logical relations

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Michael Zhang 2024-06-03 15:24:07 -04:00
parent 3ebd38caad
commit 299da9bef4

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@ -7,6 +7,7 @@
#let mapstostar = $op(arrow.r.long.bar)^*$
#let safe = $"safe"$
#let db(x) = $bracket.l.double #x bracket.r.double$
#let TODO = text(fill: red)[*TODO*]
= Mon. Jun 3 \@ 14:00
@ -140,6 +141,8 @@ The property we're interested in is: $safe(e)$
*Definition (safe).* $safe(e) :equiv forall e' . (e mapstostar e') arrow.r.double isValue(e') or exists e'' . (e' mapsto e'')$
*Definition (Semantic Type Soundness).* $dot tack.r e : tau$ then $safe(e)$
Common technique:
- Focus on the values, when do values belong to the relation, when are they safe?
@ -155,7 +158,8 @@ The body also needs to be well-typed.
$
V db(tau_1 arrow.r tau_2) = { lambda (x : tau_1) . e | forall v in V db(tau_1) . subst(x, v, e) in Epsilon db(tau_2) } \
Epsilon db(tau) = { e | forall e' . e mapstostar e' and "irreducible"(e') arrow.r.double e' in V db(tau)}
Epsilon db(tau) = { e | forall e' . e mapstostar e' and "irreducible"(e') arrow.r.double e' in V db(tau)} \
"irreducible"(e) :equiv cancel(exists) e' . e mapsto e'
$
#rect[
@ -163,3 +167,84 @@ $
Unsafe code blocks are an example of a something that may not be _syntactically well-formed_, but are still represented by logical relations because they behave the same at the boundary.
]
The difference between semantic soundness with logical relations vs. progress and preservation is that you don't need to prove anything about intermediate states of running programs.
Can't just work with closed terms because of this rule:
#tree(
axi[$Gamma , x : tau tack.r e : tau_2$],
uni[$Gamma tack.r lambda (x : tau_1) . e : tau_1 arrow.r tau_2 $]
)
(Slightly more general theorem)
#rect(width: 100%)[
*Definition (Fundamental property of logical relations).*
Theorem we are trying to prove
$
Gamma tack.r e : tau arrow.r.double Gamma tack.r.double e : tau
$
- "syntactically well typed means semantically well typed"
]
Focus on terms with open variables
Use $gamma$ as a substitution from variables to values.
#rect[$G db(Gamma)$]
- $G db(dot) = { emptyset }$
- $G #db[$Gamma , x : tau$] = { gamma[x mapsto v] | gamma in G db(Gamma) and v in V db(tau)}$
"sound inputs will give you sound outputs". This is the important case
Now define semantic soundness:
$
Gamma tack.r.double e : tau :equiv forall (gamma in G db(Gamma)) . gamma(e) in Epsilon db(tau)$
Prove:
#set enum(numbering: "a.")
1. $dot tack.r e : tau arrow.r.double e in Epsilon db(tau)$
- this cannot be proved by focusing on closed terms alone
- This is why the fundamental property must be used, in the case of $Gamma = dot$, then the substitution $gamma$ is empty and the result is $e in Epsilon db(tau)$
2. $e in Epsilon db(tau) arrow.r.double safe(e)$
#set enum(numbering: "1.")
To prove A:
_Proof._ By induction on typing derivations
-
Case example:
#tree(
axi[],
uni[$Gamma tack.r "true" : "bool"$]
)
Show $Gamma tack.r.double "true" : "bool"$. Suppose $gamma in G db(Gamma)$.
Show $gamma("true") in Epsilon db("bool") equiv "true" in Epsilon db("bool")$.
Suffices to show $"true" in V db("bool")$, which is true by definition.
False is proved similarly.
-
Case example:
#tree(
axi[$Gamma(x) = tau$],
uni[$Gamma tack.r x : tau$]
)
Show $Gamma tack.r.double x : tau$. #TODO
To prove B:
_Proof._ Suppose $e'$ s.t. $e mapstostar e'$ .
- Case : $not "irreducible" (e')$ then $exists e'' . e' mapsto e''$
- Case : $"irreducible" (e')$ then $isValue(e')$ trivially.