UniTO/tesi/tesi_unicode.org

\begin{comment}
TODO: not all todos are explicit. Check every comment section
TODO: chiedi a Gabriel se T e S vanno bene, ma prima controlla che siano coerenti
* TODO Scaletta [1/5]
  - [X] Introduction
  - [-] Background [80%]
    - [X] Low level representation
    - [X] Lambda code [0%]
    - [X] Pattern matching 
    - [X] Symbolic execution
    - [ ] Translation Validation
  - [-] Translation validation of the Pattern Matching Compiler
    - [X] Source translation
      - [X] Formal Grammar
      - [X] Compilation of source patterns
    - [ ] Target translation
      - [ ] Formal Grammar
      - [ ] Symbolic execution of the Lambda target
    - [X] Equivalence between source and target
  - [ ] Practical results 

  
\end{comment}

#+TITLE:   Translation Verification of the OCaml pattern matching compiler
#+AUTHOR:   Francesco Mecca
#+EMAIL:    me@francescomecca.eu
#+DATE:
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#+LANGUAGE: en
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#+LaTeX_HEADER: \newcommand{\semTEX}[1]{{\llbracket{#1}\rrbracket}}
#+LaTeX_HEADER: \newcommand{\EquivTEX}[3]{\mathsf{equiv}(#1, #2, #3)} % \equiv is already taken
#+LaTeX_HEADER: \newcommand{\coversTEX}[2]{#1 \mathrel{\mathsf{covers}} #2}
#+LaTeX_HEADER: \newcommand{\YesTEX}{\mathsf{Yes}}
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\section{Introduction}

This dissertation presents an algorithm for the translation validation of the OCaml pattern
matching compiler. Given a source program and its compiled version the
algorithm checks whether the two are equivalent or produce a counter
example in case of a mismatch.
For the prototype of this algorithm we have chosen a subset of the OCaml
language and implemented a prototype equivalence checker along with a
formal statement of correctness and its proof.
The prototype is to be included in the OCaml compiler infrastructure
and will aid the development.  
\begin{comment}
TODO: should this be removed?
Our equivalence algorithm works with decision trees. Source patterns are
converted into a decision tree using a matrix decomposition algorithm.
Target programs, described in the Lambda intermediate
representation language of the OCaml compiler, are turned into decision trees
by applying symbolic execution.
\end{comment}

\subsection{Motivation}
Pattern matching in computer science is a
widely employed technique for describing computation as well as
deduction. Pattern matching is central in many programming languages
such as the ML family languages, Haskell and Scala, different model
checkers, such as Murphi, and proof assistants such as Coq and
Isabelle. Recently the C++ community is considering [CIT] adding the
support for pattern matching in the compiler. The work done in this
thesis provides a general method that is agnostic with respect to the
compiler implementation and the language used. 

The work focused on the OCaml pattern matching compiler that is a
critical part of the OCaml compiler in terms of correctness because
bugs typically would result in wrong code production rather than
triggering compilation failures. Such bugs also are hard to catch by
testing because they arise in corner cases of complex patterns which
are typically not in the compiler test suite or most user programs.
\\
The OCaml core developers group considered evolving the pattern matching compiler, either by
using a new algorithm or by incremental refactoring of the current code base.
For this reason we want to verify that future implementations of the
compiler avoid the introduction of new bugs and that such
modifications don't result in a different behavior than the current one. 
\\
One possible approach is to formally verify the pattern matching compiler
implementation using a machine checked proof.
Another possibility, albeit with a weaker result, is to verify that
each source program and target program pair are semantically correct.
We chose the latter technique, translation validation because is easier to adopt in
the case of a production compiler and to integrate with an existing code base. The compiler is treated as a
black-box and proof only depends on our equivalence algorithm.
\\

\subsection{The Pattern Matching Compiler}
A pattern matching compiler turns a series of pattern matching clauses
into simple control flow structures such as \texttt{if, switch}.
For example:
\begin{lstlisting}
  match scrutinee with
  | [] -> (0, None)
  | x::[] -> (1, Some x)
  | _::y::_ -> (2, Some y)
\end{lstlisting}
Given as input to the pattern matching compiler, this snippet of code
gets translated into the Lambda intermediate representation of
the OCaml compiler. The Lambda representation of a program is shown by
calling the \texttt{ocamlc} compiler with the \texttt{-drawlambda} flag.
In this example we renamed the variables assigned in order to ease the
understanding of the tests that are performed when the code is
translated into the Lambda form.
\begin{lstlisting}
(function scrutinee
    (if scrutinee    ;;; true when scrutinee (list) not empty
    (let (tail =a (field 1 scrutinee/81))    ;;; assignment
        (if tail
        (let
            y =a (field 0 tail)) 
            ;;; y is the first element of the tail
            (makeblock 0 2 (makeblock 0 y)))
            ;;; allocate memory for tuple (2, Some y)
        (let (x =a (field 0 scrutinee))      
            ;;; x is the head of the scrutinee
            (makeblock 0 1 (makeblock 0 x))))) 
            ;;; allocate memory for tuple (1, Some x)
    [0: 0 0a])))  ;;; low level representatio of (0, None)
\end{lstlisting}

\subsection{Our approach}
Our algorithm translates both source and target programs into a common
representation that we call \texttt{decision trees}. Decision trees where chosen because
they model the space of possible values at a given branch of
execution. \\
Here are the decision trees for the source and target example program. \\

\begin{minipage}{0.5\linewidth}
  \scriptsize
\begin{verbatim}
       Switch(Root)
       /        \
     (= [])    (= ::)
     /             \
   Leaf         Switch(Root.1)
(0, None)       /         \
             (= [])      (= ::)
             /               \
          Leaf              Leaf
   [x = Root.0]         [y = Root.1.0]
   (1, Some x)          (2, Some y)
\end{verbatim}
\end{minipage}
\hfill
\begin{minipage}{0.4\linewidth}
  \scriptsize
\begin{verbatim}
       Switch(Root)
       /        \
     (= int 0)  (!= int 0)
     /             \
   Leaf         Switch(Root.1)
(mkblock 0     /       \
  0 0a)         /         \
             (= int 0)    (!= int 0)
             /               \
          Leaf              Leaf
[x = Root.0]            [y = Root.1.0]
(mkblock 0            (mkblock 0
  1 (mkblock 0 x))      2 (mkblock 0 y))
\end{verbatim}
\end{minipage}
\\
\texttt{(Root.0)} is called an \emph{accessor}, that represents the
access path to a value that can be reached by deconstructing the
scrutinee. In this example \texttt{Root.0} is the first subvalue of
the scrutinee.
\\
Target decision trees have a similar shape but the tests on the
branches are related to the low level representation of values in
Lambda code. For example, cons blocks \texttt{x::xs} or tuples
\texttt{(x,y)} are memory blocks with tag 0.
\\
The following parametrized grammar $D(\pi, e)$ describes the common structure of
source and decision trees. We denote as $\pi$ the \emph{conditions} on
each branch, and $a$ for our \emph{accessors}, which give a symbolic
description of a sub-value of the scrutinee. \\
Source conditions $\pi_S$
are just datatype constructors; target conditions $\pi_T$ are
arbitrary sets of low level OCaml values.
Expressions \texttt{$e_S$} and \texttt{$e_T$} are arbitrary OCaml
expressions that are lowered by the compiler into lambda expressions.
\begin{comment}
 possible immediate-integer or block-tag values.
\end{comment}
\begin{mathpar}
  \begin{array}{l@{~}r@{~}r@{~}l}
    \text{\emph{decision trees}} & D(\pi, e)
    & \bnfeq & \Leaf {\cle(a)} \\
    & & \bnfor & \Failure \\
    & & \bnfor & \Switch a {\Fam {i \in I} {\pi_i, D_i}} \Dfb \\
    & & \bnfor & \Guard {\cle(a)} {D_0} {D_1} \\
    & & \bnfor & Unreachable 
  \end{array}
  \quad
  \begin{array}{l@{~}r@{~}r@{~}l}
    \text{\emph{environment}} & \sigma(v)
    & \bnfeq & [x_1 \mapsto v_1, \dots, v_n \mapsto v_n] \\
    \text{\emph{closed term}} & \cle(v)
    & \bnfeq & (\sigma(v), e) \\
    \text{\emph{accessors}} & a
    & \bnfeq & \Root \;\bnfor\; a.n \quad (n \in \mathbb{N}) \\
  \end{array}
    \quad
  \begin{array}{l}
  \pi_S : \text{datatype constructors}
  \\
  \pi_T \subseteq        \{ \Int n \mid n \in \mathbb{Z} \}
                  \uplus \{ \Tag n \mid n \in \mathbb{N} \}
  \\[1em]
  a(v_S), a(v_T), D_S(v_S), D_T(v_T) \quad (\text{\emph{omitted})}
  \end{array}
\end{mathpar}
The tree $\Leaf \cle$ returns a leaf expression $e$ in a captured
environment $\sigma$ mapping variables to accessors.\\

$\Failure$ expresses a match failure that occurs when no clause matches the input
value.
\\
$\Switch a {\Fam {i \in I} {\pi_i, D_i}} \Dfb$ has one subtree $D_i$
for every head constructor that appears in the pattern matching
clauses, and a fallback case that is used when at least one variant of
the constructor doesn't appear in the clauses. The presence of the
fallback case does not imply non-exhaustive match clauses.
\begin{minipage}{0.3\linewidth}
  \scriptsize
\begin{lstlisting}
let f1 = function
| true -> 1
| false -> 0
\end{lstlisting}
\end{minipage}
\hfill\vline\hfill
\begin{minipage}{0.3\linewidth}
  \scriptsize
\begin{lstlisting}
let f1 = function
| true -> 1
| _ -> 0
\end{lstlisting}
\end{minipage}
\hfill\vline\hfill
\begin{minipage}{0.3\linewidth}
  \scriptsize
\begin{lstlisting}
let f1 = function
| true -> 1
\end{lstlisting}
\end{minipage}

\begin{minipage}{0.3\linewidth}
  \scriptsize
\begin{verbatim}
    Switch(Root)
    /          \
(Bool true)  (Bool false)
  /              \
Leaf(Int 1)  Leaf(Int 0)
\end{verbatim}
\end{minipage}
\hfill\vline\hfill
\begin{minipage}{0.3\linewidth}
  \scriptsize
\begin{verbatim}
    Switch(Root)
    /          \
(Bool true)    (`Fallback`)
  /              \
Leaf(Int 1)   Leaf(Int 0)
\end{verbatim}
\end{minipage}
\hfill\vline\hfill
\begin{minipage}{0.3\linewidth}
  \scriptsize
\begin{verbatim}
    Switch(Root)
    /          \
  (Bool true)  (`Fallback`)
  /              \
Leaf(Int 1)    Failure
\end{verbatim}
\end{minipage}
\hfill \\
As we can see from these simple examples in which we pattern match on a
boolean constructor the fallback node in the second case implicitly
covers the path in which the value is equal to false while in the
third case the failure terminal signals the presence of non-exahustive clauses.

