day 4

ahmed/day1.agda

@@ -1,34 +1,26 @@
-module ahmed.day1 where
+module Ahmed.Day1 where
 
-open import Agda.Primitive
-open import Data.Nat
-open import Data.Fin
-open import Data.Product
+open import Data.Empty
+open import Data.Unit
+open import Relation.Nullary using (Dec; yes; no)
 
-data Type : Set where
-  Bool : Type
-  _-→_ : Type → Type → Type
+data type : Set where
+  bool : type
+  _-→_ : type -> type -> type
 
-data Term (n : ℕ) : Set where
-  Var : (m : Fin n) → Term n
-  True : Term n
-  False : Term n
-  If_Then_Else_ : Term n → Term n → Term n → Term n
-  λ[_::_]_ : Term n → Type → Term n → Term n
-  _∙_ : Term n → Term n → Term n
+data term : Set where
+  true : term
 
-data isValue (n : ℕ) : Term n → Set where
-  TrueValue : isValue n True
-  FalseValue : isValue n False
-  LambdaValue : (x : Term n) → (τ : Type) → (e : Term n) → isValue n (λ[ x :: τ ] e)
+data ctx : Set where
+  nil : ctx
+  cons : ctx -> type -> ctx
 
-Value = (n : ℕ) → Σ (Term n) (isValue n)
+data substitution : Set where
+  nil : substitution
+  cons : substitution -> term -> substitution
 
-data evaluationContext : (n : ℕ) → Set where
-  dot : evaluationContext 0
-  EIf_Then_Else_ : {n : ℕ} → evaluationContext n → Term n → Term n → evaluationContext n
-  EAppLeft : {n : ℕ} → evaluationContext n → Term n → evaluationContext n
-  EAppRight : {n : ℕ} → Value → evaluationContext n → evaluationContext n
-
-data isValidValueForType (n : ℕ) (t : Type) : (e : Term n) → Set where
-  TrueV : isValidValueForType n Bool True
+isGoodSubstitution : (Γ : ctx) -> (γ : substitution) -> Set
+isGoodSubstitution nil nil = ⊤
+isGoodSubstitution nil (cons γ x) = ⊥
+isGoodSubstitution (cons Γ x) nil = ⊥
+isGoodSubstitution (cons Γ x) (cons γ x₁) = {!   !}

ahmed/day1.fst

@@ -0,0 +1,77 @@
+module Day1
+
+let var = int
+
+type ty =
+  | TyBool
+  | TyArrow of ty * ty
+
+type term =
+  | EVar of int
+  | ETrue
+  | EFalse
+  | EIfThenElse of term * term * term
+  | ELambda of ty * term
+  | EApp of term * term
+
+type value =
+  | VTrue
+  | VFalse
+  | VLambda of ty * term
+
+type evalctx =
+  | ECDot
+  | ECIfThenElse of evalctx * term * term
+  | ECAppL of evalctx * term
+  | ECAppR of value * evalctx
+
+type step : term -> term -> Type =
+  | StepIfTrue : (e1 : term) -> (e2 : term) -> step (EIfThenElse (ETrue, e1, e2)) e1
+  | StepIfFalse : (e1 : term) -> (e2 : term) -> step (EIfThenElse (EFalse, e1, e2)) e2
+
+let ctx = var -> option ty
+let empty : ctx = fun _ -> None
+let extend (t : ty) (g : ctx) : ctx = fun y -> if y = 0 then Some t else g (y - 1)
+
+noeq type typing : ctx -> term -> ty -> Type =
+  | TypeTrue : (ctx : ctx) -> typing ctx ETrue TyBool
+  | TypeFalse : (ctx : ctx) -> typing ctx EFalse TyBool
+  | TypeIfThenElse : (ctx : ctx) -> (ty : ty)
+    -> (c : term) -> (e1 : term) -> (e2 : term)
+    -> typing ctx c TyBool
+    -> typing ctx e1 ty -> typing ctx e2 ty
+    -> typing ctx (EIfThenElse (c, e1, e2)) ty
+
+let vRelation (ty : ty) (term : term) : prop =
+  match ty, term with
+  | (TyBool, ETrue) -> True
+  | (TyBool, EFalse) -> True
+  | _ -> False
+
+let irred (term : term) : prop = admit()
+
+let rec stepsToInNSteps (e1 : term) (e2 : term) (n : nat) : prop =
+  if n = 0 then False
+  else exists e' . (stepsToInNSteps e' e2 (n - 1) /\ (step e1 e'))
+
+// let stepsTo (e1 : term) (e2 : term) : prop =
+
+
+let eRelation (ty : ty) (term : term) : prop =
+  exists e' . irred e' /\ vRelation ty e'
+
+let goodRelation (g : ctx) =
+  admit()
+
+let subst (#ctx:ctx) (g : goodRelation ctx) (term : term) =
+  admit()
+
+// let typeSoundness (ctx : ctx) (term : term) (ty : ty) : prop =
+//   forall (e' : )
+
+let semanticSoundness (ctx : ctx) (term : term) (ty : ty) : prop =
+  forall (y : goodRelation ctx) . eRelation ty (subst y term)
+
+let fundamentalProperty (#ctx : ctx) (#term : term) (#ty : ty)
+  (_ : typing ctx term ty) : semanticSoundness ctx term ty =
+  admit()

