logica che palle
This commit is contained in:
Francesco Mecca
parent
0d7ed5e91d
commit
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11 changed files with 1096 additions and 305 deletions
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anno3/logica/EsameLogica_v4_soluzioni.pdf
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|
||||||
|
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|
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<recently read> &
|
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||||||
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|
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|
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|
|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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||||||
|
|
||||||
|
# TODO Scaletta <code>[1/2]</code>
|
||||||
|
|
||||||
|
- [X] Abstract
|
||||||
|
- [-] Background <code>[25%]</code>
|
||||||
|
- [X] Ocaml
|
||||||
|
- [ ] Lambda code
|
||||||
|
- [ ] Pattern matching
|
||||||
|
- [ ] Translation Verification
|
||||||
|
- [ ] Symbolic execution
|
||||||
|
|
||||||
|
\begin{abstract}
|
||||||
|
|
||||||
|
This dissertation presents an algorithm for the translation validation of the OCaml
|
||||||
|
pattern matching compiler. Given the source representation of the target program and the
|
||||||
|
target program compiled in untyped lambda form, the algoritmhm is capable of modelling
|
||||||
|
the source program in terms of symbolic constraints on it's branches and apply symbolic
|
||||||
|
execution on the untyped lambda representation in order to validate wheter the compilation
|
||||||
|
produced a valid result.
|
||||||
|
In this context a valid result means that for every input in the domain of the source
|
||||||
|
program the untyped lambda translation produces the same output as the source program.
|
||||||
|
The input of the program is modelled in terms of symbolic constraints closely related to
|
||||||
|
the runtime representation of OCaml objects and the output consists of OCaml code
|
||||||
|
blackboxes that are not evaluated in the context of the verification.
|
||||||
|
|
||||||
|
\end{abstract}
|
||||||
|
|
||||||
|
|
||||||
|
# 1. Background
|
||||||
|
|
||||||
|
|
||||||
|
## 1.1 OCaml
|
||||||
|
|
||||||
|
Objective Caml (OCaml) is a dialect of the ML (Meta-Language) family of programming
|
||||||
|
languages.
|
||||||
|
OCaml shares many features with other dialects of ML, such as SML and Caml Light,
|
||||||
|
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:
|
||||||
|
|
||||||
|
- Parametric polymorphism: in certain scenarios a function can accept more than one
|
||||||
|
|
||||||
|
type for the input parameters. For example a function that computes the lenght 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 list of integers, list of strings and in general
|
||||||
|
list 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 modelled by the use of two
|
||||||
|
|
||||||
|
algebraic operations, sum and product.
|
||||||
|
A sum type is a type that can hold of many different types of
|
||||||
|
objects, but only one at a time. For example the sum type defined
|
||||||
|
as *A + B* can hold at any moment a value of type A or a value of
|
||||||
|
type B. Sum types are also called tagged union or variants.
|
||||||
|
A product type is a type constructed as a direct product
|
||||||
|
of multiple types and contains at any moment one instance for
|
||||||
|
every type of its operands. Product types are also called tuples
|
||||||
|
or records. Algebraic data types can be recursive
|
||||||
|
in their definition and can be combined.
|
||||||
|
Moreover ML languages are functional, meaning that functions are
|
||||||
|
treated as first class citizens and variables are immutable,
|
||||||
|
although mutable statements and imperative constructs are permitted.
|
||||||
|
In addition to that OCaml features an object system, that provides
|
||||||
|
inheritance, subtyping and dynamic binding, and modules, that
|
||||||
|
provide a way to encapsulate definitions. Modules are checked
|
||||||
|
statically and can be reificated through functors.
|
||||||
|
|
||||||
|
1. TODO 1.2 Pattern matching <code>[37%]</code>
|
||||||
|
|
||||||
|
- [ ] capisci come mettere gli esempi uno accanto all'altro
|
||||||
|
|
||||||
|
Pattern matching is a widely adopted mechanism to interact with ADT.
|
||||||
|
C family languages provide branching on predicates through the use of
|
||||||
|
if statements and switch statements.
|
||||||
|
Pattern matching is a mechanism for destructuring and analyzing data
|
||||||
|
structures for the presence of values simbolically represented as
|
||||||
|
tokens. One common example of pattern matching is the use of regular
|
||||||
|
expressions on strings. OCaml provides pattern matching on ADT,
|
||||||
|
primitive data types.
|
||||||
|
|
||||||
|
- [X] Esempio enum, C e Ocaml
|
||||||
|
|
||||||
|
type color = | Red | Blue | Green
|
||||||
|
|
||||||
|
begin match color with
|
||||||
|
| Red -> print "red"
|
||||||
|
| Blue -> print "red"
|
||||||
|
| Green -> print "red"
|
||||||
|
|
||||||
|
OCaml provides tokens to express data destructoring
|
||||||
|
|
||||||
|
- [X] Esempio destructor list
|
||||||
|
|
||||||
|
|
||||||
|
begin match list with
|
||||||
|
| [ ] -> print "empty list"
|
||||||
|
| element1 :: [ ] -> print "one element"
|
||||||
|
| element1 :: element2 :: [ ] -> print "two elements"
|
||||||
|
| head :: tail-> print "head followed by many elements"
|
||||||
|
|
||||||
|
- [X] Esempio destructor tuples
|
||||||
|
|
||||||
|
|
||||||
|
begin match tuple with
|
||||||
|
| (Some _, Some _) -> print "Pair of optional types"
|
||||||
|
| (Some _, None) -> print "Pair of optional types, last null"
|
||||||
|
| (None, Some _) -> print "Pair of optional types, first null"
|
||||||
|
| (None, None) -> print "Pair of optional types, both null"
|
||||||
|
|
||||||
|
Pattern clauses can make the use of *guards* to test predicates and
|
||||||
|
variables can be binded in scope.