\\
$\Guard \cle {D_0} {D_1}$ represents a \texttt{when}-guard on a closed
expression $\cle$, expected to be of boolean type, with sub-trees
$D_0$ for the \texttt{true} case and $D_1$ for the \texttt{false}
case.
\\
We write $a(v)$ for the sub-value of the source or target value $v$
that is reachable at the accessor $a$, and $D(v)$ for the result of
running a value $v$ against a decision tree $D$.
\begin{comment}
TODO: lascio o tolgo?
 this results in
a source or target matching run $R(v)$, just like running the value
against a program.
\end{comment}
\\
To check the equivalence of a source and a target decision tree,
we proceed by case analysis.
If we have two terminals, such as leaves in the first example,
we check that the two right-hand-sides are equivalent.
If we have a node $N$ and another tree $T$ we check equivalence for
each child of $N$, which is a pair of a branch condition $\pi_i$ and a
subtree $C_i$. For every child $(\pi_i, C_i)$ we reduce $T$ by killing all
the branches that are incompatible with $\pi_i$ and check that the
reduced tree is equivalent to $C_i$.
\\
\subsection{From source programs to decision trees}
Our source language supports integers, lists, tuples and all algebraic
datatypes. Patterns support wildcards, constructors and literals,
Or-patterns such as $(p_1 | p_2)$ and pattern variables.
In particular Or-patterns provide a more compact way to group patterns
that point to the same expression.

\begin{minipage}{0.4\linewidth}
\begin{lstlisting}
match w with
| p₁ -> expr
| p₂ -> expr
| p₃ -> expr
\end{lstlisting}
\end{minipage}
\begin{minipage}{0.6\linewidth}
\begin{lstlisting}
match w with
|p₁ |p₂ |p₃ -> expr
\end{lstlisting}
\end{minipage}
We also support \texttt{when} guards, which are interesting as they
introduce the evaluation of expressions during matching. 
\\
We write 〚tₛ〛ₛ to denote the translation of the source program (the
set of pattern matching clauses) into a decision tree, computed by a matrix decomposition algorithm (each column
decomposition step gives a \texttt{Switch} node).
It satisfies the following correctness statement:
\[
\forall t_s, \forall v_s, \quad t_s(v_s) = \semTEX{t_s}_s(v_s)
\]
The correctness statement intuitively states that for every source
program, for every source value that is well-formed input to a source
program, running the program tₛ against the input value vₛ is the same
as running the compiled source program 〚tₛ〛 (that is a decision tree) against the same input
value vₛ.

\subsection{From target programs to decision trees}
The target programs include the following Lambda constructs:
\texttt{let, if, switch, Match\_failure, catch, exit, field} and
various comparison operations, guards. The symbolic execution engine
traverses the target program and builds an environment that maps
variables to accessors. It branches at every control flow statement
and emits a \texttt{Switch} node. The branch condition $\pi_i$ is expressed as
an interval set of possible values at that point.
In comparison with the source decision trees, \texttt{Unreachable}
nodes are never emitted.
\\
Guards are black boxes of OCaml code that branches the execution of
the symbolic engine. Whenever a guard is met, we emit a Guard node
that contains two subtrees, one for each boolean value that can result
from the evaluation of the \texttt{guard condition} at runtime. The
symbolic engine explores both paths because we will see later that for
the equivalence checking the computation of the guard condition can be skipped.
In comparison with the source decision trees, \texttt{Unreachable} nodes are never emitted.
\\
We write $\semTEX{t_T}_T$ to denote the translation of a target
program tₜ into a decision tree of the target program
$t_T$, satisfying the following correctness statement that is
simmetric to the correctness statement for the translation of source programs:
\[
\forall t_T, \forall v_T, \quad t_T(v_T) = \semTEX{t_T}_T(v_T)
\]

\subsection{Equivalence checking}
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.
Our algorithm respects the following correctness statement:
\begin{align*}
 \EquivTEX S {C_S} {C_T} = \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} = \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*}
* Background
** OCaml
Objective Caml (OCaml) is a dialect of the ML (Meta-Language)
family of programming that features with other dialects of ML, such
as SML and Caml Light.
The main features of ML languages are the use of the Hindley-Milner type system that
provides many advantages with respect to static type systems of traditional imperative and object
oriented language such as C, C++ and Java, such as:
    - Polymorphism: in certain scenarios a function can accept more than one
      type for the input parameters. For example a function that computes the length of a
      list doesn't need to inspect the type of the elements and for this reason
      a List.length function can accept lists of integers, lists of strings and in general
      lists of any type. JVM languages offer polymorphic functions and
      classes through subtyping at
      runtime only, while other languages such as C++ offer polymorphism through compile
      time templates and function overloading.
      With the Hindley-Milner type system each well typed function can have more than one
      type but always has a unique best type, called the /principal type/.
      For example the principal type of the List.length function is "For any /a/, function from
      list of /a/ to /int/" and /a/ is called the /type parameter/.
    - Strong typing: Languages such as C and C++ allow the programmer to operate on data
      without considering its type, mainly through pointers[CIT]. Other languages such as Swift
      and Java performs type erasure[CIT] so at runtime the type of the data can't be queried.
      In the case of programming languages using an Hindley-Milner type system the
      programmer is not allowed to operate on data by ignoring or promoting its type.
    - Type Inference: the principal type of a well formed term can be inferred without any
      annotation or declaration.
    - Algebraic data types: types that are modeled by the use of two
      algebraic operations, sum and product.
      A sum type is a type that can hold of many different types of
      objects, but only one at a time. For example the sum type defined
      as /A + B/ can hold at any moment a value of type A or a value of
      type B. Sum types are also called tagged union or variants.
      A product type is a type constructed as a direct product
      of multiple types and contains at any moment one instance for
      every type of its operands. Product types are also called tuples
      or records. Algebraic data types can be recursive
      in their definition and can be combined.
Moreover ML languages are functional, meaning that functions are
treated as first class citizens and variables are immutable,
although mutable statements and imperative constructs are permitted.
In addition to that OCaml features an object system, that provides
inheritance, subtyping and dynamic binding, and modules, that
provide a way to encapsulate definitions. Modules are checked
statically and can be reifycated through functors[CIT].

** Compiling OCaml code

The OCaml compiler provides compilation of source files in form of a bytecode executable with an
optionally embeddable interpreter or as a native executable that could
be statically linked to provide a single file executable.
Every source file is treated as a separate /compilation unit/ that is
advanced through different states.
The first stage of compilation is the parsing of the input code that
is trasformed into an untyped syntax tree. Code with syntax errors is
rejected at this stage.
After that the AST is processed by the type checker that performs
three steps at once:
- type inference, using the classical /Algorithm W/[CIT]
- perform subtyping and gathers type information from the module system
- ensures that the code obeys the rule of the OCaml type system
At this stage, incorrectly typed code is rejected. In case of success,
the untyped AST in trasformed into a /Typed Tree/.
After the  typechecker has proven that the program is type safe,
the  compiler lower the code to /Lambda/, an s-expression based
language that assumes that its input has already been proved safe[CIT].
After the  Lambda pass, the Lambda code is either translated into
bytecode or goes through a series of optimization steps performed by
the /Flambda/ optimizer[CIT] before being translated into assembly.
\begin{comment}
TODO: Talk about flambda passes?
\end{comment}

This is an overview of the different compiler steps.
[[./files/ocamlcompilation.png]]

** Memory representation of OCaml values
An usual OCaml source program contains few to none explicit type
signatures.
This is possible because of type inference that allows to annotate the
AST with type informations. However, since the OCaml typechecker guarantes that a program is well typed
before being transformed into Lambda code, values at runtime contains
only a minimal subset of type informations needed to distinguish
polymorphic values.
For runtime values, OCaml uses a uniform memory representation in
which every variable is stored as a value in a contiguous block of
memory.
Every value is a single word that is either a concrete integer or a
pointer to another block of memory, that is called /block/ or /box/.
We can abstract the type of OCaml runtime values as the following:
#+BEGIN_SRC 
type t = Constant | Block of int * t 
#+END_SRC
where a one bit tag is used to distinguish between Constant or Block.
In particular this bit of metadata is stored as the lowest bit of a
memory block.
\\
Given that all the OCaml target architectures guarantee that all
pointers are divisible by four and that means that two lowest bits are
always 00 storing this bit of metadata at the lowest bit allows an
optimization. Constant values in OCaml, such as integers, empty lists,
Unit values and constructors of arity zero (/constant/ constructors)
are unboxed at runtime while pointers are recognized by the lowest bit
set to 0.



** Lambda form compilation
\begin{comment}
https://dev.realworld.org/compiler-backend.html
CITE: realworldocaml
[CIT]
\end{comment}
A Lambda code target file is produced by the compiler and consists of a
single s-expression. Every s-expression consist of /(/, a sequence of
elements separated by a whitespace and a closing /)/.
Elements of s-expressions are:
- Atoms: sequences of ascii letters, digits or symbols
- Variables
- Strings: enclosed in double quotes and possibly escaped
- S-expressions: allowing arbitrary nesting
\\
The Lambda form is a key stage where the compiler discards type
informations and maps the original source code to the runtime memory
model described.
In this stage of the compiler pipeline pattern match statements are
analyzed and compiled into an automata.
\begin{comment}
evidenzia centralita` rispetto alla tesi
\end{comment}
#+BEGIN_SRC 
type t = | Foo | Bar | Baz | Fred

let test = function
  | Foo -> "foo"
  | Bar -> "bar"
  | Baz -> "baz"
  | Fred -> "fred"
#+END_SRC
The Lambda output for this code can be obtained by running the
compiler with the /-drawlambda/ flag or in a more compact form with
the /-dlambda/ flag:
#+BEGIN_SRC 
(setglobal Prova!
  (let
    (test/85 =
       (function param/86
         (switch* param/86
          case int 0: "foo"
          case int 1: "bar"
          case int 2: "baz"
          case int 3: "fred")))
    (makeblock 0 test/85)))
#+END_SRC
As outlined by the example, the /makeblock/ directive allows to
allocate low level OCaml values and every constant constructor
of the algebraic type /t/ is stored in memory as an integer.
The /setglobal/ directives declares a new binding in the global scope:
Every concept of modules is lost at this stage of compilation.
The pattern matching compiler uses a jump table to map every pattern
matching clauses to its target expression. Values are addressed by a
unique name.
#+BEGIN_SRC 
type t = | English of p | French of q
type p = | Foo | Bar
type q = | Tata| Titi
type t = | English of p | French of q

let test = function
  | English Foo -> "foo"
  | English Bar -> "bar"
  | French Tata -> "baz"
  | French Titi -> "fred"
#+END_SRC
In the case of types with a smaller number of variants, the pattern
matching compiler may avoid the overhead of computing a jump table.
This example also highlights the fact that non constant constructor
are mapped to cons blocks that are accessed using the /tag/ directive.
#+BEGIN_SRC 
(setglobal Prova!
  (let
    (test/89 =
       (function param/90
         (switch* param/90
          case tag 0: (if (!= (field 0 param/90) 0) "bar" "foo")
          case tag 1: (if (!= (field 0 param/90) 0) "fred" "baz"))))
    (makeblock 0 test/89)))
#+END_SRC
In the Lambda language defines several numeric types:
- integers: that us either 31 or 63 bit two's complement arithmetic
  depending on system word size, and also wrapping on overflow
- 32 bit and 64 bit integers: that use 32-bit and 64-bit two's complement arithmetic
  with wrap on overflow
- big integers: offer integers with arbitrary precision
- floats: that use IEEE754 double-precision (64-bit) arithmetic with
  the addition of the literals /infinity/, /neg_infinity/ and /nan/.
\\
The are various numeric operations:
- Arithmetic operations: +, -, *, /, % (modulo), neg (unary negation)
- Bitwise operations: &, |, ^, <<, >> (zero-shifting), a>> (sign extending)
- Numeric comparisons: <, >, <=, >=, ==

*** Functions
Functions are defined using the following syntax, and close over all
bindings in scope: (lambda (arg1 arg2 arg3) BODY)
and are applied using the following syntax: (apply FUNC ARG ARG ARG)
Evaluation is eager.

