Normalisation by evaluation

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]

Outline

Consider the simply typed lambda calculus, where types τ can be basic types (α), function types (→), or products (×), given by the following Backus–Naur Form 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 semantics, 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

See also

• MINLOG, a proof assistant that uses NBE as its rewrite engine.

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.