
171 Now our evaluation function needs two further clauses to interpret the two new expression forms `Lambda (...)` and `Apply (...)`:
173 let rec eval (t : term) (g : assignment) = match t with
175 | Lambda (arg_var, t2) -> Closure (arg_var, t2, g)
177 (* we don't handle cases where t1 doesn't evaluate to a function value *)
178 let Closure (arg_var, body, savedg) = eval t1 g
179 in let value2 = eval t2 g
180 (* evaluate body under savedg, except with arg_var bound to value2 *)
181 in let savedg' = (arg_var, value2) :: savedg
185 The complete code is available [here](/code/calculator/calc2.ml).
187 ##Adding Recursive Functions##
189 There are different ways to include recursion in our calculator. First, let's imagine our language expanded like this:
191 let x = 1 in letrec f = lambda y -> if iszero y then x else y * apply f (y - 1) in apply f 3
193 where the AST would be:
195 Let ('x', Intconstant 1,
198 If (Iszero (Variable 'y'),
200 Multiplication (Variable 'y',
202 Addition (Variable 'y', Intconstant (-1)))))),
203 Apply (Variable 'f', Intconstant 3)))
205 Here is the expanded definition for our language type:
209 | Multiplication of (term * term)
210 | Addition of (term * term)
212 | Let of (char * term * term)
214 | If of (term * term * term)
215 | Makepair of (term * term)
217 | Lambda of (char * term)
218 | Apply of (term * term)
219 | Letrec of (char * term * term)
222 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:
224 Closure ('y', body, savedg)
228 let orig_closure = Closure ('y', body, savedg)
229 in let savedg' = ('f', orig_closure) :: savedg
230 in let new_closure = Closure ('y', body, savedg')
233 Except, this isn't quite right. It's almost what we want, but not exactly. Can you see the flaw?
235 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.
237 What we really want is for `'f'` to be bound to `new_closure`, something like this:
239 let rec new_closure = Closure ('y', body, ('f', new_closure) :: savedg)
242 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:
244 let rec eval (t : term) (g : assignment) = match t with
246 | Letrec (var_to_bind, t2, t3) ->
247 (* we don't handle cases where t2 doesn't evaluate to a function value *)
248 let Closure (arg_var, body, savedg) = eval t2 g
249 in let rec new_closure = Closure (arg_var, body, (var_to_bind, new_closure) :: savedg)
250 in let g' = (var_to_bind, new_closure) :: g
254 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.
256 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:
259 | Let (var_to_bind, t2, t3) ->
260 let value2 = eval t2 g
261 in let g' = fun var -> if var = var_to_bind then value2 else g var
264 | Letrec (var_to_bind, t2, t3) ->
265 let Closure (arg_var, body, savedg) = eval t2 g
266 in let rec savedg' = fun var -> if var = var_to_bind then Closure (arg_var, body, savedg') else savedg var
267 in let g' = fun var -> if var = var_to_bind then Closure (arg_var, body, savedg') else g var
271 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.
273 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:
277 Let ('f', Lambda ('y', Variable 'f')),
282 Letrec ('f', Lambda ('y', Variable 'f')),
285 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:
287 type bound_value = Nonrecursive of expressed_value |
288 Recursive_Closure of char * char * term * assignment
289 and assignment = (char * bound_value) list
290 and expressed_value = Int of int | Bool of bool | Pair of expressed_value * expressed_value | Closure of char * term * assignment;;
293 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:
295 let rec eval (t : term) (g : assignment) = match t with
297 | Variable (var) -> (
298 (* we don't handle cases where g doesn't bind var to any value *)
299 match List.assoc var g with
300 | Nonrecursive value -> value
301 | Recursive_Closure (self_var, arg_var, body, savedg) as rec_closure ->
302 (* we update savedg to bind self_var to rec_closure here *)
303 let savedg' = (self_var, rec_closure) :: savedg
304 in Closure (arg_var, body, savedg')
306 | Let (var_to_bind, t2, t3) ->
307 (* evaluate t3 under a new assignment where var_to_bind has been bound to
308 the result of evaluating t2 under the current assignment *)
309 let value2 = eval t2 g
310 (* we have to wrap value2 in Nonrecursive *)
311 in let g' = (var_to_bind, Nonrecursive value2) :: g
314 | Lambda (arg_var, t2) -> Closure (arg_var, t2, g)
316 (* we don't handle cases where t1 doesn't evaluate to a function value *)
317 let Closure (arg_var, body, savedg) = eval t1 g
318 in let value2 = eval t2 g
319 (* evaluate body under savedg, except with arg_var bound to Nonrecursive value2 *)
320 in let savedg' = (arg_var, Nonrecursive value2) :: savedg
322 | Letrec (var_to_bind, t2, t3) ->
323 (* we don't handle cases where t2 doesn't evaluate to a function value *)
324 let Closure (arg_var, body, savedg) = eval t2 g
325 (* evaluate t3 under a new assignment where var_to_bind has been recursively bound to that function value *)
326 in let g' = (var_to_bind, Recursive_Closure (var_to_bind, arg_var, body, savedg)) :: g