*** Other atoms
The atom /let/ introduces a sequence of bindings at a smaller scope
than the global one:
(let BINDING BINDING BINDING ... BODY)

The Lambda form supports many other directives such as /strinswitch/
that is constructs aspecialized jump tables for string, integer range
comparisons and so on.
These construct are explicitely undocumented because the Lambda code
intermediate language can change across compiler releases.
\begin{comment}
Spiega che la sintassi che supporti e` quella nella BNF
\end{comment}



** Pattern matching
Pattern matching is a widely adopted mechanism to interact with ADT[CIT].
C family languages provide branching on predicates through the use of
if statements and switch statements.
Pattern matching on the other hands express predicates through
syntactic templates that also allow to bind on data structures of
arbitrary shapes. One common example of pattern matching is the use of regular
expressions on strings.  provides pattern matching on ADT and
primitive data types.
The result of a pattern matching operation is always one of:
- this value does not match this pattern
- this value matches this pattern, resulting the following bindings of
  names to values and the jump to the expression pointed at the
  pattern.

#+BEGIN_SRC 
type color = | Red | Blue | Green | Black | White

match color with
| Red -> print "red"
| Blue -> print "blue"
| Green -> print "green"
| _ -> print "white or black"
#+END_SRC
Pattern matching clauses provide tokens to express data destructoring.
For example we can examine the content of a list with pattern matching
#+BEGIN_SRC 
begin match list with
| [ ] -> print "empty list"
| element1 :: [ ] -> print "one element"
| (element1 :: element2) :: [ ] -> print "two elements"
| head :: tail-> print "head followed by many elements"
#+END_SRC
Parenthesized patterns, such as the third one in the previous example,
matches the same value as the pattern without parenthesis.
\\
The same could be done with tuples
#+BEGIN_SRC 
begin match tuple with
| (Some _, Some _) -> print "Pair of optional types"
| (Some _, None) | (None, Some _) -> print "Pair of optional types, one of which is null"
| (None, None) -> print "Pair of optional types, both null"
#+END_SRC
The pattern pattern₁ |  pattern₂ represents the logical "or" of the
two patterns, pattern₁ and pattern₂. A value matches /pattern₁ | pattern₂/ if it matches pattern₁ or pattern₂.
\\
Pattern clauses can make the use of /guards/ to test predicates and
variables can captured (binded in scope).
#+BEGIN_SRC 
begin match token_list with
| "switch"::var::"{"::rest -> ...
| "case"::":"::var::rest when is_int var -> ...
| "case"::":"::var::rest when is_string var -> ...
| "}"::[ ] -> ...
| "}"::rest -> error "syntax error: " rest
#+END_SRC
Moreover, the  pattern matching compiler emits a warning when a
pattern is not exhaustive or some patterns are shadowed by precedent ones.

** Symbolic execution
Symbolic execution is a widely used techniques in the field of
computer security.
It allows to analyze different execution paths of a program
simultanously while tracking which inputs trigger the execution of
different parts of the program.
Inputs are modelled symbolically rather than taking "concrete" values.
A symbolic execution engine keeps track of expressions and variables
in terms of these symbolic symbols and attaches logical constraints to every
branch that is being followed.
Symbolic execution engines are used to track bugs by modelling the
domain of all possible inputs of a program, detecting infeasible
paths, dead code and proving that two code segments are equivalent.
\\
Let's take as example this signedness bug that was found in the
FreeBSD kernel [CIT] and allowed, when calling the /getpeername/ function, to
read portions of kernel memory.
\begin{comment}
TODO: controlla paginazione
\end{comment}
\clearpage
#+BEGIN_SRC
int compat;
{
    struct file *fp;
    register struct socket *so;
    struct sockaddr *sa;
    int len, error;

    ...

    len = MIN(len, sa->sa_len);    /* [1] */
    error = copyout(sa, (caddr_t)uap->asa, (u_int)len);
    if (error)
        goto bad;

    ...

bad:
    if (sa)
        FREE(sa, M_SONAME);
    fdrop(fp, p);
    return (error);
}
#+END_SRC
The tree of the execution is presented below.
It is built by evaluating the function by consider the integer
variable /len/ the symbolic variable α, sa->sa_len the symbolic variable β
and π indicates the set of constraints on the symbolic variables.
The input values to the functions are identified by σ.
[[./files/symb_exec.png]]
\begin{comment}
[1]              compat (...)        { π_{α}: -∞ < α < ∞ }
                   |
[2]              min (σ₁, σ₂)        { π_{σ}: -∞ < (σ₁,σ₂) < ∞ ; π_{α}: -∞ < α < β ; π_{β}: ...}
                   |
[3]             cast(u_int) (...)    { π_{σ}: 0 ≤ (σ) < ∞ ; π_{α}: -∞ < α < β ; π_{β}: ...}
                   |   
                  ... // rest of the execution
\end{comment}
We can see that at step 3 the set of possible values of the scrutinee
α is bigger than the set of possible values of the input σ to the
/cast/ directive, that is: π_{α} ⊈ π_{σ}. For this reason the /cast/ may fail when α is /len/
negative number, outside the domain π_{σ}. In C this would trigger undefined behaviour (signed
overflow) that made the exploit possible.
***  Symbolic Execution in terms of Hoare Logic
Every step of the evaluation in a symbolic engine can be modelled as the following transition:
\[
(π_{σ}, (πᵢ)ⁱ) → (π'_{σ}, (π'ᵢ)ⁱ)
\]
if we express the π transitions as logical formulas we can model the
execution of the program in terms of Hoare Logic.
The state of the computation is a Hoare triple {P}C{Q} where P and Q are
respectively the /precondition/ and the /postcondition/ that
constitute the assertions of the program. C is the directive being
executed.
The language of the assertions P is:
| P ::= true \vert false \vert a < b \vert P_{1} \wedge P_{2} \vert P_{1} \lor P_{2} \vert \not P
where a and b are numbers.
In the Hoare rules assertions could also take the form of
| P ::= \forall i. P \vert \exists i. P \vert P_{1} \Rightarrow P_{2}
where i is a logical variable, but assertions of these kinds increases
the complexity of the symbolic engine.
\
Execution follows the following inference rules:
- Empty statement :
\begin{mathpar}
\infer{ }
{ \{P\}skip\{P\} }
\end{mathpar}
- Assignment statement : The truthness of P[a/x] is equivalent to the
  truth of {P} after the assignment.
\begin{mathpar}
\infer{ }
{ \{P[a/x]\}x:=a\{P\} }
\end{mathpar}
\\
- Composition : c₁ and c₂ are directives that are executed in order;
  {Q} is called the /mid condition/.
\begin{mathpar}
\infer { \{P\}c_1\{R\}, \{R\}c_2\{Q\} }
 {  \{P\}c₁;c₂\{Q\} }
\end{mathpar}
\\
- Conditional : 
\begin{mathpar}
\infer { \{P \wedge b \} c_1 \{Q\}, \{P\wedge\neg b\}c_2\{Q\} }
{ \{P\}\text{if b then $c_1$ else $c_2$}\{Q\} }
\end{mathpar}
\\
- Loop : {P} is the loop invariant. After the loop is finished /P/
  holds and ¬b caused the loop to end.
\begin{mathpar}
\infer { \{P \wedge b \}c\{P\} }
{ \{P\}\text{while b do c} \{P \wedge \neg b\} }
\end{mathpar}
\\
Even if the semantics of symbolic execution engines are well defined,
the user may run into different complications when applying such
analysis to non trivial codebases.[CIT]
For example, depending on the domain, loop termination is not
guaranteed. Even when termination is guaranteed, looping causes
exponential branching that may lead to path explosion or state
explosion.[CIT]
Reasoning about all possible executions of a program is not always
feasible[CIT] and in case of explosion usually symbolic execution engines
implement heuristics to reduce the size of the search space.[CIT]

** Translation Validation
Translators, such as translators and code generators, are huge pieces of
software usually consisting of multiple subsystem and
constructing an actual specification of a translator implementation for
formal validation is a very long task. Moreover, different
translators implement different algorithms, so the correctness proof of
a translator cannot be generalized and reused to prove another translator.
Translation validation is an alternative to the verification of
existing translators that consists of taking the source and the target
(compiled) program and proving /a posteriori/ their semantic equivalence.
\\
- [ ] Techniques for translation validation
- [ ] What does semantically equivalent mean
- [ ] What happens when there is no semantic equivalence
- [ ] Translation validation through symbolic execution

* Translation Validation of the Pattern Matching Compiler
** Accessors
OCaml encourages widespread usage of composite types to encapsulate
data. Composite types include /discriminated unions/, of which we have
seen different use cases, and /records/, that are a form of /product
types/ [CIT] such as /structures/ in C. \\
\begin{minipage}{0.6\linewidth}
\begin{verbatim}
struct Shape {
    int centerx;
    int centery;
    enum ShapeKind kind;
    union {
        struct { int side; };
        struct { int length, height; };
        struct { int radius; };
    };
};
\end{verbatim}
\end{minipage}
\hfill
\begin{minipage}{0.4\linewidth}
\begin{verbatim}
type shape = {
    x:int;
    y:int;
    kind:shapekind
}
and shapekind 
      | Square of int
      | Rect of int * int
      | Circle of int
\end{verbatim}
\end{minipage}
\begin{comment}
TODO: metti linea qualcosa a dividere
\end{comment}
Primitive OCaml datatypes include aggregate types in the form of
/tuples/ and /lists/. Other aggregate types are built using module
functors. [CIT]
Low level /Lambda/ untyped constructors of the form
#+BEGIN_SRC 
type t = Constant | Block of int * t 
#+END_SRC
provides enough flexibility to encode source higher kinded types.
This shouldn't surprise because the /Lambda/ language consists of
s-expressions. The /field/ operation allows to address a /Block/ value;
the expressions /(field 0 x)/ and /(field 1 x)/ are equivalent to
the Lisp primitives /(car x)/ and /(cdr x)/ respectively.
\begin{comment}
(* that can also be written *)
let value = List.cons 1 
            @@ List.cons 2
            @@ List.cons 3 [] 
\end{comment}
\begin{minipage}{0.3\linewidth}
\begin{verbatim}
let value =  1 :: 2 :: 3 :: []
\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} \\
\\
We can represent the concrete value of a higher kinded type as a flat
list of blocks.
In the prototype we call this "view" into the value of a datatype the
\emph{accessor} /a/.
| a ::= Here \vert n.a
Accessors have some resemblance with the low level /Block/ values, such
as the fact that both don't encode type informations; for example the
accessor of a list of integers is structurally equivalent to the
accessor of a tuple containing the same elements.
\\
We can intuitively think of the /accessor/ as the
access path to a value that can be reached by deconstructing the
scrutinee. 
At the source level /accessors/ are constructed by inspecting the structure of
the patterns at hand.
At the target level /accessors/ are constructed by compressing the steps taken by
the symbolic engine during the evaluation of a value.
/Accessors/ don't store any kind of information about the concrete
value of the scrutinee.
Accessors respect the following invariants:
| v(Here) = v
| K(vᵢ)ⁱ(k.a) = vₖ(a) if k ∈ [0;n[
We will see in the following chapters how at the source level and the
target level a value v_{S} and a value v_{T} can be deconstructed into
a value vector $(v_i)^{i\in{I}}$ of which we can access the root using
the $Here$ accessor and we can inspect the k-th element using an
accessor of the form $k.a$.