|
||||||
|
|
||||||
|
- [ ] Esempio binding e guards
|
||||||
|
|
||||||
|
|
||||||
|
begin match token_list with
|
||||||
|
| "switch"::var::"{"::rest ->
|
||||||
|
| "case"::":"::var::rest when is_int var ->
|
||||||
|
| "case"::":"::var::rest when is_string var ->
|
||||||
|
| "}"::[ ] -> stop ()
|
||||||
|
| "}"::rest -> error "syntax error: " rest
|
||||||
|
|
||||||
|
- [ ] Un altro esempio con destructors e tutto i lresto
|
||||||
|
|
||||||
|
In general pattern matching on primitive and algebraic data types takes the
|
||||||
|
following form.
|
||||||
|
|
||||||
|
- [ ] Esempio informale
|
||||||
|
|
||||||
|
It can be described more formally through a BNF grammar.
|
||||||
|
|
||||||
|
- [ ] BNF
|
||||||
|
|
||||||
|
- [ ] Come funziona il pattern matching?
|
||||||
|
|
||||||
|
2. TODO 1.2.1 Pattern matching compilation to lambda code
|
||||||
|
|
||||||
|
- [ ] Da tabella a matrice
|
||||||
|
|
||||||
|
Formally, pattern and values are defined as follow:
|
||||||
|
|
||||||
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||||
|
|
||||||
|
|
||||||
|
<colgroup>
|
||||||
|
<col class="org-left" />
|
||||||
|
|
||||||
|
<col class="org-left" />
|
||||||
|
</colgroup>
|
||||||
|
<thead>
|
||||||
|
<tr>
|
||||||
|
<th scope="col" class="org-left">pattern ::=</th>
|
||||||
|
<th scope="col" class="org-left">Patterns</th>
|
||||||
|
</tr>
|
||||||
|
</thead>
|
||||||
|
|
||||||
|
<tbody>
|
||||||
|
<tr>
|
||||||
|
<td class="org-left">\_</td>
|
||||||
|
<td class="org-left">wildcard</td>
|
||||||
|
</tr>
|
||||||
|
|
||||||
|
|
||||||
|
<tr>
|
||||||
|
<td class="org-left">x</td>
|
||||||
|
<td class="org-left">variable</td>
|
||||||
|
</tr>
|
||||||
|
|
||||||
|
|
||||||
|
<tr>
|
||||||
|
<td class="org-left">c(p1,p2,…,pn</td>
|
||||||
|
<td class="org-left">constructor pattern</td>
|
||||||
|
</tr>
|
||||||
|
|
||||||
|
|
||||||
|
<tr>
|
||||||
|
<td class="org-left">(p1| p2)</td>
|
||||||
|
<td class="org-left">or-pattern</td>
|
||||||
|
</tr>
|
||||||
|
</tbody>
|
||||||
|
</table>
|
||||||
|
|
||||||
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||||
|
|
||||||
|
|
||||||
|
<colgroup>
|
||||||
|
<col class="org-left" />
|
||||||
|
|
||||||
|
<col class="org-left" />
|
||||||
|
</colgroup>
|
||||||
|
<thead>
|
||||||
|
<tr>
|
||||||
|
<th scope="col" class="org-left">values ::=</th>
|
||||||
|
<th scope="col" class="org-left">Values</th>
|
||||||
|
</tr>
|
||||||
|
</thead>
|
||||||
|
|
||||||
|
<tbody>
|
||||||
|
<tr>
|
||||||
|
<td class="org-left">c(v1, v2, …, vn)</td>
|
||||||
|
<td class="org-left">constructor value</td>
|
||||||
|
</tr>
|
||||||
|
</tbody>
|
||||||
|
</table>
|
||||||
|
|
||||||
|
The entire pattern matching code can be represented as a clause matrix
|
||||||
|
that associates rows of patterns (p<sub>i,1</sub>, p<sub>i,2</sub>, …, p<sub>i,n</sub>) 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*}
|
||||||
|
|
||||||
|
Most native data types in OCaml, such as integers, tuples, lists,
|
||||||
|
records, can be seen as instances of the following definition
|
||||||
|
|
||||||
|
type t = Nil | One of int | Cons of int * t
|
||||||
|
|
||||||
|
that is a type *t* with three constructors that define its complete
|
||||||
|
signature.
|
||||||
|
Every constructor has an arity. Nil, a constructor of arity 0, is
|
||||||
|
called a constant constructor.
