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Blurb about accessibility strong normalization

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Jonathan Chan 2025-04-19 09:56:22 -04:00
parent fb9e31fddf
commit 6aa8f1049c
3 changed files with 90 additions and 12 deletions
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49
cbpv.mng
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@ -9,7 +9,7 @@
\usepackage{enumitem,float,booktabs,xspace,doi}
\usepackage{amsmath,amsthm,amssymb}
\usepackage[capitalize,nameinlink,noabbrev]{cleveref}
\usepackage{mathpartir,mathtools,stmaryrd}
\usepackage{mathpartir,mathtools,stmaryrd,latexsym}
\setlist{nosep}
\hypersetup{
@ -187,7 +187,7 @@ I present them below as judgement pseudorules
using a dashed line to indicate that they are admissible rules.
These are all proven either by construction
or by induction on the first premise.
%
\begin{mathpar}
\mprset{fraction={-~}{-~}{-}}
\drule[]{SRs-Once}
@ -481,7 +481,46 @@ Normalization holds as a corollary.
\section{Strong Normalization as an Accessibility Relation} \label{sec:sn-acc}
\drules[sn]{$[[v ∈ sn]], [[m ∈ sn]]$}{strongly normalizing terms}{Val,Com}
How do we know that our definition of strongly normal terms is correct?
What are the properties that define its correctness?
It seems to be insufficient to talk about strongly neutral and normal terms alone.
For instance, we could add another rule asserting that $[[m ∈ SN]]$ if $[[m ⤳ m]]$,
which doesn't appear to violate any of the existing properties we require.
Then $[[(λx. (force x) x) (thunk (λx. (force x) x))]]$,
which reduces to itself, would be considered a ``strongly normal'' term,
and typing rules that can type this term (perhaps with a recursive type)
would still be sound with respect to our logical relation.
To try to rule out this kind of mistake in the definition,
we might want to prove that strongly normal terms never loop,
\ie $[[m ∈ SN ∧ m ⤳* m]]$ is impossible.
However, this does not cover diverging terms that grow forever.
Furthermore, $[[m ⤳ n]]$ isn't really a full reduction relation on its own,
since it's defined mutually with strongly normal terms,
and it doesn't describe reducing anywhere other than in the head position.
The solution in POPLMark Reloaded is to prove that $\kw{SN}$ is sound
with respect to a traditional presentation of strongly normal terms
based on full small-step reduction $[[v ⇝ w]]$, $[[m ⇝ n]]$ of values and computations.
I omit here the definitions of these reductions,
but they encompass all $\beta$-reduction rules and are congruent over all subterms,
including under binders.
Traditional strongly normal terms, written $[[v ∈ sn]]$, $[[m ∈ sn]]$,
are accessibility relations with respect to small-step reduction.
Terms are inductively defined as strongly normal if they step to strongly normal terms,
so terms which don't step are trivially normal.
This rules out looping and diverging terms, since they never stop stepping.
%
% \drules[sn]{$[[v ∈ sn]], [[m ∈ sn]]$}{strongly normalizing terms}{Val,Com}
\begin{mathpar}
\fbox{$[[v ∈ sn]]$} \quad \fbox{$[[m ∈ sn]]$} \hfill \textit{strongly normalizing terms} \\
\drule[]{sn-Val}
\drule[]{sn-Com}
\end{mathpar}
\drules[sr]{$[[m ⤳' n]]$}{strong head reduction}{Thunk,Lam,Ret,Inl,Inr,App,Let}
\drules[ne]{$[[m ∈ ne]]$}{neutral computations}{Force,App,Let,Case}
@ -604,7 +643,7 @@ Normalization holds as a corollary.
since we have no congruence rule for when the head position is not neutral.
\end{proof}
\begin{lemma}[Backward closure] \label{lem:closure}
\begin{lemma}[Backward closure] \label{lem:closure-sn}
If $[[m ⤳' n]]$ and $[[n ∈ sn]]$ then $[[m ∈ sn]]$.
\end{lemma}
@ -636,7 +675,7 @@ Normalization holds as a corollary.
$[[m ∈ SNe]]$, $[[v ∈ SN]]$, $[[m ∈ SN]]$, and $[[m ⤳ n]]$.
The cases for the first three correspond to the admissible strongly normal rules,
using \nameref{lem:sound-sne} as needed,
except for the \rref*{SN-Red} case, which uses \nameref{lem:closure}.
except for the \rref*{SN-Red} case, which uses \nameref{lem:closure-sn}.
The cases for strong reduction hold by construction.
\end{proof}

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cbpv.tex
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@ -9,7 +9,7 @@
\usepackage{enumitem,float,booktabs,xspace,doi}
\usepackage{amsmath,amsthm,amssymb}
\usepackage[capitalize,nameinlink,noabbrev]{cleveref}
\usepackage{mathpartir,mathtools,stmaryrd}
\usepackage{mathpartir,mathtools,stmaryrd,latexsym}
\setlist{nosep}
\hypersetup{
@ -187,7 +187,7 @@ I present them below as judgement pseudorules
using a dashed line to indicate that they are admissible rules.
