Substitution Caches
PLAI §5 (called “deferred substitutions” there)
Evaluating using substitutions is very inefficient — at each scope, we copy a piece of the program AST. This includes all function calls which implies an impractical cost (function calls should be cheap!).
To get over this, we want to use a cache of substitutions.
Basic idea: we begin evaluating with no cached substitutions, then collect them as we encounter bindings.
[Implies another change for our evaluator: we don’t really substitute identifiers until we get to them; when we reach an identifier, it is no longer an error — we must consult the substitution cache.]
Implementation of Cache Functionality
First, we need a type for a substitution cache. For this we will use a list of lists of two elements each — a name and its value FLANG:
(define-type SubstCache = (Listof (List Symbol FLANG)))
We need to have an empty substitution cache, a way to extend it, and a way to look things up:
(define empty-subst null)
(: extend : Symbol FLANG SubstCache -> SubstCache)
;; extend a given substitution cache with a new mapping
(define (extend id expr sc)
(cons (list id expr) sc))
(: lookup : Symbol SubstCache -> FLANG)
;; lookup a symbol in a substitution cache, return the value it is
;; bound to (or throw an error if it isn't bound)
(define (lookup name sc)
(cond [(null? sc) (error 'lookup "no binding for ~s" name)]
[(eq? name (first (first sc))) (second (first sc))]
[else (lookup name (rest sc))]))
Actually, the reason to use such list of lists is that Racket has a
built-in function called assq that will do this kind of search (assq
is a search in an association list using eq? for the key comparison).
This is a version of lookup that uses assq:
(let ([cell (assq name sc)])
(if cell
(second cell)
(error 'lookup "no binding for ~s" name))))
Formal Rules for Cached Substitutions
The formal evaluation rules are now different. Evaluation carries along
a substitution cache that begins its life as empty: so eval needs an
extra argument. We begin by writing the rules that deal with the cache,
and use the above function names for simplicity — the behavior of the
three definitions can be summed up in a single rule for lookup:
lookup(x,extend(x,E,sc)) = E
lookup(x,extend(y,E,sc)) = lookup(x,sc) if `x` is not `y`
And now we can write the new rules for eval
eval({+ E1 E2},sc) = eval(E1,sc) + eval(E2,sc)
eval({- E1 E2},sc) = eval(E1,sc) - eval(E2,sc)
eval({* E1 E2},sc) = eval(E1,sc) * eval(E2,sc)
eval({/ E1 E2},sc) = eval(E1,sc) / eval(E2,sc)
eval(x,sc) = lookup(x,sc)
eval({with {x E1} E2},sc) = eval(E2,extend(x,eval(E1,sc),sc))
eval({fun {x} E},sc) = {fun {x} E}
eval({call E1 E2},sc) = eval(B,extend(x,eval(E2,sc),sc))
if eval(E1,sc) = {fun {x} B}
= error! otherwise
Note that there is no mention of subst — the whole point is that we
don’t really do substitution, but use the cache instead. The lookup
rules, and the places where extend is used replaces subst, and
therefore specifies our scoping rules.
Also note that the rule for call is still very similar to the rule for
with, but it looks like we have lost something — the interesting bit
with substituting into fun expressions.
Evaluating with Substitution Caches
Implementing the new eval is easy now — it is extended in the same
way that the formal eval rule is extended:
;; evaluates FLANG expressions by reducing them to expressions
(define (eval expr sc)
(cases expr
[(Num n) expr]
[(Add l r) (arith-op + (eval l sc) (eval r sc))]
[(Sub l r) (arith-op - (eval l sc) (eval r sc))]
[(Mul l r) (arith-op * (eval l sc) (eval r sc))]
[(Div l r) (arith-op / (eval l sc) (eval r sc))]
[(With bound-id named-expr bound-body)
(eval bound-body
(extend bound-id (eval named-expr sc) sc))]
[(Id name) (lookup name sc)]
[(Fun bound-id bound-body) expr]
[(Call fun-expr arg-expr)
(define fval (eval fun-expr sc))
(cases fval
[(Fun bound-id bound-body)
(eval bound-body (extend bound-id (eval arg-expr sc) sc))]
[else (error 'eval "`call' expects a function, got: ~s"
fval)])]))
Again, note that we don’t need subst anymore, but the rest of the code
(the data type definition, parsing, and arith-op) is exactly the same.
Finally, we need to make sure that eval is initially called with an
empty cache. This is easy to change in our main run entry point:
;; evaluate a FLANG program contained in a string
(define (run str)
(let ([result (eval (parse str) empty-subst)])
(cases result
[(Num n) n]
[else (error 'run "evaluation returned a non-number: ~s"
result)])))
The full code (including the same tests, but not including formal rules for now) follows. Note that one test does not pass.
