\begin{comment} * TODO Scaletta [1/5] - [X] Introduction - [-] Background [40%] - [ ] Lambda code [0%] - [ ] Compiler optimizations - [ ] other instructions - [X] Pattern matching - [ ] Symbolic execution - [ ] Translation Validation - [ ] Translation validation of the Pattern Matching Compiler - [ ] Source translation - [ ] Formal Grammar - [ ] Compilation of source patterns - [ ] Rest? - [ ] Target translation - [ ] Formal Grammar - [ ] Symbolic execution of the Lambda target - [ ] Equivalence between source and target - [ ] Statement of correctness - [ ] Proof of correctness - [ ] Practical results TODO: talk about compiler stuff \end{comment} #+TITLE: Translation Verification of the pattern matching compiler #+AUTHOR: Francesco Mecca #+EMAIL: me@francescomecca.eu #+DATE: #+LANGUAGE: en #+LaTeX_CLASS: article #+LaTeX_HEADER: \usepackage{algorithm} #+LaTeX_HEADER: \usepackage{comment} #+LaTeX_HEADER: \usepackage{algpseudocode} #+LaTeX_HEADER: \usepackage{amsmath,amssymb,amsthm} #+Latex_HEADER: \newtheorem{definition}{Definition} #+LaTeX_HEADER: \usepackage{graphicx} #+LaTeX_HEADER: \usepackage{listings} #+LaTeX_HEADER: \usepackage{color} #+LaTeX_HEADER: \usepackage{stmaryrd} #+LaTeX_HEADER: \newcommand{\sem}[1]{{\llbracket{#1}\rrbracket}} #+EXPORT_SELECT_TAGS: export #+EXPORT_EXCLUDE_TAGS: noexport #+OPTIONS: H:2 toc:nil \n:nil @:t ::t |:t ^:{} _:{} *:t TeX:t LaTeX:t #+STARTUP: showall \section{Introduction} This dissertation presents an algorithm for the translation validation of the OCaml pattern matching compiler. Given a source program and its compiled version the algorithm checks whether the two are equivalent or produce a counter example in case of a mismatch. For the prototype of this algorithm we have chosen a subset of the OCaml language and implemented a prototype equivalence checker along with a formal statement of correctness and its proof. The prototype is to be included in the OCaml compiler infrastructure and will aid the development. Our equivalence algorithm works with decision trees. Source patterns are converted into a decision tree using a matrix decomposition algorithm. Target programs, described in the Lambda intermediate representation language of the OCaml compiler, are turned into decision trees by applying symbolic execution. \begin{comment} \subsection{Translation validation} \end{comment} A pattern matching compiler turns a series of pattern matching clauses into simple control flow structures such as \texttt{if, switch}, for example: \begin{lstlisting} match x with | [] -> (0, None) | x::[] -> (1, Some x) | _::y::_ -> (2, Some y) \end{lstlisting} \begin{lstlisting} (if scrutinee (let (field_1 =a (field 1 scrutinee)) (if field_1 (let (field_1_1 =a (field 1 field_1) x =a (field 0 field_1)) (makeblock 0 2 (makeblock 0 x))) (let (y =a (field 0 scrutinee)) (makeblock 0 1 (makeblock 0 y))))) [0: 0 0a]) \end{lstlisting} \begin{comment} %% TODO: side by side \end{comment} The code in the right is in the Lambda intermediate representation of the OCaml compiler. The Lambda representation of a program is shown by calling the \texttt{ocamlc} compiler with \texttt{-drawlambda} flag. The OCaml pattern matching compiler is a critical part of the OCaml compiler in terms of correctness because any bug would result in wrong code production rather than triggering compilation failures. Such bugs also are hard to catch by testing because they arise in corner cases of complex patterns which are typically not in the compiler test suite. The OCaml core developers group considered evolving the pattern matching compiler, either by using a new algorithm or by incremental refactoring of its code base. For this reason we want to verify that new implementations of the compiler avoid the introduction of new bugs and that such modifications don't result in a different behavior than the current one. One possible approach is to formally verify the pattern matching compiler implementation using a machine checked proof. Another possibility, albeit with a weaker result, is to verify that each source program and target program pair are semantically correct. We chose the latter technique, translation validation because is easier to adopt in the case of a production compiler and to integrate with an existing code base. The compiler is treated as a black-box and proof only depends on our equivalence algorithm. \subsection{Our approach} %% replace common TODO Our algorithm translates both source and target programs into a common representation, decision trees. Decision trees where chosen because they model the space of possible values at a given branch of execution. Here is the decision tree for the source example program. \begin{verbatim} Node(Root) / \ (= []) (= ::) / \ Leaf Node(Root.1) (0, None) / \ (= []) (= ::) / \ Leaf Leaf (1, Some(Root.0)) (2, Some(Root.1.0)) \end{verbatim} \texttt{(Root.0)} is called an \emph{accessor}, that represents the access path to a value that can be reached by deconstructing the scrutinee. In this example \texttt{Root.0} is the first subvalue of the scrutinee. Target decision trees have a similar shape but the tests on the branches are related to the low level representation of values in Lambda code. For example, cons cells \texttt{x::xs} are blocks with tag 0. To check the equivalence of a source and a target decision tree, we proceed by case analysis. If we have two terminals, such as leaves in the previous example, we check that the two right-hand-sides are equivalent. If we have a node $N$ and another tree $T$ we check equivalence for each child of $N$, which is a pair of a branch condition $\pi_i$ and a subtree $C_i$. For every child $(\pi_i, C_i)$ we reduce $T$ by killing all the branches that are incompatible with $\pi_i$ and check that the reduced tree is equivalent to $C_i$. \subsection{From source programs to decision trees} Our source language supports integers, lists, tuples and all algebraic datatypes. Patterns support wildcards, constructors and literals, or patterns $(p_1|p_2)$ and pattern variables. We also support \texttt{when} guards. Decision trees have nodes of the form: \begin{lstlisting} type decision_tree = | Unreachable | Failure | Leaf of source_expr | Guard of source_expr * decision_tree * decision_tree | Switch of accessor * (constructor * decision_tree) list * decision_tree \end{lstlisting} In the \texttt{Switch} node we have one subtree for every head constructor that appears in the pattern matching clauses and a fallback case for other values. The branch condition $\pi_i$ expresses that the value at the switch accessor starts with the given constructor. \texttt{Failure} nodes express match failures for values that are not matched by the source clauses. \texttt{Unreachable} is used when we statically know that no value can flow to that subtree. We write $\sem{t_S}_S$ for the decision tree of the source program $t_S$, computed by a matrix decomposition algorithm (each column decomposition step gives a \texttt{Switch} node). It satisfies the following correctness statement: \[ \forall t_S, \forall v_S, \quad t_S(v_S) = \sem{t_S}_S(v_S) \] Running any source values $v_S$ against the source program gives the same result as running it against the decision tree. \subsection{From target programs to decision trees} The target programs include the following Lambda constructs: \texttt{let, if, switch, Match\_failure, catch, exit, field} and various comparison operations, guards. The symbolic execution engine traverses the target program and builds an environment that maps variables to accessors. It branches at every control flow statement and emits a Switch node. The branch condition $\pi_i$ is expressed as an interval set of possible values at that point. Guards result in branching. In comparison with the source decision trees, \texttt{Unreachable} nodes are never emitted. We write $\sem{t_T}_T$ for the decision tree of the target program $t_T$, satisfying the following correctness statement: \[ \forall t_T, \forall v_T, \quad t_T(v_T) = \sem{t_T}_T(v_T) \] \subsection{Equivalence checking} TODO * Background ** OCaml Objective Caml (OCaml) is a dialect of the ML (Meta-Language) family of programming that 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: - Polymorphism: in certain scenarios a function can accept more than one type for the input parameters. For example a function that computes the length 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 lists of integers, lists of strings and in general lists 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 modeled 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 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 reifycated through functors. ** Lambda form compilation \begin{comment} https://dev.realworld.org/compiler-backend.html \end{comment} provides compilation in form of a bytecode executable with an optionally embeddable interpreter and a native executable that could be statically linked to provide a single file executable. After the typechecker has proven that the program is type safe, the compiler lower the code to /Lambda/, an s-expression based language that assumes that its input has already been proved safe. On the /Lambda/ representation of the source program, the compiler performs a series of optimization passes before translating the lambda form to assembly code. *** OCaml Native Datatypes Most native data types in , such as integers, tuples, lists, records, can be seen as instances of the following definition #+BEGIN_SRC 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. *** Lambda form types A Lambda form target