|
||||||
|
|
||||||
|
The pattern *p* matches a value *v*, written as p ≼ v, when
|
||||||
|
one of the following rules apply
|
||||||
|
|
||||||
|
<table border="2" cellspacing="0" cellpadding="6" rules="groups" frame="hsides">
|
||||||
|
|
||||||
|
|
||||||
|
<colgroup>
|
||||||
|
<col class="org-left" />
|
||||||
|
|
||||||
|
<col class="org-left" />
|
||||||
|
|
||||||
|
<col class="org-left" />
|
||||||
|
|
||||||
|
<col class="org-left" />
|
||||||
|
</colgroup>
|
||||||
|
<tbody>
|
||||||
|
<tr>
|
||||||
|
<td class="org-left">\_</td>
|
||||||
|
<td class="org-left">≼</td>
|
||||||
|
<td class="org-left">v</td>
|
||||||
|
<td class="org-left"> </td>
|
||||||
|
</tr>
|
||||||
|
|
||||||
|
|
||||||
|
<tr>
|
||||||
|
<td class="org-left">x</td>
|
||||||
|
<td class="org-left">≼</td>
|
||||||
|
<td class="org-left">v</td>
|
||||||
|
<td class="org-left"> </td>
|
||||||
|
</tr>
|
||||||
|
|
||||||
|
|
||||||
|
<tr>
|
||||||
|
<td class="org-left">(p₁ |\\ p₂)</td>
|
||||||
|
<td class="org-left">≼</td>
|
||||||
|
<td class="org-left">v</td>
|
||||||
|
<td class="org-left">iff p₁ ≼ v or p₂ ≼ v</td>
|
||||||
|
</tr>
|
||||||
|
|
||||||
|
|
||||||
|
<tr>
|
||||||
|
<td class="org-left">c(p₁, p₂, …, pₐ)</td>
|
||||||
|
<td class="org-left">≼</td>
|
||||||
|
<td class="org-left">c(v₁, v₂, …, vₐ)</td>
|
||||||
|
<td class="org-left">iff (p₁, p₂, …, pₐ) ≼ (v₁, v₂, …, vₐ)</td>
|
||||||
|
</tr>
|
||||||
|
|
||||||
|
|
||||||
|
<tr>
|
||||||
|
<td class="org-left">(p₁, p₂, …, pₐ)</td>
|
||||||
|
<td class="org-left">≼</td>
|
||||||
|
<td class="org-left">(v₁, v₂, …, vₐ)</td>
|
||||||
|
<td class="org-left">iff pᵢ ≼ vᵢ ∀i ∈ [1..a]</td>
|
||||||
|
</tr>
|
||||||
|
</tbody>
|
||||||
|
</table>
|
||||||
|
|
||||||
|
We can also say that *v* is an *instance* of *p*.
|
||||||
|
|
||||||
|
When we consider the pattern matrix P we say that the value vector
|
||||||
|
\vv{v} = (v₁, v₂, …, vᵢ) matches the line number i in P if and only if the following two
|
||||||
|
conditions are satisfied:
|
||||||
|
|
||||||
|
- \[ p_{i,1} & p_{i,2} & \cdots & p_{i,n} \] ≼ (v₁, v₂, …, vᵢ)
|
||||||
|
- \[ ∀j < i p_{j,1} & p_{j,2} & \cdots & p_{j,n} \] ⋠ (v₁, v₂, …, vᵢ)
|
||||||
|
|
||||||
|
We can define the following three relations with respect to patterns:
|
||||||
|
|
||||||
|
- Patter p is less precise than pattern q, writtens 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
|
||||||
|
|
||||||
213
tesi/prova.org
213
tesi/prova.org
|
|
@ -1,10 +1,25 @@
|
||||||
* TODO Scaletta [1/2]
|
* TODO Scaletta [1/2]
|
||||||
- [X] Abstract
|
- [X] Abstract
|
||||||
- [ ] Introduction [0%]
|
- [-] Background [20%]
|
||||||
- [ ] Ocaml
|
- [X] Ocaml
|
||||||
- [ ] Pattern matching
|
- [ ] Lambda code [0%]
|
||||||
|
- [ ] Untyped lambda form
|
||||||
|
- [ ] OCaml specific instructions
|
||||||
|
- [-] Pattern matching [50%]
|
||||||
|
- [X] Introduzione
|
||||||
|
- [ ] Compilation to lambda form
|
||||||
- [ ] Translation Verification
|
- [ ] Translation Verification
|
||||||
- [ ] Symbolic execution
|
- [ ] Symbolic execution
|
||||||
|
- [ ] Translation verification of the Pattern Matching Compiler
|
||||||
|
- [ ] Source translation
|
||||||
|
- [ ] Formal Grammar
|
||||||
|
- [ ] Compilation of source patterns
|
||||||
|
- [ ] Target translation
|
||||||
|
- [ ] Formal Grammar
|
||||||
|
- [ ] Symbolic execution of the lambda target
|
||||||
|
- [ ] Equivalence between source and target
|
||||||
|
- [ ] Practical results
|
||||||
|
|
||||||
|
|
||||||
#+TITLE: Translation Verification of the OCaml pattern matching compiler
|
#+TITLE: Translation Verification of the OCaml pattern matching compiler
|
||||||
#+AUTHOR: Francesco Mecca
|
#+AUTHOR: Francesco Mecca
|
||||||
|
|
@ -17,6 +32,8 @@
|
||||||
#+LaTeX_HEADER: \usepackage{algorithm}
|
#+LaTeX_HEADER: \usepackage{algorithm}
|
||||||
#+LaTeX_HEADER: \usepackage{algpseudocode}
|
#+LaTeX_HEADER: \usepackage{algpseudocode}
|
||||||
#+LaTeX_HEADER: \usepackage{amsmath,amssymb,amsthm}
|
#+LaTeX_HEADER: \usepackage{amsmath,amssymb,amsthm}
|
||||||
|
#+LaTeX_HEADER: \usepackage[utf8]{inputenc}
|
||||||
|
#+LaTeX_HEADER: \usepackage[T1]{fontenc}
|
||||||
#+Latex_HEADER: \newtheorem{definition}{Definition}
|
#+Latex_HEADER: \newtheorem{definition}{Definition}
|
||||||
#+LaTeX_HEADER: \usepackage{graphicx}
|
#+LaTeX_HEADER: \usepackage{graphicx}
|
||||||
#+LaTeX_HEADER: \usepackage{listings}
|
#+LaTeX_HEADER: \usepackage{listings}
|
||||||
|
|
@ -41,15 +58,15 @@ blackboxes that are not evaluated in the context of the verification.
