prima versione per coppo

tesi/.gitignore

@@ -1,4 +1,5 @@
 tesi.tex
+.#tesi.tex
 
 ## Core latex/pdflatex auxiliary files:
 *.aux

tesi/conv.py

@@ -3,7 +3,7 @@ import re
 from sys import argv
 
 allsymbols = json.load(open('./unicode-latex.json'))
-mysymbols = ['≡', '≼', '→', '⊀', '⋠', '≺', '∀', '∈', '₂', '₁', 'ₐ', 'ₘ', 'ₙ', 'ᵢ', 'ⁱ']
+mysymbols = ['≡', '≠', '≼', '→', '←', '⊀', '⋠', '≺', '∀', '∈', '₀', '₂', '₁', '₃', 'ₐ', 'ₖ', 'ₘ', 'ₙ', 'ᵢ', 'ⁱ', '⋮']
 
 symbols = {s: allsymbols[s] for s in mysymbols}
 mathsymbols = {s: '$'+allsymbols[s]+'$' for s in symbols}
@@ -18,10 +18,11 @@ def read_by_char(fname):
         for line in fp.readlines():
             cnt += 1
             words = [w.strip() for w in line.split(' ')]
+
             if mathmode_begin.intersection(words):
                 assert mathmode == False
                 mathmode = True
-            elif mathmode_end.intersection(words):
+            if mathmode_end.intersection(words):
                 assert mathmode == True, f'Line: {words}, number: {cnt}'
                 mathmode = False
 

tesi/tesi_unicode.org

@@ -1,13 +1,14 @@
-* TODO Scaletta [1/2]
+\begin{comment}
+* TODO Scaletta [1/4]
   - [X] Abstract
-  - [-] Background [20%]
+  - [-] Background [40%]
     - [X] Ocaml
     - [ ] Lambda code [0%]
       - [ ] Untyped lambda form
       - [ ] OCaml specific instructions
-    - [-] Pattern matching [50%]
+    - [X] Pattern matching [100%]
       - [X] Introduzione
-      - [ ] Compilation to lambda form
+      - [X] Compilation to lambda form
     - [ ] Translation Verification
     - [ ] Symbolic execution
   - [ ] Translation verification of the Pattern Matching Compiler
@@ -20,6 +21,7 @@
     - [ ] Equivalence between source and target
   - [ ] Practical results 
   
+\end{comment}
 
 #+TITLE:   Translation Verification of the OCaml pattern matching compiler
 #+AUTHOR:   Francesco Mecca
@@ -28,8 +30,8 @@
 
 #+LANGUAGE: en
 #+LaTeX_CLASS: article
-#+LaTeX_HEADER: \usepackage[utf8]{inputenc} 
 #+LaTeX_HEADER: \usepackage{algorithm} 
+#+LaTeX_HEADER: \usepackage{comment} 
 #+LaTeX_HEADER: \usepackage{algpseudocode} 
 #+LaTeX_HEADER: \usepackage{amsmath,amssymb,amsthm}
 #+Latex_HEADER: \newtheorem{definition}{Definition}
@@ -102,96 +104,120 @@ inheritance, subtyping and dynamic binding, and modules, that
 provide a way to encapsulate definitions. Modules are checked
 statically and can be reificated through functors.
 
-*** TODO Pattern matching [37%] 
+*** Pattern matching
-- [ ] capisci come mettere gli esempi uno accanto all'altro
 
 Pattern matching is a widely adopted mechanism to interact with ADT.
 C family languages provide branching on predicates through the use of
 if statements and switch statements.
-Pattern matching is a mechanism for destructuring and analyzing data
+Pattern matching on the other hands express predicates through
-structures for the presence of values simbolically represented as
+syntactic templates that also allow to bind on data structures of
-tokens. One common example of pattern matching is the use of regular
+arbitrary shapes. One common example of pattern matching is the use of regular
-expressions on strings. OCaml provides pattern matching on ADT,
+expressions on strings. OCaml 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.
 