330 The complete code is available [here](/code/calculator/calc3.ml).
332 ##Adding Mutable Cells##
334 Next, we'll add mutable cells (explicit-style mutation) to our calculator, as we did in [[week9]].
336 We'll add a few more syntactic forms to the language:
340 | Multiplication of (term * term)
341 | Addition of (term * term)
343 | Let of (char * term * term)
345 | If of (term * term * term)
346 | Makepair of (term * term)
348 | Lambda of (char * term)
349 | Apply of (term * term)
350 | Letrec of (char * term * term)
353 | Setref of (term * term)
356 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`:
360 type bound_value = Nonrecursive of expressed_value |
361 Recursive_Closure of char * char * term * assignment
362 and assignment = (char * bound_value) list
363 and expressed_value = Int of int | Bool of bool | Pair of expressed_value * expressed_value | Closure of char * term * assignment | Mutcell of index;;
365 type store = expressed_value list;;
367 Our evaluation function will now expect a `store` argument as well as an `assignment`, and will return an `expressed_value * store` pair:
369 let rec eval (t : term) (g : assignment) (s : store) = match t with
370 Intconstant x -> (Int x, s)
372 | Variable (var) -> ((
373 (* we don't handle cases where g doesn't bind var to any value *)
374 match List.assoc var g with
375 | Nonrecursive value -> value
376 | Recursive_Closure (self_var, arg_var, body, savedg) as rec_closure ->
377 (* we update savedg to bind self_var to rec_closure here *)
378 let savedg' = (self_var, rec_closure) :: savedg
379 in Closure (arg_var, body, savedg')
382 | Lambda (arg_var, t2) -> (Closure (arg_var, t2, g), s)
385 also, we'll need to be sure to thread the store argument through the evaluation of any subterms, as here:
388 | Multiplication (t1, t2) ->
389 (* we don't handle cases where the subterms don't evaluate to Ints *)
390 let (Int i1, s') = eval t1 g s
391 in let (Int i2, s'') = eval t2 g s'
392 (* Multiplication (t1, t2) should evaluate to an Int *)
393 in (Int (i1 * i2), s'')
394 | Addition (t1, t2) ->
395 let (Int i1, s') = eval t1 g s
396 in let (Int i2, s'') = eval t2 g s'
397 in (Int (i1 + i2), s'')
399 | Let (var_to_bind, t2, t3) ->
400 (* evaluate t3 under a new assignment where var_to_bind has been bound to
401 the result of evaluating t2 under the current assignment *)
402 let (value2, s') = eval t2 g s
403 (* we have to wrap value2 in Nonrecursive *)
404 in let g' = (var_to_bind, Nonrecursive value2) :: g
407 (* we don't handle cases where t1 doesn't evaluate to an Int *)
408 let (Int i1, s') = eval t1 g s
409 (* Iszero t1 should evaluate to a Bool *)
410 in (Bool (i1 = 0), s')
412 | Makepair (t1, t2) ->
413 let (value1, s') = eval t1 g s
414 in let (value2, s'') = eval t2 g s'
415 in (Pair (value1, value2), s'')
417 (* we don't handle cases where t1 doesn't evaluate to a Pair *)
418 let (Pair (value1, value2), s') = eval t1 g s
422 (* we don't handle cases where t1 doesn't evaluate to a function value *)
423 let (Closure (arg_var, body, savedg), s') = eval t1 g s
424 in let (value2, s'') = eval t2 g s'
425 (* evaluate body under savedg, except with arg_var bound to Nonrecursive value2 *)
426 in let savedg' = (arg_var, Nonrecursive value2) :: savedg
427 in eval body savedg' s''
428 | Letrec (var_to_bind, t2, t3) ->
429 (* we don't handle cases where t2 doesn't evaluate to a function value *)
430 let (Closure (arg_var, body, savedg), s') = eval t2 g s
431 (* evaluate t3 under a new assignment where var_to_bind has been recursively bound to that function value *)
432 in let g' = (var_to_bind, Recursive_Closure (var_to_bind, arg_var, body, savedg)) :: g
436 The clause for `If (...)` is notable:
440 (* we don't handle cases where t1 doesn't evaluate to a boolean *)
441 let (Bool b1, s') = eval t1 g s
442 (* note we thread s' through only one of the then/else clauses *)
443 in if b1 then eval t2 g s'
