On reduction and normalization in the computational core

Abstract

We study the reduction in a λ-calculus derived from Moggi's computational one, that we call the computational core. The reduction relation consists of rules obtained by orienting three monadic laws. Such laws, in particular associativity and identity, introduce intricacies in the operational analysis. We investigate the central notions of returning a value versus having a normal form, and address the question of normalizing strategies. Our analysis relies on factorization results. * βv V ′ (for some value V ′ ) (5) Such a result (Corollary 1 in [Plo75]) comes from an analysis of the reduction properties of → βv , namely standardization.As we will see, the rules induced by associativity and identity make the behavior of the reduction-and the study of its operational properties-nontrivial in the setting of any monadic λ-calculus. The issues are inherent to the rules coming from the monadic laws ( 2) and ( 3), independently of the syntactic representation of the calculus that internalizes them. The difficulty appears clearly if we want to follow a similar route to [Plo75], as we discuss next.Reduction vs evaluation. Following [Fel88], reduction → © and evaluation → w © of λ © can be defined as the closure of the reduction rules under arbitrary and evaluation contexts, respectively. Consider the following grammars:where the hole can be filled by terms in Com only. Observe that the closure under evaluation context E is precisely weak reduction.Weak reduction of λ © , however, turns out to be non-deterministic, nonconfluent, and its normal forms are not unique. The following is a counterexample to all such properties-see Section 5 for further examples.Such an issue is not specific to the syntax of the computational core. The same phenomena show up with the let-notation. Evaluation contexts are now generated byand examples similar to the one above can be reproduced. We give the details in Example 5.3.Content and contributions. The focus of this paper is an operational analysis of two crucial properties of a term M :(i) M returns a value (i.e. M → * © unit V , for some V value). (ii) M has a normal form (i.e. M → * © N , for some N normal).The cornerstone of our analysis are factorization results (also called semistandardization in the literature): any reduction sequence can be re-organized so to first performing specific steps and then everything else.Via factorization, we establish the following key result, analogous to (5), relating reduction and evaluation:Remark 3.4 (β c and β v ). We are now in place to show how a β v -reduction is simulated by a β c -reduction, possibly with more steps. Observe that the reduction relation → © is only on computations, therefore no computation N will ever reduce to some value V ; however, this is represented by a reduction N → * © !V , where !V is the coercion of the value V into a computation. Moreover, let us assume that M → * © !λx.M ′ . We have:M N ≡ (λz.zN )M → * © 1. → w id is non-deterministic, but it is confluent.2. → w σ is non-deterministic, non-confluent and normal forms ...