X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=advanced_topics%2Fcalculator_improvements.mdwn;h=68bd39d8b44a8886bf2d0be673b33426ed12bc3e;hp=4ddff2797d5d33c48ee4c5e21e2e6e3af79e0ab6;hb=2cef07bcbf221a75e0cf0da553be6566a206d68f;hpb=bd2c74b88855aafb6a4685591eb4e401751edd60 diff --git a/advanced_topics/calculator_improvements.mdwn b/advanced_topics/calculator_improvements.mdwn index 4ddff279..68bd39d8 100644 --- a/advanced_topics/calculator_improvements.mdwn +++ b/advanced_topics/calculator_improvements.mdwn @@ -57,7 +57,7 @@ We'll switch over to using variable `g` for assignment functions, which is a con type bound_value = expressed_value;; type assignment = (char * bound_value) list;; -Here's where we should be now. We expand some of the clauses in the `eval` function for clarity: +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: type term = Constant of int | Multiplication of (term * term) @@ -79,14 +79,14 @@ Here's where we should be now. We expand some of the clauses in the `eval` funct let Int value1 = eval t1 g in let Int value2 = eval t2 g in Int (value1 + value2) - | Variable c -> - (* we don't handle cases where g doesn't bind c to any value *) - List.assoc c g - | Let (c, t1, t2) -> - (* evaluate t2 under a new assignment where c has been bound to + | Variable var -> + (* we don't handle cases where g doesn't bind var to any value *) + List.assoc var g + | Let (var_to_bind, t1, t2) -> + (* evaluate t2 under a new assignment where var_to_bind has been bound to the result of evaluating t1 under the current assignment *) let value1 = eval t1 g - in let g' = (c, value1) :: g + in let g' = (var_to_bind, value1) :: g in eval t2 g' | Iszero t1 -> (* we don't handle cases where t1 doesn't evaluate to an Int *) @@ -120,7 +120,7 @@ We can begin with our language: | Lambda of (char * term) | Apply of (term * term);; -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 expression's lexical environment. That shouldn't get shadowed by any different value `x` may have when the function value is later applied. So: +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: let x = 1 in let f = lambda y -> y + x in let x = 2 in apply f 2 @@ -143,11 +143,11 @@ Now our evaluation function needs two further clauses to interpret the two new e let rec eval (t : term) (g: assignment) = match t with ... - | Lambda(c, t1) -> Closure (c, t1, g) + | Lambda(arg_var, t1) -> Closure (arg_var, t1, g) | Apply(t1, t2) -> - let value2 = eval t2 g (* we don't handle cases where t1 doesn't evaluate to a function value *) - in let Closure (arg_var, body, savedg) = eval t1 g + let Closure (arg_var, body, savedg) = eval t1 g + in let value2 = eval t2 g (* evaluate body under savedg, except with arg_var bound to value2 *) in let savedg' = (arg_var, value2) :: savedg in eval body savedg';; @@ -157,7 +157,7 @@ Now our evaluation function needs two further clauses to interpret the two new e There are different ways to include recursion in our calculator. First, let's imagine our language expanded like this: - let x = 1 in letrec f = lambda y -> if iszero y then x else y * f (y - 1) in f 3 + let x = 1 in letrec f = lambda y -> if iszero y then x else y * apply f (y - 1) in apply f 3 where the AST would be: @@ -184,7 +184,7 @@ Here is the expanded definition for our language type: | Apply of (term * term) | Letrec of (char * term * term);; -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 `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: +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: Closure ('y', body, savedg) @@ -197,9 +197,9 @@ but instead to: Except, this isn't quite right. It's almost what we want, but not exactly. Can you see the flaw? -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 recursive calls to `f` which go more than two levels deep. +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. -What we really want is for `f` to be bound to `new_closure`, something like this: +What we really want is for `'f'` to be bound to `new_closure`, something like this: let rec new_closure = Closure ('y', body, ('f', new_closure) :: savedg) in new_closure @@ -208,11 +208,11 @@ And as a matter of fact, OCaml *does* permit us to recursively define cyclical l let rec eval (t : term) (g: assignment) = match t with ... - | Letrec (c, t1, t2) -> + | Letrec (var_to_bind, t1, t2) -> (* we don't handle cases where