** Source program
The OCaml source code of a pattern matching function has the
following form:

|match variable with
|\vert pattern₁ \to expr₁
|\vert pattern₂ when guard \to expr₂
|\vert pattern₃ as var \to expr₃ 
|⋮
|\vert pₙ \to exprₙ
\\
Patterns could or could not be exhaustive.
\\
Pattern matching code could also be written using the more compact form:
|function
|\vert pattern₁ \to expr₁
|\vert pattern₂ when guard \to expr₂
|\vert pattern₃ as var \to expr₃ 
|⋮
|\vert pₙ \to exprₙ
\begin{comment}
TODO: paginazione seria per BNF
\end{comment}
This BNF grammar describes formally the grammar of the source program:
| 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 (ε\vert{}condition)
| rexpr ::= _code ;; arbitrary code
| rule ::= wildcard\vert{}variable\vert{}constructor_pattern\vert or_pattern
| wildcard ::= "_"
| variable ::= identifier
| constructor_pattern ::= constructor (rule\vert{}ε) (assignment\vert{}ε)
| constructor ::= int\vert{}float\vert{}char\vert{}string\vert{}bool \vert{}unit
| \quad\quad\quad\quad\quad\quad\quad \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" b_guard 
| assignment ::= "as" id
| b_guard ::= ocaml_expression ;; arbitrary code
The source program is parsed using the ocaml-compiler-libs [CIT] library.
The result of parsing, when successful, results in a list of clauses
and a list of type declarations.
Every clause consists of three objects: a left-hand-side that is the
kind of pattern expressed, an option guard and a right-hand-side expression.
Patterns are encoded in the following way:
| pattern         | type        |
|-----------------+-------------|
| _               | Wildcard    |
| p₁ as x         | Assignment  |
| c(p₁,p₂,...,pₙ) | Constructor |
| (p₁\vert p₂)    | Orpat       |
\\

Once parsed, the type declarations and the list of clauses are encoded in the form of a matrix
that is later evaluated using a matrix decomposition algorithm.
\\
Patterns are of the form
| pattern         | type of pattern     |
|-----------------+---------------------|
| _               | wildcard            |
| x               | variable            |
| c(p₁,p₂,...,pₙ) | constructor pattern |
| (p₁\vert p₂)    | or-pattern          |
\\
The pattern /p/ matches a value /v/, written as p ≼ v, when one of the
following rules apply

|--------------------+---+--------------------+-------------------------------------------|
| _                  | ≼ | v                  | ∀v                                        |
| x                  | ≼ | v                  | ∀v                                        |
| (p₁ \vert p₂)      | ≼ | v                  | iff p₁ ≼ v or p₂ ≼ v                      |
| c(p₁, p₂, ..., pₐ) | ≼ | c(v₁, v₂, ..., vₐ) | iff (p₁, p₂, ..., pₐ) ≼ (v₁, v₂, ..., vₐ) |
| (p₁, p₂, ..., pₐ)  | ≼ | (v₁, v₂, ..., vₐ)  | iff pᵢ ≼ vᵢ ∀i ∈ [1..a]                   |
|--------------------+---+--------------------+-------------------------------------------|

When a value /v/ matches pattern /p/ we say that /v/ is an /instance/ of /p/.


During compilation by the translator, expressions at the
right-hand-side are compiled into
Lambda code and are referred as lambda code actions lᵢ.
\\
We define the /pattern matrix/ P as the matrix |m × n| where m is bigger
or equal than the number of clauses in the source program and n is
equal to the arity of the constructor with the gratest arity.
\begin{equation*}
P =
\begin{pmatrix}
p_{1,1} & p_{1,2} & \cdots & p_{1,n}  \\
p_{2,1} & p_{2,2} & \cdots & p_{2,n}  \\
\vdots & \vdots & \ddots & \vdots  \\
p_{m,1} & p_{m,2} & \cdots & p_{m,n} )
\end{pmatrix}
\end{equation*}
Every row of /P/ is called a pattern vector
 $\vec{p_i}$ = (p₁, p₂, ..., pₙ); in every instance of P pattern
vectors appear normalized on the length of the longest pattern vector.
Considering the pattern matrix P we say that the value vector
$\vec{v}$ = (v₁, v₂, ..., vᵢ) matches the pattern vector pᵢ in P  if and only if the following two
conditions are satisfied:
- p_{i,1}, p_{i,2}, \cdots, p_{i,n}  ≼ (v₁, v₂, ..., vₙ)
- ∀j < i p_{j,1}, p_{j,2}, \cdots, p_{j,n} ⋠ (v₁, v₂, ..., vₙ) 
In other words given the pattern vector of the i-th row pᵢ, we say
that pᵢ matches $\vec{v}$ if every
element vₖ of $\vec{v}$ is an instance of the corresponding sub-patten
p_{i,k} and none of the pattern vectors of the previous rows matches.
\\
We can define the following three relations with respect to patterns:
- Pattern p is less precise than pattern q, written p ≼ q, when all
  instances of q are instances of p
- Pattern p and q are equivalent, written p ≡ q, when their instances
  are the same
- Patterns p and q are compatible when they share a common instance
\\
Wit the support of two auxiliary functions
- tail of an ordered family
  | tail((xᵢ)^{i ∈ I}) := (xᵢ)^{i ≠ min(I)}
- first non-⊥ element of an ordered family
  | First((xᵢ)ⁱ) := ⊥ \quad \quad \quad if ∀i, xᵢ = ⊥
  | First((xᵢ)ⁱ) := x_{min{i \vert{} xᵢ ≠ ⊥}}  \quad  if ∃i, xᵢ ≠ ⊥
we now define what it means to run a pattern row against a value
vector of the same length, that is (pᵢ)ⁱ(vᵢ)ⁱ
| pᵢ                       | vᵢ                   | result_{pat}                                    |
|--------------------------+----------------------+-------------------------------------------------|
| ∅                        | (∅)                  | []                                              |
| (_, tail(pᵢ)ⁱ)           | (vᵢ)                 | tail(pᵢ)ⁱ(tail(vᵢ)ⁱ)                            |
| (x, tail(pᵢ)ⁱ)           | (vᵢ)                 | σ[x↦v₀]        if tail(pᵢ)ⁱ(tail(vᵢ)ⁱ) = σ      |
| (K(qⱼ)ʲ, tail(pᵢ)ⁱ)      | (K(v'ⱼ)ʲ,tail(vⱼ)ʲ)  | ((qⱼ)ʲ +++ tail(pᵢ)ⁱ)((v'ⱼ)ʲ +++ tail(vᵢ)ⁱ)     |
| (K(qⱼ)ʲ, tail(pᵢ)ⁱ)      | (K'(v'ₗ)ˡ,tail(vⱼ)ʲ) | ⊥              if K ≠ K'                        |
| (q₁\vert{}q₂, tail(pᵢ)ⁱ) | (vᵢ)ⁱ                | First((q₁,tail(pᵢ)ⁱ)(vᵢ)ⁱ, (q₂,tail(pᵢ)ⁱ)(vᵢ)ⁱ) |
\\
A source program tₛ is a collection of pattern clauses pointing to
blackbox /bb/ terms. Running a program tₛ against an input value vₛ, written
tₛ(vₛ) produces a result /r/ belonging to the following grammar:
| tₛ ::= (p → bb)^{i∈I}
| tₛ(vₛ) → r
| r ::= guard list * (Match (σ, bb) \vert{} NoMatch \vert{} Absurd)
\\
\begin{comment}
TODO: running a value against a tree:
| Leaf(bb) (vᵢ) := bb
| Failure (vᵢ) := Failure
| Unreachable (vᵢ) := Unreachable if source tree
| Node(a, Cᵢ, C?) := C_min{i | Cᵢ(vᵢ(a)) ≠ ⊥}
| Node (a, Cᵢ, None) :=  ⊥ if ∀Cᵢ, Cᵢ(vᵢ(a)) = ⊥ 
| Node (a, Cᵢ, C?) :=  C?(vᵢ(a)) if ∀Cᵢ, Cᵢ(vᵢ(a)) = ⊥ 
\\
\end{comment}

\begin{comment}
TODO: understand how to explain guard
\end{comment}
We can define what it means to run an input value vₛ against a source
program tₛ:
| tₛ(vₛ) := NoMatch \quad if ∀i, pᵢ(vₛ) = ⊥
| tₛ(vₛ) = { Absurd if bb_{i₀} = . (refutation clause)
| \quad \quad \quad \quad \quad Match (σ, bb_{i₀}) otherwise
| \quad \quad \quad \quad \quad where iₒ = min{i \vert{} pᵢ(vₛ) ≠ ⊥}
Expressions of type /guard/ and /bb/ 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 structural equality with the counterpart
Lambda blackboxes at the target level.
Given that this level of introspection is currently not possibile
because the OCaml compiler performs the translation of the pattern
clauses in a single pass, we
decided to restrict the structure of blackboxes to the following (valid) OCaml
code:
#+BEGIN_SRC 
external guard : 'a -> 'b = "guard"
external observe : 'a -> 'b = "observe"
#+END_SRC 
We assume the existence of these two external functions /guard/ and /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_SRC 
(* declaration of an algebraic and recursive datatype t *)
type t = K1 | K2 of t

let _ = function
  | K1 -> observe 0
  | K2 K1 -> observe 1
  | K2 x when guard x -> observe 2 (* guard inspects the x variable *)
  | K2 (K2 x) as y when guard x y -> observe 3
  | K2 _ -> observe 4
#+END_SRC
We note that the right hand side of /observe/ is just an arbitrary
value and in this case just enumerates the order in which expressions
appear.
This oversimplification of the structure of arbitrary code blackboxes
allows us to test for structural equivalence without querying the
compiler during the /TypedTree/ to /Lambda/ translation phase.
\begin{lstlisting}
let _ = function
  | K1 -> lambda₀
  | K2 K1 -> lambda₁
  | K2 x when lambda-guard₀ -> lambda₂
  | K2 (K2 x) as y when lambda-guard₁ -> lambda₃
  | K2 _ -> lambda₄
\end{lstlisting}

