468 lines
16 KiB
TeX
468 lines
16 KiB
TeX
% Created 2020-02-24 Mon 19:44
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\author{Francesco Mecca}
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\date{}
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\title{Translation Verification of the OCaml pattern matching compiler}
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\hypersetup{
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pdfauthor={Francesco Mecca},
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pdftitle={Translation Verification of the OCaml pattern matching compiler},
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pdfkeywords={},
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pdfsubject={},
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pdfcreator={Emacs 26.3 (Org mode 9.1.9)},
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pdflang={English}}
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\begin{document}
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\maketitle
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\begin{comment}
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\section{{\bfseries\sffamily TODO} Scaletta [1/4]}
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\label{sec:orgdce55fc}
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\begin{itemize}
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\item[{$\boxtimes$}] Abstract
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\item[{$\boxminus$}] Background [40\%]
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\begin{itemize}
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\item[{$\boxtimes$}] Ocaml
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\item[{$\square$}] Lambda code [0\%]
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\begin{itemize}
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\item[{$\square$}] Untyped lambda form
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\item[{$\square$}] OCaml specific instructions
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\end{itemize}
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\item[{$\boxtimes$}] Pattern matching [100\%]
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\begin{itemize}
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\item[{$\boxtimes$}] Introduzione
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\item[{$\boxtimes$}] Compilation to lambda form
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\end{itemize}
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\item[{$\square$}] Translation Verification
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\item[{$\square$}] Symbolic execution
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\end{itemize}
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\item[{$\square$}] Translation verification of the Pattern Matching Compiler
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\begin{itemize}
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\item[{$\square$}] Source translation
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\begin{itemize}
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\item[{$\square$}] Formal Grammar
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\item[{$\square$}] Compilation of source patterns
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\end{itemize}
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\item[{$\square$}] Target translation
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\begin{itemize}
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\item[{$\square$}] Formal Grammar
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\item[{$\square$}] Symbolic execution of the lambda target
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\end{itemize}
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\item[{$\square$}] Equivalence between source and target
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\end{itemize}
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\item[{$\square$}] Practical results
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\end{itemize}
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\end{comment}
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\begin{abstract}
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This dissertation presents an algorithm for the translation validation of the OCaml
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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 OCaml 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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\section{Background}
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\label{sec:orgf336654}
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\subsection{OCaml}
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\label{sec:org885a2ca}
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Objective Caml (OCaml) is a dialect of the ML (Meta-Language) family of programming
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languages.
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OCaml 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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\begin{itemize}
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\item Parametric 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
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a List.length function can accept list of integers, list of strings and in general
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list 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 \emph{principal type}.
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For example the principal type of the List.length function is "For any \emph{a}, function from
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list of \emph{a} to \emph{int}" and \emph{a} is called the \emph{type parameter}.
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\item 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.
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In the case of programming languages using an Hindley-Milner type system the
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programmer is not allowed to operate on data by ignoring or promoting its type.
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\item 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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\item Algebraic data types: types that are modelled 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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as \emph{A + B} can hold at any moment a value of type A or a value of
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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.
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\end{itemize}
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Moreover ML languages are functional, meaning that functions are
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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 OCaml 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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\begin{enumerate}
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\item Pattern matching
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\label{sec:org879bb72}
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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. OCaml 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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\begin{itemize}
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\item this value does not match this pattern”
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\item 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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\end{itemize}
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\begin{verbatim}
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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{verbatim}
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OCaml 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{verbatim}
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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{verbatim}
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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{verbatim}
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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{verbatim}
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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 \emph{guards} to test predicates and
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variables can captured (binded in scope).
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\begin{verbatim}
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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{verbatim}
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Moreover, the OCaml 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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In general pattern matching on primitive and algebraic data types takes the
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following form.
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\begin{center}
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\begin{tabular}{l}
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match variable with\\
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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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⋮\\
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\(\vert{}\) pₙ -> exprₙ\\
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\end{tabular}
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\end{center}
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It can be described more formally through a BNF grammar.
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\begin{verbatim}
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pattern ::= value-name
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| _ ;; wildcard pattern
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| constant ;; matches a constant value
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| pattern as value-name ;; binds to value-name
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| ( pattern ) ;; parenthesized pattern
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| pattern | pattern ;; or-pattern
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| constr pattern ;; variant pattern
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| [ pattern { ; pattern } [ ; ] ] ;; list patterns
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| pattern :: pattern ;; lists patterns using cons operator (::)
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| [| pattern { ; pattern } [ ; ] |] ;; array pattern
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| char-literal .. char-literal ;; match on a range of characters
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| { field [: typexpr] [= pattern] { ; field [: typexpr] [= pattern] } \
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[; _ ] [ ; ] } ;; patterns that match on records
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\end{verbatim}
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\item 1.2.1 Pattern matching compilation to lambda code
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\label{sec:org5e1ded7}
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During compilation, patterns are in the form
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\begin{center}
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\begin{tabular}{ll}
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pattern & type of pattern\\
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\hline
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\_ & wildcard\\
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x & variable\\
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c(p₁,p₂,\ldots{},pₙ) & constructor pattern\\
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(p₁\(\vert{}\) p₂) & or-pattern\\
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\end{tabular}
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\end{center}
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Expressions are compiled into lambda code and are referred as lambda
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code actions.
