UniTO/tesi/gabriel/part4.tex

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\newcommand{\EquivTEX}[3]{\mathsf{equiv}(#1, #2, #3)} % \equiv is already taken
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\begin{document}

\section{Translation validation of the Pattern Matching Compiler}
\label{sec:org3d79c1b}

\subsection{Source program}
\label{sec:orgc2c278d}
The algorithm takes as its input a source program and translates it
into an algebraic data structure which type we call \emph{decision\_tree}.

\begin{verbatim}
type decision_tree =
  | Unreachable
  | Failure
  | Leaf of source_expr
  | Guard of source_blackbox * decision_tree * decision_tree
  | Switch of accessor * (constructor * decision_tree) list * decision_tree
\end{verbatim}

Unreachable, Leaf of source\_expr and Failure are the terminals of the three.
We distinguish
\begin{itemize}
\item Unreachable: statically it is known that no value can go there
\item Failure: a value matching this part results in an error
\item Leaf: a value matching this part results into the evaluation of a
source black box of  code
\end{itemize}

The algorithm doesn't support type-declaration-based analysis 
to know the list of constructors at a given type.
Let's consider some trivial examples:

\begin{verbatim}
function true -> 1
\end{verbatim}

is translated to
\begin{center}
\begin{tabular}{l}
Switch ([(true, Leaf 1)], Failure)\\
\end{tabular}
\end{center}
while
\begin{verbatim}
function 
true -> 1 
| false -> 2
\end{verbatim}
will be translated to
\begin{center}
\begin{tabular}{l}
Switch ([(true, Leaf 1); (false, Leaf 2)])\\
\end{tabular}
\end{center}

It is possible to produce Unreachable examples by using
 refutation clauses (a "dot" in the right-hand-side)
\begin{verbatim}
function 
true -> 1 
| false -> 2 
| _ -> .
\end{verbatim}
that gets translated into
Switch ([(true, Leaf 1); (false, Leaf 2)], Unreachable)

We trust this annotation, which is reasonable as the  type-checker
verifies that it indeed holds.

Guard nodes of the tree are emitted whenever a guard is found. Guards
node contains a blackbox of  code that is never evaluated and two
branches, one that is taken in case the guard evaluates to true and
the other one that contains the path taken when the guard evaluates to
true.

\begin{comment}
[ ] Finisci con Switch
[ ] Spiega del fallback
[ ] rivedi compilazione per tenere in considerazione il tuo albero invece che le Lambda
\end{comment}

The source code of a pattern matching function has the
following form:

\begin{center}
\begin{tabular}{l}
match variable with\\
\(\vert{}\) pattern$_1$ \(\to\) expr$_1$\\
\(\vert{}\) pattern$_2$ when guard \(\to\) expr$_2$\\
\(\vert{}\) pattern$_3$ as var \(\to\) expr$_3$\\
$\vdots$\\
\(\vert{}\) p$_n$ \(\to\) expr$_n$\\
\end{tabular}
\end{center}

Patterns could or could not be exhaustive.

Pattern matching code could also be written using the more compact form:
\begin{center}
\begin{tabular}{l}
function\\
\(\vert{}\) pattern$_1$ \(\to\) expr$_1$\\
\(\vert{}\) pattern$_2$ when guard \(\to\) expr$_2$\\
\(\vert{}\) pattern$_3$ as var \(\to\) expr$_3$\\
$\vdots$\\
\(\vert{}\) p$_n$ \(\to\) expr$_n$\\
\end{tabular}
\end{center}


This BNF grammar describes formally the grammar of the source program:

\begin{center}
\begin{tabular}{l}
start ::= "match" id "with" patterns \(\vert{}\) "function" patterns\\
patterns ::= (pattern0\(\vert{}\)pattern1) pattern1+\\
;; pattern0 and pattern1 are needed to distinguish the first case in which\\
;; we can avoid writing the optional vertical line\\
pattern0 ::= clause\\
pattern1 ::= "\(\vert{}\)" clause\\
clause ::= lexpr "->" rexpr\\
lexpr ::= rule ($\varepsilon$\(\vert{}\)condition)\\
rexpr ::= \_code ;; arbitrary code\\
rule ::= wildcard\(\vert{}\)variable\(\vert{}\)constructor\_pattern\(\vert{}\)\{\}or\_pattern ;;\\
;; rules\\
wildcard ::= "\_"\\
variable ::= identifier\\
constructor\_pattern ::= constructor (rule\(\vert{}\)$\varepsilon$) (assignment\(\vert{}\)$\varepsilon$)\\
constructor ::= int\(\vert{}\)float\(\vert{}\)char\(\vert{}\)string\(\vert{}\)bool \(\vert{}\)unit\(\vert{}\)record\(\vert{}\)exn\(\vert{}\)objects\(\vert{}\)ref \(\vert{}\)list\(\vert{}\)tuple\(\vert{}\)array\(\vert{}\)variant\(\vert{}\)parameterized\_variant ;;  data types\\
or\_pattern ::=  rule ("\(\vert{}\)" wildcard\(\vert{}\)variable\(\vert{}\)constructor\_pattern)+\\
condition ::= "when" bexpr\\
assignment ::= "as" id\\
bexpr ::= \_code ;; arbitrary code\\
\end{tabular}
\end{center}

A source program t$_S$ is a collection of pattern clauses pointing to
\emph{bb} terms. Running a program t$_S$ against an input value v$_S$ produces as
a result \emph{r}:
\begin{center}
\begin{tabular}{l}
t$_S$ ::= (p $\to$ bb)\(^{\text{i$\in $I}}\)\\
p ::= \(\vert{}\) K(p$_i$)$^i$, i $\in $ I \(\vert{}\) (p\(\vert{}\)q) \(\vert{}\) n $\in $ $\mathbb{N}$\\
r ::= guard list * (Match bb \(\vert{}\) NoMatch \(\vert{}\) Absurd)\\
t$_S$(v$_S$) $\to$ r\\
\end{tabular}
\end{center}