\subsubsection{Matrix decomposition of pattern clauses}
We define a new object, the /clause matrix/ P → L of size |m x n+1| that associates
pattern vectors $\vec{p_i}$ to lambda code action lᵢ.
\begin{equation*}
P → L = 
\begin{pmatrix}
p_{1,1} & p_{1,2} & \cdots & p_{1,n} → l₁ \\
p_{2,1} & p_{2,2} & \cdots & p_{2,n} → l₂ \\
\vdots & \vdots & \ddots & \vdots →  \vdots \\
p_{m,1} & p_{m,2} & \cdots & p_{m,n} → lₘ
\end{pmatrix}
\end{equation*}
The initial input of the decomposition algorithm C consists of a vector of variables
$\vec{x}$ = (x₁, x₂, ..., xₙ)  of size /n/ where /n/ is the arity of
the type of /x/ and the /clause matrix/ P → L.
That is:

\begin{equation*}
C((\vec{x} = (x₁, x₂, ..., xₙ),
\begin{pmatrix}
p_{1,1} & p_{1,2} & \cdots & p_{1,n} → l₁ \\
p_{2,1} & p_{2,2} & \cdots & p_{2,n} → l₂ \\
\vdots & \vdots & \ddots & \vdots →  \vdots \\
p_{m,1} & p_{m,2} & \cdots & p_{m,n} → lₘ
\end{pmatrix})
\end{equation*}
\\
The base case C₀ of the algorithm is the case in which the $\vec{x}$
is an empty sequence and the result of the compilation
C₀ is l₁
\begin{equation*}
C₀((),
\begin{pmatrix}
→ l₁ \\
→ l₂ \\
→  \vdots \\
→ lₘ
\end{pmatrix})  = l₁
\end{equation*}
\\
When $\vec{x}$ ≠ () then the compilation advances using one of the
following four rules:

1) Variable rule: if all patterns of the first column of P are wildcard patterns or
   bind the value to a variable, then 

    \begin{equation*}
    C(\vec{x}, P → L) = C((x₂, x₃, ..., xₙ), P' → L')
    \end{equation*}
    where
    \begin{equation*}
    P' \to L' = 
    \begin{pmatrix}
    p_{1,2} & \cdots & p_{1,n} & → & (let & y₁ & x₁) & l₁ \\
    p_{2,2} & \cdots & p_{2,n} & → & (let & y₂ & x₁) & l₂ \\
    \vdots & \ddots & \vdots & →  & \vdots & \vdots & \vdots & \vdots \\
    p_{m,2} & \cdots & p_{m,n} & → & (let & yₘ & x₁) & lₘ
    \end{pmatrix}
    \end{equation*}

    That means in every lambda action lᵢ in the P' → L' matrix there is a binding of x₁ to the
    variable that appears on the pattern p_{i,1}. When there are
    wildcard patterns, bindings are omitted the lambda action lᵢ remains unchanged.

2) Constructor rule: if all patterns in the first column of P are
   constructors patterns of the form k(q₁, q₂, ..., q_{n'}) we define a
   new matrix, the specialized clause matrix S, by applying the
   following transformation on every row /p/:
    \begin{lstlisting}[mathescape,columns=fullflexible,basicstyle=\fontfamily{lmvtt}\selectfont,]
    for every c ∈ Set of constructors 
        for i ← 1 .. m 
            let kᵢ ← constructor_of($p_{i,1}$) 
            if kᵢ = c then 
                p ← $q_{i,1}$, $q_{i,2}$, ..., $q_{i,n'}$,  $p_{i,2}$, $p_{i,3}$, ..., $p_{i, n}$
    \end{lstlisting}
   Patterns of the form $q_{i,j}$ matches on the values of the
   constructor and we define the variables y₁, y₂, ..., yₐ so
   that the lambda action lᵢ becomes

   \begin{lstlisting}[mathescape,columns=fullflexible,basicstyle=\fontfamily{lmvtt}\selectfont,]
       (let (y₁ (field 0 x₁))
           (y₂ (field 1 x₁))
           ...
           (yₐ (field (a-1) x₁))
           lᵢ)
   \end{lstlisting}

   and the result of the compilation for the set of constructors
    ${\{c_{1}, c_{2}, ..., c_{k}\}}$ is:

   \begin{lstlisting}[mathescape,columns=fullflexible,basicstyle=\fontfamily{lmvtt}\selectfont,]
       switch x₁ with
          case c₁: l₁
          case c₂: l₂
          ...
          case cₖ: lₖ
          default: exit
   \end{lstlisting}
\\
\begin{comment}
TODO: il tre viene cambiato in uno e la lista divisa. Risolvi.
\end{comment}
3) Orpat rule: there are various strategies for dealing with
   or-patterns. The most naive one is to split the or-patterns.
   For example a row p containing an or-pattern:
   \begin{equation*}
   (p_{i,1}|q_{i,1}|r_{i,1}), p_{i,2}, ..., p_{i,m} → l
   \end{equation*}
   results in three rows added to the clause matrix
   \begin{equation*}
   p_{i,1}, p_{i,2}, ..., p_{i,m} → l \\
   \end{equation*}
   \begin{equation*}
   q_{i,1}, p_{i,2}, ..., p_{i,m} → l \\
   \end{equation*}
   \begin{equation*}
   r_{i,1}, p_{i,2}, ..., p_{i,m} → l 
   \end{equation*}
4) Mixture rule:
   When none of the previous rules apply the clause matrix ${P\space{}\to\space{}L}$ is
   split into two clause matrices, the first $P_{1}\space\to\space{}L_{1}$ that is the
   largest prefix matrix for which one of the three previous rules
   apply, and $P_{2}\space\to\space{}L_{2}$ containing the remaining rows. The algorithm is
   applied to both matrices.
It is important to note that the application of the decomposition
algorithm converges. This intuition can be verified by defining the
size of the clause matrix P → L as equal to the length of the longest
pattern vector $\vec{p_i}$ and the length of a pattern vector as the
number of symbols that appear in the clause.
While it is very easy to see that the application of rules 1) and 4)
produces new matrices of length equal or smaller than the original
clause matrix, we can show that:
- with the application of the constructor rule the pattern vector $\vec{p_i}$ loses one
  symbol after its decomposition:
  | \vert{}(p_{i,1} (q₁, q₂, ..., q_{n'}), p_{i,2}, p_{i,3}, ..., p_{i,n})\vert{} = n + n'
  | \vert{}(q_{i,1}, q_{i,2}, ..., q_{i,n'}, p_{i,2}, p_{i,3}, ..., p_{i,n})\vert{} = n + n' - 1
- with the application of the orpat rule, we add one row to the clause
  matrix P → L but the length of a row containing an
  Or-pattern decreases
\begin{equation*}
\vert{}P → L\vert{} = \big\lvert
\begin{pmatrix}
(p_{1,1}\vert{}q_{1,1}) & p_{1,2} & \cdots & p_{1,n} → l₁ \\
\vdots & \vdots & \ddots & \vdots →  \vdots \\
\end{pmatrix}\big\rvert = n + 1
\end{equation*}
\begin{equation*}
\vert{}P' → L'\vert{} = \big\lvert
\begin{pmatrix}
p_{1,1} & p_{1,2} & \cdots & p_{1,n} → l₁ \\
q_{1,1} & p_{1,2} & \cdots & p_{1,n} → l₁ \\
\vdots & \vdots & \ddots & \vdots →  \vdots \\
\end{pmatrix}\big\rvert = n
\end{equation*}
In our prototype the source matrix mₛ is defined as follows
| SMatrix mₛ := (aⱼ)^{j∈J}, ((p_{ij})^{j∈J} → bbᵢ)^{i∈I}

** Target translation

The target program of the following general form is parsed using a parser
generated by Menhir[CIT], 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[CIT] (TODO: check menhir grammar)
| start ::= sexpr
| sexpr ::= variable \vert{} string \vert{} "(" special_form ")"
| string ::= "\"" identifier "\"" ;; 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{}"=="\vert{}">="\vert{}"<="\vert{}">"\vert{}"<") sexpr digit \vert{} field ")"
| label ::= integer
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.
For performance reasons the compiler performs integer addition and
subtraction on variables that appears inside a /switch/ expression in
order to have values always start from 0.
Let's see an example of this behaviour:
#+BEGIN_SRC
let f x = match x with
  | 3 -> "3"
  | 4 -> "4"
  | 5 -> "5"
  | 6 -> "6"
  | _ -> "_"
#+END_SRC
\begin{verbatim}
(let
  (f/80 =
     (function x/81
       (catch
         (let (switcher/83 =a (-3+ x/81))
           (if (isout 3 switcher/83) (exit 1)
             (switch* switcher/83
              case int 0: "3"
              case int 1: "4"
              case int 2: "5"
              case int 3: "6")))
        with (1) "_")))
  (makeblock 0 f/80))
\end{verbatim}
The prototype takes into account such transformations and at the end
of the symbolic evaluation it traverses the result in order to "undo"
such optimization and have accessors of the variables match their
intended value directly.

** Decision Trees
We have already given the parametrized grammar for decision trees and
we will now show how a decision tree is constructed from source and
target programs.
\begin{mathpar}
  \begin{array}{l@{~}r@{~}r@{~}l}
    \text{\emph{decision trees}} & D(\pi, e)
    & \bnfeq & \Leaf {\cle(a)} \\
    & & \bnfor & \Failure \\
    & & \bnfor & \Switch a {\Fam {i \in I} {\pi_i, D_i}} \Dfb \\
    & & \bnfor & \Guard {\cle(a)} {D_0} {D_1} \\
    & & \bnfor & Unreachable  \\
    \text{\emph{accessors}} & a 
    & \bnfeq & \Root \;\bnfor\; a.n \quad (n \in \mathbb{N}) \\
  \end{array}
  \quad
  \begin{array}{l}
  \pi_S : \text{datatype constructors}
  \\
  \pi_T \subseteq        \{ \Int n \mid n \in \mathbb{Z} \}
                  \uplus \{ \Tag n \mid n \in \mathbb{N} \}
  \\[1em]
  a(v_S), a(v_T), D_S(v_S), D_T(v_T) 
  \end{array}
\end{mathpar}
While the branches of a decision tree represents intuitively the
possible paths that a program can take, branch conditions πₛ and πₜ
represents the shape of possible values that can flow along that path.