447 Now we need to formulate the clauses for evaluating the new forms `Newref (...)`, `Deref (...)`, and `Setref (...)`.
451 let (value1, s') = eval t1 g s
452 (* note that s' may be different from s, if t1 itself contained any mutation operations *)
453 (* now we want to retrieve the next free index in s' *)
454 in let new_index = List.length s'
455 (* now we want to insert value1 there; the following is an easy but inefficient way to do it *)
456 in let s'' = List.append s' [value1]
457 (* now we return a pair of a wrapped new_index, and the new store *)
458 in (Mutcell new_index, s'')
460 (* we don't handle cases where t1 doesn't evaluate to a Mutcell *)
461 let (Mutcell index1, s') = eval t1 g s
462 (* note that s' may be different from s, if t1 itself contained any mutation operations *)
463 in (List.nth s' index1, s')
465 (* we don't handle cases where t1 doesn't evaluate to a Mutcell *)
466 let (Mutcell index1, s') = eval t1 g s
467 (* note that s' may be different from s, if t1 itself contained any mutation operations *)
468 in let (value2, s'') = eval t2 g s'
469 (* now we create a list which is just like s'' except it has value2 in index1 *)
470 in let rec replace_nth lst m =
472 | [] -> failwith "list too short"
473 | x::xs when m = 0 -> value2 :: xs
474 | x::xs -> x :: replace_nth xs (m - 1)
475 in let s''' = replace_nth s'' index1
476 (* we'll arbitrarily return Int 42 as the expressed_value of a Setref operation *)
480 The complete code is available [here](/code/calculator/calc4.ml).
482 ##Adding Mutable Pairs##
484 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.
486 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.
488 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.
490 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`:
494 | Multiplication of (term * term)
495 | Addition of (term * term)
497 | Let of (char * term * term)
499 | If of (term * term * term)
500 | Makepair of (term * term)
502 | Lambda of (char * term)
503 | Apply of (term * term)
504 | Letrec of (char * term * term)
505 | Setfirst of (term * term)
508 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`:
512 type bound_value = Nonrecursive of expressed_value |
513 Recursive_Closure of char * char * term * assignment
514 and assignment = (char * bound_value) list
515 and expressed_value = Int of int | Bool of bool | Pair of index * index | Closure of char * term * assignment;;
517 type store = expressed_value list;;
519 Finally, here are the changed or added clauses to the evaluation function:
521 let rec eval (t : term) (g : assignment) (s : store) = match t with
523 | Makepair (t1, t2) ->
524 let (value1, s') = eval t1 g s
525 in let (value2, s'') = eval t2 g s'
526 (* now we want to retrieve the next free index in s'' *)
527 in let new_index = List.length s''
528 (* now we want to insert value1 and value2 there; the following is an easy but inefficient way to do it *)
529 in let s''' = List.append s'' [value1; value2]
530 in (Pair (new_index, new_index + 1), s''')
532 (* we don't handle cases where t1 doesn't evaluate to a Pair *)
533 let (Pair (index1, index2), s') = eval t1 g s
534 (* note that s' may be different from s, if t1 itself contained any mutation operations *)
535 in (List.nth s' index1, s')
537 | Setfirst (t1, t2) ->
538 (* we don't handle cases where t1 doesn't evaluate to a Pair *)
539 let (Pair (index1, index2), s') = eval t1 g s
540 (* note that s' may be different from s, if t1 itself contained any mutation operations *)
541 in let (value2, s'') = eval t2 g s'
542 (* now we create a list which is just like s'' except it has value2 in index1 *)
543 in let rec replace_nth lst m =
545 | [] -> failwith "list too short"
546 | x::xs when m = 0 -> value2 :: xs
547 | x::xs -> x :: replace_nth xs (m - 1)
548 in let s''' = replace_nth s'' index1
552 Compare these to the clauses for `Newref`, `Deref`, and `Setref` in the previous implementation.
554 The complete code is available [here](/code/calculator/calc5.ml).
556 ##Adding Implicit Mutation##
558 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.