TODO: argument on what it means to run a source program

\emph{guard} and \emph{bb} expressions are treated as blackboxes of OCaml code.
A sound approach for treating these blackboxes would be to inspect the
OCaml compiler during translation to Lambda code and extract the
blackboxes compiled in their Lambda representation.
This would allow to test for equality with the respective blackbox at
the target level.
Given that this level of introspection is currently not possibile, we
decided to restrict the structure of blackboxes to the following (valid) OCaml
code:

\begin{verbatim}
external guard : 'a -> 'b = "guard"
external observe : 'a -> 'b = "observe"
\end{verbatim}

We assume these two external functions \emph{guard} and \emph{observe} with a valid
type that lets the user pass any number of arguments to them.
All the guards are of the form \texttt{guard <arg> <arg> <arg>}, where the
<arg> are expressed using the OCaml pattern matching language.
Similarly, all the right-hand-side expressions are of the form
\texttt{observe <arg> <arg> ...} with the same constraints on arguments.

\begin{verbatim}
type t = K1 | K2 of t (* declaration of an algebraic and recursive datatype t *)

let _ = function
  | K1 -> observe 0
  | K2 K1 -> observe 1
  | K2 x when guard x -> observe 2
  | K2 (K2 x) as y when guard x y -> observe 3
  | K2 _ -> observe 4
\end{verbatim}



In our prototype we make use of accessors to encode stored values.
\begin{minipage}{0.2\linewidth}
\begin{verbatim}
let value =  1 :: 2 :: 3 :: []
(* that can also be written *)
let value = []
    |> List.cons 3 
    |> List.cons 2 
    |> List.cons 1
\end{verbatim}
\end{minipage}
\hfill
\begin{minipage}{0.5\linewidth}
\begin{verbatim}


   (field 0 x) = 1
   (field 0 (field 1 x)) = 2
   (field 0 (field 1 (field 1 x)) = 3
   (field 0 (field 1 (field 1 (field 1 x)) = []
\end{verbatim}
\end{minipage}
An \emph{accessor} \emph{a} represents the
access path to a value that can be reached by deconstructing the
scrutinee.
\begin{center}
\begin{tabular}{l}
a ::= Here \(\vert{}\) n.a\\
\end{tabular}
\end{center}
The above example, in encoded form:
\begin{verbatim}
Here = 1
1.Here = 2
1.1.Here = 3
1.1.1.Here = []
\end{verbatim}
In our prototype the source matrix m$_S$ is defined as follows
\begin{center}
\begin{tabular}{l}
SMatrix m$_S$ := (a$_j$)\(^{\text{j$\in $J}}\), ((p\(_{\text{ij}}\))\(^{\text{j$\in $J}}\) $\to$ bb$_i$)\(^{\text{i$\in $I}}\)\\
\end{tabular}
\end{center}

Source matrices are used to build source decision trees C$_S$.
A decision tree is defined as either a Leaf, a Failure terminal or
an intermediate node with different children sharing the same accessor \emph{a}
and an optional fallback.
Failure is emitted only when the patterns don't cover the whole set of
possible input values \emph{S}. The fallback is not needed when the user
doesn't use a wildcard pattern.
\%\%\% Give example of thing
\begin{center}
\begin{tabular}{l}
C$_S$ ::= Leaf bb \(\vert{}\) Switch(a, (K$_i$ $\to$ C$_i$)\(^{\text{i$\in $S}}\) , C?) \(\vert{}\) Failure \(\vert{}\) Unreachable\\
v$_S$ ::= K(v$_i$)\(^{\text{i$\in $I}}\) \(\vert{}\) n $\in $ $\mathbb{N}$\\
\end{tabular}
\end{center}
\begin{comment}
Are K and Here clear here?
\end{comment}
We say that a translation of a source program to a decision tree
is correct when for every possible input, the source program and its
respective decision tree produces the same result

\begin{center}
\begin{tabular}{l}
$\forall$v$_S$, t$_S$(v$_S$) = $\llbracket$t$_S$$\rrbracket$$_S$(v$_S$)\\
\end{tabular}
\end{center}

We define the decision tree of source programs
$\llbracket$t$_S$$\rrbracket$
in terms of the decision tree of pattern matrices
$\llbracket$m$_S$$\rrbracket$
by the following:
\begin{center}
\begin{tabular}{l}
$\llbracket$((p$_i$ $\to$ bb$_i$)\(^{\text{i$\in $I}}\)$\rrbracket$ := $\llbracket$(Here), (p$_i$ $\to$ bb$_i$)\(^{\text{i$\in $I}}\) $\rrbracket$\\
\end{tabular}
\end{center}
Decision tree computed from pattern matrices respect the following invariant:
\begin{center}
\begin{tabular}{l}
$\forall$v (v$_i$)\(^{\text{i$\in $I}}\) = v(a$_i$)\(^{\text{i$\in $I}}\) $\to$ $\llbracket$m$\rrbracket$(v) = m(v$_i$)\(^{\text{i$\in $I}}\) for m = ((a$_i$)\(^{\text{i$\in $I}}\), (r$_i$)\(^{\text{i$\in $I}}\))\\
v(Here) = v\\
K(v$_i$)$^i$(k.a) = v$_k$(a) if k $\in $ [0;n[\\
\end{tabular}
\end{center}
\begin{comment}
TODO: EXPLAIN
\end{comment}

We proceed to show the correctness of the invariant by a case
analysys.

