[TYPES] Progress + Preservation = Evaluation

William J. Bowman wjb at williamjbowman.com
Fri Jul 6 15:47:24 EDT 2018

I think the canonical reference for this is Wright Felleisen 1994, https://doi.org/10.1006/inco.1994.1093. They have a history in the introduction showing many prior versions, although most don't talk about evaluation directly.

They call "evaluation" "type soundess", in various forms, where the evaluation
function reprenents the ground truth semantics for the language, and the type
system is a syntactic proof system.
"Type soundness" is meant to indicate that the syntactic semantics (proof) is
sound w.r.t. evaluation semantics (truth).

William J. Bowman

On Fri, Jul 06, 2018 at 12:49:48PM -0300, Philip Wadler wrote:
> [ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list ]

> Everyone on this list will be familiar with Progress and Preservation
> for terms in a typed calculus.  Write ∅ ⊢ M : A to indicate that term
> M has type A for closed M.
>   Progress.  If ∅ ⊢ M : A then either M is a value or M —→ N,
>   for some term N.
>   Preservation.  If ∅ ⊢ M : A and M —→ N then ∅ ⊢ N : A.
> It is easy to combine these two proofs into an evaluator.
> Write —↠ for the transitive and reflexive closure of —→.
>   Evaluation.  If ∅ ⊢ M : A, then for every natural number n,
>   either M —↠ V, where V is a value and the reduction sequence
>   has no more than n steps, or M —↠ N, where N is not a value
>   and the reduction sequence has n steps.
> Evaluation implies that either M —↠ V or there is an infinite
> sequence M —→ M₁ —→ M₂ —→ ... that never reduces to a value;
> but this last result is not constructive, as deciding which of
> the two results holds is not decidable.
> An Agda implementation of Evaluation provides an evaluator for terms:
> given a number n it will perform up to n steps of evaluation, stopping
> early if a value is reached. This is entirely obvious (at least in
> retrospect), but I have not seen it written down anywhere. For
> instance, this approach is not exploited in Software Foundations to
> evaluate terms (it uses a normalize tactic instead).  I have used it
> in my draft textbook:
>   https:plfa.inf.ed.ac.uk
> Questions: What sources in the literature should one cite for this
> technique?  How well-known is it as folklore? Cheers, -- P
> .   \ Philip Wadler, Professor of Theoretical Computer Science,
> .   /\ School of Informatics, University of Edinburgh
> .  /  \ and Senior Research Fellow, IOHK
> . http://homepages.inf.ed.ac.uk/wadler/
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