ahmed/notes.typ

@@ -372,3 +372,106 @@ $V_k db(tau_1 arrow tau_2) = {lambda (x : tau_1) . e | forall (j lt.eq k, v in V
 $Epsilon_k db(tau) = { e | forall (j < k, e') arrow.r.double e op(mapsto)^j e' and "irreducible"(e') arrow.r.double e' in V_(k-j) db(tau)}$
 
 What $k$ doesn't matter because the entire relations are universally quantified over all $k$s.
+
+#pagebreak()
+
+== Lecture 3
+
+Step indexing history
+
+- 2001: TOPLAS Andrew Appel https://dl.acm.org/doi/pdf/10.1145/504709.504712
+- 2004: Mutable Refs
+
+https://iris-project.org/
+
+*Proof.* Prove the fundamental property for fold.
+
+Case:
+
+#tree(
+  axi[$Gamma tack.r e : subst(alpha, mu alpha . tau, tau)$],
+  uni[$Gamma tack.r "fold" e : mu alpha . tau$]
+)
+
+Show: $Gamma tack.r.double "fold" e : mu alpha . tau$
+
+- Suppose $k >= 0$. Suppose $gamma in G_k db(Gamma)$
+
+  Need to show: $gamma ("fold" e) in Epsilon_k db(mu alpha . tau)$
+
+  This is $"fold" (gamma (e)) in Epsilon_k db(mu alpha . tau)$
+
+  - Suppose $"fold" (gamma (e)) op(mapsto)^(j < k) e' and "irreducible"(e')$
+
+    We need to show $e' in V_(k- j) db(mu alpha. tau)$
+
+    By operational semantics, $gamma(e) op(mapsto)^(j_1 <= j) e_1 and "irreducible"(e_1)$
+
+    By IH, $Gamma tack.r.double e : subst(alpha,mu alpha. tau , tau)$.
+
+    Instantiate this using our $k$ and $gamma in G_k db(Gamma)$: $gamma(e) in Epsilon_k db(subst(alpha ,mu alpha. tau, tau))$
+
+    However, we said it would, so instantiate the earlier with $gamma(e) mapsto e_1$, which is irreducible after $j_1$ steps.
+
+    $e_1 in V_(k-j_1) db(subst(alpha,mu alpha. tau, tau))$
+
+    Since $e_1$ is in a value relation, it's a value.
+
+    By operational semantics, $"fold" (gamma(e)) op(mapsto)^(j_1) "fold"(v_1)$. But from earlier supposition, $j = j_1$.
+
+    Now just need to show $"fold"(v_1)$ belongs to $V_(k-j) db(mu alpha. tau)$
+
+    - Using the rule, we have to show that $forall j' < k - j$, that $v_1 in V_(j') db(subst(alpha,mu alpha. tau , tau))$
+
+Need monotonicity to prove this. If something behaves like a type for $k$ steps, it also behaves like that for fewer steps.
+
+*Lemma (monotonicity / downward closure).*
+
+$ v in V_k db(tau) => forall j <= k . v in V_j db(tau)
+$
+
+Prove this by induction on $V$.
+
+*Homework.* Do the proof of monotonicity on the function case.
+
+*Homework.* Prove the fundamental property with step indexing for lambda.
+
+=== Mutable Refs
+
+$tau ::= 1 | "bool" | tau_1 arrow tau_2 | "ref" tau$
+
+$S : "Loc" harpoon C "Val"$
+
+$e ::= angle.l angle.r | ... | "new" e | !e | e_1 := e_2$
+
+$v ::= ... | l$
+
+$E ::= ... | "new" E | ! E | E ::= e | v ::= E$
+
+--
+
+Operational semantics
+
+- $(S , l := v) mapsto (S[l mapsto v], angle.l angle.r)$
+- $(S "new" v) mapsto (S[l mapsto v])$
+
+In java, you can't change the type of a storage cell. To prevent safety problems / getting stuck
+
+=== Non-terminating program example with assignment
+
+$"let" x = "new" (lambda (z : bb(1)) . z) \
+"let" f = lambda (z : bb(1)) . (!x) z \
+x := f ; \
+f () ;
+$
+
+"Landin's Knot"
+
+"Strong update" - language allows you to update a value of a cell with a different type.
+
+Type = ${ I in P ( "CValue")}$
+Type = ${ I in "Nat" -> P("CValue")}$
+
+(CValue = Closed values)
+
+$V_(k, S) db("ref" tau) = { l | l in V_(k-1) db(tau) }$

pfenning/notes.typ

@@ -460,6 +460,8 @@ $Gamma$ can be multi-modal. This is how top-level declarations can be re-used.
 
