569 lines
21 KiB
Org Mode
569 lines
21 KiB
Org Mode
\begin{comment}
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* TODO Scaletta [1/5]
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- [X] Abstract
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- [-] Background [40%]
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- [X]
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- [ ] Lambda code [0%]
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- [ ] Compiler optimizations
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- [ ] other instructions
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- [X] Pattern matching
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- [ ] Symbolic execution
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- [ ] Translation Validation
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- [ ] Translation validation of the Pattern Matching Compiler
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- [ ] Source translation
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- [ ] Formal Grammar
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- [ ] Compilation of source patterns
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- [ ] Rest?
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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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- [ ] Equivalence between source and target
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- [ ] Proof of correctness
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- [ ] Practical results
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\end{comment}
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#+TITLE: Translation Verification of the 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: \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{graphicx}
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#+LaTeX_HEADER: \usepackage{listings}
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||
#+LaTeX_HEADER: \usepackage{color}
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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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\begin{abstract}
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This dissertation presents an algorithm for the translation validation of the
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pattern matching compiler. Given the source representation of the target program and the
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target program compiled in untyped lambda form, the algoritmhm is capable of modelling
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||
the source program in terms of symbolic constraints on it's branches and apply symbolic
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execution on the untyped lambda representation in order to validate wheter the compilation
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||
produced a valid result.
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||
In this context a valid result means that for every input in the domain of the source
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||
program the untyped lambda translation produces the same output as the source program.
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||
The input of the program is modelled in terms of symbolic constraints closely related to
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||
the runtime representation of objects and the output consists of OCaml code
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blackboxes that are not evaluated in the context of the verification.
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\end{abstract}
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* Background
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||
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**
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Objective Caml () is a dialect of the ML (Meta-Language) family of programming
|
||
languages.
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||
shares many features with other dialects of ML, such as SML and Caml Light,
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||
The main features of ML languages are the use of the Hindley-Milner type system that
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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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||
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
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||
lists of any type. Such languages offer polymorphic functions through subtyping at
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||
runtime only, while other languages such as C++ offer polymorphism through compile
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||
time templates and function overloading.
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||
With the Hindley-Milner type system each well typed function can have more than one
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||
type but always has a unique best type, called the /principal type/.
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||
For example the principal type of the List.length function is "For any /a/, function from
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||
list of /a/ to /int/" and /a/ is called the /type parameter/.
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||
- Strong typing: Languages such as C and C++ allow the programmer to operate on data
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||
without considering its type, mainly through pointers. Other languages such as C#
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||
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.
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||
- 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
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||
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
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||
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.
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||
A product type is a type constructed as a direct product
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||
of multiple types and contains at any moment one instance for
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||
every type of its operands. Product types are also called tuples
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||
or records. Algebraic data types can be recursive
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||
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,
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||
although mutable statements and imperative constructs are permitted.
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||
In addition to that features an object system, that provides
|
||
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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||
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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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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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||
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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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lambda form to assembly code.
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*** datatypes
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||
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Most native data types in , such as integers, tuples, lists,
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records, can be seen as instances of the following definition
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#+BEGIN_SRC
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type t = Nil | One of int | Cons of int * t
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||
#+END_SRC
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||
that is a type /t/ with three constructors that define its complete
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||
signature.
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||
Every constructor has an arity. Nil, a constructor of arity 0, is
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||
called a constant constructor.
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||
|
||
*** Lambda form types
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||
A lambda form target file produced by the compiler consists of a
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||
single s-expression. Every s-expression consist of /(/, a sequence of
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||
elements separated by a whitespace and a closing /)/.
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||
Elements of s-expressions are:
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||
- Atoms: sequences of ascii letters, digits or symbols
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||
- Variables
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||
- Strings: enclosed in double quotes and possibly escaped
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- S-expressions: allowing arbitrary nesting
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There are several numeric types:
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- integers: that us either 31 or 63 bit two's complement arithmetic
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depending on system word size, and also wrapping on overflow
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- 32 bit and 64 bit integers: that use 32-bit and 64-bit two's complement arithmetic
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with wrap on overflow
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- big integers: offer integers with arbitrary precision
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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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||
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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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- Numeric comparisons: <, >, <=, >=, ==
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||
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*** Functions
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Functions are defined using the following syntax, and close over all
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bindings in scope: (lambda (arg1 arg2 arg3) BODY)
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and are applied using the following syntax: (apply FUNC ARG ARG ARG)
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Evaluation is eager.