These are all proven either by construction
or by induction on the first premise.
%
\begin{mathpar}
\mprset{fraction={-~}{-~}{-}}
\drule[]{SRs-Once}
@ -481,7 +481,46 @@ Normalization holds as a corollary.
\section{Strong Normalization as an Accessibility Relation} \label{sec:sn-acc}
\drules[sn]{$ \ottnt{v} \in \kw{sn} , \ottnt{m} \in \kw{sn} $}{strongly normalizing terms}{Val,Com}
How do we know that our definition of strongly normal terms is correct?
What are the properties that define its correctness?
It seems to be insufficient to talk about strongly neutral and normal terms alone.
For instance, we could add another rule asserting that $ \ottnt{m} \in \kw{SN} $ if $ \ottnt{m} \leadsto \ottnt{m} $,
which doesn't appear to violate any of the existing properties we require.
Then $ ( \lambda \ottmv{x} \mathpunct{.} ( \kw{force} \gap \ottmv{x} ) \gap \ottmv{x} ) \gap ( \kw{thunk} \gap ( \lambda \ottmv{x} \mathpunct{.} ( \kw{force} \gap \ottmv{x} ) \gap \ottmv{x} ) ) $,
which reduces to itself, would be considered a ``strongly normal'' term,
and typing rules that can type this term (perhaps with a recursive type)
would still be sound with respect to our logical relation.
To try to rule out this kind of mistake in the definition,
we might want to prove that strongly normal terms never loop,
\ie $ \ottnt{m} \in \kw{SN} \wedge \ottnt{m} \leadsto^* \ottnt{m} $ is impossible.
However, this does not cover diverging terms that grow forever.
Furthermore, $ \ottnt{m} \leadsto \ottnt{n} $ isn't really a full reduction relation on its own,
since it's defined mutually with strongly normal terms,
and it doesn't describe reducing anywhere other than in the head position.
The solution in POPLMark Reloaded is to prove that $\kw{SN}$ is sound
with respect to a traditional presentation of strongly normal terms
based on full small-step reduction $ \ottnt{v} \rightsquigarrow \ottnt{w} $, $ \ottnt{m} \rightsquigarrow \ottnt{n} $ of values and computations.
I omit here the definitions of these reductions,
but they encompass all $\beta$-reduction rules and are congruent over all subterms,
including under binders.
Traditional strongly normal terms, written $ \ottnt{v} \in \kw{sn} $, $ \ottnt{m} \in \kw{sn} $,
are accessibility relations with respect to small-step reduction.
Terms are inductively defined as strongly normal if they step to strongly normal terms,
so terms which don't step are trivially normal.
This rules out looping and diverging terms, since they never stop stepping.
%
% \drules[sn]{$ \ottnt{v} \in \kw{sn} , \ottnt{m} \in \kw{sn} $}{strongly normalizing terms}{Val,Com}
\begin{mathpar}
\fbox{$ \ottnt{v} \in \kw{sn} $} \quad \fbox{$ \ottnt{m} \in \kw{sn} $} \hfill \textit{strongly normalizing terms} \\
\drule[]{sn-Val}
\drule[]{sn-Com}
\end{mathpar}
\drules[sr]{$ \ottnt{m} \leadsto_{ \kw{sn} } \ottnt{n} $}{strong head reduction}{Thunk,Lam,Ret,Inl,Inr,App,Let}
\drules[ne]{$ \ottnt{m} \in \kw{ne} $}{neutral computations}{Force,App,Let,Case}
@ -604,7 +643,7 @@ Normalization holds as a corollary.
since we have no congruence rule for when the head position is not neutral.
\end{proof}
\begin{lemma}[Backward closure] \label{lem:closure}
\begin{lemma}[Backward closure] \label{lem:closure-sn}
If $ \ottnt{m} \leadsto_{ \kw{sn} } \ottnt{n} $ and $ \ottnt{n} \in \kw{sn} $ then $ \ottnt{m} \in \kw{sn} $.
\end{lemma}
@ -636,7 +675,7 @@ Normalization holds as a corollary.
$ \ottnt{m} \in \kw{SNe} $, $ \ottnt{v} \in \kw{SN} $, $ \ottnt{m} \in \kw{SN} $, and $ \ottnt{m} \leadsto \ottnt{n} $.
The cases for the first three correspond to the admissible strongly normal rules,
using \nameref{lem:sound-sne} as needed,
except for the \rref*{SN-Red} case, which uses \nameref{lem:closure}.
except for the \rref*{SN-Red} case, which uses \nameref{lem:closure-sn}.
The cases for strong reduction hold by construction.
\end{proof}