\subsubsection{From source programs to decision trees}
Let's consider some trivial examples:
| function true -> 1
is translated to
|Switch ([(true, Leaf 1)], Failure)
while
| function 
| \vert{} true -> 1 
| \vert{} false -> 2
will be translated to

|Switch ([(true, Leaf 1); (false, Leaf 2)])
It is possible to produce Unreachable examples by using
 refutation clauses (a "dot" in the right-hand-side)
|function 
|\vert{} true -> 1 
|\vert{} false -> 2 
|\vert{} _ -> .
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.
We'll see that while we can build Unreachable nodes from source
programs, in the lambda target there isn't a construct equivalent to the refutation clause.
\\
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
false.
\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

| ∀vₛ, tₛ(vₛ) = 〚tₛ〛ₛ(vₛ)

We define the decision tree of source programs
〚tₛ〛
in terms of the decision tree of pattern matrices
〚mₛ〛
by the following:
| 〚((pᵢ → bbᵢ)^{i∈I}〛 := 〚(Here), (pᵢ → bbᵢ)^{i∈I} 〛
Decision tree computed from pattern matrices respect the following invariant:
| ∀v (vᵢ)^{i∈I} = v(aᵢ)^{i∈I} → m(vᵢ)^{i∈I} = 〚m〛(v) for m = ((aᵢ)^{i∈I}, (rᵢ)^{i∈I})
The invariant conveys the fact that OCaml pattern matching values can
be deconstructed into a value vector and if we can correctly inspect
the value vector elements using the accessor notation we can build a
decision tree 〚m〛 from a pattern matrix that are equivalent when run
against the value at hand.

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

Base cases:
1. [| ∅, (∅ → bbᵢ)ⁱ |] ≡ Leaf bbᵢ where i := min(I), that is a
   decision tree [|m|] defined by an empty accessor and empty
   patterns pointing to blackboxes /bbᵢ/. This respects the invariant
   because a source matrix in the case of empty rows returns
   the first expression and $(Leaf bb)(v) := Match\enspace bb$
2. [| (aⱼ)ʲ, ∅ |] ≡ Failure, as it is the case with the matrix
   decomposition algorithm
Regarding non base cases:
Let's first define some auxiliary functions
- The index family of a constructor: $Idx(K) := [0; arity(K)[$
- head of an ordered family (we write x for any object here, value,
  pattern etc.): $head((x_i)^{i \in I}) = x_{min(I)}$
- tail of an ordered family: $tail((x_i)^{i \in I}) := (x_i)^{i \ne min(I)}$
- head constructor of a value or pattern:
  | constr(K(xᵢ)ⁱ) = K
  | constr(_) = ⊥
  | constr(x) = ⊥
- first non-⊥ element of an ordered family:
  | First((xᵢ)ⁱ) := ⊥ \quad \quad \quad if ∀i, xᵢ = ⊥
  | First((xᵢ)ⁱ) := x_min{i \vert{} xᵢ ≠ ⊥}  \quad  if ∃i, xᵢ ≠ ⊥
- definition of group decomposition:
| let constrs((pᵢ)^{i ∈ I}) = { K \vert{} K = constr(pᵢ), i ∈ I }
| let Groups(m) where m = ((aᵢ)ⁱ ((pᵢⱼ)ⁱ → eⱼ)ⁱʲ) =
| \quad \quad let (Kₖ)ᵏ = constrs(pᵢ₀)ⁱ in
| \quad \quad ( Kₖ →
| \quad \quad \quad \quad ((a₀.ₗ)ˡ +++ tail(aᵢ)ⁱ)
| \quad \quad \quad \quad (
| \quad \quad \quad \quad if pₒⱼ is Kₖ(qₗ) then
| \quad \quad \quad \quad \quad \quad (qₗ)ˡ +++ tail(pᵢⱼ)ⁱ  → eⱼ
| \quad \quad \quad \quad if pₒⱼ is _ then
| \quad \quad \quad \quad \quad \quad (_)ˡ +++ tail(pᵢⱼ)ⁱ   → eⱼ
| \quad \quad \quad \quad else ⊥
| \quad \quad \quad \quad )ʲ
| \quad \quad ), (
| \quad \quad \quad \quad tail(aᵢ)ⁱ, (tail(pᵢⱼ)ⁱ  → eⱼ if p₀ⱼ is _ else ⊥)ʲ
| \quad \quad  )
Groups(m) is an auxiliary function that decomposes 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_{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 Kₖ, then running (v_i)^i against m is the same
as running it against the submatrix m_{Kₖ}; otherwise (when its head
constructor is not one of (Kₖ)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) = (kᵣ \to mᵣ)^k, m_{wild}
    For any value vector $(v_i)^l$ such that $v_0 = k(v'_l)^l$ for some
    constructor k,
    we have:
    | if k = kₖ \text{ for some k then} 
    | \quad    m(vᵢ)ⁱ = mₖ((v_{l}')ˡ +++ (v_{i})^{i∈I\DZ}) 
    |  \text{else} 
    |   \quad   m(vᵢ)ⁱ = m_{wild}(vᵢ)^{i∈I\DZ}

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

*** Proof:
Let /m/ be a matrix ((aᵢ)ⁱ, ((pᵢⱼ)ⁱ → eⱼ)ʲ) with 
| Groups(m) = (Kₖ → mₖ)ᵏ, m_{wild}
Below we are going to assume that m is a simplified matrix such that
the first row does not contain an or-pattern or a binding to a
variable.
\\
Let (vᵢ)ⁱ be an input matrix with v₀ = Kᵥ(v'_{l})ˡ for some constructor Kᵥ.
We have to show that:
- if Kₖ = Kᵥ for some Kₖ ∈ constrs(p₀ⱼ)ʲ, then
  | m(vᵢ)ⁱ = mₖ((v'ₗ)ˡ +++ tail(vᵢ)ⁱ)
- otherwise
    m(vᵢ)ⁱ = m_{wild}(tail(vᵢ)ⁱ)
Let us call (rₖⱼ) the j-th row of the submatrix mₖ, and rⱼ_{wild}
the j-th row of the wildcard submatrix m_{wild}.
\\
Our goal contains same-behavior equalities between matrices, for
a fixed input vector (vᵢ)ⁱ. It suffices to show same-behavior
equalities between each row of the matrices for this input
vector. We show that for any j,
- if Kₖ = Kᵥ for some Kₖ ∈ constrs(p₀ⱼ)ʲ, then
  | (pᵢⱼ)ⁱ(vᵢ)ⁱ = rₖⱼ((v'ₗ)ˡ +++ tail(vᵢ)ⁱ
- otherwise
  | (pᵢⱼ)ⁱ(vᵢ)ⁱ = rⱼ_{wild} tail(vᵢ)ⁱ
In the first case (Kᵥ is Kₖ for some Kₖ ∈ constrs(p₀ⱼ)ʲ), we
have to prove that
| (pᵢⱼ)ⁱ(vᵢ)ⁱ = rₖⱼ((v'ₗ)ˡ +++ tail(vᵢ)ⁱ
By definition of mₖ we know that rₖⱼ is equal to
| if pₒⱼ is Kₖ(qₗ) then
| \quad  (qₗ)ˡ +++ tail(pᵢⱼ)ⁱ  → eⱼ
| if pₒⱼ is _ then
| \quad  (_)ˡ +++ tail(pᵢⱼ)ⁱ   → eⱼ
| else ⊥
\begin{comment}
Maybe better as a table?
| pₒⱼ    | rₖⱼ                       |
|--------+---------------------------|
| Kₖ(qₗ) | (qₗ)ˡ +++ tail(pᵢⱼ)ⁱ → eⱼ |
| _      | (_)ˡ +++ tail(pᵢⱼ)ⁱ → eⱼ  |
| else   | ⊥                         |
\end{comment}
By definition of (pᵢ)ⁱ(vᵢ)ⁱ we know that (pᵢⱼ)ⁱ(vᵢ) is equal to
| (K(qⱼ)ʲ, tail(pᵢⱼ)ⁱ) (K(v'ₗ)ˡ,tail(vᵢ)ⁱ)  := ((qⱼ)ʲ +++ tail(pᵢⱼ)ⁱ)((v'ₗ)ˡ +++ tail(vᵢ)ⁱ)
| (_, tail(pᵢⱼ)ⁱ)      (vᵢ)                 := tail(pᵢⱼ)ⁱ(tail(vᵢ)ⁱ)
| (K(qⱼ)ʲ, tail(pᵢⱼ)ⁱ) (K'(v'ₗ)ˡ,tail(vⱼ)ʲ) := ⊥ if K ≠ K'

We prove this first case by a second case analysis on p₀ⱼ.
\\
TODO
\begin{comment}
GOAL:
| (pᵢⱼ)ⁱ(vᵢ)ⁱ = rₖⱼ((v'ₗ)ˡ +++ tail(vᵢ)ⁱ
case analysis on pₒⱼ:
| pₒⱼ    | pᵢⱼ                   |
|--------+-----------------------|
| Kₖ(qₗ) | (qₗ)ˡ +++ tail(pᵢⱼ)ⁱ  |
| else   |  ⊥                    |
that is exactly the same as  rₖⱼ :=  (qₗ)ˡ +++ tail(pᵢⱼ)ⁱ  → eⱼ 
But I am just putting together the definition given above (in the pdf).
\\

Second case, below:
the wildcard matrix r_{jwild} is 
| tail(aᵢ)ⁱ, (tail(pᵢⱼ)ⁱ  → eⱼ if p₀ⱼ is _ else ⊥)ʲ
\\
case analysis on p₀ⱼ:
| pₒⱼ    | (pᵢⱼ)ⁱ                |
|--------+-----------------------|
| Kₖ(qₗ) | (qₗ)ˡ +++ tail(pᵢⱼ)ⁱ  | not possible because we are in the second case
| _      | (_)ˡ +++ tail(pᵢⱼ)ⁱ   |
| else   |  ⊥                    |
\\

Both seems too trivial to be correct.