|
||||||
|
|
||||||
\end{abstract}
|
\end{abstract}
|
||||||
|
|
||||||
* Introduction
|
* 1. Background
|
||||||
|
|
||||||
** TODO OCaml
|
** 1.1 OCaml
|
||||||
Objective Caml (OCaml) is a dialect of the ML (Meta-Language) family of programming
|
Objective Caml (OCaml) is a dialect of the ML (Meta-Language) family of programming
|
||||||
languages.
|
languages.
|
||||||
OCaml shares many features with other dialects of ML, such as SML and Caml Light,
|
OCaml shares many 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
|
The main features of ML languages are the use of the Hindley-Milner type system that
|
||||||
provides with respect to static type systems of traditional imperative and/or object
|
provides many advantages with respect to static type systems of traditional imperative and object
|
||||||
oriented language such as C, C++ and Java many advantages such as:
|
oriented language such as C, C++ and Java, such as:
|
||||||
- Parametric polymorphism: in certain scenarios a function can accept more than one
|
- Parametric polymorphism: in certain scenarios a function can accept more than one
|
||||||
type for the input parameters. For example a function that computes the lenght of a
|
type for the input parameters. For example a function that computes the lenght of a
|
||||||
list doesn't need to inspect the type of the elements of the list and for this reason
|
list doesn't need to inspect the type of the elements of the list and for this reason
|
||||||
|
|
@ -68,3 +85,185 @@ blackboxes that are not evaluated in the context of the verification.
|
||||||
programmer is not allowed to operate on data by ignoring or promoting its type.
|
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
|
- Type Inference: the principal type of a well formed term can be inferred without any
|
||||||
annotation or declaration.
|
annotation or declaration.
|
||||||
|
- Algebraic data types: types that are modelled by the use of two
|
||||||
|
algebraic operations, sum and product.
|
||||||
|
A sum type is a type that can hold of many different types of
|
||||||
|
objects, but only one at a time. For example the sum type defined
|
||||||
|
as /A + B/ can hold at any moment a value of type A or a value of
|
||||||
|
type B. Sum types are also called tagged union or variants.
|
||||||
|
A product type is a type constructed as a direct product
|
||||||
|
of multiple types and contains at any moment one instance for
|
||||||
|
every type of its operands. Product types are also called tuples
|
||||||
|
or records. Algebraic data types can be recursive
|
||||||
|
in their definition and can be combined.
|
||||||
|
Moreover ML languages are functional, meaning that functions are
|
||||||
|
treated as first class citizens and variables are immutable,
|
||||||
|
although mutable statements and imperative constructs are permitted.
|
||||||
|
In addition to that OCaml features an object system, that provides
|
||||||
|
inheritance, subtyping and dynamic binding, and modules, that
|
||||||
|
provide a way to encapsulate definitions. Modules are checked
|
||||||
|
statically and can be reificated through functors.
|
||||||
|
|
||||||
|
*** TODO 1.2 Pattern matching [37%]
|
||||||
|
- [ ] capisci come mettere gli esempi uno accanto all'altro
|
||||||
|
|
||||||
|
Pattern matching is a widely adopted mechanism to interact with ADT.
|
||||||
|
C family languages provide branching on predicates through the use of
|
||||||
|
if statements and switch statements.
|
||||||
|
Pattern matching is a mechanism for destructuring and analyzing data
|
||||||
|
structures for the presence of values simbolically represented as
|
||||||
|
tokens. One common example of pattern matching is the use of regular
|
||||||
|
expressions on strings. OCaml provides pattern matching on ADT,
|
||||||
|
primitive data types.
|
||||||
|
|
||||||
|
- [X] Esempio enum, C e Ocaml
|
||||||
|
#+BEGIN_SRC ocaml
|
||||||
|
type color = | Red | Blue | Green
|
||||||
|
|
||||||
|
begin match color with
|
||||||
|
| Red -> print "red"
|
||||||
|
| Blue -> print "red"
|
||||||
|
| Green -> print "red"
|
||||||
|
|
||||||
|
#+END_SRC
|
||||||
|
|
||||||
|
OCaml provides tokens to express data destructoring
|
||||||
|
|
||||||
|
|
||||||
|
- [X] Esempio destructor list
|
||||||
|
#+BEGIN_SRC ocaml
|
||||||
|
|
||||||
|
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
|
||||||
|
|
||||||
|
- [X] Esempio destructor tuples
|
||||||
|
#+BEGIN_SRC ocaml
|
||||||
|
|
||||||
|
begin match tuple with
|
||||||
|
| (Some _, Some _) -> print "Pair of optional types"
|
||||||
|
| (Some _, None) -> print "Pair of optional types, last null"
|
||||||
|
| (None, Some _) -> print "Pair of optional types, first null"
|
||||||
|
| (None, None) -> print "Pair of optional types, both null"
|
||||||
|
#+END_SRC
|
||||||
|
|
||||||
|
Pattern clauses can make the use of /guards/ to test predicates and
|
||||||
|
variables can be binded in scope.