 https://www.cs.cmu.edu/~fp/papers/tocl07.pdf
 
+https://arxiv.org/pdf/2402.01428v1
+
 Need to enforce independence, that $Gamma tack.r e : A_m$ means $Gamma >= m$
 
 Pointer:

silva/notes.typ

@@ -472,6 +472,8 @@ Show these are the same by applying denesting and sliding.
 
 === Net KAT
 
+https://www.cs.cornell.edu/~jnfoster/papers/frenetic-netkat.pdf
+
 Net KAT is a special KAT. Think of networks as boxes the only thing they can do is get packets in, change something about the packet, and then sends the packet on.
 
 Packets are records of fields $f_1, ..., f_k$ mapping to values $v_1, ..., v_k$. Program actions ($p = f arrow.l n$) and tests ($f = n$) come in pairs.
@@ -491,3 +493,95 @@ For example:
 
   Using KAT, you can perform network reachability.
 ]
+
+#pagebreak()
+
+== Lecture 4
+
+NetKAT
+
+- $f <- n$
+- $f = n$
+- "dup" (prev. known as observation) - takes a snapshot of what the packet looked like, so you can remember where it went throughout its lifetime
+  - (sw $<- 2 dot$ dup $dot$ sw $<- 5$)
+  - make claims like "an SSH packet never visited switch 3" by checking the trace
+
+$db(.) : P -> 2^P$
+
+- $db(f <- n)(pi) = { subst(f, n, pi) }$
+- $db(f = n)(pi) = cases(emptyset "if" pi(f) eq.not n, {pi} "else")$
+- $db(e + e')(pi) = db(e)(pi) union db(e')(pi)$
+- $db(e\;f)(pi) = union.big_(pi' in db(e)(pi)) db(e')(pi')$
+
+#let At = "At"
+#let dup = "dup"
+For NetKAT, $At tilde.equiv P $
+
+=== Packet axioms
+
+NetKat = KAT + Packet axioms:
+
+- $f <- n ; f <- m equiv f <- m$
+- $f <- n ; g <- m equiv g <- m ; f <- n$
+- $f <- n ; g = m equiv g = m ; f <- n$
+- $f <- n ; f = n equiv f <- n$
+- $f = n ; f <- n equiv f = n$
+- $sum_(n in "Val") f = n equiv 1$
+  - Says that every field has a value in it, but cannot have 2 values
+  - This is where finiteness of "Val" is important
+- $f = n ; f = m equiv 0$
+- $f = n ; dup equiv dup ; f = n$
+
+With these axioms, you can prove:
+
+$ db(e) = db(f) <=> e equiv f $
+
+=== Match Action Table
+
+Example:
+
+#table(
+  columns: (auto, auto),
+  [Test], [Action],
+  [type = SSH], [drop],
+  [dst = 3], [pt $<-$ 5 + pt $<-$ 3],
+)
+
+Each of these can be written down as a NetKAT expression. This table is a big sum of tests and actions
+
+The NetKAT expression corresponding to its behavior is
+
+$ "sw" = k dot sum_e t_e ; a_e $
+
+(This is an oversimplification, since the test order matters)
+
+=== Network definition using NetKAT
+
+- $"link"_(k j) = "sw" = k ; "pt" = 1 ; "sw" <- j ; "pt" <- 2$
+- $"topology" = sum_"link" l_(k j)$
+- $"switch" = sum_"sw" s_k$
+- $"network" = ("topology" ; "switch")^*$
+
+This is the definition of the entire network
+
+=== Networking queries
+
+==== Reachability queries
+
+Ask for $"sw" = A ; ... ; "sw" = B$
+
+Either A and B are directly connected by the topology, or take several steps and get to B.
+
+$ "sw" = A ; "top" ; ("switch" ; "top")^* ; "sw" = B $
+
+If this expression is $equiv 0$, then I know that A is not connected to B.
+
+==== Forwarding Loop
+
+This is when a packet goes in a cycle between nodes. Normally this is stopped with a TTL, which is decreased, and when it reaches 0 then the packet is ejected.
+
+Instead, with NetKAT you can check if you have a forwarding loop.
+
+$ alpha ; ("top" ; "switch") ; ("switch" ; "top")^* ; alpha $
+
+- Temporal NetKAT for temporal properties https://www.cs.princeton.edu/~dpw/papers/temporal-netkat-pldi-2016.pdf