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The entire pattern matching code can be represented as a clause matrix
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that associates rows of patterns (p\(_{\text{i,1}}\), p\(_{\text{i,2}}\), \ldots{}, p\(_{\text{i,n}}\)) to
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lambda code action lⁱ
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\begin{equation*}
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(P → L) =
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\begin{pmatrix}
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p_{1,1} & p_{1,2} & \cdots & p_{1,n} & → l₁ \\
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p_{2,1} & p_{2,2} & \cdots & p_{2,n} & → l₂ \\
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\vdots & \vdots & \ddots & \vdots & → \vdots \\
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p_{m,1} & p_{m,2} & \cdots & p_{m,n} & → lₘ
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\end{pmatrix}
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\end{equation*}
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Most native data types in OCaml, such as integers, tuples, lists,
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records, can be seen as instances of the following definition
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\begin{verbatim}
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type t = Nil | One of int | Cons of int * t
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\end{verbatim}
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that is a type \emph{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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The pattern \emph{p} matches a value \emph{v}, written as p ≼ v, when one of the
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following rules apply
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\begin{center}
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\begin{tabular}{llll}
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\hline
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\_ & ≼ & v & ∀v\\
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x & ≼ & v & ∀v\\
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(p₁ \(\vert{}\)$\backslash$ p₂) & ≼ & v & iff p₁ ≼ v or p₂ ≼ v\\
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c(p₁, p₂, \ldots{}, pₐ) & ≼ & c(v₁, v₂, \ldots{}, vₐ) & iff (p₁, p₂, \ldots{}, pₐ) ≼ (v₁, v₂, \ldots{}, vₐ)\\
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(p₁, p₂, \ldots{}, pₐ) & ≼ & (v₁, v₂, \ldots{}, vₐ) & iff pᵢ ≼ vᵢ ∀i ∈ [1..a]\\
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\hline
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\end{tabular}
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\end{center}
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When a value \emph{v} matches pattern \emph{p} we say that \emph{v} is an \emph{instance} of \emph{p}.
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Considering the pattern matrix P we say that the value vector
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\(\vec{v}\) = (v₁, v₂, \ldots{}, vᵢ) matches the line number i in P if and only if the following two
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conditions are satisfied:
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\begin{itemize}
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\item p\(_{\text{i,1}}\), p\(_{\text{i,2}}\), \(\cdots{}\), p\(_{\text{i,n}}\) ≼ (v₁, v₂, \ldots{}, vᵢ)
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\item ∀j < i p\(_{\text{j,1}}\), p\(_{\text{j,2}}\), \(\cdots{}\), p\(_{\text{j,n}}\) ⋠ (v₁, v₂, \ldots{}, vᵢ)
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\end{itemize}
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We can define the following three relations with respect to patterns:
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\begin{itemize}
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\item Patter p is less precise than pattern q, written p ≼ q, when all
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instances of q are instances of p
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\item Pattern p and q are equivalent, written p ≡ q, when their instances
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are the same
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\item Patterns p and q are compatible when they share a common instance
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\end{itemize}
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\begin{enumerate}
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\item Initial state of the compilation
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\label{sec:org886c2ed}
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Given a source of the following form:
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\begin{center}
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\begin{tabular}{l}
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match variable with\\
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\(\vert{}\) pattern₁ -> e₁\\
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\(\vert{}\) pattern₂ -> e₂\\
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⋮\\
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\(\vert{}\) pₘ -> eₘ\\
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\end{tabular}
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\end{center}
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the initial input of the algorithm C consists of a vector of variables
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\(\vec{x}\) = (x₁, x₂, \ldots{}, xₙ) of size \emph{n} where \emph{n} is the arity of
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the type of \emph{x} and a clause matrix P → L of width n and height m.
