DeepAI AI Chat
Log In Sign Up

Normalization by Evaluation for Call-by-Push-Value and Polarized Lambda-Calculus

by   Andreas Abel, et al.
Göteborgs universitet

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.


page 1

page 2

page 3

page 4


Probabilistic Rewriting: Relations between Normalization, Termination, and Unique Normal Forms

We investigate how techniques from Rewrite Theory can help us to study c...

Functional Pearl: The Distributive λ-Calculus

We introduce a simple extension of the λ-calculus with pairs—called the ...

Normalization by Evaluation for the Lambek Calculus

The syntactic calculus of Lambek is a deductive system for the multiplic...

The Power of Tightness for Call-By-Push-Value

We propose tight type systems for Call-by-Name (CBN) and Call-by-Value (...

Deductive Systems and Coherence for Skew Prounital Closed Categories

In this paper, we develop the proof theory of skew prounital closed cate...

Two Decreasing Measures for Simply Typed Lambda-Terms (Extended Version)

This paper defines two decreasing measures for terms of the simply typed...

Strategies for Asymptotic Normalization

We present a technique to study normalizing strategies when termination ...