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Evaluation

While the lambda calculus is exceedingly simple, there is a great deal of variety in ways to evaluate and implement the reduction of lambda expressions. The different models for evaluation are evaluation stratgies.

There is a bifurcation between two points in the design space: strict and non-strict evaluation. An evaluation strategy is strict if the arguments to a lambda expression are necessarily evaluated before a lambda is reduced. A language in which the arguments are not necessarily evaluated before a lambda is reduced is non-strict.

Alternatively expressed, diverging terms are represented by the bottom value, written as \(\bot\). A function \(f\) is non-strict if:

\[ f \bot \neq \bot \]

Evaluation Models

There are many different models, and various hybrids thereof. We will consider three dominant models:

  • Call-by-value: arguments are evaluated before a function is entered
  • Call-by-name: arguments passed unevaluated
  • Call-by-need: arguments passed unevaluated but an expression is only evaluated once and shared upon subsequent references

Given an expression \(f x\) the reduction in different evaluation models proceeds differently:

Call-by-value:

  1. Evaluate \(x\) to \(v\)
  2. Evaluate \(f\) to \(\lambda y. e\)
  3. Evaluate \([y/v]e\)

Call-by-name:

  1. Evaluate \(f\) to \(\lambda y. e\)
  2. Evaluate \([y/x]e\)

Call-by-need:

  1. Allocate a thunk \(v\) for \(x\)
  2. Evaluate \(f\) to \(\lambda y. e\)
  3. Evaluate \([y/v]e\)

Terms that have a normal form in one model, may or may not have a normal form in another. In call-by-need and call-by-name evaluation diverging terms are not necessarily evaluated before entry, so some terms that have a normal form in these models may diverge under call-by-value.

Call-by-value

Call by value is an extremely common evaluation model. Many programming languages both imperative and functional use this evaluation strategy. The essence of call-by-value is that there are two categories of expressions: terms and values. Values are lambda expressions and other terms which are in normal form and cannot be reduced further. All arguments to a function will be reduced to normal form before they are bound inside the lambda and reduction only proceeds once the arguments are reduced.

For a simple arithmetic expression, the reduction proceeds as follows. Notice how the subexpression (2 + 2) is evaluated to normal form before being bound.

(λx. λy. y x) (2 + 2) λx. x + 1 
=> (λy. y 4) λx. x + 1
=> (λy. x + 1) 4
=> 4 + 1
=> 5

Naturally there now are two evaluation rules for applications.

\[ \begin{array}{cl} \displaystyle \frac{e_1 \to e_1'}{e_1 e_2 \to e_1' e_2} & \trule{E-App1} \\ \\ \displaystyle \frac{e_2 \to e_2'}{v_1 e_2 \to v_1 e_2'} & \trule{E-App2} \\ \\ \displaystyle {(\lambda x . e) v \to [x / v] e } & \trule{E-AppLam} \\ \\ \end{array} \]

Call-by-value interpreter

For a simple little lambda calculus the call-by-value interpreter is quite simple. Part of the runtime evaluation of lambda calculus involves the creation of closures, environments which hold the local variables in scope. In our little language there are two possible values which reduction may converge on, VInt and VClosure.

data Expr
  = Var Int
  | Lam Expr
  | App Expr Expr
  | Lit Int
  | Prim PrimOp Expr Expr
  deriving Show

data PrimOp = Add | Mul
  deriving Show

data Value
  = VInt Int
  | VClosure Expr Env
  deriving Show

type Env = [Value]

emptyEnv :: Env
emptyEnv = []

The evaluator function simply maps the local scope and a term to the final value. Whenever a variable is referred to it is looked up in the environment. Whenever a lambda is entered it extends the environment with the local scope of the closure.

eval :: Env -> Expr -> Value
eval env term = case term of
  Var n -> env !! n
  Lam a -> VClosure a env
  App a b ->
    let VClosure c env' = eval env a in
    let v = eval env b in
    eval (v : env') c

  Lit n -> VInt n
  Prim p a b -> (evalPrim p) (eval env a) (eval env b)

evalPrim :: PrimOp -> Value -> Value -> Value
evalPrim Add (VInt a) (VInt b) = VInt (a + b)
evalPrim Mul (VInt a) (VInt b) = VInt (a * b)

Call-by-name

In call-by-name evaluation, the arguments to lambda expressions are substituted as is, evaluation simply proceeds from left to right substituting the outermost lambda or reducing a value. If a substituted expression is not used it is never evaluated.

\[ \begin{array}{cl} \displaystyle \frac{e_1 \to e_1'}{e_1 e_2 \to e_1' e_2} & \trule{E-App} \\ \\ \displaystyle {(\lambda x . e) v \to [x / v] e } & \trule{E-AppLam} \\ \\ \end{array} \]

For example, the same expression we looked at for call-by-value has the same normal form but arrives at it by a different sequence of reductions:

(λx. λy. y x) (2 + 2) λx. x + 1 
=> (λy.y (2 + 2)) λx. x + 1
=> (λx.x + 1) (2 + 2)
=> (2 + 2) + 1
=> 4 + 1
=> 5

Call-by-name is non-strict, although very few languages use this model, Frege being the most notable example.

