From 8bd79b8e39eaed63e5e71e95536f611a7fc16c78 Mon Sep 17 00:00:00 2001 From: Francesco Mecca Date: Fri, 21 Feb 2020 11:29:04 +0100 Subject: [PATCH] logica che palle --- anno3/logica/EsameLogica_v4_soluzioni.pdf | Bin 0 -> 186811 bytes tesi/prova.aux | 18 +- tesi/prova.bbl | 0 tesi/prova.blg | 48 --- tesi/prova.log | 491 +++++++++++++--------- tesi/prova.md | 320 ++++++++++++++ tesi/prova.org | 257 +++++++++-- tesi/prova.out | 7 +- tesi/prova.pdf | Bin 133009 -> 45830 bytes tesi/prova.tex | 260 +++++++++++- tesi/referenze/opt-pat.pdf | Bin 0 -> 234995 bytes 11 files changed, 1096 insertions(+), 305 deletions(-) create mode 100644 anno3/logica/EsameLogica_v4_soluzioni.pdf delete mode 100644 tesi/prova.bbl delete mode 100644 tesi/prova.blg create mode 100644 tesi/prova.md create mode 100644 tesi/referenze/opt-pat.pdf diff --git a/anno3/logica/EsameLogica_v4_soluzioni.pdf b/anno3/logica/EsameLogica_v4_soluzioni.pdf new file mode 100644 index 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Misplaced alignment tab character &. +l.289 \item \[ ∀j < i p_{j,1} & + p_{j,2} & \cdots & p_{j,n} \] ⋠ (v₁, v₂, \ld... + +? +! Misplaced alignment tab character &. +l.289 \item \[ ∀j < i p_{j,1} & p_{j,2} & + \cdots & p_{j,n} \] ⋠ (v₁, v₂, \ld... + +? +! Misplaced alignment tab character &. + & + +l.289 \item \[ ∀j < i p_{j,1} & p_{j,2} & \cdots & + p_{j,n} \] ⋠ (v₁, v₂, \ld... + +? +Missing character: There is no ∀ in font cmr10! +Missing character: There is no ⋠ in font ec-lmr10! +Missing character: There is no ₁ in font ec-lmr10! +Missing character: There is no ₂ in font ec-lmr10! +Missing character: There is no ᵢ in font ec-lmr10! +Missing character: There is no ≼ in font ec-lmr10! +Missing character: There is no ≡ in font ec-lmr10! +Missing character: There is no ₁ in font ec-lmr10! +Missing character: There is no ₁ in font ec-lmr10! +Missing character: There is no ₂ in font ec-lmr10! +Missing character: There is no ₂ in font ec-lmr10! +Missing character: There is no ₘ in font ec-lmr10! +Missing character: There is no ₘ in font ec-lmr10! +! Undefined control sequence. +l.324 \vv + {x} = (x₁, x₂, \ldots{}, xₙ) of size n +? +Missing character: There is no ₁ in font ec-lmr10! +Missing character: There is no ₂ in font ec-lmr10! +Missing character: There is no ₙ in font ec-lmr10! +Missing character: There is no → in font ec-lmr10! +Missing character: There is no → in font cmr10! +Missing character: There is no ₁ in font cmr10! +Missing character: There is no → in font cmr10! +Missing character: There is no ₂ in font cmr10! +Missing character: There is no → in font cmr10! +Missing character: There is no → in font cmr10! +Missing character: There is no ₘ in font cmmi10! +Missing character: There is no → in font cmr10! +Package atveryend Info: Empty hook `BeforeClearDocument' on input line 336. +[6] +Package atveryend Info: Empty hook `AfterLastShipout' on input line 336. (./prova.aux) -Package atveryend Info: Executing hook `AtVeryEndDocument' on input line 98. -Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 98. +Package atveryend Info: Empty hook `AtVeryEndDocument' on input line 336. +Package atveryend Info: Executing hook `AtEndAfterFileList' on input line 336. Package rerunfilecheck Warning: File `prova.out' has changed. @@ -659,36 +752,20 @@ Package rerunfilecheck Warning: File `prova.out' has changed. (rerunfilecheck) or use package `bookmark'. Package rerunfilecheck Info: Checksums for `prova.out': -(rerunfilecheck) Before: -(rerunfilecheck) After: 7405D30B8D200DFB9FDA8AFBCA5DCCFE;160. +(rerunfilecheck) Before: 25CD49CE93FE2EBDE0A77084F183FACA +(rerunfilecheck) After: 4C10F3A28DB5B56905D83B509FE58CA0. LaTeX Warning: Label(s) may have changed. 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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 [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 + + 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: + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
