1773 lines
72 KiB
Org Mode
1773 lines
72 KiB
Org Mode
\begin{comment}
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TODO: not all todos are explicit. Check every comment section
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TODO: chiedi a Gabriel se T e S vanno bene, ma prima controlla che siano coerenti
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* TODO Scaletta [1/5]
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- [X] Introduction
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- [-] Background [80%]
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- [X] Low level representation
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- [X] Lambda code [0%]
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- [X] Pattern matching
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- [X] Symbolic execution
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- [ ] Translation Validation
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- [-] Translation validation of the Pattern Matching Compiler
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- [X] Source translation
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- [X] Formal Grammar
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- [X] Compilation of source patterns
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- [ ] Target translation
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- [ ] Formal Grammar
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- [ ] Symbolic execution of the Lambda target
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- [X] Equivalence between source and target
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- [ ] Practical results
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\end{comment}
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||
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#+TITLE: Translation Verification of the OCaml pattern matching compiler
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#+AUTHOR: Francesco Mecca
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#+EMAIL: me@francescomecca.eu
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#+DATE:
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#+LANGUAGE: en
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#+LaTeX_CLASS: article
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#+LaTeX_HEADER: \linespread{1.25}
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#+LaTeX_HEADER: \usepackage{algorithm}
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#+LaTeX_HEADER: \usepackage{comment}
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#+LaTeX_HEADER: \usepackage{algpseudocode}
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#+LaTeX_HEADER: \usepackage{amsmath,amssymb,amsthm}
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#+LaTeX_HEADER: \newtheorem{definition}{Definition}
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#+LaTeX_HEADER: \usepackage{mathpartir}
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#+LaTeX_HEADER: \usepackage{graphicx}
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#+LaTeX_HEADER: \usepackage{listings}
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#+LaTeX_HEADER: \usepackage{color}
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#+LaTeX_HEADER: \usepackage{stmaryrd}
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#+LaTeX_HEADER: \newcommand{\semTEX}[1]{{\llbracket{#1}\rrbracket}}
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#+LaTeX_HEADER: \newcommand{\EquivTEX}[3]{\mathsf{equiv}(#1, #2, #3)} % \equiv is already taken
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#+LaTeX_HEADER: \newcommand{\coversTEX}[2]{#1 \mathrel{\mathsf{covers}} #2}
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#+LaTeX_HEADER: \newcommand{\YesTEX}{\mathsf{Yes}}
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#+LaTeX_HEADER: \newcommand{\DZ}{\backslash\text{\{0\}}}
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#+LaTeX_HEADER: \newcommand{\NoTEX}[2]{\mathsf{No}(#1, #2)}
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#+LaTeX_HEADER: \usepackage{comment}
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#+LaTeX_HEADER: \usepackage{mathpartir}
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#+LaTeX_HEADER: \usepackage{stmaryrd} % llbracket, rrbracket
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#+LaTeX_HEADER: \usepackage{listings}
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#+LaTeX_HEADER: \usepackage{notations}
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#+LaTeX_HEADER: \lstset{
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#+LaTeX_HEADER: mathescape=true,
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#+LaTeX_HEADER: language=[Objective]{Caml},
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#+LaTeX_HEADER: basicstyle=\ttfamily,
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#+LaTeX_HEADER: extendedchars=true,
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#+LaTeX_HEADER: showstringspaces=false,
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#+LaTeX_HEADER: aboveskip=\smallskipamount,
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#+LaTeX_HEADER: % belowskip=\smallskipamount,
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#+LaTeX_HEADER: columns=fullflexible,
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#+LaTeX_HEADER: moredelim=**[is][\color{blue}]{/*}{*/},
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#+LaTeX_HEADER: moredelim=**[is][\color{green!60!black}]{/!}{!/},
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#+LaTeX_HEADER: moredelim=**[is][\color{orange}]{/(}{)/},
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#+LaTeX_HEADER: moredelim=[is][\color{red}]{/[}{]/},
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#+LaTeX_HEADER: xleftmargin=1em,
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#+LaTeX_HEADER: }
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#+LaTeX_HEADER: \lstset{aboveskip=0.4ex,belowskip=0.4ex}
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#+EXPORT_SELECT_TAGS: export
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#+EXPORT_EXCLUDE_TAGS: noexport
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#+OPTIONS: H:2 toc:nil \n:nil @:t ::t |:t ^:{} _:{} *:t TeX:t LaTeX:t
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#+STARTUP: showall
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\section{Introduction}
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This dissertation presents an algorithm for the translation validation of the OCaml pattern
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matching compiler. Given a source program and its compiled version the
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algorithm checks whether the two are equivalent or produce a counter
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example in case of a mismatch.