(define-type FLANG
[Num Number]
[Add FLANG FLANG]
[Sub FLANG FLANG]
[Mul FLANG FLANG]
[Div FLANG FLANG]
[Id Symbol]
[With Symbol FLANG FLANG]
[Fun Symbol FLANG]
[Call FLANG FLANG])
(: parse-sexpr : Sexpr -> FLANG)
;; parses s-expressions into FLANGs
(define (parse-sexpr sexpr)
(match sexpr
[(number: n) (Num n)]
[(symbol: name) (Id name)]
[(cons 'with more)
(match sexpr
[(list 'with (list (symbol: name) named) body)
(With name (parse-sexpr named) (parse-sexpr body))]
[else (error 'parse-sexpr "bad `with' syntax in ~s" sexpr)])]
[(cons 'fun more)
(match sexpr
[(list 'fun (list (symbol: name)) body)
(Fun name (parse-sexpr body))]
[else (error 'parse-sexpr "bad `fun' syntax in ~s" sexpr)])]
[(list '+ lhs rhs) (Add (parse-sexpr lhs) (parse-sexpr rhs))]
[(list '- lhs rhs) (Sub (parse-sexpr lhs) (parse-sexpr rhs))]
[(list '* lhs rhs) (Mul (parse-sexpr lhs) (parse-sexpr rhs))]
[(list '/ lhs rhs) (Div (parse-sexpr lhs) (parse-sexpr rhs))]
[(list 'call fun arg)
(Call (parse-sexpr fun) (parse-sexpr arg))]
[else (error 'parse-sexpr "bad syntax in ~s" sexpr)]))
(: parse : String -> FLANG)
;; parses a string containing a FLANG expression to a FLANG AST
(define (parse str)
(parse-sexpr (string->sexpr str)))
;; a type for substitution caches:
(define-type SubstCache = (Listof (List Symbol FLANG)))
(: empty-subst : SubstCache)
(define empty-subst null)
(: extend : Symbol FLANG SubstCache -> SubstCache)
;; extend a given substitution cache with a new mapping
(define (extend name val sc)
(cons (list name val) sc))
(: lookup : Symbol SubstCache -> FLANG)
;; lookup a symbol in a substitution cache, return the value it is
;; bound to (or throw an error if it isn't bound)
(define (lookup name sc)
(let ([cell (assq name sc)])
(if cell
(second cell)
(error 'lookup "no binding for ~s" name))))
(: Num->number : FLANG -> Number)
;; convert a FLANG number to a Racket one
(define (Num->number e)
(cases e
[(Num n) n]
[else (error 'arith-op "expected a number, got: ~s" e)]))
(: arith-op : (Number Number -> Number) FLANG FLANG -> FLANG)
;; gets a Racket numeric binary operator, and uses it within a FLANG
;; `Num' wrapper
(define (arith-op op val1 val2)
(Num (op (Num->number val1) (Num->number val2))))
(: eval : FLANG SubstCache -> FLANG)
;; evaluates FLANG expressions by reducing them to expressions
(define (eval expr sc)
(cases expr
[(Num n) expr]
[(Add l r) (arith-op + (eval l sc) (eval r sc))]
[(Sub l r) (arith-op - (eval l sc) (eval r sc))]
[(Mul l r) (arith-op * (eval l sc) (eval r sc))]
[(Div l r) (arith-op / (eval l sc) (eval r sc))]
[(With bound-id named-expr bound-body)
(eval bound-body
(extend bound-id (eval named-expr sc) sc))]
[(Id name) (lookup name sc)]
[(Fun bound-id bound-body) expr]
[(Call fun-expr arg-expr)
(define fval (eval fun-expr sc))
(cases fval
[(Fun bound-id bound-body)
(eval bound-body (extend bound-id (eval arg-expr sc) sc))]
[else (error 'eval "`call' expects a function, got: ~s"
fval)])]))
(: run : String -> Number)
;; evaluate a FLANG program contained in a string
(define (run str)
(let ([result (eval (parse str) empty-subst)])
(cases result
[(Num n) n]
[else (error 'run "evaluation returned a non-number: ~s"
result)])))
;; tests
(test (run "{call {fun {x} {+ x 1}} 4}")
=> 5)
(test (run "{with {add3 {fun {x} {+ x 3}}}
{call add3 1}}")
=> 4)
(test (run "{with {add3 {fun {x} {+ x 3}}}
{with {add1 {fun {x} {+ x 1}}}
{with {x 3}
{call add1 {call add3 x}}}}}")
=> 7)
(test (run "{with {identity {fun {x} x}}
{with {foo {fun {x} {+ x 1}}}
{call {call identity foo} 123}}}")
=> 124)
(test (run "{call {with {x 3}
{fun {y} {+ x y}}}
4}")
=> 7)
(test (run "{with {f {with {x 3} {fun {y} {+ x y}}}}
{with {x 100}
{call f 4}}}")
=> 7)
(test (run "{with {x 3}
{with {f {fun {y} {+ x y}}}
{with {x 5}
{call f 4}}}}")
=> "???")
(test (run "{call {call {fun {x} {call x 1}}
{fun {x} {fun {y} {+ x y}}}}
123}")
=> 124)