-- [X] Esempio enum, C e Ocaml
 #+BEGIN_SRC ocaml
-type color = | Red | Blue | Green
+type color = | Red | Blue | Green | Black | White
 
-begin match color with
+match color with
 | Red -> print "red"
 | Blue -> print "red"
 | Green -> print "red"
-
+| _ -> print "white or black"
 #+END_SRC
 
-OCaml provides tokens to express data destructoring
+OCaml provides tokens to express data destructoring.
+For example we can examine the content of a list with patten matching
 
-
-- [X] Esempio destructor list
 #+BEGIN_SRC ocaml
 
 begin match list with
 | [ ] -> print "empty list"
 | element1 :: [ ] -> print "one element"
-| element1 :: element2 :: [ ] -> print "two elements"
+| (element1 :: element2) :: [ ] -> print "two elements"
 | head :: tail-> print "head followed by many elements"
 #+END_SRC
 
-- [X] Esempio destructor tuples
+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 ocaml
 
 begin match tuple with
 | (Some _, Some _) -> print "Pair of optional types"
-| (Some _, None) -> print "Pair of optional types, last null"
+| (Some _, None) | (None, Some _) -> print "Pair of optional types, one of which is null"
-| (None, Some _) -> print "Pair of optional types, first null"
 | (None, None) -> print "Pair of optional types, both null"
 #+END_SRC
 
-Pattern clauses can make the use of /guards/ to test predicates and
+The pattern pattern₁ |  pattern₂ represents the logical "or" of the
-variables can be binded in scope.
+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).
 
-- [ ] Esempio binding e guards
 #+BEGIN_SRC ocaml
 
 begin match token_list with
-| "switch"::var::"{"::rest ->
+| "switch"::var::"{"::rest -> ...
-| "case"::":"::var::rest when is_int var ->
+| "case"::":"::var::rest when is_int var -> ...
-| "case"::":"::var::rest when is_string var ->
+| "case"::":"::var::rest when is_string var -> ...
-| "}"::[ ] -> stop ()
+| "}"::[ ] -> ...
 | "}"::rest -> error "syntax error: " rest
 
 #+END_SRC
 
-- [ ] Un altro esempio con destructors e tutto i lresto
+Moreover, the OCaml pattern matching compiler emits a warning when a
+pattern is not exhaustive or some patterns are shadowed by precedent ones.
 
 In general pattern matching on primitive and algebraic data types takes the
 following form.
 
-- [ ] Esempio informale
+|match variable with
+|\vert pattern₁ -> expr₁
+|\vert pattern₂ when guard -> expr₂
+|\vert pattern₃ as var -> expr₃ 
+|⋮
+|\vert pₙ -> exprₙ
 
 It can be described more formally through a BNF grammar.
 
-- [ ] BNF
+#+BEGIN_SRC bnf
 
-- [ ] Come funziona il pattern matching?
+pattern ::= value-name  
+| _   ;; wildcard pattern
+| constant  ;; matches a constant value
+| pattern as  value-name  ;; binds to value-name
+| ( pattern ) ;; parenthesized pattern
+| pattern |  pattern ;; or-pattern 
+| constr  pattern ;; variant pattern
+| [ pattern  { ; pattern }  [ ; ] ] ;; list patterns 
+| pattern ::  pattern   ;; lists patterns using cons operator (::)
+| [| pattern  { ; pattern }  [ ; ] |] ;; array pattern 
+| char-literal ..  char-literal   ;; match on a range of characters
+| { field  [: typexpr]  [= pattern] { ; field  [: typexpr]  [= pattern] } \
+  [; _ ] [ ; ] } ;; patterns that match on records
 