Base cases:
\begin{enumerate}
\item\relax [| $\varnothing$, ($\varnothing$ $\to$ bb$_i$)$^i$ |] $\equiv$ Leaf bb$_i$ where i := min(I), that is a
decision tree [|ms|] defined by an empty accessor and empty
patterns pointing to blackboxes \emph{bb$_i$}. This respects the invariant
because a source matrix in the case of empty rows returns
the first expression and (Leaf bb)(v) := Match bb
\item\relax [| (a$_j$)$^j$, $\varnothing$ |] $\equiv$ Failure
\end{enumerate}

Regarding non base cases:
Let's first define
\begin{center}
\begin{tabular}{l}
let Idx(k) := [0; arity(k)[\\
let First($\varnothing$) := $\bot$\\
let First((a$_j$)$^j$) := a\(_{\text{min(j$\in $J$\ne$$\varnothing$)}}\)\\
\end{tabular}
\end{center}
\[
m := ((a_i)^i ((p_{ij})^i \to e_j)^{ij})
\]
\[
 (k_k)^k := headconstructor(p_{i0})^i
\]
\begin{equation}
 Groups(m) := ( k_k \to  ((a)_{0.l})^{l  \in  Idx(k_k)} +++ (a_i)^{i \in I\DZ}), \\
    ( if p_{0j} is k(q_l) then \\
        (q_l)^{l \in Idx(k_k)} +++ (p_{ij})^{i \in I\DZ} \to e_j \\
     if p_{0j} is \_ then \\
        (\_)^{l \in Idx(k_k)} +++ (p_{ij})^{i \in I\DZ}   \to e_j \\
     else \bot  )^j ), \\
  ((a_i)^{i \in I\DZ}, ((p_{ij})^{i \in I\DZ}  \to e_j if p_{0j} is \_ else \bot)^{j \in J}) 
\end{equation}

Groups(m) is an auxiliary function that source a matrix m into
submatrices, according to the head constructor of their first pattern.
Groups(m) returns one submatrix m\_r for each head constructor k that
occurs on the first row of m, plus one "wildcard submatrix" m\(_{\text{wild}}\)
that matches on all values that do not start with one of those head
constructors.

Intuitively, m is equivalent to its decomposition in the following
sense: if the first pattern of an input vector (v$_i$)$^i$ starts with one
of the head constructors k, then running (v$_i$)$^i$ against m is the same
as running it against the submatrix m$_k$; otherwise (its head
constructor is none of the k) it is equivalent to running it against
the wildcard submatrix.

We formalize this intuition as follows:
Lemma (Groups):
Let \[m\] be a matrix with \[Groups(m) = (k_r \to m_r)^k, m_{wild}\].
For any value vector \((v_i)^l\) such that \(v_0 = k(v'_l)^l\) for some
constructor k,
we have:
\begin{center}
\begin{tabular}{l}
if k = k$_k$ \text{ for some k then}\\
\quad    m(v$_i$)$^i$ = m$_k$((v\(_{\text{l}}\)')$^l$ \sout{+} (v\(_{\text{i}}\))\(^{\text{i$\in $I}\DZ}\))\\
\text{else}\\
\quad   m(v$_i$)$^i$ = m\(_{\text{wild}}\)(v$_i$)\(^{\text{i$\in $I}\DZ}\)\\
\end{tabular}
\end{center}

\begin{comment}
TODO: fix \{0}
\end{comment}

\begin{enumerate}
\item Proof:
\label{sec:org160a47b}
Let \(m\) be a matrix with \[Group(m) = (k_r \to m_r)^k, m_{wild}\]
Let \((v_i)^i\) be an input matrix with \(v_0 = k(v'_l)^l\) for some k.
We proceed by case analysis:
\begin{itemize}
\item either k is one of the k$_k$ for some k
\item or k is none of the (k$_k$)$^k$
\end{itemize}

Both m(v$_i$)$^i$ and m$_k$(v$_k$)$^k$ are defined as the first matching result of
a family over each row r$_j$ of a matrix 

We know, from the definition of
Groups(m), that m$_k$ is 
\begin{center}
\begin{tabular}{l}
((a)\{0.l\})\(^{\text{l$\in $Idx(k$_k$)}}\) \sout{+} (a$_i$)\(^{\text{i$\in $I}\DZ}\)),\\
(\\
\quad   if p\(_{\text{0j}}\) is k(q$_l$) then\\
\quad   \quad    (q$_l$)$^l$ \sout{+} (p\(_{\text{ij}}\))\(^{\text{i$\in $I}\DZ\ }\) $\to$ e$_j$\\
\quad   if p\(_{\text{0j}}\) is \_ then\\
\quad  \quad   (\_)$^l$ \sout{+} (p\(_{\text{ij}}\))\(^{\text{i$\in $I}\DZ}\)   $\to$ e$_j$\\
\quad    else $\bot$\\
)\(^{\text{j$\in $J}}\)\\
\end{tabular}
\end{center}