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||
For the prototype of this algorithm we have chosen a subset of the OCaml
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||
language and implemented a prototype equivalence checker along with a
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||
formal statement of correctness and its proof.
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||
The prototype is to be included in the OCaml compiler infrastructure
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||
and will aid the development.
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||
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||
Our equivalence algorithm works with decision trees. Source patterns are
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||
converted into a decision tree using a matrix decomposition algorithm.
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||
Target programs, described in the Lambda intermediate
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representation language of the OCaml compiler, are turned into decision trees
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||
by applying symbolic execution.
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||
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||
\subsection{Motivation}
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Pattern matching in computer science is widely
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employed technique for describing computation as well as deduction.
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Pattern matching is central in many programming languages, such as the
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ML family languages, Haskell and Scala, model checkers, such as Murphi
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||
and proof assistants such as Coq and Isabelle. The work done in this
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||
thesis provides a general method that can be applied to compilers such
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||
as the Kotlin compiler and the C++ compiler that are considering to
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||
implement pattern matching as a first class citizen in the language.
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||
|
||
The work focused on the OCaml pattern matching compiler that is a
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critical part of the OCaml compiler in terms of correctness because
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||
bugs typically would result in wrong code production rather than
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||
triggering compilation failures. Such bugs also are hard to catch by
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||
testing because they arise in corner cases of complex patterns which
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||
are typically not in the compiler test suite or most user programs.
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||
|
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The OCaml core developers group considered evolving the pattern matching compiler, either by
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||
using a new algorithm or by incremental refactoring of its code base.
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||
For this reason we want to verify that new implementations of the
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||
compiler avoid the introduction of new bugs and that such
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modifications don't result in a different behavior than the current one.
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||
|
||
One possible approach is to formally verify the pattern matching compiler
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implementation using a machine checked proof.
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Another possibility, albeit with a weaker result, is to verify that
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each source program and target program pair are semantically correct.
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||
We chose the latter technique, translation validation because is easier to adopt in
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||
the case of a production compiler and to integrate with an existing code base. The compiler is treated as a
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black-box and proof only depends on our equivalence algorithm.
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||
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||
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||
\subsection{The Pattern Matching Compiler}
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||
A pattern matching compiler turns a series of pattern matching clauses
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||
into simple control flow structures such as \texttt{if, switch}, for example:
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||
\begin{lstlisting}
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||
match scrutinee with
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||
| [] -> (0, None)
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||
| x::[] -> (1, Some x)
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| _::y::_ -> (2, Some y)
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\end{lstlisting}
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||
Given as input to the pattern matching compiler, this snippet of code
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||
gets translated into the Lambda intermediate representation of
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the OCaml compiler. The Lambda representation of a program is shown by
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||
calling the \texttt{ocamlc} compiler with \texttt{-drawlambda} flag.
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||
In this example we renamed the variables assigned in order to ease the
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||
understanding of the tests that are performed when the code is
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translated into the Lambda form.
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||
code phase.
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\begin{lstlisting}
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||
(function scrutinee
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||
(if scrutinee ;;; true when scrutinee (list) not empty
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||
(let (tail =a (field 1 scrutinee/81)) ;;; assignment
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||
(if tail
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||
(let
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||
y =a (field 0 tail))
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||
;;; y is the first element of the tail
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(makeblock 0 2 (makeblock 0 y)))
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||
;;; allocate memory for tuple (2, Some y)
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(let (x =a (field 0 scrutinee))
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||
;;; x is the head of the scrutinee
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||
(makeblock 0 1 (makeblock 0 x)))))
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||
;;; allocate memory for tuple (1, Some x)
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[0: 0 0a]))) ;;; low level representatio of (0, None)
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\end{lstlisting}
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||
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||
\subsection{Our approach}
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||
Our algorithm translates both source and target programs into a common
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representation, decision trees. Decision trees where chosen because
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||
they model the space of possible values at a given branch of
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execution.
