coppo #2
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Francesco Mecca
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2 changed files with 28 additions and 22 deletions
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tesi/.gitignore
vendored
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tesi/.gitignore
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@ -1,6 +1,9 @@
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tesi.tex
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.#tesi.tex
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tesi_unicode.tex
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## Core latex/pdflatex auxiliary files:
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*.aux
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*.lof
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@ -49,8 +49,13 @@ TODO: talk about compiler stuff
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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 wheter the two are equivalent or produce a counter
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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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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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@ -58,7 +63,9 @@ 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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\begin{comment}
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\subsection{Translation validation}
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\end{comment}
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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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@ -79,7 +86,9 @@ into simple control flow structures such as \texttt{if, switch}, for example:
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(makeblock 0 1 (makeblock 0 y)))))
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[0: 0 0a])
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\end{lstlisting}
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\begin{comment}
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%% TODO: side by side
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\end{comment}
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The code in the right is in 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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@ -92,10 +101,10 @@ corner cases of complex patterns which are typically not in the
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compiler test suite.
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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 refactorings of its codebase.
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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 behaviour than the current one.
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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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@ -103,12 +112,12 @@ 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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blackbox and proof only depends on our equivalence algorithm.
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black-box and proof only depends on our equivalence algorithm.
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\subsection{Our approach}
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%% replace common TODO
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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 choosen because
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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 is the decision tree for the source example program.
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@ -144,12 +153,6 @@ 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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For the prototype we have choosen a simple 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 proof sketches.
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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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\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, or
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@ -186,7 +189,7 @@ same result as running it against the decision tree.
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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 comparation operations, guards. The symbolic execution engine
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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 Switch node. The branch condition $\pi_i$ is expressed as
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@ -214,7 +217,7 @@ The main features of ML languages are the use of the Hindley-Milner type system
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provides many advantages with respect to static type systems of traditional imperative and object
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oriented language such as C, C++ and Java, such as:
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- Polymorphism: in certain scenarios a function can accept more than one
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type for the input parameters. For example a function that computes the lenght of a
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type for the input parameters. For example a function that computes the length of a
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list doesn't need to inspect the type of the elements of the list and for this reason
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a List.length function can accept lists of integers, lists of strings and in general
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lists of any type. Such languages offer polymorphic functions through subtyping at
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@ -231,7 +234,7 @@ oriented language such as C, C++ and Java, such as:
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programmer is not allowed to operate on data by ignoring or promoting its type.
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- Type Inference: the principal type of a well formed term can be inferred without any
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annotation or declaration.
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- Algebraic data types: types that are modelled by the use of two
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- Algebraic data types: types that are modeled by the use of two
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algebraic operations, sum and product.
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A sum type is a type that can hold of many different types of
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objects, but only one at a time. For example the sum type defined
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@ -248,14 +251,14 @@ although mutable statements and imperative constructs are permitted.
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In addition to that features an object system, that provides
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inheritance, subtyping and dynamic binding, and modules, that
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provide a way to encapsulate definitions. Modules are checked
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statically and can be reificated through functors.
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statically and can be reifycated through functors.
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** Lambda form compilation
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\begin{comment}
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https://dev.realworld.org/compiler-backend.html
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\end{comment}
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provides compilation in form of a byecode executable with an
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provides compilation in form of a bytecode executable with an
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optionally embeddable interpreter and a native executable that could
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be statically linked to provide a single file executable.
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@ -263,7 +266,7 @@ After the typechecker has proven that the program is type safe,
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the compiler lower the code to /Lambda/, an s-expression based
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language that assumes that its input has already been proved safe.
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On the /Lambda/ representation of the source program, the compiler
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performes a series of optimization passes before translating the
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performs a series of optimization passes before translating the
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lambda form to assembly code.
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*** OCaml Native Datatypes
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@ -298,7 +301,7 @@ There are several numeric types:
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- floats: that use IEEE754 double-precision (64-bit) arithmetic with
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the addition of the literals /infinity/, /neg_infinity/ and /nan/.
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The are varios numeric operations defined:
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The are various numeric operations defined:
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- Arithmetic operations: +, -, *, /, % (modulo), neg (unary negation)
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- Bitwise operations: &, |, ^, <<, >> (zero-shifting), a>> (sign extending)
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@ -349,7 +352,7 @@ match color with
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#+END_SRC
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provides tokens to express data destructoring.
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For example we can examine the content of a list with patten matching
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For example we can examine the content of a list with pattern matching
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#+BEGIN_SRC
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@ -758,7 +761,7 @@ following four rules:
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\end{equation*}
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4) Mixture rule:
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When none of the previous rules apply the clause matrix P → L is
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splitted into two clause matrices, the first P₁ → L₁ that is the
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split into two clause matrices, the first P₁ → L₁ that is the
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largest prefix matrix for which one of the three previous rules
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apply, and P₂ → L₂ containing the remaining rows. The algorithm is
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applied to both matrices.
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