By definition, m(v$_i$)$^i$ is
\begin{center}
\begin{tabular}{l}
m(v$_i$)$^i$ = First(r$_j$(v$_i$)$^i$)$^j$ for m = ((a$_i$)$^i$, (r$_j$)$^j$)\\
(p$_i$)$^i$ (v$_i$)$^i$  =  \{\\
\quad if p$_0$ = k(q$_l$)$^l$, v$_0$ = k'(v'$_k$)$^k$, k=Idx(k') and l=Idx(k)\\
\quad \quad  if k $\ne$ k' then $\bot$\\
if k = k' then ((q$_l$)$^l$ \sout{+} (p$_i$)\(^{\text{i$\in $I}\DZ}\)) ((v'$_k$)$^k$ \sout{+} (v$_i$)\(^{\text{i$\in $I}\DZ}\) )\\
if p$_0$ = (q$_1$\(\vert{}\)q$_2$) then\\
First(  (q$_1$p$_i$\(^{\text{i$\in $I}\DZ}\)) v$_i$\(^{\text{i$\in $I}\DZ}\), (q$_2$p$_i$\(^{\text{i$\in $I }\DZ}\)) v$_i$\(^{\text{i$\in $I}\DZ}\))\}\\
\end{tabular}
\end{center}

For this reason, if we can prove that 
\begin{center}
\begin{tabular}{l}
$\forall$j,  r$_j$(v$_i$)$^i$ = r'$_j$((v'$_k$)$^k$ ++ (v$_i$)$^i$)\\
\end{tabular}
\end{center}
it follows that 
\begin{center}
\begin{tabular}{l}
m(v$_i$)$^i$ = m$_k$((v'$_k$)$^k$ ++ (v$_i$)$^i$)\\
\end{tabular}
\end{center}
from the above definition.

We can also show that a$_i$ = (a\(_{\text{0.l}}\))$^l$ \sout{+} a\(_{\text{i$\in $I}\DZ}\) because v(a$_0$) = K(v(a)\{0.l\})$^l$)
\end{enumerate}



\subsection{Target translation}
\label{sec:orgd2ea1d0}

The target program of the following general form is parsed using a parser
generated by Menhir, a LR(1) parser generator for the OCaml programming language.
Menhir compiles LR(1) a grammar specification, in this case a subset
of the Lambda intermediate language, down to OCaml code. 
This is the grammar of the target language (TODO: check menhir grammar)
\begin{center}
\begin{tabular}{l}
start ::= sexpr\\
sexpr ::= variable \(\vert{}\) string \(\vert{}\) "(" special\_form ")"\\
string ::= "$\backslash$"" identifier "$\backslash$"" ;; string between doublequotes\\
variable ::= identifier\\
special\_form ::=  let\(\vert{}\)catch\(\vert{}\)if\(\vert{}\)switch\(\vert{}\)switch-star\(\vert{}\)field\(\vert{}\)apply\(\vert{}\)isout\\
let ::= "let" assignment sexpr ;; (assignment sexpr)+ outside of pattern match code\\
assignment ::= "function" variable variable+ ;; the first variable is the identifier of the function\\
field ::= "field" digit variable\\
apply ::= ocaml\_lambda\_code ;; arbitrary code\\
catch ::= "catch" sexpr with sexpr\\
with ::= "with" "(" label ")"\\
exit ::= "exit" label\\
switch-star ::= "switch*" variable case*\\
switch::= "switch" variable case* "default:" sexpr\\
case ::= "case" casevar ":" sexpr\\
casevar ::= ("tag"\(\vert{}\)"int") integer\\
if ::= "if" bexpr sexpr sexpr\\
bexpr ::= "("  ("!="\(\vert{}\)"\texttt{="\textbackslash{}vert\{\}">}"\(\vert{}\)"<="\(\vert{}\)">"\(\vert{}\)"<") sexpr digit \(\vert{}\) field ")"\\
label ::= integer\\
\end{tabular}
\end{center}
The prototype doesn't support strings.

The AST built by the parser is traversed and evaluated by the symbolic
execution engine.
Given that the target language supports jumps in the form of "catch - exit"
blocks the engine tries to evaluate the instructions inside the blocks
and stores the result of the partial evaluation into a record.
When a jump is encountered, the information at the point allows to
finalize the evaluation of the jump block.
In the environment the engine also stores bindings to values and
functions.
Integer additions and subtractions are simplified in a second pass.
The result of the symbolic evaluation is a target decision tree C$_T$
\begin{center}
\begin{tabular}{l}
C$_T$ ::= Leaf bb \(\vert{}\) Switch(a, ($\pi$$_i$ $\to$ C$_i$)\(^{\text{i$\in $S}}\) , C?) \(\vert{}\) Failure\\
v$_T$ ::= Cell(tag $\in $ $\mathbb{N}$, (v$_i$)\(^{\text{i$\in $I}}\)) \(\vert{}\) n $\in $ $\mathbb{N}$\\
\end{tabular}
\end{center}
Every branch of the decision tree is "constrained" by a domain
\begin{center}
\begin{tabular}{l}
Domain $\pi$ = \{ n\(\vert{}\)n$\in $$\mathbb{N}$ x n\(\vert{}\)n$\in $Tag$\subseteq$$\mathbb{N}$ \}\\
\end{tabular}
\end{center}
Intuitively, the $\pi$ condition at every branch tells us the set of
possible values that can "flow" through that path.
$\pi$ conditions are refined by the engine during the evaluation; at the
root of the decision tree the domain corresponds to the set of
possible values that the type of the function can hold.
C? is the fallback node of the tree that is taken whenever the value
at that point of the execution can't flow to any other subbranch.
Intuitively, the $\pi$\(_{\text{fallback}}\) condition of the C? fallback node is
\begin{center}
\begin{tabular}{l}
$\pi$\(_{\text{fallback}}\) = ¬\bigcup\limits\(_{\text{i$\in $I}}\)$\pi$$_i$\\
\end{tabular}
\end{center}
The fallback node can be omitted in the case where the domain of the
children nodes correspond to set of possible values pointed by the
accessor at that point of the execution
\begin{center}
\begin{tabular}{l}
domain(v$_S$(a)) = \bigcup\limits\(_{\text{i$\in $I}}\)$\pi$$_i$\\
\end{tabular}
\end{center}
We say that a translation of a target program to a decision tree
is correct when for every possible input, the target program and its
respective decision tree produces the same result
\begin{center}
\begin{tabular}{l}
$\forall$v$_T$, t$_T$(v$_T$) = $\llbracket$t$_T$$\rrbracket$$_T$(v$_T$)\\
\end{tabular}
\end{center}