Call-by-need

Call-by-need is a special type of non-strict evaluation in which unevaluated expressions are represented by suspensions or thunks which are passed into a function unevaluated and only evaluated when needed or forced. When the thunk is forced the representation of the thunk is updated with the computed value and is not recomputed upon further reference.

The thunks for unevaluated lambda expressions are allocated when evaluated, and the resulting computed value is placed in the same reference so that subsequent computations share the result. If the argument is never needed it is never computed, which results in a trade-off between space and time.

Since the evaluation of subexpression does not follow any pre-defined order, any impure functions with side-effects will be evaluated in an unspecified order. As a result call-by-need can only effectively be implemented in a purely functional setting.

type Thunk = () -> IO Value

data Value
  = VBool Bool
  | VInt Integer
  | VClosure (Thunk -> IO Value)
update :: IORef Thunk -> Value -> IO ()
update ref v = do
  writeIORef ref (\() -> return v)
  return ()
force :: IORef Thunk -> IO Value
force ref = do
  th <- readIORef ref
  v <- th ()
  update ref v
  return v
mkThunk :: Env -> String -> Expr -> (Thunk -> IO Value)
mkThunk env x body = \a -> do
  a' <- newIORef a
  eval ((x, a') : env) body
eval :: Env -> Expr -> IO Value
eval env ex = case ex of
  EVar n -> do
    th <- lookupEnv env n
    v <- force th
    return v

  ELam x e -> return $ VClosure (mkThunk env x e)

  EApp a b -> do
    VClosure c <- eval env a
    c (\() -> eval env b)

  EBool b -> return $ VBool b
  EInt n  -> return $ VInt n
  EFix e  -> eval env (EApp e (EFix e))

For example, in this model the following program will not diverge since the omega combinator passed into the constant function is not used and therefore the argument is not evaluated.

omega = (\x -> x x) (\x -> x x)
test1 = (\y -> 42) omega
omega :: Expr
omega = EApp (ELam "x" (EApp (EVar "x") (EVar "x")))
             (ELam "x" (EApp (EVar "x") (EVar "x")))

test1 :: IO Value
test1 = eval [] $ EApp (ELam "y" (EInt 42)) omega

Higher Order Interpreters

Higher Order Abstract Syntax (HOAS)

Haskell being a rich language has a variety of extensions that, among other things, allow us to map lambda expressions in our defined language directly onto lambda expressions in Haskell. In this case we will use a GADT to embed a Haskell expression inside our expression type.

{-# LANGUAGE GADTs #-}

data Expr a where
  Lift :: a                       -> Expr a
  Tup  :: Expr a -> Expr b        -> Expr (a, b)
  Lam  :: (Expr a -> Expr b)      -> Expr (a -> b)
  App  :: Expr (a -> b) -> Expr a -> Expr b
  Fix  :: Expr (a -> a)           -> Expr a

The most notable feature of this encoding is that there is no distinct constructor for variables. Instead they are simply values in the host language. Some example expressions:

id :: Expr (a -> a)
id = Lam (\x -> x)

tr :: Expr (a -> b -> a)
tr = Lam (\x -> (Lam (\y -> x))) 

fl :: Expr (a -> b -> b)
fl = Lam (\x -> (Lam (\y -> y))) 

Our evaluator then simply uses Haskell for evaluation.

eval :: Expr a -> a
eval (Lift v)    = v
eval (Tup e1 e2) = (eval e1, eval e2)
eval (Lam f)     = \x -> eval (f (Lift x))
eval (App e1 e2) = (eval e1) (eval e2)
eval (Fix f)     = (eval f) (eval (Fix f))

Some examples of use:

fact :: Expr (Integer -> Integer)
fact =
  Fix (
    Lam (\f ->
      Lam (\y ->
        Lift (
          if eval y == 0
          then 1
          else eval y * (eval f) (eval y - 1)))))

test :: Integer
test = eval fact 10

main :: IO ()
main = print test

Several caveats must be taken when working with HOAS. First of all, it takes more work to transform expressions in this form since in order to work with the expression we would need to reach under the lambda binder of a Haskell function itself. Since all the machinery is wrapped up inside of Haskell's implementation even simple operations like pretty printing and writing transformation passes can be more difficult. This form is a good form for evaluation, but not for transformation.

Parametric Higher Order Abstract Syntax (PHOAS)

A slightly different form of HOAS called PHOAS uses a lambda representation parameterized over the binder type under an existential type.