\end{comment}

In the second case (Kᵥ is distinct from Kₖ for all Kₖ ∈ constrs(pₒⱼ)ʲ),
we have to prove that
| (pᵢⱼ)ⁱ(vᵢ)ⁱ = rⱼ_{wild} tail(vᵢ)ⁱ

# TODO

\subsubsection{From target programs to target decision trees}
# TODO: replace C with D
Symbolic Values during symbolic evaluation have the following form
| vₜ ::= Block(tag ∈ ℕ, (vᵢ)^{i∈I}) \vert n ∈ ℕ
The result of the symbolic evaluation is a target decision tree Dₜ
| Dₜ ::= Leaf bb \vert Switch(a, (πᵢ → Dᵢ)^{i∈S} , D?) \vert Failure
Every branch of the decision tree is "constrained" by a domain πₜ that
intuitively tells us the set of
possible values that can "flow" through that path.
| Domain π = { n\vert{}n∈ℕ x n\vert{}n∈Tag⊆ℕ }
# TODO: copy the definition from above
πₜ 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.
D? 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 π_{fallback} condition of the D? fallback node is
| π_{fallback} = ¬\bigcup\limits_{i∈I}πᵢ
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
| domain(vₛ(a)) = \bigcup\limits_{i∈I}πᵢ
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
| ∀vₜ, tₜ(vₜ) = 〚tₜ〛ₜ(vₜ)



** Equivalence checking
\subsubsection{Introductory remarks}
We assume as given an equivalence relation $\erel {e_S} {e_T}$ on
expressions (that are contained in the leaves of the decision trees).
As highlighted before, in the prototype we use a simple structural
equivalence between $observe$ expressions but ideally we could query
the OCaml compiler and compare the blackboxes in Lambda form.
We already defined what it means to run a value vₛ against a program
tₛ:
| tₛ(vₛ) := NoMatch \quad if ∀i, pᵢ(vₛ) = ⊥
| tₛ(vₛ) = { Absurd if bb_{i₀} = . (refutation clause)
| \quad \quad \quad \quad \quad Match (σ, bb_{i₀}) otherwise
| \quad \quad \quad \quad \quad where iₒ = min{i \vert{} pᵢ(vₛ) ≠ ⊥}
and simmetrically, to run a value vₜ against a target program tₜ:
# TODO t_t todo.1 
We denote as $\vrel {v_S} {v_T}$ the equivalence relation between a
source value vₛ and a target value vₜ. The equivalence relation is proven by
structural induction.
\begin{mathpar}
  \infer[integers]
  {\quad}
  {\vrel {i} {(i)}}

  \infer[boolean]
  {\quad}
  {\vrel {false} {(0)}}

  \infer[boolean]
  {\quad}
  {\vrel {true} {(1)}}

  \infer[unit value]
  {\quad}
  {\vrel {()} {(0)}}

  \infer[Empty List]
  {\quad}
  {\vrel {[]} {(0)}}

  \infer[List]
  {\vrel {v_S} {v_T} \\ \vrel {v'_S} {v'_T}}
  {\vrel {[v_S; v'_{S}]} {(block \enspace v_T \enspace v'_{T})}}

  \infer[Tuple]
  {\vrel {v_S} {v_T} \\ \vrel {v'_S} {v'_T}}
  {\vrel {(v_S; v'_{S})} {(block \enspace v_T \enspace v'_{T})}}

  \infer[Record]
  {\vrel {v_S} {v_T} \\ \vrel {v'_S} {v'_T}}
  {\vrel {\{v_S; v'_{S}\}} {(block \enspace v_T \enspace v'_{T})}}
  
  \infer[constant constructor]
  {K_i \in Variant }
  {\vrel {K_i} {(i)}}

  \infer[Variant]
  {K_i \in Variant \\ \vrel {v_S} {v_T}}
  {\vrel {K_i v_S} {(block \enspace (tag \enspace i) \enspace v_T)}}
\end{mathpar}
The relation $\vrel {v_S} {v_T}$ captures our knowledge of the
OCaml value representation, for example it relates the empty list
constructor \texttt{[]} to $\Int 0$. We can then define \emph{closed}
expressions $\cle$, pairing a (source or target) expression with the
environment σ captured by a program, and what it means to
``run'' a value against a program or a decision, written $t(v)$ and
$D(v)$, which returns a trace $(\cle_1, \dots, \cle_n)$ of the
executed guards and a \emph{matching result} $r$.
\begin{mathpar}
  \erel {e_S} {e_T} \; (\text{\emph{assumed}})

  t_S(v_S),\ t_T(v_T),\ \vrel {v_S} {v_T} \; 

  \resrel {r_S} {r_T}, \runrel {R_S} {R_T} \; (\text{\emph{simple}})
\\
  \begin{array}{l@{~}r@{~}r@{~}l}
    \text{\emph{environment}} & \sigma(v)
    & \bnfeq & [x_1 \mapsto v_1, \dots, v_n \mapsto v_n] \\
    \text{\emph{closed term}} & \cle(v)
    & \bnfeq & (\sigma(v), e) \\
  \end{array}
\quad
  \begin{array}{l@{~}r@{~}r@{~}l}
    \text{\emph{matching result}} & r(v)
    & \bnfeq & \NoMatch \bnfor \Match {\cle(v)} \\
    \text{\emph{matching run}} & R(v)
    & \bnfeq & (\cle(v)_1, \dots, \cle(v)_n), r(v) \\
  \end{array}
\\
  \infer
  {\forall x,\ \vrel {\sigma_S(x)} {\sigma_T(x)}}
  {\envrel {\sigma_S} {\sigma_T}}

  \infer
  {\envrel {\sigma_S} {\sigma_T} \\ \erel {e_S} {e_T}}
  {\clerel {(\sigma_S, e_S)} {(\sigma_T, e_T)}}

  \infer
  {\forall {\vrel {v_S} {v_T}},\quad \runrel {t_S(v_S)} {t_T(v_T)}}
  {\progrel {t_S} {t_T}}
\end{mathpar}
Once formulated in this way, our equivalence algorithm must check the
natural notion of input-output equivalence for matching programs,
captured by the relation $\progrel {t_S} {t_T}$.

\subsubsection{The equivalence checking algorithm}

\begin{comment}
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:
% TODO: we have to define what \covers mean for readers to understand the specifications
% (or we use a simplifying lie by hiding \covers in the statements).
\begin{align*}
 \SimpleEquiv S {D_S} {D_T} = \Yes \;\land\; \covers {D_T} S
 & \implies
 \forall {\vrel {v_S} {v_T}} \in S,\; \runrel {D_S(v_S)} {D_T(v_T)}
\\
 \SimpleEquiv S {D_S} {D_T} = \No {v_S} {v_T} \;\land\; \covers {D_T} S
 & \implies
 \vrel {v_S} {v_T} \in S \;\land\; {\nparamrel{run} {D_S(v_S)} {D_T(v_T)}}
\end{align*}
This algorithm $\SimpleEquiv S {D_S} {D_T}$ is in fact exactly
a decision procedure for the provability of the judgment
$(\Equivrel S {D_S} {D_T} \emptyqueue)$, defined below in the general
form $(\Equivrel S {D_S} {D_T} G)$ where $G$ is a \emph{guard queue},
indicating an imbalance between the guards observed in the source tree
and in the target tree.
\end{comment}

During the equivalence checking phase we traverse the two trees,
recursively checking equivalence of pairs of subtrees. When we
traverse a branch condition, we learn a condition on an accessor that
restricts the set of possible input values that can flow in the
corresponding subtree. We represent this in our algorithm as an
\emph{input domain} $S$ of possible values (a mapping from accessors
to target domains).

The equivalence checking algorithm $\SimpleEquiv S {D_S} {D_T}$ takes
an input domain \emph{S} and a pair of source and target decision
trees. In case the two trees are not equivalent, it returns a counter
example.

It is defined exactly as a decision procedure for the provability of
the judgment $(\Equivrel S {D_S} {D_T} \emptyqueue)$, defined below in
the general form $(\Equivrel S {D_S} {D_T} G)$ where $G$ is a
\emph{guard queue}, indicating an imbalance between the guards
observed in the source tree and in the target tree. (For clarity of
exposition, the inference rules do not explain how we build the
counter-example.)

\begin{mathpar}
  \begin{array}{l@{~}r@{~}l}
    & & \text{\emph{input space}} \\
    S & \subseteq & \{ (v_S, v_T) \mid \vrel {v_S} {v_T} \} \\
  \end{array}

  \begin{array}{l@{~}r@{~}l}
    & & \text{\emph{boolean result}} \\
    b & \in & \{ 0, 1 \} \\
  \end{array}

  \begin{array}{l@{~}r@{~}l}
    & & \text{\emph{guard queues}} \\
    G & \bnfeq & (t_1 = b_1), \dots, (t_n = b_n) \\
  \end{array}
\end{mathpar}

The algorithm proceeds by case analysis. Inference rules are shown.
If $S$ is empty the results is $\YesTEX$.
\begin{comment}
OLD INFERENCE RULES:
\begin{mathpar}
  \infer{ }
  {\EquivTEX \emptyset {C_S} {C_T} G}
\end{mathpar}

If the two decision trees are both terminal nodes the algorithm checks
for content equality.
\begin{mathpar}
  \infer{ }
  {\EquivTEX S \Failure \Failure \emptyqueue}
\\
  \infer
  {\trel {t_S} {t_T}}
  {\EquivTEX S {\Leaf {t_S}} {\Leaf {t_T}} \emptyqueue}

\end{mathpar}
\end{comment}
\begin{mathpar}
  \infer[empty]{ }
  {\Equivrel \emptyset {D_S} {D_T} G}

  \infer{ }
  {\Equivrel S \Failure \Failure \emptyqueue}

  \infer
  {\erel {t_S} {t_T}}
  {\Equivrel S {\Leaf {t_S}} {\Leaf {t_T}} \emptyqueue}
\end{mathpar}

If the source decision tree (left hand side) is a terminal while the
target decisiorn tree (right hand side) is not, the algorithm proceeds
by \emph{explosion} of the right hand side. Explosion means that every
child of the right hand side is tested for equality against the left
hand side.

\begin{mathpar}
  \infer[explode-right]
  {D_S \in {\Leaf t, \Failure}
   \\
   \forall i,\; \Equivrel {(S \cap a \in \pi_i)} {D_S} {D_i} G
   \\
   \Equivrel {(S \cap a \notin \Fam i {\pi_i})} {D_S} \Dfb G}
  {\Equivrel S
    {D_S} {\Switch a {\Fam i {\pi_i} {D_i}} \Dfb} G}
\end{mathpar}

\begin{comment}
% TODO: [Gabriel] in practice the $dom_S$ are constructors and the
% $dom_T$ are domains. Do we want to hide this fact, or be explicit
% about it? (Maybe we should introduce explicitly/clearly the idea of
% target domains at some point).
\end{comment}

When the left hand side is not a terminal, the algorithm explodes the
left hand side while trimming every right hand side subtree. Trimming
a left hand side tree on an interval set $dom_S$ computed from the right hand
side tree constructor means mapping every branch condition $dom_T$ (interval set of
possible values) on the left to the intersection of $dom_T$ and $dom_S$ when
the accessors on both side are equal, and removing the branches that
result in an empty intersection. If the accessors are different,
\emph{$dom_T$} is left unchanged.

\begin{mathpar}
  \infer[explode-right]
  {D_S \in {\Leaf t, \Failure}
   \\
   \forall i,\; \Equivrel {(S \cap a \in \pi_i)} {D_S} {D_i} G
   \\
   \Equivrel {(S \cap a \notin \Fam i {\pi_i})} {D_S} \Dfb G}
  {\Equivrel S
    {D_S} {\Switch a {\Fam i {\pi_i} {D_i}} \Dfb} G}
\end{mathpar}

The equivalence checking algorithm deals with guards by storing a
queue. A guard blackbox is pushed to the queue whenever the algorithm
encounters a Guard node on the right, while it pops a blackbox from
the queue whenever a Guard node appears on the left hand side.
The algorithm stops with failure if the popped blackbox and the and
blackbox on the left hand Guard node are different, otherwise in
continues by exploding to two subtrees, one in which the guard
condition evaluates to true, the other when it evaluates to false.
Termination of the algorithm is successful only when the guards queue
is empty.
\begin{mathpar}
  \infer
  {\Equivrel S {D_0} {D_T} {G, (e_S = 0)}
   \\
   \Equivrel S {D_1} {D_T} {G, (e_S = 1)}}
  {\Equivrel S
    {\Guard {e_S} {D_0} {D_1}} {D_T} G}

