From 286cda4043c2b721f5e6036057ab157e7d26cd60 Mon Sep 17 00:00:00 2001 From: Francesco Mecca Date: Wed, 27 May 2020 18:49:35 +0200 Subject: [PATCH] small changes related to abstract --- tesi/tesi_unicode.org | 357 +++++++++++++++++++++++++++++------------- tesi/tesi_unicode.pdf | Bin 252707 -> 0 bytes todo.org | 42 +++-- 3 files changed, 277 insertions(+), 122 deletions(-) delete mode 100644 tesi/tesi_unicode.pdf diff --git a/tesi/tesi_unicode.org b/tesi/tesi_unicode.org index a9a6cc8..f8e3633 100644 --- a/tesi/tesi_unicode.org +++ b/tesi/tesi_unicode.org @@ -237,7 +237,7 @@ match w with \begin{minipage}{0.6\linewidth} \begin{lstlisting} match w with -| p₁|p₂|p₃ -> expr +|p₁ |p₂ |p₃ -> expr \end{lstlisting} \end{minipage} We also support \texttt{when} guards, which are interesting as they @@ -559,8 +559,8 @@ type color = | Red | Blue | Green | Black | White match color with | Red -> print "red" -| Blue -> print "red" -| Green -> print "red" +| Blue -> print "blue" +| Green -> print "green" | _ -> print "white or black" #+END_SRC @@ -751,60 +751,6 @@ existing translators that consists of taking the source and the target * Translation validation of the Pattern Matching Compiler ** Source program -Our algorithm takes as its input a source program and translates it -into an algebraic data structure which type we call /decision_tree/. - -#+BEGIN_SRC -type decision_tree = - | Unreachable - | Failure - | Leaf of source_expr - | Guard of source_blackbox * decision_tree * decision_tree - | Switch 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 -Our algorithm doesn't support type-declaration-based analysis -to know the list of constructors at a given type. -Let's consider some trivial examples: - -| function true -> 1 -is translated to -|Switch ([(true, Leaf 1)], Failure) -while -| function -| \vert{} true -> 1 -| \vert{} false -> 2 -will be translated to - -|Switch ([(true, Leaf 1); (false, Leaf 2)]) -It is possible to produce Unreachable examples by using - refutation clauses (a "dot" in the right-hand-side) -|function -|\vert{} true -> 1 -|\vert{} false -> 2 -|\vert{} _ -> . -that gets translated into -| Switch ([(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. - -\begin{comment} -[ ] Finisci con Switch -[ ] Spiega del fallback -\end{comment} The source code of a pattern matching function has the following form: @@ -1200,17 +1146,243 @@ Here = 1 In our prototype the source matrix mₛ is defined as follows | SMatrix mₛ := (aⱼ)^{j∈J}, ((p_{ij})^{j∈J} → bbᵢ)^{i∈I} -Source matrices are used to build source decision trees Cₛ. +** Target translation + +The target 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 a subset +of the Lambda intermediate language, down to OCaml code. +This is the grammar of the target language (TODO: check menhir grammar) +| start ::= sexpr +| sexpr ::= variable \vert{} string \vert{} "(" special_form ")" +| string ::= "\"" identifier "\"" ;; string between doublequotes +| variable ::= identifier +| special_form ::= let\vert{}catch\vert{}if\vert{}switch\vert{}switch-star\vert{}field\vert{}apply\vert{}isout +| let ::= "let" assignment sexpr ;; (assignment sexpr)+ outside of pattern match code +| assignment ::= "function" variable variable+ ;; the first variable is the identifier of the function +| field ::= "field" digit variable +| apply ::= ocaml_lambda_code ;; arbitrary code +| catch ::= "catch" sexpr with sexpr +| with ::= "with" "(" label ")" +| exit ::= "exit" label +| switch-star ::= "switch*" variable case* +| switch::= "switch" variable case* "default:" sexpr +| case ::= "case" casevar ":" sexpr +| casevar ::= ("tag"\vert{}"int") integer +| if ::= "if" bexpr sexpr sexpr +| bexpr ::= "(" ("!