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Here are the decision trees for the source and target example program.
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||
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||
\begin{minipage}{0.5\linewidth}
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\scriptsize
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||
\begin{verbatim}
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||
Switch(Root)
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||
/ \
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(= []) (= ::)
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/ \
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Leaf Switch(Root.1)
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(0, None) / \
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(= []) (= ::)
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||
/ \
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||
Leaf Leaf
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||
[x = Root.0] [y = Root.1.0]
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||
(1, Some x) (2, Some y)
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||
\end{verbatim}
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||
\end{minipage}
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||
\hfill
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||
\begin{minipage}{0.4\linewidth}
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||
\scriptsize
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||
\begin{verbatim}
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||
Switch(Root)
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||
/ \
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||
(= int 0) (!= int 0)
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||
/ \
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Leaf Switch(Root.1)
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||
(mkblock 0 / \
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||
0 0a) / \
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||
(= int 0) (!= int 0)
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||
/ \
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||
Leaf Leaf
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||
[x = Root.0] [y = Root.1.0]
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(mkblock 0 (mkblock 0
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1 (mkblock 0 x)) 2 (mkblock 0 y))
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||
\end{verbatim}
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||
\end{minipage}
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||
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||
\texttt{(Root.0)} is called an \emph{accessor}, that represents the
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||
access path to a value that can be reached by deconstructing the
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||
scrutinee. In this example \texttt{Root.0} is the first subvalue of
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||
the scrutinee.
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||
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||
Target decision trees have a similar shape but the tests on the
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||
branches are related to the low level representation of values in
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Lambda code. For example, cons cells \texttt{x::xs} or tuples
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\texttt{(x,y)} are blocks with tag 0.
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||
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To check the equivalence of a source and a target decision tree,
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we proceed by case analysis.
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||
If we have two terminals, such as leaves in the previous example,
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||
we check that the two right-hand-sides are equivalent.
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If we have a node $N$ and another tree $T$ we check equivalence for
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||
each child of $N$, which is a pair of a branch condition $\pi_i$ and a
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||
subtree $C_i$. For every child $(\pi_i, C_i)$ we reduce $T$ by killing all
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||
the branches that are incompatible with $\pi_i$ and check that the
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||
reduced tree is equivalent to $C_i$.
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||
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||
\subsection{From source programs to decision trees}
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||
Our source language supports integers, lists, tuples and all algebraic
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||
datatypes. Patterns support wildcards, constructors and literals,
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||
Or-patterns such as $(p_1 | p_2)$ and pattern variables.
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||
In particular Or-patterns provide a more compact way to group patterns
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||
that point to the same expression.
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||
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\begin{minipage}{0.4\linewidth}
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||
\begin{lstlisting}
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||
match w with
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| p₁ -> expr
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| p₂ -> expr
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| p₃ -> expr
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\end{lstlisting}
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||
\end{minipage}
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||
\begin{minipage}{0.6\linewidth}
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||
\begin{lstlisting}
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||
match w with
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| p₁|p₂|p₃ -> expr
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\end{lstlisting}
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\end{minipage}
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||
We also support \texttt{when} guards, which are interesting as they
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introduce the evaluation of expressions during matching.
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This is the type definition of decision tree as they are used in the
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prototype implementation:
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\begin{lstlisting}
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type decision_tree =
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| Unreachable
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| Failure
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| Leaf of source_expr
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| Guard of source_expr * decision_tree * decision_tree
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||
| Switch of accessor * (constructor * decision_tree) list * decision_tree
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||
\end{lstlisting}
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||
In the \texttt{Switch} node we have one subtree for every head constructor
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that appears in the pattern matching clauses and a fallback case for
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other values. The branch condition $\pi_i$ expresses that the value at the
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switch accessor starts with the given constructor.
