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@@ -6,6 +6,7 @@
 
 #let tru = $"true"$
 #let fls = $"false"$
+#let star(x) = $#x^*$
 
 == Lecture 1
 
@@ -329,4 +330,164 @@ Subset of $K$: $(B, 0, 1, +, ;, overline((..)))$ is a boolean algebra. _Sub-alge
 
 This is only true of language without effects or concurrency.
 
-- Threads can be reasoned about in a partially distributive commutative lattice
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+- Threads can be reasoned about in a partially distributive commutative lattice
+
+#image("lec2.jpg")
+
+#pagebreak()
+
+== Lecture 3
+
+- _what is the semantics of $K A(T)$_?
+- Examples (such as how to tell if while loops are the same)
+- Net KAT (syntax and examples)
+
+Two types of semantics for Kleene algebra are the denotational semantics and the operational semantics.
+
+Having a conversion between both representations lets you pick the best of both worlds.
+
+- Denotational better for describing the gist of the language
+- Operational better for implementation
+
+=== KAT syntax
+
+- $e ::= 0 | 1 | p in P | e + e | e ; e | star(e) | b in B$
+  - b is read "assert b", a kind of test
+- $b ::= t in T | 0 | 1 | b or b | b and b | overline(b)$
+  - split alphabet $A$ into two parts, $T union.plus P$ where $T$ describes the basic tests
+  - this is embedded in the programs part
+
+After writing down assertion, everything after the assertion can be assumed to be true.
+
+If you have $alpha :equiv t_1 overline(t_2) t_3 ... overline(t_1) t_2 overline(t_3) ...$ including all basic tests $t$ and their complements this is considered a _full test_.
+
+For example, if $t_1 t_2 overline(t_1) overline(t_2)$, then there are 4 options for $alpha$:
+
+- $alpha_0 :equiv t_1 t_2$
+- $alpha_1 :equiv t_1 overline(t_2)$
+- $alpha_2 :equiv overline(t_1) t_2$
+- $alpha_3 :equiv overline(t_1) overline(t_2)$
+
+Any other expression is a subset of this full test, $2^T$. So the semantics of booleans is $db(b) equiv 2^(2^T)$
+
+Interleave programs between atoms, used to be $star(A)$, now $star((A t ; P)) ; A t$. There is an option of where to refine the $P$.  So now we have: 
+
+#rect[$db(e) : 2^star((A t ; P)) ; A t$]
+
+Also, $db(b) = {alpha | alpha <= b}$ (boolean satisfiability). For example, $t_1 t_2 <= t_1$ and $t_1 t_2 <= t_2$ but it doesn't $<= overline(t_2)$.
+
+Also $db(p) = {alpha p beta | alpha_1 beta in A t}$. With $p$ uninterpreted, we only know that $p$ can transform any $alpha$ into any $beta$
+
+- $db(e + f) equiv db(e) union db(f)$
+- $db(e \; f) equiv db(e) diamond.small db(f)$
+  - $alpha_0 p_0 alpha_k diamond.small beta_0 ...$ the $alpha_k diamond.small beta_0$ must match.
+    Otherwise the program cannot continue; this operation is undefined.
+    Then it deletes the repeated atoms so it is the same.
+    (Because $alpha$ and $beta$ are atoms, they are not expressions that can contain other atoms so it's based on exact matching)
+- Star is the same as before, but also using the diamond
+
+=== KAT Example
+
+For $ifthenelse(b,p,q)$ which is $b;p+overline(b);q$:
+
+$&db(ifthenelse(b,p,q)) \
+equiv &db(b\;p) union db(overline(b\;q)) \
+equiv & db(b) diamond.small db(p) union ... \
+equiv & {alpha | alpha <= b } diamond.small {alpha p beta | alpha beta in A t} union ... \
+equiv & {alpha p beta | alpha <= b} union {alpha q beta | alpha <= overline(b)}
+$
+
+The semantics of this expression has 2 types of traces.
+One where I start in b, where $p$ is executed.
+Another is where I started in a state where the condition $b$ is false, so I executed the $q$ branch.
+
+For $"while" b "do" p$ which is $star((b ; p)) ; overline(b)$
+
+$&db("while" b "do" p) \
+equiv & { beta , alpha_0 p beta , alpha_0 p alpha_1 p beta , ... | alpha_i <= b, beta <= overline(b)}$
+
+For traces that don't terminate: $"while true do skip" equiv 0$. (there are alternate semantics where you record infinite traces) With non-termination, your observation power is 0.
+
+=== Connection to Hoare triples
+
+For ${b} C {c}$, the validity of the Hoare triple is the same thing as the KAT equation $b C overline(c) equiv 0$ (filtering out all the postconditions that render the condition false would result in empty)
+
+Another one is $b C <= C c$. This is equivalent to the previous one #TODO Show this
+
+=== Exercise solutions
+
+$star(x + y) equiv star((star(x) y)) star(x)$
+
+Need to prove it in two steps, $<=$ and $>=$:
+
+- Use the fixpoint rule: $1 <= star((star(x) y)) star(x)$.
+  - This is true because $1 <= star(x)$ for any $x$
+- $x star(x) star((y star(x))) <= star(x) star((y star(x)))$.
+  - This uses the fact that $x star(x) <= star(x)$.
+    - This is true because unfold the star to get $x star(x) <= 1 + x star(x)$
+- $star((y star(x))) star((y star(x))) <= star((y star(x))) <= star(x) star((y star(x)))$
+  - $1 <= star(x)$, so add this to both sides
+
+- Taking the sum of the previous 3 things: $1 + (x+y) star(x) star((y star(x))) <= star(x) star((y star(x)))$
+  - let $cal(x) = star(x) star((y star(x)))$. THen $1 + (x + y) cal(x) <= cal(x)$
+  - Apply fix point rule here: $star((x + y)) <= star(x) star((y star(x)))$
+
+=== Summary
+
+- Soundness + Completeness (KA + BA) are all you need to prove that 2 programs are equivalent
+- Automaton model is where transitions are (atoms ; programs) and the final state is what atoms to validate. This is known as a KAT automaton.
+- Equivalece of these automata is decidable, using BDDs (Pous), matrices (Kozen) and these are all $in$ PSPACE.
+  - Some of these algorithms are very efficient despite PSPACE and they use the Union-find datastructure (Hopcroft, Tarjan)
+  - $O(n log_c n)$ where $c$ is the inverse of the ackermann function, so the log part is _very_ small (in fact the original conjecture was that it was linear)
+  - Improvement "Coinduction up-to" on NFA, DFA, Brz. derivatives, can improve the theoretical limit (?)
+
+=== While loop example
+
+#table(
+  columns: (1fr, 1fr),
+
+  ```
+  while a do
+    p;
+    while c do
+      q;
+  ```,
+  
+  ```
+  if b then
+    p;
+    while b or c do
+      if c then
+        q;
+      else
+        p;
+  ```,
+
+  $star((b p star((c q)) overline(c))) b$,
+
+  $b ; p ; star(((b + c) (c q + overline(c) p))) overline(b) overline(c) + overline(b)$
+)
+
+Show these are the same by applying denesting and sliding.
+
+=== Net KAT
+
+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.
+
+For example:
+
+#quote(block: true)[
+
+  Sw - switch \
+  Type - { SSH, ... } \
+  Pt - port \
+  Dst - destination
+
+  $("Sw" = 2); ("Type" = "SSH"); ("Pt" <- 4) ; ("Dst" <- "10.2.10.2")$
+
+  For any packet that comes in, it filters using the test. If they pass the test, then you can set the fields on them.
+
+  Using KAT, you can perform network reachability.
+]
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