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||
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*** Bindings
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The atom /let/ introduces a sequence of bindings:
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(let BINDING BINDING BINDING ... BODY)
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*** Other atoms
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TODO: if, switch, stringswitch...
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TODO: magari esempi
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** Pattern matching
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Pattern matching is a widely adopted mechanism to interact with ADT.
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C family languages provide branching on predicates through the use of
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if statements and switch statements.
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Pattern matching on the other hands express predicates through
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syntactic templates that also allow to bind on data structures of
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arbitrary shapes. One common example of pattern matching is the use of regular
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||
expressions on strings. provides pattern matching on ADT and
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primitive data types.
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The result of a pattern matching operation is always one of:
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- this value does not match this pattern”
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- this value matches this pattern, resulting the following bindings of
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names to values and the jump to the expression pointed at the
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pattern.
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#+BEGIN_SRC
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type color = | Red | Blue | Green | Black | White
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match color with
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| Red -> print "red"
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| Blue -> print "red"
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| Green -> print "red"
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| _ -> print "white or black"
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#+END_SRC
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||
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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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#+BEGIN_SRC
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begin match list with
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| [ ] -> print "empty list"
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| element1 :: [ ] -> print "one element"
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| (element1 :: element2) :: [ ] -> print "two elements"
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| head :: tail-> print "head followed by many elements"
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#+END_SRC
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Parenthesized patterns, such as the third one in the previous example,
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matches the same value as the pattern without parenthesis.
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The same could be done with tuples
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#+BEGIN_SRC
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begin match tuple with
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| (Some _, Some _) -> print "Pair of optional types"
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| (Some _, None) | (None, Some _) -> print "Pair of optional types, one of which is null"
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| (None, None) -> print "Pair of optional types, both null"
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#+END_SRC
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The pattern pattern₁ | pattern₂ represents the logical "or" of the
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two patterns pattern₁ and pattern₂. A value matches pattern₁ |
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pattern₂ if it matches pattern₁ or pattern₂.
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Pattern clauses can make the use of /guards/ to test predicates and
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variables can captured (binded in scope).
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#+BEGIN_SRC
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begin match token_list with
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| "switch"::var::"{"::rest -> ...
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| "case"::":"::var::rest when is_int var -> ...
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||
| "case"::":"::var::rest when is_string var -> ...
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| "}"::[ ] -> ...
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| "}"::rest -> error "syntax error: " rest
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#+END_SRC
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Moreover, the pattern matching compiler emits a warning when a
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pattern is not exhaustive or some patterns are shadowed by precedent ones.
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** Symbolic execution
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** Translation validation
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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
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||
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.
|
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Translation validation is an alternative to the verification of
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existing translators that consists of taking the source and the target
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(compiled) program and proving /a posteriori/ their semantic equivalence.
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- [ ] Techniques for translation validation
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- [ ] What does semantically equivalent mean
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- [ ] What happens when there is no semantic equivalence
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- [ ] Translation validation through symbolic execution
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** Translation validation of the Pattern Matching Compiler
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*** Source program
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The algorithm takes as its input a source program and translates it
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into an algebraic data structure called /constraint_tree/.
|
||
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#+BEGIN_SRC
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type constraint_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_blackbox * constraint_tree * constraint_tree
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||
| Node of accessor * (constructor * constraint_tree) list * constraint_tree
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#+END_SRC
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Unreachable, Leaf of source_expr and Failure are the terminals of the three.
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We distinguish
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- Unreachable: statically it is known that no value can go there
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- Failure: a value matching this part results in an error
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- Leaf: a value matching this part results into the evaluation of a
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source blackbox of code
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|
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The algorithm doesn't support type-declaration-based analysis
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to know the list of constructors at a given type.
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Let's consider some trivial examples:
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#+BEGIN_SRC
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function true -> 1
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#+END_SRC
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[ ] Converti a disegni
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||
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Is translated to
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|Node ([(true, Leaf 1)], Failure)
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while
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#+BEGIN_SRC
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function
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true -> 1
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| false -> 2
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#+END_SRC
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will give
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||
|Node ([(true, Leaf 1); (false, Leaf 2)])
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||
|
||
It is possible to produce Unreachable examples by using
|
||
refutation clauses (a "dot" in the right-hand-side)
|
||
#+BEGIN_SRC
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function
|
||
true -> 1
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||
| false -> 2
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||
| _ -> .
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||
#+END_SRC
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that gets translated into
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Node ([(true, Leaf 1); (false, Leaf 2)], Unreachable)
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We trust this annotation, which is reasonable as the type-checker
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verifies that it indeed holds.