-*** TODO 1.2.1 Pattern matching compilation to lambda code
+#+END_SRC
 
-- [ ] Da tabella a matrice
 
-Formally, pattern are defined as follows:
+*** 1.2.1 Pattern matching compilation to lambda code
-| pattern         | Patterns            |
+
+During compilation, patterns are in the form
+| pattern         | type of pattern    |
 |-----------------+---------------------|
 | _               | wildcard            |
 | x               | variable            |
 | c(p₁,p₂,...,pₙ) | constructor pattern |
 | (p₁\vert p₂)    | or-pattern          |
 
-Values are defined as follows:
+Expressions are compiled into lambda code and are referred as lambda
-| values              | Values            |
+code actions.
-|---------------------+-------------------|
-| c(v₁,  v₂, ..., vₙ) | constructor value |
 
 The entire pattern matching code can be represented as a clause matrix
 that associates rows of patterns (p_{i,1}, p_{i,2}, ..., p_{i,n}) to
@@ -220,7 +246,6 @@ called a constant constructor.
 The pattern /p/ matches a value /v/, written as p ≼ v, when one of the
 following rules apply
 
-|                    |   |                    |                                           |
 |--------------------+---+--------------------+-------------------------------------------|
 | _                  | ≼ | v                  | ∀v                                        |
 | x                  | ≼ | v                  | ∀v                                        |
@@ -229,9 +254,9 @@ following rules apply
 | (p₁, p₂, ..., pₐ)  | ≼ | (v₁, v₂, ..., vₐ)  | iff pᵢ ≼ vᵢ ∀i ∈ [1..a]                   |
 |--------------------+---+--------------------+-------------------------------------------|
 
-We can also say that /v/ is an /instance/ of /p/.
+When a value /v/ matches pattern /p/ we say that /v/ is an /instance/ of /p/.
 
-When we consider the pattern matrix P we say that the value vector
+Considering the pattern matrix P we say that the value vector
 $\vec{v}$ = (v₁, v₂, ..., vᵢ) matches the line number i in P  if and only if the following two
 conditions are satisfied:
 - p_{i,1}, p_{i,2}, \cdots, p_{i,n}  ≼ (v₁, v₂, ..., vᵢ)
@@ -244,28 +269,129 @@ We can define the following three relations with respect to patterns:
   are the same
 - Patterns p and q are compatible when they share a common instance
 
-** 1.2.1.1 Initial state of the compilation
+**** Initial state of the compilation
 
 Given a source of the following form:
 
-#+BEGIN_SRC ocaml
-match x with 
-| p₁ -> e₁
-| p₂ -> e₂
-...
-| pₘ -> eₘ
-#+END_SRC ocaml
 
-the initial input of the algorithm consists of a vector of variables
+|match variable with
-$\vec{x}$ = (x₁, x₂, ..., xₙ)  of size n
+|\vert pattern₁ -> e₁
-and a clause matrix P → L  of width n and height m.
+|\vert pattern₂ -> e₂
+|⋮
+|\vert pₘ -> eₘ
+
+the initial input of the 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 a clause matrix P → L of width n and height m.
+That is:
 
 \begin{equation*}
-(P → L) =
+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 \\
+\vdots & \vdots & \ddots & \vdots →  \vdots \\
-p_{m,1} & p_{m,2} & \cdots & p_{m,n} → lₘ
+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 empty, that is $\vec{x}$ = (), then 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*}
+    \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 there is a binding of x₁ to the
+    variable that appears on the pattern $p_{i,1}. Bindings are omitted
+    for wildcard patterns and the lambda action lᵢ remains unchanged.
+
+2) Constructor rule: if all patterns in the first column of P are
+   constructors patterns of the form k(q₁, q₂, ..., qₙ) we define a
+   new matrix, the specialized clause matrix S, by applying the
+   following transformation on every row /p/:
+    \begin{lstlisting}[mathescape,columns=fullflexible,basicstyle=\fontfamily{lmvtt}\selectfont,]
+    for every c ∈ Set of constructors 
+        for i ← 1 .. m 
+            let kᵢ ← constructor_of($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 new fresh 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
+   {c₁, c₂, ..., cₖ} 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}
+
+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} → lᵢ
+   \end{equation*}
+   results in three rows added to the clause matrix
+   \begin{equation*}
+   p_{i,1}, p_{i,2}, ..., p_{i,m} → lᵢ \\
+   \end{equation*}
+   \begin{equation*}
+   q_{i,1}, p_{i,2}, ..., p_{i,m} → lᵢ \\
+   \end{equation*}
+   \begin{equation*}
+   r_{i,1}, p_{i,2}, ..., p_{i,m} → lᵢ 
+   \end{equation*}
+4) Mixture rule:
+   When none of the previous rules apply the clause matrix P → L is
+   splitted into two clause matrices, the first P₁ → L₁ that is the
+   largest prefix matrix for which one of the three previous rules
+   apply, and P₂ → L₂ containing the remaining rows. The algorithm is
+   applied to both matrices.
+
+\begin{comment}
+#+BEGIN_COMMENT
+CITE paper?
+#+END_COMMENT’ 
+\end{comment}