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\texttt{Failure} nodes express match failures for values that are not
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matched by the source clauses.
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\texttt{Unreachable} is used when we statically know that no value
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can flow to that subtree.
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||
We write 〚tₛ〛ₛ to denote the translation of the source program (the
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set of pattern matching clauses) into a decision tree, computed by a matrix decomposition algorithm (each column
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||
decomposition step gives a \texttt{Switch} node).
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||
It satisfies the following correctness statement:
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||
\[
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\forall t_s, \forall v_s, \quad t_s(v_s) = \semTEX{t_s}_s(v_s)
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||
\]
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The correctness statement intuitively states that for every source
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program, for every source value that is well-formed input to a source
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||
program, running the program tₛ against the input value vₛ is the same
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as running the compiled source program 〚tₛ〛 (that is a decision tree) against the same input
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value vₛ".
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||
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||
\subsection{From target programs to decision trees}
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||
The target programs include the following Lambda constructs:
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||
\texttt{let, if, switch, Match\_failure, catch, exit, field} and
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||
various comparison operations, guards. The symbolic execution engine
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||
traverses the target program and builds an environment that maps
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||
variables to accessors. It branches at every control flow statement
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||
and emits a \texttt{Switch} node. The branch condition $\pi_i$ is expressed as
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||
an interval set of possible values at that point.
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In comparison with the source decision trees, \texttt{Unreachable}
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nodes are never emitted.
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Guards result in branching. In comparison with the source decision
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trees, \texttt{Unreachable} nodes are never emitted.
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||
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||
We write $\semTEX{t_T}_T$ to denote the translation of a target
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program tₜ into a decision tree of the target program
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$t_T$, satisfying the following correctness statement that is
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||
simmetric to the correctness statement for the translation of source programs:
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||
\[
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||
\forall t_T, \forall v_T, \quad t_T(v_T) = \semTEX{t_T}_T(v_T)
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||
\]
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\subsection{Equivalence checking}
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||
The equivalence checking algorithm takes as input a domain of
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||
possible values \emph{S} and a pair of source and target decision trees and
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||
in case the two trees are not equivalent it returns a counter example.
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||
Our algorithm respects the following correctness statement:
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||
\begin{comment}
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% TODO: we have to define what \coversTEX mean for readers to understand the specifications
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||
% (or we use a simplifying lie by hiding \coversTEX in the statements).
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||
\end{comment}
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||
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\begin{align*}
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\EquivTEX S {C_S} {C_T} = \YesTEX \;\land\; \coversTEX {C_T} S
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||
& \implies
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||
\forall v_S \approx v_T \in S,\; C_S(v_S) = C_T(v_T)
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\\
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\EquivTEX S {C_S} {C_T} = \NoTEX {v_S} {v_T} \;\land\; \coversTEX {C_T} S
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||
& \implies
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||
v_S \approx v_T \in S \;\land\; C_S(v_S) \neq C_T(v_T)
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||
\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 of the list and for this reason
|
||
a List.length function can accept lists of integers, lists of strings and in general
|
||
lists of any type. Such languages offer polymorphic functions 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. Other languages such as C#
|
||
and Go allow type erasure 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 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.
|
||
|
||
** 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/
|
||
- 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.
|
||
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 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 /cell/ or /box/.
|
||
We can abstract the type of OCaml runtime values as the following:
|
||
#+BEGIN_SRC
|
||
type t = Constant | Cell of int * t
|
||
#+END_SRC
|
||
where a one bit tag is used to distinguish between Constant or Cell.
|
||
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
|
||
\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 /-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 is responsible
|
||
for allocating low level OCaml values and every constant constructor
|
||
ot 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 cell 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 are 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 defined:
|
||
|
||
- 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.
|
||
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 "red"
|
||
| Green -> print "red"
|
||
| _ -> print "white or black"
|
||
#+END_SRC
|
||
|
||
provides 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 and allowed, when calling the getpeername function, to
|
||
read portions of kernel memory.