|
||||||
|
|
||||||
|
- [ ] Esempio binding e guards
|
||||||
|
#+BEGIN_SRC ocaml
|
||||||
|
|
||||||
|
begin match token_list with
|
||||||
|
| "switch"::var::"{"::rest ->
|
||||||
|
| "case"::":"::var::rest when is_int var ->
|
||||||
|
| "case"::":"::var::rest when is_string var ->
|
||||||
|
| "}"::[ ] -> stop ()
|
||||||
|
| "}"::rest -> error "syntax error: " rest
|
||||||
|
|
||||||
|
#+END_SRC
|
||||||
|
|
||||||
|
- [ ] Un altro esempio con destructors e tutto i lresto
|
||||||
|
|
||||||
|
In general pattern matching on primitive and algebraic data types takes the
|
||||||
|
following form.
|
||||||
|
|
||||||
|
- [ ] Esempio informale
|
||||||
|
|
||||||
|
It can be described more formally through a BNF grammar.
|
||||||
|
|
||||||
|
- [ ] BNF
|
||||||
|
|
||||||
|
- [ ] Come funziona il pattern matching?
|
||||||
|
|
||||||
|
*** TODO 1.2.1 Pattern matching compilation to lambda code
|
||||||
|
|
||||||
|
- [ ] Da tabella a matrice
|
||||||
|
|
||||||
|
Formally, pattern are defined as follows:
|
||||||
|
| pattern ::= | Patterns |
|
||||||
|
|----------------+---------------------|
|
||||||
|
| _ | wildcard |
|
||||||
|
| x | variable |
|
||||||
|
| c(p₁,p₂,...,pₙ | constructor pattern |
|
||||||
|
| (p₁\vert p₂) | or-pattern |
|
||||||
|
|
||||||
|
Values are defined as follows:
|
||||||
|
| values ::= | Values |
|
||||||
|
|---------------------+-------------------|
|
||||||
|
| c(v₁, v₂, ..., vₙ) | constructor value |
|
||||||
|
|
||||||
|
The entire pattern matching code can be represented as a clause matrix
|
||||||
|
that associates rows of patterns (p_{i,1}, p_{i,2}, ..., p_{i,n}) to
|
||||||
|
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*}
|
||||||
|
|
||||||
|
Most native data types in OCaml, such as integers, tuples, lists,
|
||||||
|
records, can be seen as instances of the following definition
|
||||||
|
|
||||||
|
#+BEGIN_SRC ocaml
|
||||||
|
type t = Nil | One of int | Cons of int * t
|
||||||
|
#+END_SRC
|
||||||
|
that is a type /t/ with three constructors that define its complete
|
||||||
|
signature.
|
||||||
|
Every constructor has an arity. Nil, a constructor of arity 0, is
|
||||||
|
called a constant constructor.
|
||||||
|
|
||||||
|
The pattern /p/ matches a value /v/, written as p ≼ v, when
|
||||||
|
one of the following rules apply
|
||||||
|
|
||||||
|
| _ | ≼ | v | |
|
||||||
|
| x | ≼ | 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] |
|
||||||
|
We can also say that /v/ is an /instance/ of /p/.
|
||||||
|
|
||||||
|
When we consider the pattern matrix P we say that the value vector
|
||||||
|
\vv{v} = (v₁, v₂, ..., vᵢ) matches the line number i in P if and only if the following two
|
||||||
|
conditions are satisfied:
|
||||||
|
- \[ p_{i,1} & p_{i,2} & \cdots & p_{i,n} \] ≼ (v₁, v₂, ..., vᵢ)
|
||||||
|
- \[ ∀j < i p_{j,1} & p_{j,2} & \cdots & p_{j,n} \] ⋠ (v₁, v₂, ..., vᵢ)
|
||||||
|
|
||||||
|
We can define the following three relations with respect to patterns:
|
||||||
|
- Patter p is less precise than pattern q, writtens 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
|
||||||
|
|
||||||
|
** 1.2.1.1 Initial state of the compilation
|
||||||
|
|
||||||
|
Given a source of the following form:
|
||||||
|
|
||||||
|
#+BEGIN_SRC ocaml
|
||||||
|
match x with
|
||||||
|
| p₁ -> e₁
|
||||||
|
| p₂ -> e₂
|
||||||
|
...
|
||||||
|
| pₘ -> eₘ
|
||||||
|
#+END_SRC ocaml
|
||||||
|
|
||||||
|
the initial input of the algorithm consists of a vector of variables
|
||||||
|
\vv{x} = (x₁, x₂, ..., xₙ) of size n
|
||||||
|
and a clause matrix P → L of width n and height m.
|
||||||
|
|
||||||
|
\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*}
|
||||||
|
|
|
||||||
|
|
@ -1,3 +1,4 @@
|
||||||
\BOOKMARK [1][-]{section.1}{TODO Scaletta [1/2]}{}% 1
|
\BOOKMARK [1][-]{section.1}{TODO\040Scaletta\040[1/2]}{}% 1
|
||||||
\BOOKMARK [1][-]{section.2}{Introduction}{}% 2
|
\BOOKMARK [1][-]{section.2}{1.\040Background}{}% 2
|
||||||
\BOOKMARK [2][-]{subsection.2.1}{TODO OCaml}{section.2}% 3
|
\BOOKMARK [2][-]{subsection.2.1}{1.1\040OCaml}{section.2}% 3
|
||||||
|
\BOOKMARK [2][-]{subsection.2.2}{1.2.1.1\040Initial\040state\040of\040the\040compilation}{section.2}% 4
|
||||||
|
|
|
||||||
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BIN
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Binary file not shown.
