# [TYPES] Progress + Preservation = Evaluation

Sam Lindley Sam.Lindley at ed.ac.uk
Fri Jul 6 18:23:18 EDT 2018

```You should certainly cite Conor McBride's MPC 2015 paper
"Turing-Completeness Totally Free":

https://personal.cis.strath.ac.uk/conor.mcbride/TotallyFree.pdf

Sam

On 06/07/18 17:49, 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
>
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>

--
The University of Edinburgh is a charitable body, registered in
Scotland, with registration number SC005336.

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