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+#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. +] \ No newline at end of file

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