oplss2024/ahmed/notes.typ

#import "../common.typ": *
#import "@preview/prooftrees:0.1.0": *
#show: doc => conf("Logical Relations", doc)

#let safe = $"safe"$

== Lecture 1

=== Logical relations

Logical relations are used to prove things about programs

- Unary equivalence
  - Example: Prove termination of the STLC
  - Example: Prove type soundness / safety (these lectures)

- Binary equivalence
  - Example: are two programs going to behave the same when you put them in some different context? (*contextual equivalence*)

-
  Representation independence

  - Example: suppose u implement a stack and have 2 different implementations. Prove that it doesn't matter which implementation you use.
  - Parametricity for existential types "there exists a stack such that these properties are true"

- Parametricity
- Security levels of data in a security programming language
  - Establish that high-security data never flows to low-security observers (*non-interference*)

You can also relate source terms to target terms in different languages.
This can be used to prove compiler correctness.

=== Type safety / soundness

Unary realizability relations for FFI. "all well-behaved terms from language A can be used in language B without causing problems with either language"

=== STLC

Statics

- $tau ::= "bool" | tau_1 arrow.r tau_2$
- $e ::= x | "true" | "false" | ifthenelse(e, e_1, e_2) | lambda (x : tau) . e | e_1 " " e_2$
- $v ::= "true" | "false" | lambda (x:tau) . e$
- $E ::= [dot] | "if" E "then" e_1 "else" e_2 | E " " e_2 | v " " E$

Operational semantics


#rect[$e mapsto e'$]

#tree(
  axi[],
  uni[$ifthenelse("true", e_1, e_2) mapsto e_1$]
)

#tree(
  axi[],
  uni[$ifthenelse("false", e_1, e_2) mapsto e_2$]
)

#tree(
  axi[],
  uni[$(lambda (x:tau) e) v mapsto subst(x, v, e)$]
)

#tree(
  axi[$e mapsto e'$],
  uni[$E[e] mapsto E[e']$]
)

Contexts:

$Gamma ::= dot | Gamma , x : A$

#rect[$Gamma tack.r e : tau$]

#tree(
  axi[],
  uni[$Gamma tack.r "true" : "bool"$]
)

#tree(
  axi[],
  uni[$Gamma tack.r "false" : "bool"$]
)

#tree(
  axi[$Gamma tack.r e : "bool"$],
  axi[$Gamma tack.r e_1 : tau$],
  axi[$Gamma tack.r e_2 : tau$],
  tri[$Gamma tack.r ifthenelse(e, e_1, e_2) : tau$]
)

#tree(
  axi[$Gamma(x) = tau$],
  uni[$Gamma tack.r x : tau$]
)

#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  $]
)

#tree(
  axi[$Gamma tack.r e_1 : tau_2 arrow.r tau$],
  axi[$Gamma tack.r e_2 : tau_2$],
  bin[$Gamma tack.r e_1 e_2 : tau$],
)

#quote(block: true, attribution: [Milner])[
  Well-typed programs do not go wrong
]

What does "going wrong" mean? It means getting stuck. You can't take another step and it's not a value.

*Definition (Type Soundness).* If $dot tack.r e : tau$, then
$forall(e' . e mapstostar e')$ either $isValue(e')$ or $exists e'' . e' mapsto e''$

*Definition (Progress).* If $dot tack.r e : tau$ then either $isValue(e)$ or $exists e' . (e mapsto e')$.

*Definition (Preservation).* If $dot tack.r e : tau$ and $e mapsto e'$ then $dot tack.r e' : tau$.

Progress and preservation are a _technique_ for proving type soundness, by just using the two functions over and over.

=== Logical relations

$ P_tau(e) $

A unary relation on any expression $e$ and its corresponding type $tau$.
(There are also binary relations $R_tau (e_1, e_2)$)

The property we're interested in is: $safe(e)$

- If $dot tack.r e:tau$ then $safe(e)$

*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? 
- Then check expressions. When do expressions belong to the relation, and when are they safe?

$ V db(tau_1) = { v | ... } \
V db("bool") = { "true" , "false" } \
V db(tau_1 arrow.r tau_2) = { lambda (x : tau_1) . e | ...  }
$ 

Can't have just arbitrary $lambda (x : tau_1) . e$ under $V db(tau_1 arrow.r tau_2)$.
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)} \
"irreducible"(e) :equiv cancel(exists) e' . e mapsto e'
$

#rect[
*Example of code from Rust.*
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.

#pagebreak()

== Lecture 2

#set enum(numbering: "A.")

Recall the lemmas we're proving:

1. $dot tack.r e : tau arrow.r.double e in Epsilon db(tau)$
2. $e in Epsilon db(tau) arrow.r.double safe(e)$

#set enum(numbering: "1.")

