Normalisation by evaluation

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In programming language semantics, normalisation by evaluation (NBE) is a style of obtaining the normal form of terms in the λ calculus by appealing to their denotational semantics. A term is first interpreted into a denotational model of the λ-term structure, and then a canonical (β-normal and η-long) representative is extracted by reifying the denotation. Such an essentially semantic approach differs from the more traditional syntactic description of normalisation as a reductions in a term rewrite system where β-reductions are allowed deep inside λ-terms.

NBE was first described for the simply typed lambda calculus.[1] It has since been extended both to weaker type systems such as the untyped lambda calculus[2] using a domain theoretic approach, and to richer type systems such as several variants of Martin-Löf type theory.[3][4]

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[edit] Outline

Consider the simply typed lambda calculus, where types τ can be basic types (α), function types (→), or products (×), given by the following BNF grammar (→ associating to the right, as usual):

(Types) τ ::= α | τ1 → τ2 | τ1 × τ2

These can be implemented as a datatype in the meta-language; for example, for Standard ML, we might use:

datatype ty = Basic of string
            | Arrow of ty * ty
            | Prod of ty * ty

Terms are defined at two levels.[5] The lower syntactic level (sometimes called the dynamic level) is the representation that one intends to normalise.

(Syntax Terms) s,t,… ::= var x | lam (x, t) | app (s, t) | pair (s, t) | fst t | snd t

Here lam/app (resp. pair/fst,snd) are the intro/elim forms for → (resp. ×), and x are variables. These terms are intended to be implemented as a first-order in the meta-language:

datatype tm = var of string
            | lam of string * tm | app of tm * tm
            | pair of tm * tm | fst of tm | snd of tm

The denotational semantics of (closed) terms in the meta-language interprets the constructs of the syntax in terms of features of the meta-language; thus, lam is interpreted as abstraction, app as application, etc. The semantic objects constructed are as follows:

(Semantic Terms) S,T,… ::= LAM (λx. S x) | PAIR (S, T) | SYN t

Note that there are no variables or elimination forms in the semantics; they are represented simply as syntax. These semantic objects are represented by the following datatype:

datatype sem = LAM of (sem -> sem)
             | PAIR of sem * sem
             | SYN of tm

There are a pair of type-indexed functions that move back and forth between the syntactic and semantic layer. The first function, usually written ↑τ, reflects the term syntax into the semantcs, while the second reifies the semantics as a syntactic term (written as ↓τ). Their definitions are mutually recursive as follows:

\begin{align} \uparrow_{\alpha} t &= \mathbf{SYN}\ t \\ \uparrow_{\tau_1 \to \tau_2} v &= \mathbf{LAM} (\lambda S.\ \uparrow_{\tau_2} (\mathbf{app}\ (v, \downarrow^{\tau_1} S))) \\ \uparrow_{\tau_1 \times \tau_2} v &= \mathbf{PAIR} (\uparrow_{\tau_1} (\mathbf{fst}\ v), \uparrow_{\tau_2} (\mathbf{snd}\ v)) \\[1ex] \downarrow^{\alpha} (\mathbf{SYN}\ t) &= t \\ \downarrow^{\tau_1 \to \tau_2} (\mathbf{LAM}\ S) &= \mathbf{lam}\ (x, \downarrow^{\tau_2} (S\ (\uparrow_{\tau_1} (\mathbf{var}\ x)))) \text{ where } x \text{ is fresh} \\ \downarrow^{\tau_1 \times \tau_2} (\mathbf{PAIR}\ (S, T)) &= \mathbf{pair}\ (\downarrow^{\tau_1} S, \downarrow^{\tau_2} T) \end{align}

These definitions are easily implemented in the meta-language:

(* reflect : ty -> tm -> sem *)
fun reflect (Arrow (a, b)) t =
      LAM (fn S => reflect b (app t (reify a S)))
  | reflect (Prod (a, b)) t =
      PAIR (reflect a (fst t)) (reflect b (snd t))
  | reflect (Basic _) t =
      SYN t

