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-(let BINDING BINDING BINDING \ldots{} BODY) - -\item Other atoms -\label{sec:org58bd28f} -TODO: if, switch, stringswitch\ldots{} -TODO: magari esempi -\end{enumerate} - - -\subsection{Pattern matching} -\label{sec:org5d3b2f5} - -Pattern matching is a widely adopted mechanism to interact with ADT. -C family languages provide branching on predicates through the use of -if statements and switch statements. -Pattern matching on the other hands express predicates through -syntactic templates that also allow to bind on data structures of -arbitrary shapes. One common example of pattern matching is the use of regular -expressions on strings. provides pattern matching on ADT and -primitive data types. -The result of a pattern matching operation is always one of: -\begin{itemize} -\item this value does not match this pattern” -\item this value matches this pattern, resulting the following bindings of -names to values and the jump to the expression pointed at the -pattern. -\end{itemize} - -\begin{verbatim} -type color = | Red | Blue | Green | Black | White - -match color with -| Red -> print "red" -| Blue -> print "red" -| Green -> print "red" -| _ -> print "white or black" -\end{verbatim} - - provides tokens to express data destructoring. -For example we can examine the content of a list with patten matching - -\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} - -Parenthesized patterns, such as the third one in the previous example, -matches the same value as the pattern without parenthesis. - -The same could be done with tuples -\begin{verbatim} - -begin match tuple with -| (Some _, Some _) -> print "Pair of optional types" -| (Some _, None) | (None, Some _) -> print "Pair of optional types, one of which is null" -| (None, None) -> print "Pair of optional types, both null" -\end{verbatim} - -The pattern pattern₁ | pattern₂ represents the logical "or" of the -two patterns pattern₁ and pattern₂. A value matches pattern₁ | -pattern₂ if it matches pattern₁ or pattern₂. - -Pattern clauses can make the use of \emph{guards} to test predicates and -variables can captured (binded in scope). - -\begin{verbatim} - -begin match token_list with -| "switch"::var::"{"::rest -> ... -| "case"::":"::var::rest when is_int var -> ... -| "case"::":"::var::rest when is_string var -> ... -| "}"::[ ] -> ... -| "}"::rest -> error "syntax error: " rest - -\end{verbatim} - -Moreover, the pattern matching compiler emits a warning when a -pattern is not exhaustive or some patterns are shadowed by precedent ones. - -\subsection{Symbolic execution} -\label{sec:orge2e0205} - -\subsection{Translation validation} -\label{sec:orgbafe7bc} -Translators, such as translators and code generators, are huge pieces of -software usually consisting of multiple subsystem and -constructing an actual specification of a translator implementation for -formal validation is a very long task. Moreover, different -translators implement different algorithms, so the correctness proof of -a translator cannot be generalized and reused to prove another translator. -Translation validation is an alternative to the verification of -existing translators that consists of taking the source and the target -(compiled) program and proving \emph{a posteriori} their semantic equivalence. - -\begin{itemize} -\item[{$\square$}] Techniques for translation validation -\item[{$\square$}] What does semantically equivalent mean -\item[{$\square$}] What happens when there is no semantic equivalence -\item[{$\square$}] Translation validation through symbolic execution -\end{itemize} - -\subsection{Translation validation of the Pattern Matching Compiler} -\label{sec:org24ee133} - -\begin{enumerate} -\item Source program -\label{sec:org8c21912} -The algorithm takes as its input a source program and translates it -into an algebraic data structure called \emph{constraint\_tree}. - -\begin{verbatim} -type constraint_tree = - | Unreachable - | Failure - | Leaf of source_expr - | Guard of source_blackbox * constraint_tree * constraint_tree - | Node of accessor * (constructor * constraint_tree) list * constraint_tree -\end{verbatim} - -Unreachable, Leaf of source\_expr and Failure are the terminals of the three. -We distinguish -\begin{itemize} -\item Unreachable: statically it is known that no value can go there -\item Failure: a value matching this part results in an error -\item Leaf: a value matching this part results into the evaluation of a -source blackbox of code -\end{itemize} - -The algorithm doesn't support type-declaration-based analysis -to know the list of constructors at a given type. -Let's consider some trivial examples: - -\begin{verbatim} -function true -> 1 -\end{verbatim} - -[ ] Converti a disegni - -Is translated to -\begin{center} -\begin{tabular}{l} -Node ([(true, Leaf 1)], Failure)\\ -\end{tabular} -\end{center} -while -\begin{verbatim} -function -true -> 1 -| false -> 2 -\end{verbatim} -will give -\begin{center} -\begin{tabular}{l} -Node ([(true, Leaf 1); (false, Leaf 2)])\\ -\end{tabular} -\end{center} - -It is possible to produce Unreachable examples by using - refutation clauses (a "dot" in the right-hand-side) -\begin{verbatim} -function -true -> 1 -| false -> 2 -| _ -> . -\end{verbatim} -that gets translated into -Node ([(true, Leaf 1); (false, Leaf 2)], Unreachable) - -We trust this annotation, which is reasonable as the type-checker -verifies that it indeed holds. - -Guard nodes of the tree are emitted whenever a guard is found. Guards -node contains a blackbox of code that is never evaluated and two -branches, one that is taken in case the guard evaluates to true and -the other one that contains the path taken when the guard evaluates to -true. - -[ ] Finisci con Node -[ ] Spiega del fallback -[ ] rivedi compilazione per tenere in considerazione il tuo albero invece che le lambda -[ ] Specifica che stesso algoritmo usato per compilare a lambda, piu` optimizations - -The source code of a pattern matching function in has the -following form: - -\begin{center} -\begin{tabular}{l} -match variable with\\ -\(\vert{}\) pattern₁ -> expr₁\\ -\(\vert{}\) pattern₂ when guard -> expr₂\\ -\(\vert{}\) pattern₃ as var -> expr₃\\ -⋮\\ -\(\vert{}\) pₙ -> exprₙ\\ -\end{tabular} -\end{center} - -and can include any expression that is legal for the compiler, -such as "when" conditions and assignments. Patterns could or could not -be exhaustive. - -Pattern matching code could also be written using the more compact form: -\begin{center} -\begin{tabular}{l} -function\\ -\(\vert{}\) pattern₁ -> expr₁\\ -\(\vert{}\) pattern₂ when guard -> expr₂\\ -\(\vert{}\) pattern₃ as var -> expr₃\\ -⋮\\ -\(\vert{}\) pₙ -> exprₙ\\ -\end{tabular} -\end{center} - - -This BNF grammar describes formally the grammar of the source program: - -\begin{verbatim} -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{verbatim} - -\begin{comment} -Check that it is still this -\end{comment} - -Patterns are of the form -\begin{center} -\begin{tabular}{ll} -pattern & type of pattern\\ -\hline -\_ & wildcard\\ -x & variable\\ -c(p₁,p₂,\ldots{},pₙ) & constructor pattern\\ -(p₁\(\vert{}\) p₂) & or-pattern\\ -\end{tabular} -\end{center} - -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\(_{\text{i,1}}\), 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*} - -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} -\hline -\_ & ≼ & v & ∀v\\ -x & ≼ & v & ∀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]\\ -\hline -\end{tabular} -\end{center} - -When a value \emph{v} matches pattern \emph{p} we say that \emph{v} is an \emph{instance} of \emph{p}. - -Considering the pattern matrix P we say that the value vector -\(\vec{v}\) = (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\(_{\text{i,1}}\), p\(_{\text{i,2}}\), \(\cdots{}\), p\(_{\text{i,n}}\) ≼ (v₁, v₂, \ldots{}, vᵢ) -\item ∀j < i p\(_{\text{j,1}}\), p\(_{\text{j,2}}\), \(\cdots{}\), p\(_{\text{j,n}}\) ⋠ (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, written 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} - -\begin{enumerate} -\item Initial state of the compilation -\label{sec:org9a7b624} - -Given a source of the following form: - - -\begin{center} -\begin{tabular}{l} -match variable with\\ -\(\vert{}\) pattern₁ -> e₁\\ -\(\vert{}\) pattern₂ -> e₂\\ -⋮\\ -\(\vert{}\) pₘ -> eₘ\\ -\end{tabular} -\end{center} - -the initial input of the algorithm C consists of a vector of variables -\(\vec{x}\) = (x₁, x₂, \ldots{}, xₙ) of size \emph{n} where \emph{n} is the arity of -the type of \emph{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: - -\begin{enumerate} -\item 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\(_{\text{i,1}}\). Bindings are omitted -for wildcard patterns and the lambda action lᵢ remains unchanged. - -\item Constructor rule: if all patterns in the first column of P are -constructors patterns of the form k(q₁, q₂, \ldots{}, qₙ) we define a -new matrix, the specialized clause matrix S, by applying the -following transformation on every row \emph{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, \ldots{}, yₐ so -that the lambda action lᵢ becomes -\end{enumerate} - -\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, \ldots{}, 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} - -\begin{enumerate} -\item 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*} -\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}