{-# LANGUAGE RankNTypes #-}

data ExprP a
  = VarP a
  | AppP (ExprP a) (ExprP a)
  | LamP (a -> ExprP a)
  | LitP Integer

newtype Expr = Expr { unExpr :: forall a . ExprP a }

The lambda in our language is simply a lambda within Haskell. As an example, the usual SK combinators would be written as follows:

i :: ExprP a
i = LamP (\a -> VarP a)

k :: ExprP a
k = LamP (\x -> LamP (\y -> VarP x))

s :: ExprP a
s = LamP (\x -> LamP (\y -> LamP (\z -> AppP (AppP (VarP x) (VarP z)) (AppP (VarP y) (VarP z)))))

Evaluation will result in a runtime Value type, just as before with our outer interpreters. We will use several "extractor" function which use incomplete patterns under the hood. The model itself does not prevent malformed programs from blowing up here, and so it is necessary to guarantee that the program is sound before evaluation. Normally this would be guaranteed at a higher level by a typechecker before even reaching this point.

data Value
  = VLit Integer
  | VFun (Value -> Value)

fromVFun :: Value -> (Value -> Value)
fromVFun val = case val of
  VFun f -> f
  _      -> error "not a function"

fromVLit :: Value -> Integer
fromVLit val = case val of
  VLit n -> n
  _      -> error "not a integer"

Evaluation simply exploits the fact that nestled up under our existential type is just a Haskell function and so we get all the name capture, closures and binding machinery for free. The evaluation logic for PHOAS model is extremely short.

eval :: Expr -> Value
eval e = ev (unExpr e) where
  ev (LamP f)      = VFun(ev . f)
  ev (VarP v)      = v
  ev (AppP e1 e2)  = fromVFun (ev e1) (ev e2)
  ev (LitP n)      = VLit n

Consider the S K K = I example again and check the result:

skk :: ExprP a
skk = AppP (AppP s k) k

example :: Integer
example = fromVLit $ eval $ Expr (AppP skk (LitP 3))

We will use this evaluation technique extensively in writing interpreters for our larger languages. It is an extremely convenient and useful method for writing interpreters in Haskell.

Embedding IO

As mentioned before, effects are first class values in Haskell.

In Haskell we don't read from a file directly, but create a value that represents reading from a file. This allows us to very cleanly model an interpreter for our language inside of Haskell by establishing a mapping between the base operations of our language and existing function implementations of the standard operations in Haskell, and using monadic operations to build up a pure effectful computation as a result of interpretation. After evaluation, we finally lift the resulting IO value into Haskell and execute the results. This fits in nicely with the PHOAS model and allows us to efficiently implement a fully-fledged interpreter for our language with remarkably little code, simply by exploiting Haskell's implementation.

To embed IO actions inside of our interpreter we create a distinct VEffect value that will build up a sequenced IO computation during evaluation. This value will be passed off to Haskell and reified into real world effects.

data ExprP a
  = VarP a
  | GlobalP Name
  | AppP (ExprP a) (ExprP a)
  | LamP (a -> ExprP a)
  | LitP Char
  | EffectP a

data Value
  = VChar Char
  | VFun (Value -> Value)
  | VEffect (IO Value)
  | VUnit

fromVEff :: Value -> (IO Value)
fromVEff val = case val of
  VEffect f -> f
  _         -> error "not a effect"
eval :: Expr -> Value
eval e = ev (unExpr e) where
  ev (LamP f)      = VFun(ev . f)
  ev (AppP e1 e2)  = fromVFun (ev e1) (ev e2)
  ev (LitP n)      = VChar n
  ev (EffectP v)   = v
  ev (VarP v)      = v
  ev (GlobalP op)  = prim op

-- Lift an effect from our language into Haskell IO.
run :: Expr -> IO ()
run f = void (fromVEff (eval f))

The prim function will simply perform a lookup on the set of builtin operations, which we'll define with a bit of syntactic sugar for wrapping up Haskell functions.

unary :: (Value -> Value) -> Value
unary f = lam $ \a -> f a

binary :: (Value -> Value -> Value) -> Value
binary f = lam $ \a ->
           lam $ \b -> f a b

prim :: Name -> Value
prim op = case op of
 "putChar#" -> unary $ \x ->
    VEffect $ do
      putChar (fromVChar x)
      return VUnit

 "getChar#" -> VEffect $ do
      val <- getChar
      return (VChar val)

 "bindIO#"   -> binary $ \x y -> bindIO x y
 "returnIO#" -> unary  $ \x   -> returnIO x
 "thenIO#"   -> binary $ \x y -> thenIO x y

For example thenIO# sequences effects in our language will simply squash two VEffect objects into one composite effect building up a new VEffect value that is using Haskell's monadic sequencing on the internal IO value.

bindIO :: Value -> Value -> Value
bindIO (VEffect f) (VFun g) = VEffect (f >>= fromVEff . g)

thenIO :: Value -> Value -> Value
thenIO (VEffect f) (VEffect g) = VEffect (f >> g)

returnIO :: Value -> Value
returnIO a = VEffect $ return a

Effectively we're just recreating the same conceptual relationship that Haskell IO has with its runtime, but instead our host language uses Haskell as the runtime!

Full Source

Evaluation

Higher Order Interpreters