\subsection{Equivalence checking}
\label{sec:org33b978b}

The equivalence checking algorithm takes as input a domain of
possible values \emph{S} and a pair of source and target decision trees and
in case the two trees are not equivalent it returns a counter example.
The algorithm respects the following correctness statement:

\begin{comment}
TODO: we have to define what \coversTEX mean for readers to understand the specifications
(or we use a simplifying lie by hiding \coversTEX in the statements).
\end{comment}

\begin{align*}
 \EquivTEX S {C_S} {C_T} \emptyqueue = \YesTEX \;\land\; \coversTEX {C_T} S
 & \implies
 \forall v_S \approx v_T \in S,\; C_S(v_S) = C_T(v_T)
\\
 \EquivTEX S {C_S} {C_T} \emptyqueue = \NoTEX {v_S} {v_T} \;\land\; \coversTEX {C_T} S
 & \implies
 v_S \approx v_T \in S \;\land\; C_S(v_S) \neq C_T(v_T)
\end{align*}
Our equivalence-checking algorithm \(\EquivTEX S {C_S} {C_T} G\) is
a exactly decision procedure for the provability of the judgment
\((\EquivTEX  S {C_S} {C_T} G)\), defined below.
\begin{mathpar}
  \begin{array}{l@{~}r@{~}l}
    & & \text{\emph{constraint trees}} \\
    C & \bnfeq & \Leaf t \\
    & \bnfor & \Failure \\
    & \bnfor & \Switch a {\Fam i {\pi_i, C_i}} \Cfb \\
    & \bnfor & \Guard t {C_0} {C_1} \\
  \end{array}

  \begin{array}{l@{~}r@{~}l}
    & & \text{\emph{boolean result}} \\
    b & \in & \{ 0, 1 \} \\[0.5em]
    & & \text{\emph{guard queues}} \\
    G & \bnfeq & (t_1 = b_1), \dots, (t_n = b_n) \\
  \end{array}

  \begin{array}{l@{~}r@{~}l}
    & & \text{\emph{input space}} \\
    S & \subseteq & \{ (v_S, v_T) \mid \vrel {v_S} {v_T} \} \\
  \end{array}
  \\
  \infer{ }
  {\EquivTEX \emptyset {C_S} {C_T} G}

  \infer{ }
  {\EquivTEX S \Failure \Failure \emptyqueue}

  \infer
  {\trel {t_S} {t_T}}
  {\EquivTEX S {\Leaf {t_S}} {\Leaf {t_T}} \emptyqueue}

  \infer
  {\forall i,\;
    \EquivTEX
      {(S \land a = K_i)}
      {C_i} {\trim {C_T} {a = K_i}} G
   \\
   \EquivTEX
     {(S \land a \notin \Fam i {K_i})}
     \Cfb {\trim {C_T} {a \notin \Fam i {K_i}}} G
  }
  {\EquivTEX S
    {\Switch a {\Fam i {K_i, C_i}} \Cfb} {C_T} G}

\begin{comment}
  % TODO explain somewhere why the judgment is not symmetric:
  % we avoid defining trimming on source trees, which would
  % require more expressive conditions than just constructors
\end{comment}
  \infer
  {C_S \in {\Leaf t, \Failure}
   \\
   \forall i,\; \EquivTEX  {(S \land a \in D_i)} {C_S} {C_i} G
   \\
   \EquivTEX {(S \land a \notin \Fam i {D_i})} {C_S} \Cfb G}
  {\EquivTEX S
    {C_S} {\Switch a {\Fam i {D_i} {C_i}} \Cfb} G}

  \infer
  {\EquivTEX S {C_0} {C_T} {G, (t_S = 0)}
   \\
   \EquivTEX S {C_1} {C_T} {G, (t_S = 1)}}
  {\EquivTEX S
    {\Guard {t_S} {C_0} {C_1}} {C_T} G}

  \infer
  {\trel {t_S} {t_T}
   \\
   \EquivTEX S {C_S} {C_b} G}
  {\EquivTEX S
    {C_S} {\Guard {t_T} {C_0} {C_1}} {(t_S = b), G}}
\end{mathpar}
Running a program t$_S$ or its translation $\llbracket$t$_S$$\rrbracket$ against an input v$_S$
produces as a result \emph{r} in the following way:
\begin{center}
\begin{tabular}{l}
( $\llbracket$t$_S$$\rrbracket$$_S$(v$_S$) $\equiv$ C$_S$(v$_S$) ) $\to$ r\\
t$_S$(v$_S$) $\to$ r\\
\end{tabular}
\end{center}
Likewise
\begin{center}
\begin{tabular}{l}
( $\llbracket$t$_T$$\rrbracket$$_T$(v$_T$) $\equiv$ C$_T$(v$_T$) ) $\to$ r\\
t$_T$(v$_T$) $\to$ r\\
result r ::= guard list * (Match blackbox \(\vert{}\) NoMatch \(\vert{}\) Absurd)\\
guard ::= blackbox.\\
\end{tabular}
\end{center}
Having defined equivalence between two inputs of which one is
expressed in the source language and the other in the target language,
v$_S$ $\simeq$ v$_T$, we can define the equivalence between a couple of programs or
a couple of decision trees 
\begin{center}
\begin{tabular}{l}
t$_S$ $\simeq$ t$_T$ := $\forall$v$_S$$\simeq$v$_T$, t$_S$(v$_S$) = t$_T$(v$_T$)\\
C$_S$ $\simeq$ C$_T$ := $\forall$v$_S$$\simeq$v$_T$, C$_S$(v$_S$) = C$_T$(v$_T$)\\
\end{tabular}
\end{center}
The result of the proposed equivalence algorithm is \emph{Yes} or \emph{No(v$_S$,
v$_T$)}. In particular, in the negative case, v$_S$ and v$_T$ are a couple of
possible counter examples for which the decision trees produce a
different result.