|
||
#+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 when the function is evaluated considering
|
||
/int len/ our symbolic variable α, sa->sa_len as symbolic variable β
|
||
and π as the set of constraints on a symbolic variable:
|
||
[[./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 exploitation possible.
|
||
|
||
Every step of evaluation can be modelled as the following transition:
|
||
\[
|
||
(π_{σ}, (πᵢ)ⁱ) → (π'_{σ}, (π'ᵢ)ⁱ)
|
||
\]
|
||
if we express the π constraints as logical formulas we can model the
|
||
execution of the program in terms of Hoare Logic.
|
||
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 rules of Hoare logic:
|
||
- 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 /midcondition/.
|
||
\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\not 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.
|
||
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.
|
||
Reasoning about all possible executions of a program is not always
|
||
feasible and in case of explosion usually symbolic execution engines
|
||
implement heuristics to reduce the size of the search space.
|
||
|
||
** 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
|
||
|
||
** Source program
|
||
Our algorithm takes as its input a source program and translates it
|
||
into an algebraic data structure which type we call /decision_tree/.
|
||
|
||
#+BEGIN_SRC
|
||
type decision_tree =
|
||
| Unreachable
|
||
| Failure
|
||
| Leaf of source_expr
|
||
| Guard of source_blackbox * decision_tree * decision_tree
|
||
| Switch of accessor * (constructor * decision_tree) list * decision_tree
|
||
#+END_SRC
|
||
|
||
Unreachable, Leaf of source_expr and Failure are the terminals of the three.
|
||
We distinguish
|
||
- Unreachable: statically it is known that no value can go there
|
||
- Failure: a value matching this part results in an error
|
||
- Leaf: a value matching this part results into the evaluation of a
|
||
source black box of code
|
||
Our algorithm doesn't support type-declaration-based analysis
|
||
to know the list of constructors at a given type.
|
||
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.
|
||
|
||
Guard nodes of the tree are emitted whenever a guard is found. Guards
|
||
node contains a blackbox of code that is never evaluated and two
|
||
branches, one that is taken in case the guard evaluates to true and
|
||
the other one that contains the path taken when the guard evaluates to
|
||
true.
|
||
|
||
\begin{comment}
|
||
[ ] Finisci con Switch
|
||
[ ] Spiega del fallback
|
||
\end{comment}
|
||
|
||
The 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
|
||
|
||
|
||
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\vert{}record\vert{}exn\vert{}objects\vert{}ref \vert{}list\vert{}tuple\vert{}array\vert{}variant\vert{}parameterized_variant ;; data types
|
||
| or_pattern ::= rule ("\vert{}" wildcard\vert{}variable\vert{}constructor_pattern)+
|
||
| condition ::= "when" bexpr
|
||
| assignment ::= "as" id
|
||
| bexpr ::= _code ;; arbitrary code
|
||
The source program is parsed using the ocaml-compiler-libs 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 translators, 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 x n| where m 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, ..., vᵢ)
|
||
- ∀j < i p_{j,1}, p_{j,2}, \cdots, p_{j,n} ⋠ (v₁, v₂, ..., vᵢ)
|
||
|
||
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
|
||
/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:
|
||
| First((xᵢ)ⁱ) := x_min{i \vert{} xᵢ ≠ ⊥} \quad if ∃i, xᵢ ≠ ⊥
|
||
| 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) ≠ ⊥}
|
||
/guard/ and /bb/ expressions are treated as blackboxes of OCaml code.
|
||
A sound approach for treating these blackboxes would be to inspect the
|
||
OCaml compiler during translation to Lambda code and extract the
|
||
blackboxes compiled in their Lambda representation.
|
||
This would allow to test for equality with the respective blackbox at
|
||
the target level.
|
||
Given that this level of introspection is currently not possibile, we
|
||
decided to restrict the structure of blackboxes to the following (valid) OCaml
|
||
code:
|
||
|
||
#+BEGIN_SRC
|
||
external guard : 'a -> 'b = "guard"
|
||
external observe : 'a -> 'b = "observe"
|
||
#+END_SRC
|
||
|
||
We assume 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.