tesi/tesi_unicode.tex

@@ -1,4 +1,4 @@
-% Created 2020-02-24 Mon 14:37
+% Created 2020-02-24 Mon 19:44
 % Intended LaTeX compiler: pdflatex
 \documentclass[11pt]{article}
 \usepackage[utf8]{inputenc}
@@ -14,8 +14,8 @@
 \usepackage{amssymb}
 \usepackage{capt-of}
 \usepackage{hyperref}
-\usepackage[utf8]{inputenc}
 \usepackage{algorithm}
+\usepackage{comment}
 \usepackage{algpseudocode}
 \usepackage{amsmath,amssymb,amsthm}
 \newtheorem{definition}{Definition}
@@ -35,11 +35,12 @@
 \begin{document}
 
 \maketitle
-\section{{\bfseries\sffamily TODO} Scaletta [1/2]}
+\begin{comment}
-\label{sec:org8ebbc9e}
+\section{{\bfseries\sffamily TODO} Scaletta [1/4]}
+\label{sec:orgdce55fc}
 \begin{itemize}
 \item[{$\boxtimes$}] Abstract
-\item[{$\boxminus$}] Background [20\%]
+\item[{$\boxminus$}] Background [40\%]
 \begin{itemize}
 \item[{$\boxtimes$}] Ocaml
 \item[{$\square$}] Lambda code [0\%]
@@ -47,10 +48,10 @@
 \item[{$\square$}] Untyped lambda form
 \item[{$\square$}] OCaml specific instructions
 \end{itemize}
-\item[{$\boxminus$}] Pattern matching [50\%]
+\item[{$\boxtimes$}] Pattern matching [100\%]
 \begin{itemize}
 \item[{$\boxtimes$}] Introduzione
-\item[{$\square$}] Compilation to lambda form
+\item[{$\boxtimes$}] Compilation to lambda form
 \end{itemize}
 \item[{$\square$}] Translation Verification
 \item[{$\square$}] Symbolic execution
@@ -69,11 +70,10 @@
 \end{itemize}
 \item[{$\square$}] Equivalence between source and target
 \end{itemize}
-\begin{itemize}
 \item[{$\square$}] Practical results
 \end{itemize}
-\end{itemize}
 
+\end{comment}
 
 \begin{abstract}
 
@@ -92,10 +92,10 @@ blackboxes that are not evaluated in the context of the verification.
 \end{abstract}
 
 \section{Background}
-\label{sec:org73f49d4}
+\label{sec:orgf336654}
 
 \subsection{OCaml}
-\label{sec:orga0f2160}
+\label{sec:org885a2ca}
 Objective Caml (OCaml) is a dialect of the ML (Meta-Language) family of programming
 languages.
 OCaml shares many features with other dialects of ML, such as SML and Caml Light,
@@ -142,108 +142,122 @@ provide a way to encapsulate definitions. Modules are checked
 statically and can be reificated through functors.
 