t1 doesn't evaluate to a function value *) let Closure (arg_var, body, savedg) = eval t1 g - in let rec new_closure = Closure (arg_var, body, (c, new_closure) :: savedg) - in let g' = (c, new_closure) :: g + in let rec new_closure = Closure (arg_var, body, (var_to_bind, new_closure) :: savedg) + in let g' = (var_to_bind, new_closure) :: g in eval t2 g';; 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. @@ -220,15 +220,15 @@ However, this is a somewhat exotic ability in a programming language, so it woul 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: - | Let (c, t1, t2) -> + | Let (var_to_bind, t1, t2) -> let value1 = eval t1 g - in let g' = fun var -> if var = c then value1 else g var + in let g' = fun var -> if var = var_to_bind then value1 else g var in eval t2 g' ... - | Letrec (c, t1, t2) -> + | Letrec (var_to_bind, t1, t2) -> let Closure (arg_var, body, savedg) = eval t1 g - in let rec savedg' = fun var -> if var = c then Closure (arg_var, body, savedg') else savedg var - in let g' = fun var -> if var = c then Closure (arg_var, body, savedg') else g var + in let rec savedg' = fun var -> if var = var_to_bind then Closure (arg_var, body, savedg') else savedg var + in let g' = fun var -> if var = var_to_bind then Closure (arg_var, body, savedg') else g var in eval t2 g';; 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. @@ -245,7 +245,7 @@ The way we'll do this is that, when we bind a value to a variable, we'll keep tr Letrec ('f', Lambda ('y', Variable 'f')), ...) -In the first case, an application of `f` to any argument should evaluate to `Int 1`; in the second case, it should evaluate to the same function closure that `f` evaluates to. We'll keep track of which way a variable was bound by expanding our `bound_value` type: +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: type expressed_value = Int of int | Bool of bool | Closure of char * term * assignment and bound_value = Nonrecursive of expressed_value | @@ -253,40 +253,40 @@ In the first case, an application of `f` to any argument should evaluate to `Int and assignment = (char * bound_value) list;; -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 of what 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* `f`. The result will look like this: +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: let rec eval (t : term) (g: assignment) = match t with ... - | Variable c -> ( - (* we don't handle cases where g doesn't bind c to any value *) - match List.assoc c g with + | Variable var -> ( + (* we don't handle cases where g doesn't bind var to any value *) + match List.assoc var g with | Nonrecursive value -> value | Recursive_Closure (self_var, arg_var, body, savedg) as rec_closure -> (* we update savedg to bind self_var to rec_closure here *) let savedg' = (self_var, rec_closure) :: savedg in Closure (arg_var, body, savedg') ) - | Let (c, t1, t2) -> - (* evaluate t2 under a new assignment where c has been bound to + | Let (var_to_bind, t1, t2) -> + (* evaluate t2 under a new assignment where var_to_bind has been bound to the result of evaluating t1 under the current assignment *) let value1 = eval t1 g (* we have to wrap value1 in Nonrecursive *) - in let g' = (c, Nonrecursive value1) :: g + in let g' = (var_to_bind, Nonrecursive value1) :: g in eval t2 g' ... - | Lambda(c, t1) -> Closure (c, t1, g) + | Lambda(arg_var, t1) -> Closure (arg_var, t1, g) | Apply(t1, t2) -> - let value2 = eval t2 g (* we don't handle cases where t1 doesn't evaluate to a function value *) - in let Closure (arg_var, body, savedg) = eval t1 g + let Closure (arg_var, body, savedg) = eval t1 g + in let value2 = eval t2 g (* evaluate body under savedg, except with arg_var bound to Nonrecursive value2 *) in let savedg' = (arg_var, Nonrecursive value2) :: savedg in eval body savedg' - | Letrec (c, t1, t2) -> + | Letrec (var_to_bind, t1, t2) -> (* we don't handle cases where t1 doesn't evaluate to a function value *) let Closure (arg_var, body, savedg) = eval t1 g - (* evaluate t2 under a new assignment where c has been recursively bound to that function value *) - in let g' = (c, Recursive_Closure(c, arg_var, body, savedg)) :: g + (* evaluate t2 under a new assignment where var_to_bind has been recursively bound to that function value *) + in let g' = (var_to_bind, Recursive_Closure(var_to_bind, arg_var, body, savedg)) :: g in eval t2 g';;