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. 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.
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
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
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:
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:
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)
- 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
- Berger, Ulrich; Schwichtenberg, Helmut (1991). "An inverse of the evaluation functional for typed λ-calculus". LICS.
- Filinski, Andrzej; Rohde, Henning Korsholm (2004). "A denotational account of untyped normalization by evaluation". FOSSACS.
- Abel, Andreas; Aehlig, Klaus; Dybjer, Peter (2007). "Normalization by Evaluation for Martin-Löf Type Theory with One Universe" (PDF). MFPS.
- Abel, Andreas; Coquand, Thierry; Dybjer, Peter (2007). "Normalization by Evaluation for Martin-Löf Type Theory with Typed Equality Judgements" (PDF). LICS.
- Gratzer, Daniel; Sterling, Jon; Birkedal, Lars (2019). "Implementing a Modal Dependent Type Theory" (PDF). ICFP.
- Danvy, Olivier (1996). "Type-directed partial evaluation" (gzipped PostScript). POPL: 242–257.