 \begin{enumerate}
-\item {\bfseries\sffamily TODO} Pattern matching [37\%]
+\item Pattern matching
-\label{sec:org8bcefd9}
+\label{sec:org879bb72}
-\begin{itemize}
-\item[{$\square$}] capisci come mettere gli esempi uno accanto all'altro
-\end{itemize}
 
 Pattern matching is a widely adopted mechanism to interact with ADT.
 C family languages provide branching on predicates through the use of
 if statements and switch statements.
-Pattern matching is a mechanism for destructuring and analyzing data
+Pattern matching on the other hands express predicates through
-structures for the presence of values simbolically represented as
+syntactic templates that also allow to bind on data structures of
-tokens. One common example of pattern matching is the use of regular
+arbitrary shapes. One common example of pattern matching is the use of regular
-expressions on strings. OCaml provides pattern matching on ADT,
+expressions on strings. OCaml provides pattern matching on ADT and
 primitive data types.
-
+The result of a pattern matching operation is always one of:
 \begin{itemize}
-\item[{$\boxtimes$}] Esempio enum, C e Ocaml
+\item this value does not match this pattern”
+\item this value matches this pattern, resulting the following bindings of
+names to values and the jump to the expression pointed at the
+pattern.
 \end{itemize}
-\begin{verbatim}
-type color = | Red | Blue | Green
 
-begin match color with
+\begin{verbatim}
+type color = | Red | Blue | Green | Black | White
+
+match color with
 | Red -> print "red"
 | Blue -> print "red"
 | Green -> print "red"
-
+| _ -> print "white or black"
 \end{verbatim}
 
-OCaml provides tokens to express data destructoring
+OCaml provides tokens to express data destructoring.
+For example we can examine the content of a list with patten matching
 
-
-\begin{itemize}
-\item[{$\boxtimes$}] Esempio destructor list
-\end{itemize}
 \begin{verbatim}
 
 begin match list with
 | [ ] -> print "empty list"
 | element1 :: [ ] -> print "one element"
-| element1 :: element2 :: [ ] -> print "two elements"
+| (element1 :: element2) :: [ ] -> print "two elements"
 | head :: tail-> print "head followed by many elements"
 \end{verbatim}
 
-\begin{itemize}
+Parenthesized patterns, such as the third one in the previous example,
-\item[{$\boxtimes$}] Esempio destructor tuples
+matches the same value as the pattern without parenthesis.
-\end{itemize}
+
+The same could be done with tuples
 \begin{verbatim}
 
 begin match tuple with
 | (Some _, Some _) -> print "Pair of optional types"
-| (Some _, None) -> print "Pair of optional types, last null"
+| (Some _, None) | (None, Some _) -> print "Pair of optional types, one of which is null"
-| (None, Some _) -> print "Pair of optional types, first null"
 | (None, None) -> print "Pair of optional types, both null"
 \end{verbatim}
 
-Pattern clauses can make the use of \emph{guards} to test predicates and
+The pattern pattern₁ |  pattern₂ represents the logical "or" of the
-variables can be binded in scope.
+two patterns pattern₁ and pattern₂. A value matches pattern₁ |
+pattern₂ if it matches pattern₁ or pattern₂.
+
+Pattern clauses can make the use of \emph{guards} to test predicates and
+variables can captured (binded in scope).
 
-\begin{itemize}
-\item[{$\square$}] Esempio binding e guards
-\end{itemize}
 \begin{verbatim}
 
 begin match token_list with
-| "switch"::var::"{"::rest ->
+| "switch"::var::"{"::rest -> ...
-| "case"::":"::var::rest when is_int var ->
+| "case"::":"::var::rest when is_int var -> ...
-| "case"::":"::var::rest when is_string var ->
+| "case"::":"::var::rest when is_string var -> ...
-| "}"::[ ] -> stop ()
+| "}"::[ ] -> ...
 | "}"::rest -> error "syntax error: " rest
 
 \end{verbatim}
 
-\begin{itemize}
+Moreover, the OCaml pattern matching compiler emits a warning when a
-\item[{$\square$}] Un altro esempio con destructors e tutto i lresto
+pattern is not exhaustive or some patterns are shadowed by precedent ones.
-\end{itemize}
 
 In general pattern matching on primitive and algebraic data types takes the
 following form.
 