260
tesi/prova.tex
260
tesi/prova.tex
|
|
@ -1,4 +1,4 @@
|
||||||
% Created 2020-02-17 Mon 17:30
|
% Created 2020-02-19 Wed 16:42
|
||||||
% Intended LaTeX compiler: pdflatex
|
% Intended LaTeX compiler: pdflatex
|
||||||
\documentclass[11pt]{article}
|
\documentclass[11pt]{article}
|
||||||
\usepackage[utf8]{inputenc}
|
\usepackage[utf8]{inputenc}
|
||||||
|
|
@ -18,6 +18,8 @@
|
||||||
\usepackage{algorithm}
|
\usepackage{algorithm}
|
||||||
\usepackage{algpseudocode}
|
\usepackage{algpseudocode}
|
||||||
\usepackage{amsmath,amssymb,amsthm}
|
\usepackage{amsmath,amssymb,amsthm}
|
||||||
|
\usepackage[utf8]{inputenc}
|
||||||
|
\usepackage[T1]{fontenc}
|
||||||
\newtheorem{definition}{Definition}
|
\newtheorem{definition}{Definition}
|
||||||
\usepackage{graphicx}
|
\usepackage{graphicx}
|
||||||
\usepackage{listings}
|
\usepackage{listings}
|
||||||
|
|
@ -36,12 +38,13 @@
|
||||||
|
|
||||||
\maketitle
|
\maketitle
|
||||||
\section{{\bfseries\sffamily TODO} Scaletta [1/2]}
|
\section{{\bfseries\sffamily TODO} Scaletta [1/2]}
|
||||||
\label{sec:org5a6f376}
|
\label{sec:org6c3943f}
|
||||||
\begin{itemize}
|
\begin{itemize}
|
||||||
\item[{$\boxtimes$}] Abstract
|
\item[{$\boxtimes$}] Abstract
|
||||||
\item[{$\square$}] Introduction [0\%]
|
\item[{$\boxminus$}] Background [25\%]
|
||||||
\begin{itemize}
|
\begin{itemize}
|
||||||
\item[{$\square$}] Ocaml
|
\item[{$\boxtimes$}] Ocaml
|
||||||
|
\item[{$\square$}] Lambda code
|
||||||
\item[{$\square$}] Pattern matching
|
\item[{$\square$}] Pattern matching
|
||||||
\item[{$\square$}] Translation Verification
|
\item[{$\square$}] Translation Verification
|
||||||
\item[{$\square$}] Symbolic execution
|
\item[{$\square$}] Symbolic execution
|
||||||
|
|
@ -64,19 +67,20 @@ blackboxes that are not evaluated in the context of the verification.
|
||||||
|
|
||||||
\end{abstract}
|
\end{abstract}
|
||||||
|
|
||||||
\section{Introduction}
|
\section{1. Background}
|
||||||
\label{sec:orgef00ecd}
|
\label{sec:orgad08c62}
|
||||||
|
|
||||||
\subsection{{\bfseries\sffamily TODO} OCaml}
|
\subsection{1.1 OCaml}
|
||||||
\label{sec:org5659ec2}
|
\label{sec:orgca8d200}
|
||||||
Objective Caml (OCaml) is a dialect of the ML (Meta-Language) family of programming
|
Objective Caml (OCaml) is a dialect of the ML (Meta-Language) family of programming
|
||||||
languages.
|
languages.
|
||||||
OCaml shares many features with other dialects of ML, such as SML and Caml Light,
|
OCaml shares many 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
|
The main features of ML languages are the use of the Hindley-Milner type system that
|
||||||
provides with respect to static type systems of traditional imperative and/or object
|
provides many advantages with respect to static type systems of traditional imperative and object
|
||||||
oriented language such as C, C++ and Java many advantages such as:
|
oriented language such as C, C++ and Java, such as:
|
||||||
\begin{itemize}
|
\begin{itemize}
|
||||||
\item Parametric polymorphism: in certain scenarios a function can accept more than one
|
\item Parametric polymorphism: in certain scenarios a function can accept more than one
|
||||||
|
\end{itemize}
|
||||||
type for the input parameters. For example a function that computes the lenght of a
|
type for the input parameters. For example a function that computes the lenght of a
|
||||||
list doesn't need to inspect the type of the elements of the list and for this reason
|
list doesn't need to inspect the type of the elements of the list and for this reason
|
||||||
a List.length function can accept list of integers, list of strings and in general
|
a List.length function can accept list of integers, list of strings and in general
|
||||||
|
|
@ -87,12 +91,246 @@ With the Hindley-Milner type system each well typed function can have more than
|
||||||
type but always has a unique best type, called the \emph{principal type}.
|
type but always has a unique best type, called the \emph{principal type}.
|
||||||
For example the principal type of the List.length function is "For any \emph{a}, function from
|
For example the principal type of the List.length function is "For any \emph{a}, function from
|
||||||
list of \emph{a} to \emph{int}" and \emph{a} is called the \emph{type parameter}.
|
list of \emph{a} to \emph{int}" and \emph{a} is called the \emph{type parameter}.
|
||||||
|
\begin{itemize}
|
||||||
\item Strong typing: Languages such as C and C++ allow the programmer to operate on data
|
\item Strong typing: Languages such as C and C++ allow the programmer to operate on data
|
||||||
|
\end{itemize}
|
||||||
without considering its type, mainly through pointers. Other languages such as C\#
|
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.
|
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
|
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.
|
programmer is not allowed to operate on data by ignoring or promoting its type.