To prove lemma A, we must prove the fundamental property: $Gamma tack.r e:tau arrow.r.double Gamma tack.r.double e:tau$. "Syntactically well-typed implies semantic well-typed"

$ Gamma tack.r.double e : tau :equiv forall (gamma in G db(Gamma)) . gamma(e) in Epsilon db(tau) $

==== Proving A case for $lambda$

Case:

#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  $]
)

*Show:* $Gamma tack.r.double lambda (x : tau_1) . e : tau_1 arrow tau_2$.

Suppose $gamma in G db(Gamma)$. Then we ned to *show*: 

$ &gamma (lambda (x : tau_1). e) in Epsilon db(tau_1 arrow tau_2) \
equiv &lambda (x:tau_1).gamma(e) in Epsilon db(tau_1 arrow tau_2)
$

Suffices to show: $lambda (x:tau_1) gamma(e) in V db(tau_1 arrow tau_2)$.

Suppose $v in V db(tau_1)$. Then we need to show: 

$ &subst(x, v, gamma(e)) in Epsilon db(tau_2) \
equiv & gamma[x mapsto v] (e) in Epsilon db(tau_2)$

By the induction hypothesis, $Gamma,x:tau_1 tack.r e:tau_2$, so $Gamma,x:tau_1 tack.r.double e:tau_2$.

Because $gamma[x mapsto v] in G db(#[$Gamma, x : tau_1$])$, we can instantiate $Gamma,x:tau_1$ with $gamma[x mapsto v]$. (basically just translating it over to $gamma$)

Therefore, $gamma[x mapsto v](e) in Epsilon db(tau_2)$

#TODO Do the app case at home.

=== Adding recursive types

$ Omega :equiv (lambda x . x " " x) (lambda x . x " " x) $

Cannot be type-checked in STLC, but adding recursive types allows this to be typechecked.

$ tau ::= 1 | "bool" | tau_1 arrow tau_2 | alpha | mu alpha . tau $

For example:

- $"tree" = mu alpha . 1 + ("int" times alpha times alpha)$
- $"intlist" = mu alpha . 1 + ("int" times alpha)$

$ e ::= ... | "fold"(e) | "unfold"(e) $

#tree(
  axi[$Gamma tack.r e : subst(alpha, mu alpha . tau, tau)$],
  uni[$Gamma tack.r "fold"(e) : mu alpha . tau $],
)

#tree(
  axi[$Gamma tack.r e : mu alpha . tau $],
  uni[$Gamma tack.r "unfold"(e) : subst(alpha, mu alpha . tau, tau)$],
)

*NOTE*: Fold usually needs to be annotated with $mu alpha . tau$ in order to guide type checking.

=== Type-checking $Omega$

$ Omega :equiv (lambda (x : ?) . x " " x) (lambda (x : ?) . x " " x) $

With inductive types, we can give the second $x$ some type $alpha$.
Then the first $x$ that calls it would be $alpha arrow tau$ for some $tau$.

$ "SA" : (mu alpha . alpha arrow tau) arrow tau \
"SA" :equiv lambda (x : mu alpha . alpha arrow tau) . ("unfold" x) x$

Then re-define $Omega$ to be:

$ Omega' :equiv "SA" ("fold" "SA") $

=== Proving soundness for lambda calculus with recursive types with logical relations

$ v ::= ... | "fold"(v) $

*NOTE:* Unfold will not be a value.

$ E ::= ... | "fold" E $

New reduction rule:

$ "unfold"("fold" v) mapsto v $

$V db(mu alpha . tau) = { "fold"(v) | forall v in V db(subst(alpha, mu alpha. tau, tau))  }$

This is *not* a well-founded relation, because $V$ would depend on itself. This was an open problem (ATTAPL exercise) for a long time.

==== Step Indexing

This shows that this is well founded.

"If i were to run my program for $k$ steps, then I would not be able to tell that $V$ isn't well-founded"

#rect[$V_k db(tau)$]

$V_k db("bool") = {"true" , "false"}$

$V_k db(mu alpha . tau) = {"fold" v | v in V_(k-1) db(subst(alpha , mu alpha. tau , tau))}$

- Doing $"unfold"("fold" x) mapsto x$ "consumes" a step of $k$, so that's why $v$ already consumes a step.

$V_k db(tau_1 arrow tau_2) = {lambda (x : tau_1) . e | forall (j lt.eq k, v in V_j db(tau_1)) arrow.r.double subst(x, v, e) in Epsilon_j db(tau_2)}$

- Think of this as a timeline, starting with $k$ steps left
- $j+1$ is the number of steps left when you get to the application of the lambda
- Use 1 step to do the application, resulting in $j$ steps
- Eventually reach 0 steps left.

$Epsilon_k db(tau) = { e | forall (j < k, e') arrow.r.double e op(mapsto)^j e' and "irreducible"(e') arrow.r.double e' in V_(k-j) db(tau)}$

What $k$ doesn't matter because the entire relations are universally quantified over all $k$s.