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That is:
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\begin{equation*}
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C((\vec{x} = (x₁, x₂, ..., xₙ),
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\begin{pmatrix}
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p_{1,1} & p_{1,2} & \cdots & p_{1,n} → l₁ \\
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p_{2,1} & p_{2,2} & \cdots & p_{2,n} → l₂ \\
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\vdots & \vdots & \ddots & \vdots → \vdots \\
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p_{m,1} & p_{m,2} & \cdots & p_{m,n} → lₘ)
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\end{pmatrix}
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\end{equation*}
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The base case C₀ of the algorithm is the case in which the \(\vec{x}\)
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is empty, that is \(\vec{x}\) = (), then the result of the compilation
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C₀ is l₁
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\begin{equation*}
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C₀((),
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\begin{pmatrix}
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→ l₁ \\
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→ l₂ \\
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→ \vdots \\
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→ lₘ
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\end{pmatrix})
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) = l₁
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\end{equation*}
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When \(\vec{x}\) ≠ () then the compilation advances using one of the
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following four rules:
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\begin{enumerate}
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\item Variable rule: if all patterns of the first column of P are wildcard patterns or
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bind the value to a variable, then
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\begin{equation*}
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C(\vec{x}, P → L) = C((x₂, x₃, ..., xₙ), P' → L')
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\end{equation*}
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where
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\begin{equation*}
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\begin{pmatrix}
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p_{1,2} & \cdots & p_{1,n} & → & (let & y₁ & x₁) & l₁ \\
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p_{2,2} & \cdots & p_{2,n} & → & (let & y₂ & x₁) & l₂ \\
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\vdots & \ddots & \vdots & → & \vdots & \vdots & \vdots & \vdots \\
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p_{m,2} & \cdots & p_{m,n} & → & (let & yₘ & x₁) & lₘ
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\end{pmatrix}
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\end{equation*}
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That means in every lambda action lᵢ there is a binding of x₁ to the
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variable that appears on the pattern \$p\(_{\text{i,1}}\). Bindings are omitted
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for wildcard patterns and the lambda action lᵢ remains unchanged.
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\item Constructor rule: if all patterns in the first column of P are
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constructors patterns of the form k(q₁, q₂, \ldots{}, qₙ) we define a
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new matrix, the specialized clause matrix S, by applying the
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following transformation on every row \emph{p}:
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\begin{lstlisting}[mathescape,columns=fullflexible,basicstyle=\fontfamily{lmvtt}\selectfont,]
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for every c ∈ Set of constructors
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for i ← 1 .. m
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let kᵢ ← constructor_of($p_{i,1}$)
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if kᵢ = c then
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p ← $q_{i,1}$, $q_{i,2}$, ..., $q_{i,n'}$, $p_{i,2}$, $p_{i,3}$, ..., $p_{i, n}$
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\end{lstlisting}
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Patterns of the form \(q_{i,j}\) matches on the values of the
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constructor and we define new fresh variables y₁, y₂, \ldots{}, yₐ so
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that the lambda action lᵢ becomes
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\end{enumerate}
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\begin{lstlisting}[mathescape,columns=fullflexible,basicstyle=\fontfamily{lmvtt}\selectfont,]
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(let (y₁ (field 0 x₁))
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(y₂ (field 1 x₁))
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...
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(yₐ (field (a-1) x₁))
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lᵢ)
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\end{lstlisting}
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and the result of the compilation for the set of constructors
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\{c₁, c₂, \ldots{}, cₖ\} is:
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\begin{lstlisting}[mathescape,columns=fullflexible,basicstyle=\fontfamily{lmvtt}\selectfont,]
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switch x₁ with
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case c₁: l₁
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case c₂: l₂
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...
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case cₖ: lₖ
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default: exit
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\end{lstlisting}
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\begin{enumerate}
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\item Orpat rule: there are various strategies for dealing with
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or-patterns. The most naive one is to split the or-patterns.
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For example a row p containing an or-pattern:
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\begin{equation*}
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(p_{i,1}|q_{i,1}|r_{i,1}), p_{i,2}, ..., p_{i,m} → lᵢ
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\end{equation*}
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results in three rows added to the clause matrix
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\begin{equation*}
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p_{i,1}, p_{i,2}, ..., p_{i,m} → lᵢ \\
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\end{equation*}
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\begin{equation*}
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q_{i,1}, p_{i,2}, ..., p_{i,m} → lᵢ \\
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\end{equation*}
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\begin{equation*}
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r_{i,1}, p_{i,2}, ..., p_{i,m} → lᵢ
|
||
\end{equation*}
|
||
\item Mixture rule:
|
||
When none of the previous rules apply the clause matrix P → L is
|
||
splitted into two clause matrices, the first P₁ → L₁ that is the
|
||
largest prefix matrix for which one of the three previous rules
|
||
apply, and P₂ → L₂ containing the remaining rows. The algorithm is
|
||
applied to both matrices.
|
||
\end{enumerate}
|
||
|
||
\begin{comment}
|
||
#+BEGIN_COMMENT
|
||
CITE paper?
|
||
#+END_COMMENT’
|
||
\end{comment}
|
||
\end{enumerate}
|
||
\end{enumerate}
|
||
\end{document}
|