|
||||||
|
\begin{itemize}
|
||||||
\item Type Inference: the principal type of a well formed term can be inferred without any
|
\item Type Inference: the principal type of a well formed term can be inferred without any
|
||||||
annotation or declaration.
|
|
||||||
\end{itemize}
|
\end{itemize}
|
||||||
|
annotation or declaration.
|
||||||
|
\begin{itemize}
|
||||||
|
\item Algebraic data types: types that are modelled by the use of two
|
||||||
|
\end{itemize}
|
||||||
|
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 \emph{A + B} can hold at any moment a value of type A or a value of
|
||||||
|
type B. Sum types are also called tagged union or variants.
|
||||||
|
A product type is a type constructed as a direct product
|
||||||
|
of multiple types and contains at any moment one instance for
|
||||||
|
every type of its operands. Product types are also called tuples
|
||||||
|
or records. Algebraic data types can be recursive
|
||||||
|
in their definition and can be combined.
|
||||||
|
Moreover ML languages are functional, meaning that functions are
|
||||||
|
treated as first class citizens and variables are immutable,
|
||||||
|
although mutable statements and imperative constructs are permitted.
|
||||||
|
In addition to that OCaml features an object system, that provides
|
||||||
|
inheritance, subtyping and dynamic binding, and modules, that
|
||||||
|
provide a way to encapsulate definitions. Modules are checked
|
||||||
|
statically and can be reificated through functors.
|
||||||
|
|
||||||
|
\begin{enumerate}
|
||||||
|
\item {\bfseries\sffamily TODO} 1.2 Pattern matching [37\%]
|
||||||
|
\label{sec:org83a2e95}
|
||||||
|
\begin{itemize}
|
||||||
|
\item[{$\square$}] capisci come mettere gli esempi uno accanto all'altro
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
Pattern matching is a widely adopted mechanism to interact with ADT.
|
||||||
|
C family languages provide branching on predicates through the use of
|
||||||
|
if statements and switch statements.
|
||||||
|
Pattern matching is a mechanism for destructuring and analyzing data
|
||||||
|
structures for the presence of values simbolically represented as
|
||||||
|
tokens. One common example of pattern matching is the use of regular
|
||||||
|
expressions on strings. OCaml provides pattern matching on ADT,
|
||||||
|
primitive data types.
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item[{$\boxtimes$}] Esempio enum, C e Ocaml
|
||||||
|
\end{itemize}
|
||||||
|
\begin{verbatim}
|
||||||
|
type color = | Red | Blue | Green
|
||||||
|
|
||||||
|
begin match color with
|
||||||
|
| Red -> print "red"
|
||||||
|
| Blue -> print "red"
|
||||||
|
| Green -> print "red"
|
||||||
|
|
||||||
|
\end{verbatim}
|
||||||
|
|
||||||
|
OCaml provides tokens to express data destructoring
|
||||||
|
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item[{$\boxtimes$}] Esempio destructor list
|
||||||
|
\end{itemize}
|
||||||
|
\begin{verbatim}
|
||||||
|
|
||||||
|
begin match list with
|
||||||
|
| [ ] -> print "empty list"
|
||||||
|
| element1 :: [ ] -> print "one element"
|
||||||
|
| element1 :: element2 :: [ ] -> print "two elements"
|
||||||
|
| head :: tail-> print "head followed by many elements"
|
||||||
|
\end{verbatim}
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item[{$\boxtimes$}] Esempio destructor tuples
|
||||||
|
\end{itemize}
|
||||||
|
\begin{verbatim}
|
||||||
|
|
||||||
|
begin match tuple with
|
||||||
|
| (Some _, Some _) -> print "Pair of optional types"
|
||||||
|
| (Some _, None) -> print "Pair of optional types, last null"
|
||||||
|
| (None, Some _) -> print "Pair of optional types, first null"
|
||||||
|
| (None, None) -> print "Pair of optional types, both null"
|
||||||
|
\end{verbatim}
|
||||||
|
|
||||||
|
Pattern clauses can make the use of \emph{guards} to test predicates and
|
||||||
|
variables can be binded in scope.
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item[{$\square$}] Esempio binding e guards
|
||||||
|
\end{itemize}
|
||||||
|
\begin{verbatim}
|
||||||
|
|
||||||
|
begin match token_list with
|
||||||
|
| "switch"::var::"{"::rest ->
|
||||||
|
| "case"::":"::var::rest when is_int var ->
|
||||||
|
| "case"::":"::var::rest when is_string var ->
|
||||||
|
| "}"::[ ] -> stop ()
|
||||||
|
| "}"::rest -> error "syntax error: " rest
|
||||||
|
|
||||||
|
\end{verbatim}
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item[{$\square$}] Un altro esempio con destructors e tutto i lresto
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
In general pattern matching on primitive and algebraic data types takes the
|
||||||
|
following form.
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item[{$\square$}] Esempio informale
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
It can be described more formally through a BNF grammar.
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item[{$\square$}] BNF
|
||||||
|
|
||||||
|
\item[{$\square$}] Come funziona il pattern matching?