In the presence of guards we can say that two results are
equivalent modulo the guards queue, written \emph{r$_1$ $\simeq$gs r$_2$}, when:
\begin{center}
\begin{tabular}{l}
(gs$_1$, r$_1$) $\simeq$gs (gs$_2$, r$_2$)  $\Leftrightarrow$  (gs$_1$, r$_1$) = (gs$_2$ ++ gs, r$_2$)\\
\end{tabular}
\end{center}
We say that C$_T$ covers the input space \emph{S}, written
\emph{covers(C$_T$, S)} when every value v$_S$$\in $S is a valid input to the
decision tree C$_T$. (TODO: rephrase)
Given an input space \emph{S} and a couple of decision trees, where
the target decision tree C$_T$ covers the input space \emph{S}, we say that
the two decision trees are equivalent when:
\begin{center}
\begin{tabular}{l}
equiv(S, C$_S$, C$_T$, gs) = Yes $\wedge$ covers(C$_T$, S) $\to$ $\forall$v$_S$$\simeq$v$_T$ $\in $ S, C$_S$(v$_S$) $\simeq$gs C$_T$(v$_T$)\\
\end{tabular}
\end{center}
Similarly we say that a couple of decision trees in the presence of
an input space \emph{S} are \emph{not} equivalent when:
\begin{center}
\begin{tabular}{l}
equiv(S, C$_S$, C$_T$, gs) = No(v$_S$,v$_T$) $\wedge$ covers(C$_T$, S) $\to$ v$_S$$\simeq$v$_T$ $\in $ S $\wedge$ C$_S$(v$_S$) $\ne$gs C$_T$(v$_T$)\\
\end{tabular}
\end{center}
Corollary: For a full input space \emph{S}, that is the universe of the
target program we say:
\begin{center}
\begin{tabular}{l}
equiv(S, $\llbracket$t$_S$$\rrbracket$$_S$, $\llbracket$t$_T$$\rrbracket$$_T$, $\varnothing$) = Yes  $\Leftrightarrow$  t$_S$ $\simeq$ t$_T$\\
\end{tabular}
\end{center}


\begin{comment}
TODO: put ^i$\in $I where needed
\end{comment}
\subsubsection{The trimming lemma}
The trimming lemma allows to reduce the size of a decision tree given
an accessor \emph{a} $\to$ $\pi$ relation (TODO: expand)
\begin{center}
\begin{tabular}{l}
$\forall$v$_T$ $\in $ (a$\to$$\pi$), C$_T$(v$_T$) = C\(_{\text{t/a$\to$$\pi$}}\)(v$_T$)\\
\end{tabular}
\end{center}
We prove this by induction on C$_T$:

\begin{itemize}
\item C$_T$ = Leaf\(_{\text{bb}}\): when the decision tree is a leaf terminal, the result of trimming on a Leaf is the Leaf itself
\begin{center}
\begin{tabular}{l}
Leaf\(_{\text{bb/a$\to$$\pi$}}\)(v) = Leaf\(_{\text{bb}}\)(v)\\
\end{tabular}
\end{center}
\item The same applies to Failure terminal
\begin{center}
\begin{tabular}{l}
Failure\(_{\text{/a$\to$$\pi$}}\)(v) = Failure(v)\\
\end{tabular}
\end{center}
\item When C$_T$ = Switch(b, ($\pi$$\to$C$_i$)$^i$)\(_{\text{/a$\to$$\pi$}}\) then we look at the accessor
\emph{a} of the subtree C$_i$ and we define $\pi$$_i$' = $\pi$$_i$ if a$\ne$b else $\pi$$_i$$\cap$$\pi$ Trimming
a switch node yields the following result:
\begin{center}
\begin{tabular}{l}
Switch(b, ($\pi$$\to$C$_i$)\(^{\text{i$\in $I}}\))\(_{\text{/a$\to$$\pi$}}\) :=  Switch(b, ($\pi$'$_i$$\to$C\(_{\text{i/a$\to$$\pi$}}\))\(^{\text{i$\in $I}}\))\\
\end{tabular}
\end{center}
\end{itemize}
\begin{comment}
TODO: understand how to properly align lists
check that every list is aligned
\end{comment}
For the trimming lemma we have to prove that running the value v$_T$ against
the decision tree C$_T$ is the same as running v$_T$ against the tree
C\(_{\text{trim}}\) that is the result of the trimming operation on C$_T$
\begin{center}
\begin{tabular}{l}
C$_T$(v$_T$) = C\(_{\text{trim}}\)(v$_T$) = Switch(b, ($\pi$$_i$'$\to$C\(_{\text{i/a$\to$$\pi$}}\))\(^{\text{i$\in $I}}\))(v$_T$)\\
\end{tabular}
\end{center}
We can reason by first noting that when v$_T$$\notin$(b$\to$$\pi$$_i$)$^i$ the node must be a Failure node.
In the case where $\exists$k \(\vert{}\) v$_T$$\in $(b$\to$$\pi$$_k$) then we can prove that
\begin{center}
\begin{tabular}{l}
C\(_{\text{k/a$\to$$\pi$}}\)(v$_T$) = Switch(b, ($\pi$$_i$'$\to$C\(_{\text{i/a$\to$$\pi$}}\))\(^{\text{i$\in $I}}\))(v$_T$)\\
\end{tabular}
\end{center}
because when a $\ne$ b then $\pi$$_k$'= $\pi$$_k$ and this means that v$_T$$\in $$\pi$$_k$'
while when a = b  then $\pi$$_k$'=($\pi$$_k$$\cap$$\pi$) and v$_T$$\in $$\pi$$_k$' because:
\begin{itemize}
\item by the hypothesis, v$_T$$\in $$\pi$
\item we are in the case where v$_T$$\in $$\pi$$_k$
\end{itemize}
So v$_T$ $\in $ $\pi$$_k$' and by induction
\begin{center}
\begin{tabular}{l}
C$_k$(v$_T$) = C\(_{\text{k/a$\to$$\pi$}}\)(v$_T$)\\
\end{tabular}
\end{center}
We also know that $\forall$v$_T$$\in $(b$\to$$\pi$$_k$) $\to$ C$_T$(v$_T$) = C$_k$(v$_T$) 
By putting together the last two steps, we have proven the trimming
lemma.