-\begin{itemize}
+\begin{center}
-\item[{$\square$}] Esempio informale
+\begin{tabular}{l}
-\end{itemize}
+match variable with\\
+\(\vert{}\) pattern₁ -> expr₁\\
+\(\vert{}\) pattern₂ when guard -> expr₂\\
+\(\vert{}\) pattern₃ as var -> expr₃\\
+⋮\\
+\(\vert{}\) pₙ -> exprₙ\\
+\end{tabular}
+\end{center}
 
 It can be described more formally through a BNF grammar.
 
-\begin{itemize}
+\begin{verbatim}
-\item[{$\square$}] BNF
 
-\item[{$\square$}] Come funziona il pattern matching?
+pattern ::= value-name  
-\end{itemize}
+| _   ;; wildcard pattern
+| constant  ;; matches a constant value
+| pattern as  value-name  ;; binds to value-name
+| ( pattern ) ;; parenthesized pattern
+| pattern |  pattern ;; or-pattern 
+| constr  pattern ;; variant pattern
+| [ pattern  { ; pattern }  [ ; ] ] ;; list patterns 
+| pattern ::  pattern   ;; lists patterns using cons operator (::)
+| [| pattern  { ; pattern }  [ ; ] |] ;; array pattern 
+| char-literal ..  char-literal   ;; match on a range of characters
+| { field  [: typexpr]  [= pattern] { ; field  [: typexpr]  [= pattern] } \
+  [; _ ] [ ; ] } ;; patterns that match on records
 
-\item {\bfseries\sffamily TODO} 1.2.1 Pattern matching compilation to lambda code
+\end{verbatim}
-\label{sec:orgd51f0ff}
 
-\begin{itemize}
-\item[{$\square$}] Da tabella a matrice
-\end{itemize}
 
-Formally, pattern are defined as follows:
+\item 1.2.1 Pattern matching compilation to lambda code
+\label{sec:org5e1ded7}
+
+During compilation, patterns are in the form
 \begin{center}
 \begin{tabular}{ll}
-pattern & Patterns\\
+pattern & type of pattern\\
 \hline
 \_ & wildcard\\
 x & variable\\
@@ -252,14 +266,8 @@ c(p₁,p₂,\ldots{},pₙ) & constructor pattern\\
 \end{tabular}
 \end{center}
 
-Values are defined as follows:
+Expressions are compiled into lambda code and are referred as lambda
-\begin{center}
+code actions.
-\begin{tabular}{ll}
-values & Values\\
-\hline
-c(v₁,  v₂, \ldots{}, vₙ) & constructor value\\
-\end{tabular}
-\end{center}
 
 The entire pattern matching code can be represented as a clause matrix
 that associates rows of patterns (p\(_{\text{i,1}}\), p\(_{\text{i,2}}\), \ldots{}, p\(_{\text{i,n}}\)) to
@@ -290,7 +298,6 @@ following rules apply
 
 \begin{center}
 \begin{tabular}{llll}
- &  &  & \\
 \hline
 \_ & ≼ & v & ∀v\\
 x & ≼ & v & ∀v\\
@@ -301,9 +308,9 @@ c(p₁, p₂, \ldots{}, pₐ) & ≼ & c(v₁, v₂, \ldots{}, vₐ) & iff (p₁,
 \end{tabular}
 \end{center}
 
-We can also say that \emph{v} is an \emph{instance} of \emph{p}.
+When a value \emph{v} matches pattern \emph{p} we say that \emph{v} is an \emph{instance} of \emph{p}.
 