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
\item {\bfseries\sffamily TODO} 1.2.1 Pattern matching compilation to lambda code
|
||||||
|
\label{sec:org2127ffb}
|
||||||
|
|
||||||
|
\begin{itemize}
|
||||||
|
\item[{$\square$}] Da tabella a matrice
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
Formally, pattern are defined as follows:
|
||||||
|
\begin{center}
|
||||||
|
\begin{tabular}{ll}
|
||||||
|
pattern ::= & Patterns\\
|
||||||
|
\hline
|
||||||
|
\_ & wildcard\\
|
||||||
|
x & variable\\
|
||||||
|
c(p₁,p₂,\ldots{},pₙ & constructor pattern\\
|
||||||
|
(p₁\(\vert{}\) p₂) & or-pattern\\
|
||||||
|
\end{tabular}
|
||||||
|
\end{center}
|
||||||
|
|
||||||
|
Values are defined as follows:
|
||||||
|
\begin{center}
|
||||||
|
\begin{tabular}{ll}
|
||||||
|
values ::= & Values\\
|
||||||
|
\hline
|
||||||
|
c(v₁, v₂, \ldots{}, vₙ) & constructor value\\
|
||||||
|
\end{tabular}
|
||||||
|
\end{center}
|
||||||
|
|
||||||
|
The entire pattern matching code can be represented as a clause matrix
|
||||||
|
that associates rows of patterns (p\(_{\text{i,1}}\), p\(_{\text{i,2}}\), \ldots{}, p\(_{\text{i,n}}\)) to
|
||||||
|
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*}
|
||||||
|
|
||||||
|
Most native data types in OCaml, such as integers, tuples, lists,
|
||||||
|
records, can be seen as instances of the following definition
|
||||||
|
|
||||||
|
\begin{verbatim}
|
||||||
|
type t = Nil | One of int | Cons of int * t
|
||||||
|
\end{verbatim}
|
||||||
|
that is a type \emph{t} with three constructors that define its complete
|
||||||
|
signature.
|
||||||
|
Every constructor has an arity. Nil, a constructor of arity 0, is
|
||||||
|
called a constant constructor.
|
||||||
|
|
||||||
|
The pattern \emph{p} matches a value \emph{v}, written as p ≼ v, when
|
||||||
|
one of the following rules apply
|
||||||
|
|
||||||
|
\begin{center}
|
||||||
|
\begin{tabular}{llll}
|
||||||
|
\_ & ≼ & v & \\
|
||||||
|
x & ≼ & v & \\
|
||||||
|
(p₁ \(\vert{}\)$\backslash$ p₂) & ≼ & v & iff p₁ ≼ v or p₂ ≼ v\\
|
||||||
|
c(p₁, p₂, \ldots{}, pₐ) & ≼ & c(v₁, v₂, \ldots{}, vₐ) & iff (p₁, p₂, \ldots{}, pₐ) ≼ (v₁, v₂, \ldots{}, vₐ)\\
|
||||||
|
(p₁, p₂, \ldots{}, pₐ) & ≼ & (v₁, v₂, \ldots{}, vₐ) & iff pᵢ ≼ vᵢ ∀i ∈ [1..a]\\
|
||||||
|
\end{tabular}
|
||||||
|
\end{center}
|
||||||
|
We can also say that \emph{v} is an \emph{instance} of \emph{p}.
|
||||||
|
|
||||||
|
When we consider the pattern matrix P we say that the value vector
|
||||||
|
\vv{v} = (v₁, v₂, \ldots{}, vᵢ) matches the line number i in P if and only if the following two
|
||||||
|
conditions are satisfied:
|
||||||
|
\begin{itemize}
|
||||||
|
\item \[ p_{i,1} & p_{i,2} & \cdots & p_{i,n} \] ≼ (v₁, v₂, \ldots{}, vᵢ)
|
||||||
|
\item \[ ∀j < i p_{j,1} & p_{j,2} & \cdots & p_{j,n} \] ⋠ (v₁, v₂, \ldots{}, vᵢ)
|
||||||
|
\end{itemize}
|
||||||
|
|
||||||
|
We can define the following three relations with respect to patterns:
|
||||||
|
\begin{itemize}
|
||||||
|
\item Patter p is less precise than pattern q, writtens p ≼ q when all
|
||||||
|
instances of q are instances of p
|
||||||
|
\item Pattern p and q are equivalent, written p ≡ q, when their instances
|
||||||
|
are the same
|
||||||
|
\item Patterns p and q are compatible when they share a common instance
|
||||||
|
\end{itemize}
|
||||||
|
\end{enumerate}
|
||||||
|
|
||||||
|
\subsection{1.2.1.1 Initial state of the compilation}
|
||||||
|
\label{sec:org0b0c254}
|
||||||
|
|
||||||
|
Given a source of the following form:
|
||||||
|
|
||||||
|
\#+BEGIN\_SRC ocaml
|
||||||
|
match x with
|
||||||
|
\begin{center}
|
||||||
|
\begin{tabular}{l}
|
||||||
|
p₁ -> e₁\\
|
||||||
|
p₂ -> e₂\\
|
||||||
|
\end{tabular}
|
||||||
|
\end{center}
|
||||||
|
\ldots{}
|
||||||
|
\begin{center}
|
||||||
|
\begin{tabular}{l}
|
||||||
|
pₘ -> eₘ\\
|
||||||
|
\end{tabular}
|
||||||
|
\end{center}
|
||||||
|
\#+END\_SRC ocaml
|
||||||
|
|
||||||
|
the initial input of the algorithm consists of a vector of variables
|
||||||
|
\vv{x} = (x₁, x₂, \ldots{}, xₙ) of size n
|
||||||
|
and a clause matrix P → L of width n and height m.
|
||||||
|
|
||||||
|
\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*}
|
||||||
\end{document}
|
\end{document}
|
||||||
|
|
|
||||||
BIN
tesi/referenze/opt-pat.pdf
Normal file
BIN
tesi/referenze/opt-pat.pdf
Normal file
Binary file not shown.
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Reference in a new issue