560 Our language for the present implementation will be the language for the calculator with recursive functions, with one added syntactic form, `Change (...)`:
564 | Multiplication of (term * term)
565 | Addition of (term * term)
567 | Let of (char * term * term)
569 | If of (term * term * term)
570 | Makepair of (term * term)
572 | Lambda of (char * term)
573 | Apply of (term * term)
574 | Letrec of (char * term * term)
575 | Change of (char * term * term)
578 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:
582 type bound_value = index;;
583 type assignment = (char * bound_value) list;;
584 type expressed_value = Int of int | Bool of bool | Pair of expressed_value * expressed_value | Closure of char * term * assignment;;
586 type store = expressed_value list;;
588 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`:
590 let rec eval (t : term) (g : assignment) (s : store) = match t with
593 (* we don't handle cases where g doesn't bind var to any value *)
594 let index = List.assoc var g
595 (* get value stored at location index in s *)
596 in let value = List.nth s index
600 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`:
603 | Let (var_to_bind, t2, t3) ->
604 let (value2, s') = eval t2 g s
605 (* note that s' may be different from s, if t2 itself contained any mutation operations *)
606 (* get next free index in s' *)
607 in let new_index = List.length s'
608 (* now we want to insert value2 there; the following is an easy but inefficient way to do it *)
609 in let s'' = List.append s' [value2]
610 (* bind var_to_bind to location new_index in the store *)
611 in let g' = ((var_to_bind, new_index) :: g)
615 (* we don't handle cases where t1 doesn't evaluate to a function value *)
616 let (Closure (arg_var, body, savedg), s') = eval t1 g s
617 in let (value2, s'') = eval t2 g s'
618 (* evaluate body under savedg, except with arg_var bound to a new location containing value2 *)
619 in let new_index = List.length s''
620 in let s''' = List.append s'' [value2]
621 in let savedg' = (arg_var, new_index) :: savedg
622 in eval body savedg' s'''
625 `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:
628 | Letrec (var_to_bind, t2, t3) ->
629 (* we don't handle cases where t2 doesn't evaluate to a function value *)
630 let (Closure (arg_var, body, savedg), s') = eval t2 g s
631 in let new_index = List.length s'
632 in let savedg' = (var_to_bind, new_index) :: savedg
633 in let new_closure = Closure (arg_var, body, savedg')
634 in let s'' = List.append s' [new_closure]
635 in let g' = (var_to_bind, new_index) :: g
639 Finally, here is the clause for `Change (...)`, which takes over the role earlier played by `Setref`:
642 | Change (var, t2, t3) ->
643 (* we don't handle cases where g doesn't bind var to any value *)
644 let index = List.assoc var g
645 in let (value2, s') = eval t2 g s
646 (* note that s' may be different from s, if t2 itself contained any mutation operations *)
647 (* now we create a list which is just like s' except it has value2 at index *)
648 in let rec replace_nth lst m =
650 | [] -> failwith "list too short"
651 | x::xs when m = 0 -> value2 :: xs
652 | x::xs -> x :: replace_nth xs (m - 1)
653 in let s'' = replace_nth s' index
654 (* evaluate t3 using original assignment function and new store *)
658 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.
660 The complete code is available [here](/code/calculator/calc6.ml).
662 ##Adding Aliasing and Passing by Reference##
666 | Multiplication of (term * term)
667 | Addition of (term * term)
669 | Let of (char * term * term)
671 | If of (term * term * term)
672 | Makepair of (term * term)
674 | Lambda of (char * term)
675 | Apply of (term * term)
676 | Letrec of (char * term * term)
677 | Change of (char * term * term)
678 | Alias of (char * char * term)
679 | Applyalias of (term * char)
682 let rec eval (t : term) (g : assignment) (s : store) = match t with
684 | Alias (var_to_bind, orig_var, t3) ->
685 (* we don't handle cases where g doesn't bind orig_var to any value *)
686 let index = List.assoc orig_var g
687 (* bind var_to_bind to the same index in the store *)
688 in let g' = ((var_to_bind, index) :: g)
690 | Applyalias (t1, var) ->
691 (* we don't handle cases where t1 doesn't evaluate to a function value *)
692 let (Closure (arg_var, body, savedg), s') = eval t1 g s
693 (* we don't handle cases where g doesn't bind var to any value *)
694 in let index = List.assoc var g
695 (* evaluate body under savedg, except with arg_var bound to existing index *)
696 in let savedg' = (arg_var, index) :: savedg
697 in eval body savedg' s'
700 The complete code is available [here](/code/calculator/calc7.ml).