\begin{comment}
TODO: what should I say about covering??? I swap $\pi$ and $\pi$' 
Covering lemma:
        $\forall$a,$\pi$ covers(C$_T$,S) $\to$ covers(C_{t/a$\to$$\pi$}, (S$\cap$a$\to$$\pi$))
            U$_i$$\pi$$^i$ $\approx$ U$_i$$\pi$'$\cap$(a$\to$$\pi$) $\approx$ (U$_i$$\pi$')$\cap$(a$\to$$\pi$) %%


%%%%%%% Also: Should I swap $\pi$ and $\pi$'  ?
\end{comment}

\subsubsection{Equivalence checking}
The equivalence checking algorithm takes as parameters an input space
\emph{S}, a source decision tree \emph{C$_S$} and a target decision tree \emph{C$_T$}:
\begin{center}
\begin{tabular}{l}
equiv(S, C$_S$, C$_T$) $\to$ Yes \(\vert{}\) No(v$_S$, v$_T$)\\
\end{tabular}
\end{center}

When the algorithm returns Yes and the input space is covered by \emph{C$_S$}
we can say that the couple of decision trees are the same for
every couple of source value \emph{v$_S$} and target value \emph{v$_T$} that are equivalent.
\begin{comment}
Define "covered"
Is it correct to say the same? How to correctly distinguish in words $\simeq$ and = ?
\end{comment}
\begin{center}
\begin{tabular}{l}
equiv(S, C$_S$, C$_T$) = Yes and cover(C$_T$, S) $\to$ $\forall$ v$_S$ $\simeq$ v$_T$$\in $S $\wedge$ C$_S$(v$_S$) = C$_T$(v$_T$)\\
\end{tabular}
\end{center}
In the case where the algorithm returns No we have at least a couple
of counter example values v$_S$ and v$_T$ for which the two decision trees
outputs a different result.
\begin{center}
\begin{tabular}{l}
equiv(S, C$_S$, C$_T$) = No(v$_S$,v$_T$) and cover(C$_T$, S) $\to$ $\forall$ v$_S$ $\simeq$ v$_T$$\in $S $\wedge$ C$_S$(v$_S$) $\ne$ C$_T$(v$_T$)\\
\end{tabular}
\end{center}
We define  the following
\begin{center}
\begin{tabular}{l}
Forall(Yes) = Yes\\
Forall(Yes::l) = Forall(l)\\
Forall(No(v$_S$,v$_T$)::\_) = No(v$_S$,v$_T$)\\
\end{tabular}
\end{center}
There exists and are injective:
\begin{center}
\begin{tabular}{l}
int(k) $\in $ $\mathbb{N}$ (arity(k) = 0)\\
tag(k) $\in $ $\mathbb{N}$ (arity(k) > 0)\\
$\pi$(k) = \{n\(\vert{}\) int(k) = n\} x \{n\(\vert{}\) tag(k) = n\}\\
\end{tabular}
\end{center}
where k is a constructor.

\begin{comment}
TODO: explain:
$\forall$v$\in $a$\to$$\pi$, C_{/a$\to$$\pi$}(v) = C(v)
\end{comment}