(* reify   : ty -> sem -> tm *)
and reify (Arrow (a, b)) (LAM S) =
      let x = fresh_var () in
        Lam (x, reify b (S (reflect a (var x))))
      end
  | reify (Prod (a, b)) (PAIR S T) =
      Pair (reify a S, reify b T)
  | reify (Basic _) (SYN t) = t

By induction on the structure of types, it follows that if the semantic object S denotes a well-typed term s of type τ, then reifying the object (i.e., ↓τ S) produces the β-normal η-long form of s. All that remains is, therefore, to construct the initial semantic interpretation S from a syntactic term s. This operation, written ∥s∥Γ, where Γ is a context of bindings, proceeds by induction solely on the term structure:

\begin{align} \| \mathbf{var}\ x \|_\Gamma &= \Gamma(x) \\ \| \mathbf{lam}\ (x, s) \|_\Gamma &= \mathbf{LAM}\ (\lambda S.\ \| s \|_{\Gamma, x \mapsto S}) \\ \| \mathbf{app}\ (s, t) \|_\Gamma &= S\ (\|t\|_\Gamma) \text{ where } \|s\|_\Gamma = \mathbf{LAM}\ S \\ \| \mathbf{pair}\ (s, t) \|_\Gamma &= \mathbf{PAIR}\ (\|s\|_\Gamma, \|t\|_\Gamma) \\ \| \mathbf{fst}\ s \|_\Gamma &= S \text{ where } \|s\|_\Gamma = \mathbf{PAIR}\ (S, T) \\ \| \mathbf{snd}\ t \|_\Gamma &= T \text{ where } \|t\|_\Gamma = \mathbf{PAIR}\ (S, T) \end{align}

In the implementation:

(* meaning : ctx -> tm -> sem *)
fun meaning G t =
      case t of
        var x => lookup G x
      | lam (x, s) => LAM (fn S => meaning (add G (x, S)) s)
      | app (s, t) => (case meaning G s of
                         LAM S => S (meaning G t))
      | pair (s, t) => PAIR (meaning G s, meaning G t)
      | fst s => (case meaning G s of
                    PAIR (S, T) => S)
      | snd t => (case meaning G t of
                    PAIR (S, T) => T)

Note that there are many non-exhaustive cases; however, if applied to a closed well-typed term, none of these missing cases are ever encountered. The NBE operation on closed terms is then:

(* nbe : ty -> tm -> tm *)
fun nbe a t = reify a (meaning empty t)

As an example of its use, consider the syntactic term SKK defined below:

val K = lam ("x", lam ("y", var "x"))
val S = lam ("x", lam ("y", lam ("z", app (app (var "x", var "z"), app (var "y", var "z")))))
val SKK = app (app (S, K), K)

This is the well-known encoding of the identity function in combinatory logic. Normalising it at an identity type produces:

- nbe (Arrow (Basic "a", Basic "a")) SKK;
val it = lam ("v0",var "v0") : tm

The result is actually in η-long form, as can be easily seen by normalizing it at a different identity type:

- nbe (Arrow (Arrow (Basic "a", Basic "b"), Arrow (Basic "a", Basic "b"))) SKK;
val it = lam ("v1",lam ("v2",app (var "v1",var "v2"))) : tm

[edit] Correctness

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[edit] Extensions

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[edit] See also

[edit] References

  1. ^ Berger, Ulrich; Schwichtenberg, Helmut (1991). "An inverse of the evaluation functional for typed λ-calculus". LICS. 
  2. ^ Filinski, Andrzej; Rohde, Henning Korsholm (2004). "A denotational account of untyped normalization by evaluation" (PDF). FOSSACS. 
  3. ^ Abel, Andreas; Aehlig, Klaus; Dybjer, Peter (2007). "Normalization by Evaluation for Martin-Löf Type Theory with One Universe" (PDF). MFPS. 
  4. ^ Abel, Andreas; Coquand, Thierry; Dybjer, Peter (2007). "Normalization by Evaluation for Martin-Löf Type Theory with Typed Equality Judgements" (PDF). LICS. 
  5. ^ Danvy, Olivier (1996). "Type-directed partial evaluation" (gzipped PostScript). POPL: 242–257.