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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.[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][5][6]
^Berger, Ulrich; Schwichtenberg, Helmut (1991). "An inverse of the evaluation functional for typed λ-calculus". LICS.
^Filinski, Andrzej; Rohde, Henning Korsholm (2005). "A denotational account of untyped normalization by evaluation". Foundations of Software Science and Computation Structures (FOSSACS). Vol. 10. doi:10.7146/brics.v12i4.21870.
^Coquand, Thierry; Dybjer, Peter (1997). "Intuitionistic model constructions and normalization proofs". Mathematical Structure in Computer Science. 7 (1): 75–94. doi:10.1017/S0960129596002150.
^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.
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