|
||
Even if this is an oversimplification of the problem for the
|
||
prototype, it must be noted that at the compiler level we have the
|
||
possibility to compile the pattern clauses in two separate steps so
|
||
that the guards and right-hand-side expressions are semantically equal
|
||
to their counterparts at the target program level.
|
||
\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*}
|
||
\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, ..., 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 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
|
||
split 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.
|
||
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 we make use of accessors to encode stored values.
|
||
|
||
\begin{minipage}{0.6\linewidth}
|
||
\begin{verbatim}
|
||
|
||
let value = 1 :: 2 :: 3 :: []
|
||
(* that can also be written *)
|
||
let value = []
|
||
|> List.cons 3
|
||
|> List.cons 2
|
||
|> List.cons 1
|
||
\end{verbatim}
|
||
\end{minipage}
|
||
\hfill
|
||
\begin{minipage}{0.5\linewidth}
|
||
\begin{verbatim}
|
||
|
||
|
||
(field 0 x) = 1
|
||
(field 0 (field 1 x)) = 2
|
||
(field 0 (field 1 (field 1 x)) = 3
|
||
(field 0 (field 1 (field 1 (field 1 x)) = []
|
||
\end{verbatim}
|
||
\end{minipage}
|
||
An \emph{accessor} /a/ represents the
|
||
access path to a value that can be reached by deconstructing the
|
||
scrutinee.
|
||
| a ::= Here \vert n.a
|
||
The above example, in encoded form:
|
||
\begin{verbatim}
|
||
Here = 1
|
||
1.Here = 2
|
||
1.1.Here = 3
|
||
1.1.1.Here = []
|
||
\end{verbatim}
|
||
In our prototype the source matrix mₛ is defined as follows
|
||
| SMatrix mₛ := (aⱼ)^{j∈J}, ((p_{ij})^{j∈J} → bbᵢ)^{i∈I}
|
||
|
||
Source matrices are used to build source decision trees Cₛ.
|
||
A decision tree is defined as either a Leaf, a Failure terminal or
|
||
an intermediate node with different children sharing the same accessor /a/
|
||
and an optional fallback.
|
||
Failure is emitted only when the patterns don't cover the whole set of
|
||
possible input values /S/. The fallback is not needed when the user
|
||
doesn't use a wildcard pattern.
|
||
%%% Give example of thing
|
||
| Cₛ ::= Leaf bb \vert Switch(a, (Kᵢ → Cᵢ)^{i∈S} , C?) \vert Failure \vert Unreachable
|
||
| vₛ ::= K(vᵢ)^{i∈I} \vert n ∈ ℕ
|
||
\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〛(v) = m(vᵢ)^{i∈I} for m = ((aᵢ)^{i∈I}, (rᵢ)^{i∈I})
|
||
| v(Here) = v
|
||
| K(vᵢ)ⁱ(k.a) = vₖ(a) if k ∈ [0;n[
|
||
\begin{comment}
|
||
TODO: EXPLAIN
|
||
\end{comment}
|
||
|
||
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 [|ms|] 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 bb
|
||
2. [| (aⱼ)ʲ, ∅ |] ≡ Failure
|
||
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((xᵢ)^{i ∈ I}) = x_min(I)
|
||
- tail of an ordered family
|
||
| tail((xᵢ)^{i ∈ I}) := (xᵢ)^{i ≠ 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 (its head
|
||
constructor ∉ (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
|
||
|
||
|
||
** Target translation
|
||
|
||
The target program of the following general form is parsed using a parser
|
||
generated by Menhir, a LR(1) parser generator for the OCaml programming language.
|
||
Menhir compiles LR(1) a grammar specification, in this case a subset
|
||
of the Lambda intermediate language, down to OCaml code.
|
||
This is the grammar of the target language (TODO: check menhir grammar)
|
||
| 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.
|
||
Integer additions and subtractions are simplified in a second pass.