We proceed by case analysis:
\begin{comment}
I start numbering from zero to leave the numbers as they were on the blackboard, were we skipped some things
I think the unreachable case should go at the end.
\end{comment}
\begin{enumerate}
\item in case of unreachable:
\end{enumerate}
\begin{center}
\begin{tabular}{l}
C$_S$(v$_S$) = Absurd(Unreachable) $\ne$ C$_T$(v$_T$) $\forall$v$_S$,v$_T$\\
\end{tabular}
\end{center}
\begin{enumerate}
\item In the case of an empty input space
\begin{center}
\begin{tabular}{l}
equiv($\varnothing$, C$_S$, C$_T$) := Yes\\
\end{tabular}
\end{center}
and that is trivial to prove because there is no pair of values (v$_S$, v$_T$) that could be
tested against the decision trees.
In the other subcases S is always non-empty.
\item When there are \emph{Failure} nodes at both sides the result is \emph{Yes}:
\begin{center}
\begin{tabular}{l}
equiv(S, Failure, Failure) := Yes\\
\end{tabular}
\end{center}
Given that $\forall$v, Failure(v) = Failure, the statement holds.
\item When we have a Leaf or a Failure at the left side:
\begin{center}
\begin{tabular}{l}
equiv(S, Failure as C$_S$, Switch(a, ($\pi$$_i$ $\to$ C$_T$$_i$)\(^{\text{i$\in $I}}\))) :=  Forall(equiv( S$\cap$a$\to$$\pi$(k$_i$)), C$_S$, C$_T$$_i$)\(^{\text{i$\in $I}}\))\\
equiv(S, Leaf bb$_S$ as C$_S$, Switch(a, ($\pi$$_i$ $\to$ C$_T$$_i$)\(^{\text{i$\in $I}}\))) :=  Forall(equiv( S$\cap$a$\to$$\pi$(k$_i$)), C$_S$, C$_T$$_i$)\(^{\text{i$\in $I}}\))\\
\end{tabular}
\end{center}
The algorithm either returns Yes for every sub-input space S$_i$ := S$\cap$(a$\to$$\pi$(k$_i$)) and
subtree C$_T$$_i$
\begin{center}
\begin{tabular}{l}
equiv(S$_i$, C$_S$, C$_T$$_i$) = Yes $\forall$i\\
\end{tabular}
\end{center}
or we have a counter example v$_S$, v$_T$ for which
\begin{center}
\begin{tabular}{l}
v$_S$$\simeq$v$_T$$\in $S$_k$ $\wedge$ c$_S$(v$_S$) $\ne$ C$_T$$_k$(v$_T$)\\
\end{tabular}
\end{center}
then because 
\begin{center}
\begin{tabular}{l}
v$_T$$\in $(a$\to$$\pi$$_k$) $\to$ C$_T$(v$_T$) = C$_T$$_k$(v$_T$) ,\\
v$_S$$\simeq$v$_T$$\in $S $\wedge$ C$_S$(v$_S$)$\ne$C$_T$(v$_T$)\\
\end{tabular}
\end{center}
we can say that
\begin{center}
\begin{tabular}{l}
equiv(S$_i$, C$_S$, C$_T$$_i$) = No(v$_S$, v$_T$) for some minimal k$\in $I\\
\end{tabular}
\end{center}
\item When we have a Switch on the right we define $\pi$\(_{\text{fallback}}\) as the domain of
values not covered but the union of the constructors k$_i$
\begin{center}
\begin{tabular}{l}
$\pi$\(_{\text{fallback}}\) = ¬\bigcup\limits\(_{\text{i$\in $I}}\)$\pi$(k$_i$)\\
\end{tabular}
\end{center}
The algorithm proceeds by trimming
\begin{center}
\begin{tabular}{l}
equiv(S, Switch(a, (k$_i$ $\to$ C$_S$$_i$)\(^{\text{i$\in $I}}\), C\(_{\text{sf}}\)), C$_T$) :=\\
Forall(equiv( S$\cap$(a$\to$$\pi$(k$_i$)\(^{\text{i$\in $I}}\)), C$_S$$_i$, C\(_{\text{t/a$\to$$\pi$(k$_i$)}}\))\(^{\text{i$\in $I}}\) \sout{+}   equiv(S$\cap$(a$\to$$\pi$$_n$), C$_S$, C\(_{\text{a$\to$$\pi$}_{\text{fallback}}}\)))\\
\end{tabular}
\end{center}
 The statement still holds and we show this by first analyzing the
\emph{Yes} case:
\begin{center}
\begin{tabular}{l}
Forall(equiv( S$\cap$(a$\to$$\pi$(k$_i$)\(^{\text{i$\in $I}}\)), C$_S$$_i$, C\(_{\text{t/a$\to$$\pi$(k$_i$)}}\))\(^{\text{i$\in $I}}\) = Yes\\
\end{tabular}
\end{center}
The constructor k is either included in the set of constructors k$_i$:
\begin{center}
\begin{tabular}{l}
k \(\vert{}\) k$\in $(k$_i$)$^i$ $\wedge$ C$_S$(v$_S$) = C$_S$$_i$(v$_S$)\\
\end{tabular}
\end{center}
We also know that 
\begin{center}
\begin{tabular}{l}
(1) C$_S$$_i$(v$_S$) = C\(_{\text{t/a$\to$$\pi$$_i$}}\)(v$_T$)\\
(2) C\(_{\text{T/a$\to$$\pi$$_i$}}\)(v$_T$) = C$_T$(v$_T$)\\
\end{tabular}
\end{center}
(1) is true by induction and (2) is a consequence of the trimming lemma.
Putting everything together:
\begin{center}
\begin{tabular}{l}
C$_S$(v$_S$) = C$_S$$_i$(v$_S$) = C\(_{\text{T/a$\to$$\pi$$_i$}}\)(v$_T$) = C$_T$(v$_T$)\\
\end{tabular}
\end{center}

When the k$\notin$(k$_i$)$^i$ [TODO]

The auxiliary Forall function returns \emph{No(v$_S$, v$_T$)} when, for a minimum  k,
\begin{center}
\begin{tabular}{l}
equiv(S$_k$, C$_S$$_k$, C\(_{\text{T/a$\to$$\pi$$_k$}}\) = No(v$_S$, v$_T$)\\
\end{tabular}
\end{center}
Then we can say that 
\begin{center}
\begin{tabular}{l}
C$_S$$_k$(v$_S$) $\ne$ C\(_{\text{t/a$\to$$\pi$$_k$}}\)(v$_T$)\\
\end{tabular}
\end{center}
that is enough for proving that 
\begin{center}
\begin{tabular}{l}
C$_S$$_k$(v$_S$) $\ne$ (C\(_{\text{t/a$\to$$\pi$$_k$}}\)(v$_T$) = C$_T$(v$_T$))\\
\end{tabular}
\end{center}
\end{enumerate}
\end{document}