  \infer
  {\erel {e_S} {e_T}
   \\
   \Equivrel S {D_S} {D_b} G}
  {\Equivrel S
    {D_S} {\Guard {e_T} {D_0} {D_1}} {(e_S = b), G}}

\end{mathpar}
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 by the previous inference rules.
Running a program tₛ or its translation 〚tₛ〛 against an input vₛ
produces as a result /r/ in the following way:
| ( 〚tₛ〛ₛ(vₛ) ≡ Cₛ(vₛ) ) → r
| tₛ(vₛ) → r
Likewise
| ( 〚tₜ〛ₜ(vₜ) ≡ Cₜ(vₜ) ) → r
| tₜ(vₜ) → r
| result r ::= guard list * (Match blackbox \vert{} NoMatch \vert{} Absurd)
| guard ::= blackbox.
Having defined equivalence between two inputs of which one is
expressed in the source language and the other in the target language,
vₛ ≃ vₜ, we can define the equivalence between a couple of programs or
a couple of decision trees 
| tₛ ≃ tₜ := ∀vₛ≃vₜ, tₛ(vₛ) = tₜ(vₜ)
| Cₛ ≃ Cₜ := ∀vₛ≃vₜ, Cₛ(vₛ) = Cₜ(vₜ)
The result of the proposed equivalence algorithm is /Yes/ or /No(vₛ,
vₜ)/. In particular, in the negative case, vₛ and vₜ 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 /r₁ ≃gs r₂/, when:
| (gs₁, r₁) ≃gs (gs₂, r₂)  ⇔  (gs₁, r₁) = (gs₂ ++ gs, r₂)
We say that Cₜ covers the input space /S/, written
/covers(Cₜ, S)/ when every value vₛ∈S is a valid input to the
decision tree Cₜ. (TODO: rephrase)
Given an input space /S/ and a couple of decision trees, where
the target decision tree Cₜ covers the input space /S/ we can define equivalence:
| equiv(S, Cₛ, Cₜ, gs) = Yes ∧ covers(Cₜ, S) → ∀vₛ≃vₜ ∈ S, Cₛ(vₛ) ≃gs Cₜ(vₜ)
Similarly we say that a couple of decision trees in the presence of
an input space /S/ are /not/ equivalent in the following way:
| equiv(S, Cₛ, Cₜ, gs) = No(vₛ,vₜ) ∧ covers(Cₜ, S) → vₛ≃vₜ ∈ S ∧ Cₛ(vₛ) ≠gs Cₜ(vₜ)
Corollary: For a full input space /S/, that is the universe of the
target program:
| equiv(S, 〚tₛ〛ₛ, 〚tₜ〛ₜ, ∅) = Yes  ⇔  tₛ ≃ tₜ


\begin{comment}
TODO: put ^i∈I where needed
\end{comment}
\subsubsection{The trimming lemma}
The trimming lemma allows to reduce the size of a decision tree given
an accessor /a/ → π relation (TODO: expand)
| ∀vₜ ∈ (a→π), Cₜ(vₜ) = C_{t/a→π}(vₜ)
We prove this by induction on Cₜ:

- Cₜ = Leaf_{bb}: when the decision tree is a leaf terminal, the result of trimming on a Leaf is the Leaf itself
    | Leaf_{bb/a→π}(v) = Leaf_{bb}(v)
- The same applies to Failure terminal
    | Failure_{/a→π}(v) = Failure(v)
- When Cₜ = Switch(b, (π→Cᵢ)ⁱ)_{/a→π} then we look at the accessor
    /a/ of the subtree Cᵢ and we define πᵢ' = πᵢ if a≠b else πᵢ∩π Trimming
    a switch node yields the following result:
    | Switch(b, (π→Cᵢ)^{i∈I})_{/a→π} :=  Switch(b, (π'ᵢ→C_{i/a→π})^{i∈I})
\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 vₜ against
the decision tree Cₜ is the same as running vₜ against the tree
C_{trim} that is the result of the trimming operation on Cₜ
| Cₜ(vₜ) = C_{trim}(vₜ) = Switch(b, (πᵢ'→C_{i/a→π})^{i∈I})(vₜ)
We can reason by first noting that when vₜ∉(b→πᵢ)ⁱ the node must be a Failure node.
In the case where ∃k \vert{} vₜ∈(b→πₖ) then we can prove that
| C_{k/a→π}(vₜ) = Switch(b, (πᵢ'→C_{i/a→π})^{i∈I})(vₜ)
because when a ≠ b then πₖ'= πₖ and this means that vₜ∈πₖ'
while when a = b  then πₖ'=(πₖ∩π) and vₜ∈πₖ' because:
- by the hypothesis, vₜ∈π
- we are in the case where vₜ∈πₖ
So vₜ ∈ πₖ' and by induction
| Cₖ(vₜ) = C_{k/a→π}(vₜ)
We also know that ∀vₜ∈(b→πₖ) → Cₜ(vₜ) = Cₖ(vₜ) 
By putting together the last two steps, we have proven the trimming
lemma.

\begin{comment}
TODO: what should I say about covering??? I swap π and π' 
Covering lemma:
        ∀a,π covers(Cₜ,S) → covers(C_{t/a→π}, (S∩a→π))
            Uᵢπⁱ ≈ Uᵢπ'∩(a→π) ≈ (Uᵢπ')∩(a→π) %%


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

\subsubsection{Equivalence checking}
The equivalence checking algorithm takes as parameters an input space
/S/, a source decision tree /Cₛ/ and a target decision tree /Cₜ/:
| equiv(S, Cₛ, Cₜ) → Yes \vert{} No(vₛ, vₜ)
\\
When the algorithm returns Yes and the input space is covered by /Cₛ/
we can say that the couple of decision trees are the same for
every couple of source value /vₛ/ and target value /vₜ/ that are equivalent.
\begin{comment}
Define "covered"
Is it correct to say the same? How to correctly distinguish in words ≃ and = ?
\end{comment}
| equiv(S, Cₛ, Cₜ) = Yes and cover(Cₜ, S) → ∀ vₛ ≃ vₜ∈S ∧ Cₛ(vₛ) = Cₜ(vₜ)
In the case where the algorithm returns No we have at least a couple
of counter example values vₛ and vₜ for which the two decision trees
outputs a different result.
| equiv(S, Cₛ, Cₜ) = No(vₛ,vₜ) and cover(Cₜ, S) → ∀ vₛ ≃ vₜ∈S ∧ Cₛ(vₛ) ≠ Cₜ(vₜ)
We define  the following
| Forall(Yes) = Yes
| Forall(Yes::l) = Forall(l)
| Forall(No(vₛ,vₜ)::_) = No(vₛ,vₜ)
There exists and are injective:
| int(k) ∈ ℕ (arity(k) = 0) 
| tag(k) ∈ ℕ (arity(k) > 0) 
| π(k) = {n\vert int(k) = n} x {n\vert tag(k) = n}
where k is a constructor.
\\
\begin{comment}
TODO: explain:
∀v∈a→π, C_{/a→π}(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}
0. in case of unreachable: 
| Cₛ(vₛ) = Absurd(Unreachable) ≠ Cₜ(vₜ) ∀vₛ,vₜ
1. In the case of an empty input space
   | equiv(∅, Cₛ, Cₜ) := Yes
   and that is trivial to prove because there is no pair of values (vₛ, vₜ) that could be
   tested against the decision trees.
   In the other subcases S is always non-empty.
2. When there are /Failure/ nodes at both sides the result is /Yes/:
   |equiv(S, Failure, Failure) := Yes
   Given that ∀v, Failure(v) = Failure, the statement holds.
3. When we have a Leaf or a Failure at the left side:
   | equiv(S, Failure as Cₛ, Switch(a, (πᵢ → Cₜᵢ)^{i∈I})) :=  Forall(equiv( S∩a→π(kᵢ)), Cₛ, Cₜᵢ)^{i∈I})
   | equiv(S, Leaf bbₛ as Cₛ, Switch(a, (πᵢ → Cₜᵢ)^{i∈I})) :=  Forall(equiv( S∩a→π(kᵢ)), Cₛ, Cₜᵢ)^{i∈I})
   Our algorithm either returns Yes for every sub-input space Sᵢ := S∩(a→π(kᵢ)) and
   subtree Cₜᵢ
   | equiv(Sᵢ, Cₛ, Cₜᵢ) = Yes ∀i
   or we have a counter example vₛ, vₜ for which
   | vₛ≃vₜ∈Sₖ ∧ cₛ(vₛ) ≠ Cₜₖ(vₜ)
   then because 
   | vₜ∈(a→πₖ) → Cₜ(vₜ) = Cₜₖ(vₜ) ,
   | vₛ≃vₜ∈S ∧ Cₛ(vₛ)≠Cₜ(vₜ) 
   we can say that
   | equiv(Sᵢ, Cₛ, Cₜᵢ) = No(vₛ, vₜ) for some minimal k∈I
4. When we have a Switch on the right we define π_{fallback} as the domain of
   values not covered but the union of the constructors kᵢ
   | π_{fallback} = ¬\bigcup\limits_{i∈I}π(kᵢ)
   Our algorithm proceeds by trimming
   | equiv(S, Switch(a, (kᵢ → Cₛᵢ)^{i∈I}, C_{sf}), Cₜ) := 
   | Forall(equiv( S∩(a→π(kᵢ)^{i∈I}), Cₛᵢ, C_{t/a→π(kᵢ)})^{i∈I} +++   equiv(S∩(a→πₙ), Cₛ, C_{a→π_{fallback}}))
    The statement still holds and we show this by first analyzing the
   /Yes/ case:
    | Forall(equiv( S∩(a→π(kᵢ)^{i∈I}), Cₛᵢ, C_{t/a→π(kᵢ)})^{i∈I} = Yes
    The constructor k is either included in the set of constructors kᵢ:
    | k \vert k∈(kᵢ)ⁱ ∧ Cₛ(vₛ) = Cₛᵢ(vₛ)
    We also know that 
    | (1) Cₛᵢ(vₛ) = C_{t/a→πᵢ}(vₜ)      
    | (2) C_{T/a→πᵢ}(vₜ) = Cₜ(vₜ)      
    (1) is true by induction and (2) is a consequence of the trimming lemma.
    Putting everything together:
    | Cₛ(vₛ) = Cₛᵢ(vₛ) = C_{T/a→πᵢ}(vₜ) = Cₜ(vₜ)
            
    When the k∉(kᵢ)ⁱ [TODO]

    The auxiliary Forall function returns /No(vₛ, vₜ)/ when, for a minimum  k,
    | equiv(Sₖ, Cₛₖ, C_{T/a→πₖ} = No(vₛ, vₜ)
    Then we can say that 
    | Cₛₖ(vₛ) ≠ C_{t/a→πₖ}(vₜ)
    that is enough for proving that 
    | Cₛₖ(vₛ) ≠ (C_{t/a→πₖ}(vₜ) = Cₜ(vₜ))