pattern ::=Patterns
\_wildcard
xvariable
c(p1,p2,…,pnconstructor pattern
(p1| p2)or-pattern
+ + + + + + + + + + + + + + + + + + + + + + +
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 (pi,1, pi,2, …, pi,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 + + 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 + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
\_v 
xv 
(p₁ |\\ p₂)viff 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 + diff --git a/tesi/prova.org b/tesi/prova.org index f46f443..5dd2bbc 100644 --- a/tesi/prova.org +++ b/tesi/prova.org @@ -1,10 +1,25 @@ * TODO Scaletta [1/2] - [X] Abstract - - [ ] Introduction [0%] - - [ ] Ocaml - - [ ] Pattern matching + - [-] Background [20%] + - [X] Ocaml + - [ ] Lambda code [0%] + - [ ] Untyped lambda form + - [ ] OCaml specific instructions + - [-] Pattern matching [50%] + - [X] Introduzione + - [ ] Compilation to lambda form - [ ] Translation Verification - [ ] 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 #+AUTHOR: Francesco Mecca @@ -17,6 +32,8 @@ #+LaTeX_HEADER: \usepackage{algorithm} #+LaTeX_HEADER: \usepackage{algpseudocode} #+LaTeX_HEADER: \usepackage{amsmath,amssymb,amsthm} +#+LaTeX_HEADER: \usepackage[utf8]{inputenc} +#+LaTeX_HEADER: \usepackage[T1]{fontenc} #+Latex_HEADER: \newtheorem{definition}{Definition} #+LaTeX_HEADER: \usepackage{graphicx} #+LaTeX_HEADER: \usepackage{listings} @@ -41,30 +58,212 @@ blackboxes that are not evaluated in the context of the verification. \end{abstract} -* Introduction +* 1. Background -** TODO 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 with respect to static type systems of traditional imperative and/or object - oriented language such as C, C++ and Java many advantages 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. +** 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. + +*** 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*} diff --git a/tesi/prova.out b/tesi/prova.out index 946788b..ac99723 100644 --- a/tesi/prova.out +++ b/tesi/prova.out @@ -1,3 +1,4 @@ -\BOOKMARK [1][-]{section.1}{TODO Scaletta [1/2]}{}% 1 -\BOOKMARK [1][-]{section.2}{Introduction}{}% 2 -\BOOKMARK 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+\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} diff --git a/tesi/referenze/opt-pat.pdf b/tesi/referenze/opt-pat.pdf new file mode 100644 index 0000000000000000000000000000000000000000..bc0b518efffe71786ff9f78957f7ccb5d98132b4 GIT binary patch literal 234995 zcma&tLy#{(lOXW!er?;fZQHip|F&P-wr$(CUfZ^9+xyM#>}f7@%8EQzRApr3FOp1A zRGf~9o)wyG^eV0pnvIZ=(B9YznwOVB*38cQuLU9de=^Dp;+8gl&724s#BGfJnu(g3 z*qfU1^FurTbuu%ug?7(w(Y1}+5_f`sa5OpMpbZ2X<+@V50z6&^0ww%qmk%(Te9*rM zIzrgOx`D01a*}mw)+X_z`ejz1x{)9BM*FpvTOl3sy9}KZ!B|%xK&k0$5y!@YnWxIPw~@p#rdW&q={uR ze>t6kZ`Q1I?>S}`&v&6=IOZN%s@5fPR&kn_!r6$0oUhh1-X>IM0=&(FT*3J}j=!k&{wDPH|R=tPt6% zCGR&bO}4#T={|eQK5Y|6^b54u7j}!~3ah${=)vj?Qn7bTA1|71VFKe!`rWhXB>I^n zgf6ayUXNbKz8H?F6p3q(P?uc=+ttCkF7*4&hEbxKSnT?+lZUWpw-!K{iGz_#uus?5 z%L@N_HGfDt0Cv99QPt7Tdv?(MAk%*`FP5{aFzk2q6$U#xJn!4AU7rsa9)9(Eyx@L4 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