Normalization by evaluation for call-by-push-value and polarized lambda calculus
Paper in proceeding, 2019

We observe that normalization by evaluation for simply-typed lambda-calculus with weak coproducts can be carried out in a weak bi-cartesian closed category of presheaves equipped with a monad that allows us to perform case distinction on neutral terms of sum type. The placement of the monad influences the normal forms we obtain: for instance, placing the monad on coproducts gives us eta-long beta-pi normal forms where pi refers to permutation of case distinctions out of elimination positions. We further observe that placing the monad on every coproduct is rather wasteful, and an optimal placement of the monad can be determined by considering polarized simple types inspired by focalization. Polarization classifies types into positive and negative, and it is sufficient to place the monad at the embedding of positive types into negative ones. We consider two calculi based on polarized types: pure call-by-push-value (CBPV) and polarized lambda-calculus, the natural deduction calculus corresponding to focalized sequent calculus. For these two calculi, we present algorithms for normalization by evaluation. We further discuss different implementations of the monad and their relation to existing normalization proofs for lambda-calculus with sums. Our developments have been partially formalized in the Agda proof assistant.

Intuitionistic Propositional Logic

Evaluation

Monad

Polarized Logic

Normalization

Semantics

Lambda-Calculus

Author

Andreas Abel

University of Gothenburg

Christian Sattler

University of Nottingham

ACM International Conference Proceeding Series

a3

21st International Symposium on Principles and Practice of Declarative Programming, PPDP 2019
Porto, Portugal,

Subject Categories

Algebra and Logic

Computational Mathematics

Geometry

DOI

10.1145/3354166.3354168

More information

Latest update

1/17/2023