file produced by the compiler consists of a single s-expression. Every s-expression consist of /(/, a sequence of elements separated by a whitespace and a closing /)/. Elements of s-expressions are: - Atoms: sequences of ascii letters, digits or symbols - Variables - Strings: enclosed in double quotes and possibly escaped - S-expressions: allowing arbitrary nesting There are several numeric types: - integers: that us either 31 or 63 bit two's complement arithmetic depending on system word size, and also wrapping on overflow - 32 bit and 64 bit integers: that use 32-bit and 64-bit two's complement arithmetic with wrap on overflow - big integers: offer integers with arbitrary precision - floats: that use IEEE754 double-precision (64-bit) arithmetic with the addition of the literals /infinity/, /neg_infinity/ and /nan/. The are various numeric operations defined: - Arithmetic operations: +, -, *, /, % (modulo), neg (unary negation) - Bitwise operations: &, |, ^, <<, >> (zero-shifting), a>> (sign extending) - Numeric comparisons: <, >, <=, >=, == *** Functions Functions are defined using the following syntax, and close over all bindings in scope: (lambda (arg1 arg2 arg3) BODY) and are applied using the following syntax: (apply FUNC ARG ARG ARG) Evaluation is eager. begin{comment} *** Bindings The atom /let/ introduces a sequence of bindings: (let BINDING BINDING BINDING ... BODY) *** Other atoms TODO: if, switch, stringswitch... TODO: magari esempi end{comment} ** Pattern matching 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: - this value does not match this pattern” - this value matches this pattern, resulting the following bindings of names to values and the jump to the expression pointed at the pattern. #+BEGIN_SRC type color = | Red | Blue | Green | Black | White match color with | Red -> print "red" | Blue -> print "red" | Green -> print "red" | _ -> print "white or black" #+END_SRC provides tokens to express data destructoring. For example we can examine the content of a list with pattern matching #+BEGIN_SRC 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 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_SRC 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_SRC 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 /guards/ to test predicates and variables can captured (binded in scope). #+BEGIN_SRC 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_SRC Moreover, the pattern matching compiler emits a warning when a pattern is not exhaustive or some patterns are shadowed by precedent ones. ** Symbolic execution TODO ** Translation validation 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 /a posteriori/ their semantic equivalence. - [ ] Techniques for translation validation - [ ] What does semantically equivalent mean - [ ] What happens when there is no semantic equivalence - [ ] Translation validation through symbolic execution * Translation validation of the Pattern Matching Compiler ** Source program The algorithm takes as its input a source program and translates it into an algebraic data structure called /decision_tree/. #+BEGIN_SRC type decision_tree = | Unreachable | Failure | Leaf of source_expr | Guard of source_blackbox * decision_tree * decision_tree | Node of accessor * (constructor * decision_tree) list * decision_tree #+END_SRC Unreachable, Leaf of source_expr and Failure are the terminals of the three. We distinguish - Unreachable: statically it is known that no value can go there - Failure: a value matching this part results in an error - Leaf: a value matching this part results into the evaluation of a source black box of code 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_SRC function true -> 1 #+END_SRC [ ] Converti a disegni Is translated to |Node ([(true, Leaf 1)], Failure) while #+BEGIN_SRC function true -> 1 | false -> 2 #+END_SRC will give |Node ([(true, Leaf 1); (false, Leaf 2)]) It is possible to produce Unreachable examples by using refutation clauses (a "dot" in the right-hand-side) #+BEGIN_SRC function true -> 1 | false -> 2 | _ -> . #+END_SRC 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 has the following form: |match variable with |\vert pattern₁ -> expr₁ |\vert pattern₂ when guard -> expr₂ |\vert pattern₃ as var -> expr₃ |⋮ |\vert pₙ -> exprₙ and can include any expression that is legal for the OCaml compiler, such as /when/ guards and assignments. Patterns could or could not be exhaustive. Pattern matching code could also be written using the more compact form: |function |\vert pattern₁ -> expr₁ |\vert pattern₂ when guard -> expr₂ |\vert pattern₃ as var -> expr₃ |⋮ |\vert pₙ -> exprₙ This BNF grammar describes formally the grammar of the source program: #+BEGIN_SRC bnf 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 ::= rule ("|" wildcard|variable|constructor_pattern)+ condition ::= "when" bexpr assignment ::= "as" id bexpr ::= _code ;; arbitrary code #+END_SRC \begin{comment} Check that it is still coherent to this bnf \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 \subsubsection{Parsing of the source program} The source program of the following general form is parsed using a parser generated by Menhir, a LR(1) parser generator for the OCaml programming language. Menhir compiles LR(1) a grammar specification, in this case the OCaml pattern matching grammar, down to OCaml code. |match variable with |\vert pattern₁ -> e₁ |\vert pattern₂ -> e₂ |⋮ |\vert pₘ -> eₘ The result of parsing, when successful, results in a list of clauses and a list of type declarations. Every clause consists of three objects: a left-hand-side that is the kind of pattern expressed, an option guard and a right-hand-side expression. Patterns are encoded in the following way: | pattern | type | |-----------------+-------------| | _ | Wildcard | | p₁ as x | Assignment | | c(p₁,p₂,...,pₙ) | Constructor | | (p₁\vert p₂) | Orpat | Guards and right-hand-sides are treated as a blackbox of OCaml code. A sound approach for treating these blackbox would be to inspect the OCaml compiler during translation to Lambda code and extract the blackboxes compiled in their Lambda representation. This would allow to test for equality with the respective blackbox at the target level. Given that this level of introspection is currently not possibile, we decided to restrict the structure of blackboxes to the following (valid) OCaml code: #+BEGIN_SRC external guard : 'a -> 'b = "guard" external observe : 'a -> 'b = "observe" #+END_SRC We assume these two external functions /guard/ and /observe/ with a valid type that lets the user pass any number of arguments to them. All the guards are of the form \texttt{guard }, where the are expressed using the OCaml pattern matching language. Similarly, all the right-hand-side expressions are of the form \texttt{observe ...} with the same constraints on arguments. #+BEGIN_SRC type t = Z | S of t let _ = function | Z -> observe 0 | S Z -> observe 1 | S x when guard x -> observe 2 | S (S x) as y when guard x y -> observe 3 | S _ -> observe 4 #+END_SRC Once parsed, the type declarations and the list of clauses are encoded in the form of a matrix that is later evaluated using a matrix decomposition algorithm. \subsubsection{Matrix decomposition of pattern clauses} The initial input of the decomposition 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 split 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. * Correctness of the algorithm Running a program tₛ or its translation 〚tₛ〛 against an input vₛ produces as a result a result /r/ in the following way: | ( 〚tₛ〛ₛ(vₛ) = Cₛ(vₛ) ) → r | tₛ(vₛ) → r Likewise | ( 〚tₜ〛ₜ(vₜ) = Cₜ(vₜ) ) → r | tₜ(vₜ) → r where result r ::= guard list * (Match blackbox | NoMatch | Absurd) and guard ::= blackbox. Having defined equivalence between two inputs of which one is expressed in the source language and the other in the target language vₛ ≃ vₜ (TODO define, this talks about the representation of source values in the target) we can define the equivalence between a couple of programs or a couple ofconstraint trees | tₛ ≃ tₜ := ∀vₛ≃vₜ, tₛ(vₛ) = tₜ(vₜ) | Cₛ ≃ Cₜ := ∀vₛ≃vₜ, Cₛ(vₛ) = Cₜ(vₜ) The proposed equivalence algorithm that works on a couple of constraint trees is returns either /Yes/ or /No(vₛ, vₜ)/ where vₛ and vₜ are a couple of possible counter examples for which the constraint trees produce a different result. ** Statements Theorem. We say that a translation of a source program to a constraint tree is correct when for every possible input, the source program and its respective constraint tree produces the same result | ∀vₛ, tₛ(vₛ) = 〚tₛ〛ₛ(vₛ) Likewise, for the target language: | ∀vₜ, tₜ(vₜ) = 〚tₜ〛ₜ(vₜ) Definition: in the presence of guards we can say that two results are equivalent modulo the guards queue, written /r₁ ≃gs r₂/, when: | (gs₁, r₁) ≃gs (gs₂, r₂) ⇔ (gs₁, r₁) = (gs₂ ++ gs, r₂) Definition: we say that Cₜ covers the input space /S/, written /covers(Cₜ, S) when every value vₛ∈S is a valid input to the constraint tree Cₜ. (TODO: rephrase) Theorem: Given an input space /S/ and a couple of constraint trees, where the target constraint tree Cₜ covers the input space /S/, we say that the two constraint trees are equivalent when: | (equiv S Cₛ Cₜ gs = Yes) ∧ covers(Cₜ, S) ⇒ ∀vₛ≃vₜ ∈ S, Cₛ(vₛ) ≃gs Cₜ(vₜ) Similarly we say that a couple of constraint trees in the presence of an input space /S/ are /not/ equivalent when: | (equiv S Cₛ Cₜ gs = No(vₛ,vₜ) ∧ covers(Cₜ, S) ⇒ vₛ≃vₜ ∈ S ∧ Cₛ(vₛ) ≠gs Cₜ(vₜ) Corollary: For a full input space /S/, that is the universe of the target program we say: | (equiv S [|tₛ|]ₛ [|tₜ|]ₜ ∅ = Yes) ⇔ tₛ ≃ tₜ