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Joined 2026-09-21 Message
Incoming links: Recursive closure
Use programming closures that receive evaluated arguments and return a computation in the chosen computational monad. A sufficient core of the semantic value domain is:
data Value = IntVal Integer
| BoolVal Bool
| Function ([Value] -> M Value)Additional constructors may represent the other built-in values. Assume a semantic environment abstract data type with operations
lookupEnv :: Env -> Ident -> Value and extend :: Env -> [(Ident,Value)] -> Env. The latter simultaneously adds the listed bindings, shadowing old bindings. Parameter names in one binding list are distinct, and calls have the expected arity.Define programming closure construction separately; it does not run the body:
close :: Expr -> Env -> Value
close (Lambda xs body) rho =
Function (\args -> eval body (extend rho (zip xs args)))
apply :: Value -> [Value] -> M Value
apply (Function f) args = f argsThe four expression forms have the following semantic equations. Function position is evaluated first, followed by argument expressions from left to right:
eval :: Expr -> Env -> M Value
eval (Variable x) rho = result (lookupEnv rho x)
eval (Apply e es) rho =
bind (eval e rho) (\f ->
bind (evalArgs es rho) (\args -> apply f args))
eval e@(Lambda xs body) rho = result (close e rho)
eval (Let d body) rho =
bind (defn d rho) (\rho' -> eval body rho')
evalArgs :: [Expr] -> Env -> M [Value]
evalArgs [] rho = result []
evalArgs (e:es) rho =
bind (eval e rho) (\v ->
bind (evalArgs es rho) (\vs -> result (v:vs)))The two definition forms are:
defn :: Defn -> Env -> M Env
defn (Val x e) rho =
bind (eval e rho) (\v -> result (extend rho [(x,v)]))
defn (Rec x e@(Lambda xs body)) rho = result rho'
where
rho' = extend rho [(x,v)]
v = close e rho'The ordinary definition evaluates its right-hand side in the old semantic environment; its own name is not yet bound there. The recursive definition creates a recursive closure in the extended semantic environment containing itself. Because the right-hand side is a lambda abstraction, constructing that programming closure does not evaluate its body. Haskell's lazy evaluation can tie this recursive data structure without requiring any extra fixed-point operation from the arbitrary computational monad.
Lexical scope is explicit: calling a programming closure extends the defining semantic environment, and its parameters shadow bindings in that semantic environment. The six requested constructs are therefore covered by
Add the abstract syntax tree alternative
Fix Ident Expr. Its syntactic sugar expansion and the semantic equations from part (a) givewhere and . The final step uses left identity from the monad laws and the lookup rule for the newly bound identifier.eval (Fix x e@(Lambda xs body)) rho = result v
where
rho' = extend rho [(x,v)]
v = close e rho'Equivalently, spelling out programming closure construction,The result is a recursive closure, not the result of calling its body. Its self-binding is visible inside that body but does not extend the caller's surrounding semantic environment.
The eight-page University of Oxford computer science Finals paper dated Thursday 11 June 2015, 2:30 pm–4:00 pm. The released examination archive supplied by the University of Oxford contains it as
2015-popl.pdf. Printed pp. 2–3 concern monad transformers; pp. 4–6 concern continuations, coroutines and semaphores; pp. 7–8 concern a monadic interpreter and recursive closures.