|
||
The result of the symbolic evaluation is a target decision tree Cₜ
|
||
| Cₜ ::= Leaf bb \vert Switch(a, (πᵢ → Cᵢ)^{i∈S} , C?) \vert Failure
|
||
| vₜ ::= Cell(tag ∈ ℕ, (vᵢ)^{i∈I}) \vert n ∈ ℕ
|
||
Every branch of the decision tree is "constrained" by a domain
|
||
| Domain π = { n\vert{}n∈ℕ x n\vert{}n∈Tag⊆ℕ }
|
||
Intuitively, the π condition at every branch tells us the set of
|
||
possible values that can "flow" through that path.
|
||
π conditions are refined by the engine during the evaluation; at the
|
||
root of the decision tree the domain corresponds to the set of
|
||
possible values that the type of the function can hold.
|
||
C? is the fallback node of the tree that is taken whenever the value
|
||
at that point of the execution can't flow to any other subbranch.
|
||
Intuitively, the π_{fallback} condition of the C? 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
|
||
The equivalence checking algorithm takes as input a domain of
|
||
possible values \emph{S} and a pair of source and target decision trees and
|
||
in case the two trees are not equivalent it returns a counter example.
|
||
The algorithm respects the following correctness statement:
|
||
\begin{comment}
|
||
% TODO: we have to define what \covers mean for readers to understand the specifications
|
||
% (or we use a simplifying lie by hiding \covers in the statements).
|
||
\end{comment}
|
||
\begin{align*}
|
||
\SimpleEquiv S {C_S} {C_T} = \Yes \;\land\; \covers {C_T} S
|
||
& \implies
|
||
\forall {\vrel {v_S} {v_T}} \in S,\; \runrel {C_S(v_S)} {C_T(v_T)}
|
||
\\
|
||
\SimpleEquiv S {C_S} {C_T} = \No {v_S} {v_T} \;\land\; \covers {C_T} S
|
||
& \implies
|
||
\vrel {v_S} {v_T} \in S \;\land\; {\nparamrel{run} {C_S(v_S)} {C_T(v_T)}}
|
||
\end{align*}
|
||
This algorithm $\SimpleEquiv S {C_S} {C_T}$ is in fact exactly
|
||
a decision procedure for the provability of the judgment
|
||
$(\Equivrel S {C_S} {C_T} \emptyqueue)$, defined below in the general
|
||
form $(\Equivrel S {C_S} {C_T} G)$ where $G$ is a \emph{guard queue},
|
||
indicating an imbalance between the guards observed in the source tree
|
||
and in the target tree.
|
||
\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{ }
|
||
{\Equivrel \emptyset {C_S} {C_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
|
||
{C_S \in {\Leaf t, \Failure}
|
||
\\
|
||
\forall i,\; \EquivTEX {(S \land a \in D_i)} {C_S} {C_i} G
|
||
\\
|
||
\EquivTEX {(S \land a \notin \Fam i {D_i})} {C_S} \Cfb G}
|
||
{\EquivTEX S
|
||
{C_S} {\Switch a {\Fam i {D_i} {C_i}} \Cfb} G}
|
||
\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
|
||
{\forall i,\;
|
||
\EquivTEX
|
||
{(S \land a = K_i)}
|
||
{C_i} {\trim {C_T} {a = K_i}} G
|
||
\\
|
||
\EquivTEX
|
||
{(S \land a \notin \Fam i {K_i})}
|
||
\Cfb {\trim {C_T} {a \notin \Fam i {K_i}}} G
|
||
}
|
||
{\EquivTEX S
|
||
{\Switch a {\Fam i {K_i, C_i}} \Cfb} {C_T} G}
|
||
\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
|
||
{\EquivTEX S {C_0} {C_T} {G, (t_S = 0)}
|
||
\\
|
||
\EquivTEX S {C_1} {C_T} {G, (t_S = 1)}}
|
||
{\EquivTEX S
|
||
{\Guard {t_S} {C_0} {C_1}} {C_T} G}
|
||
|
||
\infer
|
||
{\trel {t_S} {t_T}
|
||
\\
|
||
\EquivTEX S {C_S} {C_b} G}
|
||
{\EquivTEX S
|
||
{C_S} {\Guard {t_T} {C_0} {C_1}} {(t_S = b), G}}
|
||
\end{mathpar}
|
||
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))
|