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|
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Guard nodes of the tree are emitted whenever a guard is found. Guards
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node contains a blackbox of code that is never evaluated and two
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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.
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[ ] Finisci con Node
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[ ] Spiega del fallback
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[ ] rivedi compilazione per tenere in considerazione il tuo albero invece che le lambda
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[ ] Specifica che stesso algoritmo usato per compilare a lambda, piu` optimizations
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The source code of a pattern matching function in has the
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following form:
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||
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||
|match variable with
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|\vert pattern₁ -> expr₁
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||
|\vert pattern₂ when guard -> expr₂
|
||
|\vert pattern₃ as var -> expr₃
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||
|⋮
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|\vert pₙ -> exprₙ
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||
|
||
and can include any expression that is legal for the compiler,
|
||
such as "when" conditions and assignments. Patterns could or could not
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be exhaustive.
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||
|
||
Pattern matching code could also be written using the more compact form:
|
||
|function
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||
|\vert pattern₁ -> expr₁
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|\vert pattern₂ when guard -> expr₂
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|\vert pattern₃ as var -> expr₃
|
||
|⋮
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|\vert pₙ -> exprₙ
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||
|
||
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||
This BNF grammar describes formally the grammar of the source program:
|
||
|
||
#+BEGIN_SRC bnf
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||
start ::= "match" id "with" patterns | "function" patterns
|
||
patterns ::= (pattern0|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 ::= "|" clause
|
||
clause ::= lexpr "->" rexpr
|
||
|
||
lexpr ::= rule (ε|condition)
|
||
rexpr ::= _code ;; arbitrary code
|
||
|
||
rule ::= wildcard|variable|constructor_pattern|or_pattern ;;
|
||
|
||
;; rules
|
||
wildcard ::= "_"
|
||
variable ::= identifier
|
||
constructor_pattern ::= constructor (rule|ε) (assignment|ε)
|
||
|
||
constructor ::= int|float|char|string|bool
|
||
|unit|record|exn|objects|ref
|
||
|list|tuple|array
|
||
|variant|parameterized_variant ;; data types
|
||
|
||
or_pattern ::= wildcard|variable|constructor_pattern ("|" wildcard|variable|constructor_pattern)+
|
||
|
||
condition ::= "when" bexpr
|
||
assignment ::= "as" id
|
||
bexpr ::= _code ;; arbitrary code
|
||
#+END_SRC
|
||
|
||
\begin{comment}
|
||
Check that it is still this
|
||
\end{comment}
|
||
|
||
Patterns are of the form
|
||
| pattern | type of pattern |
|
||
|-----------------+---------------------|
|
||
| _ | wildcard |
|
||
| x | variable |
|
||
| c(p₁,p₂,...,pₙ) | constructor pattern |
|
||
| (p₁\vert p₂) | or-pattern |
|
||
|
||
During compilation by the translators expressions are compiled into
|
||
lambda code and are referred as lambda code actions lᵢ.
|
||
|
||
The entire pattern matching code is 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*}
|
||
|
||
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/.
|
||
|
||
Considering the pattern matrix P we say that the value vector
|
||
$\vec{v}$ = (v₁, v₂, ..., vᵢ) matches the line number i in P if and only if the following two
|
||
conditions are satisfied:
|
||
- p_{i,1}, p_{i,2}, \cdots, p_{i,n} ≼ (v₁, v₂, ..., vᵢ)
|
||
- ∀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, 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
|
||
|
||
**** Initial state of the compilation
|
||
|
||
Given a source of the following form:
|
||
|
||
|
||
|match variable with
|
||
|\vert pattern₁ -> e₁
|
||
|\vert pattern₂ -> e₂
|
||
|⋮
|
||
|\vert pₘ -> eₘ
|
||
|
||
the initial input of the algorithm C consists of a vector of variables
|
||
$\vec{x}$ = (x₁, x₂, ..., xₙ) of size /n/ where /n/ is the arity of
|
||
the type of /x/ and a clause matrix P → L of width n and height m.