-When we consider the pattern matrix P we say that the value vector
+Considering the pattern matrix P we say that the value vector
 \(\vec{v}\) = (v₁, v₂, \ldots{}, vᵢ) matches the line number i in P  if and only if the following two
 conditions are satisfied:
 \begin{itemize}
@@ -319,40 +326,143 @@ instances of q are instances of p
 are the same
 \item Patterns p and q are compatible when they share a common instance
 \end{itemize}
-\end{enumerate}
 
-\subsection{1.2.1.1 Initial state of the compilation}
+\begin{enumerate}
-\label{sec:org333d6ea}
+\item Initial state of the compilation
+\label{sec:org886c2ed}
 
 Given a source of the following form:
 
-\#+BEGIN\_SRC ocaml
-match x with 
-\begin{center}
-\begin{tabular}{l}
-p₁ -> e₁\\
-p₂ -> e₂\\
-\end{tabular}
-\end{center}
-\ldots{}
-\begin{center}
-\begin{tabular}{l}
-pₘ -> eₘ\\
-\end{tabular}
-\end{center}
-\#+END\_SRC ocaml
 
-the initial input of the algorithm consists of a vector of variables
+\begin{center}
-\(\vec{x}\) = (x₁, x₂, \ldots{}, xₙ)  of size n
+\begin{tabular}{l}
-and a clause matrix P → L  of width n and height m.
+match variable with\\
+\(\vert{}\) pattern₁ -> e₁\\
+\(\vert{}\) pattern₂ -> e₂\\
+⋮\\
+\(\vert{}\) pₘ -> eₘ\\
+\end{tabular}
+\end{center}
+
+the initial input of the algorithm C consists of a vector of variables
+\(\vec{x}\) = (x₁, x₂, \ldots{}, xₙ)  of size \emph{n} where \emph{n} is the arity of
+the type of \emph{x} and a clause matrix P → L of width n and height m.
+That is:
 
 \begin{equation*}
-(P → L) =
+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 \\
+\vdots & \vdots & \ddots & \vdots →  \vdots \\
-p_{m,1} & p_{m,2} & \cdots & p_{m,n} → lₘ
+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 empty, that is \(\vec{x}\) = (), then 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:
+
+\begin{enumerate}
+\item 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*}
+\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 there is a binding of x₁ to the
+variable that appears on the pattern \$p\(_{\text{i,1}}\). Bindings are omitted
+for wildcard patterns and the lambda action lᵢ remains unchanged.
+
+\item Constructor rule: if all patterns in the first column of P are
+constructors patterns of the form k(q₁, q₂, \ldots{}, qₙ) we define a
+new matrix, the specialized clause matrix S, by applying the
+following transformation on every row \emph{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 new fresh variables y₁, y₂, \ldots{}, yₐ so
+that the lambda action lᵢ becomes
+\end{enumerate}
+
+\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
+\{c₁, c₂, \ldots{}, cₖ\} 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{enumerate}
+\item 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} → lᵢ
+\end{equation*}
+results in three rows added to the clause matrix
+\begin{equation*}
+p_{i,1}, p_{i,2}, ..., p_{i,m} → lᵢ \\
+\end{equation*}
+\begin{equation*}
+q_{i,1}, p_{i,2}, ..., p_{i,m} → lᵢ \\
+\end{equation*}
+\begin{equation*}
+r_{i,1}, p_{i,2}, ..., p_{i,m} → lᵢ 
+\end{equation*}
+\item Mixture rule:
+When none of the previous rules apply the clause matrix P → L is
+splitted into two clause matrices, the first P₁ → L₁ that is the
+largest prefix matrix for which one of the three previous rules
+apply, and P₂ → L₂ containing the remaining rows. The algorithm is
+applied to both matrices.
+\end{enumerate}
+
+\begin{comment}
+#+BEGIN_COMMENT
+CITE paper?
+#+END_COMMENT’ 
+\end{comment}
+\end{enumerate}
+\end{enumerate}
 \end{document}