Normalisation by evaluation

In programming language semantics, normalisation by evaluation (NBE) is a method 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, reduction-free, approach differs from the more traditional syntactic, reduction-based, description of normalisation as 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. It has since been extended both to weaker type systems such as the untyped lambda calculus using a domain theoretic approach, and to richer type systems such as several variants of Martin-Löf type theory.

In programming language semantics, normalisation by evaluation (NBE) is a method 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, reduction-free, approach differs from the more traditional syntactic, reduction-based, description of normalisation as 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. It has since been extended both to weaker type systems such as the untyped lambda calculus using a domain theoretic approach, and to richer type systems such as several variants of Martin-Löf type theory. (Types) τ ::= α | τ 1 → τ 2 | τ 1 × τ 2 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. 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 datatype 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: (* fresh_var : unit -> string *) val variable_ctr = ref ~1 fun fresh_var () = (variable_ctr := 1 + !variable_ctr; "v" ^ Int.toString (!variable_ctr)) (* 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 val 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: datatype ctx = empty | add of ctx * (string * sem) (* lookup : ctx -> string -> sem *) fun lookup (add (remdr, (y, value))) x = if x = y then value else lookup remdr x (* 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

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