|
||
That is:
|
||
|
||
\begin{equation*}
|
||
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 empty, that is $\vec{x}$ = (), then the result of the compilation
|
||
C₀ is l₁
|
||
\begin{equation*}
|
||
C₀((),
|
||
\begin{pmatrix}
|
||
→ l₁ \\
|
||
→ l₂ \\
|
||
→ \vdots \\
|
||
→ lₘ
|
||
\end{pmatrix})
|
||
) = l₁
|
||
\end{equation*}
|
||
|
||
When $\vec{x}$ ≠ () then the compilation advances using one of the
|
||
following four rules:
|
||
|
||
1) Variable rule: if all patterns of the first column of P are wildcard patterns or
|
||
bind the value to a variable, then
|
||
|
||
\begin{equation*}
|
||
C(\vec{x}, P → L) = C((x₂, x₃, ..., xₙ), P' → L')
|
||
\end{equation*}
|
||
where
|
||
\begin{equation*}
|
||
\begin{pmatrix}
|
||
p_{1,2} & \cdots & p_{1,n} & → & (let & y₁ & x₁) & l₁ \\
|
||
p_{2,2} & \cdots & p_{2,n} & → & (let & y₂ & x₁) & l₂ \\
|
||
\vdots & \ddots & \vdots & → & \vdots & \vdots & \vdots & \vdots \\
|
||
p_{m,2} & \cdots & p_{m,n} & → & (let & yₘ & x₁) & lₘ
|
||
\end{pmatrix}
|
||
\end{equation*}
|
||
|
||
That means in every lambda action lᵢ there is a binding of x₁ to the
|
||
variable that appears on the pattern $p_{i,1}. Bindings are omitted
|
||
for wildcard patterns and the lambda action lᵢ remains unchanged.
|
||
|
||
2) Constructor rule: if all patterns in the first column of P are
|
||
constructors patterns of the form k(q₁, q₂, ..., qₙ) we define a
|
||
new matrix, the specialized clause matrix S, by applying the
|
||
following transformation on every row /p/:
|
||
\begin{lstlisting}[mathescape,columns=fullflexible,basicstyle=\fontfamily{lmvtt}\selectfont,]
|
||
for every c ∈ Set of constructors
|
||
for i ← 1 .. m
|
||
let kᵢ ← constructor_of($p_{i,1}$)
|
||
if kᵢ = c then
|
||
p ← $q_{i,1}$, $q_{i,2}$, ..., $q_{i,n'}$, $p_{i,2}$, $p_{i,3}$, ..., $p_{i, n}$
|
||
\end{lstlisting}
|
||
Patterns of the form $q_{i,j}$ matches on the values of the
|
||
constructor and we define new fresh variables y₁, y₂, ..., yₐ so
|
||
that the lambda action lᵢ becomes
|
||
|
||
\begin{lstlisting}[mathescape,columns=fullflexible,basicstyle=\fontfamily{lmvtt}\selectfont,]
|
||
(let (y₁ (field 0 x₁))
|
||
(y₂ (field 1 x₁))
|
||
...
|
||
(yₐ (field (a-1) x₁))
|
||
lᵢ)
|
||
\end{lstlisting}
|
||
|
||
and the result of the compilation for the set of constructors
|
||
{c₁, c₂, ..., cₖ} is:
|
||
|
||
\begin{lstlisting}[mathescape,columns=fullflexible,basicstyle=\fontfamily{lmvtt}\selectfont,]
|
||
switch x₁ with
|
||
case c₁: l₁
|
||
case c₂: l₂
|
||
...
|
||
case cₖ: lₖ
|
||
default: exit
|
||
\end{lstlisting}
|
||
|
||
3) Orpat rule: there are various strategies for dealing with
|
||
or-patterns. The most naive one is to split the or-patterns.
|
||
For example a row p containing an or-pattern:
|
||
\begin{equation*}
|
||
(p_{i,1}|q_{i,1}|r_{i,1}), p_{i,2}, ..., p_{i,m} → lᵢ
|
||
\end{equation*}
|
||
results in three rows added to the clause matrix
|
||
\begin{equation*}
|
||
p_{i,1}, p_{i,2}, ..., p_{i,m} → lᵢ \\
|
||
\end{equation*}
|
||
\begin{equation*}
|
||
q_{i,1}, p_{i,2}, ..., p_{i,m} → lᵢ \\
|
||
\end{equation*}
|
||
\begin{equation*}
|
||
r_{i,1}, p_{i,2}, ..., p_{i,m} → lᵢ
|
||
\end{equation*}
|
||
4) Mixture rule:
|
||
When none of the previous rules apply the clause matrix P → L is
|
||
splitted into two clause matrices, the first P₁ → L₁ that is the
|
||
largest prefix matrix for which one of the three previous rules
|
||
apply, and P₂ → L₂ containing the remaining rows. The algorithm is
|
||
applied to both matrices.
|
||
|
||
\begin{comment}
|
||
#+BEGIN_COMMENT
|
||
CITE paper?
|
||
#+END_COMMENT’
|
||
\end{comment}
|