="\vert{}"=="\vert{}">="\vert{}"<="\vert{}">"\vert{}"<") sexpr digit \vert{} field ")" +| label ::= integer +The prototype doesn't support strings. + +The AST built by the parser is traversed and evaluated by the symbolic +execution engine. +Given that the target language supports jumps in the form of "catch - exit" +blocks the engine tries to evaluate the instructions inside the blocks +and stores the result of the partial evaluation into a record. +When a jump is encountered, the information at the point allows to +finalize the evaluation of the jump block. +In the environment the engine also stores bindings to values and +functions. +For performance reasons the compiler performs integer addition and +subtraction on variables that appears inside a /switch/ expression in +order to have values always start from 0. +Let's see an example of this behaviour: +#+BEGIN_SRC +let f x = match x with + | 3 -> "3" + | 4 -> "4" + | 5 -> "5" + | 6 -> "6" + | _ -> "_" +#+END_SRC +#+BEGIN_SRC +(setglobal Prova! + (let + (f/80 = + (function x/81[int] + (catch + (let (switcher/83 =a (-3+ x/81)) + (if (isout 3 switcher/83) (exit 1) + (switch* switcher/83 + case int 0: "3" + case int 1: "4" + case int 2: "5" + case int 3: "6"))) + with (1) "_"))) + (makeblock 0 f/80))) +#+END_SRC +The prototype takes into account such transformations and at the end +of the symbolic evaluation it traverses the result in order to "undo" +such optimization and have accessors of the variables match their +intended value directly. + +** Decision Trees +For the source language we use Source matrices to build source +decision trees Cₛ, +while in the case of the target language the decision tree Cₜ is the +result of the symbolic execution on the lambda code. +We give the following definitions. +| Source and target programs: tₛ and tₜ +| Source and target expressions: eₛ and eₜ +A source program $t_S$ is a list of pattern-matching +clauses (with an optional \texttt{when}-guard) of the form +\lstinline{| $p$ (when $e_S$)? -> $e_S$}, and a target program $t_T$ +is a series of control-flow conditionals (\texttt{if, switch, + Match\_failure, catch, exit}), value access (\texttt{field}), +variable bindings (\texttt{let}) and comparison operators in Lambda +code, with arbitrary Lambda-code expressions $e_T$ at the leaves. + +We assume given an equivalence relation $\erel {e_S} {e_T}$ on +expressions (that are contained in the leaves of the decision trees). +In our translation-validation setting, it suffices to +relate each source expression to its compiled form -- the compiler +computes this relation. We have to lift this relation on leaves +into an equivalence procedure for (pattern-matching) programs. + +We already defined what it means to run a value vₛ against a program +tₛ: +| tₛ(vₛ) := NoMatch \quad if ∀i, pᵢ(vₛ) = ⊥ +| tₛ(vₛ) = { Absurd if bb_{i₀} = . (refutation clause) +| \quad \quad \quad \quad \quad Match (σ, bb_{i₀}) otherwise +| \quad \quad \quad \quad \quad where iₒ = min{i \vert{} pᵢ(vₛ) ≠ ⊥} +and simmetrically, to run a value vₜ against a target program tₜ: +# TODO t_t +Starting from this we can define a +a relation $\vrel {v_S} {v_T}$ to relate a source to a target value +in terms of equivalence between the two results: +\begin{mathpar} + \erel {e_S} {e_T} \; (\text{\emph{assumed}}) + + t_S(v_S),\ t_T(v_T),\ \vrel {v_S} {v_T} \; (\text{\emph{omitted}}) + + \resrel {r_S} {r_T}, \runrel {R_S} {R_T} \; (\text{\emph{simple}}) +\\ + \begin{array}{l@{~}r@{~}r@{~}l} + \text{\emph{environment}} & \sigma(v) + & \bnfeq & [x_1 \mapsto v_1, \dots, v_n \mapsto v_n] \\ + \text{\emph{closed term}} & \cle(v) + & \bnfeq & (\sigma(v), e) \\ + \end{array} +\quad + \begin{array}{l@{~}r@{~}r@{~}l} + \text{\emph{matching result}} & r(v) + & \bnfeq & \NoMatch \bnfor \Match {\cle(v)} \\ + \text{\emph{matching run}} & R(v) + & \bnfeq & (\cle(v)_1, \dots, \cle(v)_n), r(v) \\ + \end{array} +\\ + \infer + {\forall x,\ \vrel {\sigma_S(x)} {\sigma_T(x)}} + {\envrel {\sigma_S} {\sigma_T}} + + \infer + {\envrel {\sigma_S} {\sigma_T} \\ \erel {e_S} {e_T}} + {\clerel {(\sigma_S, e_S)} {(\sigma_T, e_T)}} + + \infer + {\forall {\vrel {v_S} {v_T}},\quad \runrel {t_S(v_S)} {t_T(v_T)}} + {\progrel t_S t_T} +\end{mathpar} + +This relation captures our knowledge of the +OCaml value representation, for example it relates the empty list +constructor \texttt{[]} to $\Int 0$. We can then define \emph{closed} +expressions $\cle$, pairing a (source or target) expression with the +environment $\sigma$ captured by a program, and what it means to +``run'' a value against a program or a decision, written $t(v)$ and +$D(v)$, which returns a trace $(\cle_1, \dots, \cle_n)$ of the +executed guards and a \emph{matching result} $r$. + +Once formulated in this way, our equivalence algorithm must check the +natural notion of input-output equivalence for matching programs, +captured by the relation $\progrel {t_S} {t_T}$. +# TODO eq_muovi + +\subsection{Grammar of the decision trees} A decision tree is defined as either a Leaf, a Failure terminal or an intermediate node with different children sharing the same accessor /a/ and an optional fallback. -Failure is emitted only when the patterns don't cover the whole set of -possible input values /S/. The fallback is not needed when the user -doesn't use a wildcard pattern. -%%% Give example of thing -| Cₛ ::= Leaf bb \vert Switch(a, (Kᵢ → Cᵢ)^{i∈S} , C?) \vert Failure \vert Unreachable -| vₛ ::= K(vᵢ)^{i∈I} \vert n ∈ ℕ -\begin{comment} + +The parametrized grammar $D(\pi, e)$ describes the common structure of +source and decision trees. We use $\pi$ for the \emph{conditions} on +each branch, and $a$ for our \emph{accessors}, which give a symbolic +description of a sub-value of the scrutinee. Source conditions $\pi_S$ +are just datatype constructors; target conditions $\pi_T$ are +arbitrary sets of possible immediate-integer or block-tag values. +% +\begin{mathpar} + \begin{array}{l@{~}r@{~}r@{~}l} + \text{\emph{decision trees}} & D(\pi, e) + & \bnfeq & \Leaf {\cle(a)} \\ + & & \bnfor & \Failure \\ + & & \bnfor & \Switch a {\Fam {i \in I} {\pi_i, D_i}} \Dfb \\ + & & \bnfor & \Guard {\cle(a)} {D_0} {D_1} \\ + & & \bnfor & Unreachable \\ + \text{\emph{accessors}} & a + & \bnfeq & \Root \;\bnfor\; a.n \quad (n \in \mathbb{N}) \\ + \end{array} + \quad + \begin{array}{l} + \pi_S : \text{datatype constructors} + \\ + \pi_T \subseteq \{ \Int n \mid n \in \mathbb{Z} \} + \uplus \{ \Tag n \mid n \in \mathbb{N} \} + \\[1em] + a(v_S), a(v_T), D_S(v_S), D_T(v_T) \quad (\text{\emph{omitted})} + \end{array} +\end{mathpar} +The tree $\Leaf \cle$ returns a leaf expression $e$ in a captured +environment $\sigma$ mapping variables to accessors. +% +$\Failure$ expresses match failure, when no clause matches the input +value. +% +$\Switch a {\Fam {i \in I} {\pi_i, D_i}} \Dfb$ has one subtree $D_i$ +for every head constructor that appears in the pattern matching +clauses, and a fallback case for the constructors. +% +$\Guard \cle {D_0} {D_1}$ represents a \texttt{when}-guard on a closed +expression $\cle$, expected to be of boolean type, with sub-trees +$D_0$ for the \texttt{true} case and $D_1$ for the \texttt{false} +case. + +We write $a(v)$ for the sub-value of the (source or target) value $v$ +that is reachable at the accessor $a$, and $D(v)$ for the result of +running a value $v$ against a decision tree $D$; this results in +a (source or target) matching run $R(v)$, just like running the value +against a program. + +\subsection{From source programs to decision trees} +Let's consider some trivial examples: +| function true -> 1 +is translated to +|Switch ([(true, Leaf 1)], Failure) +while +| function +| \vert{} true -> 1 +| \vert{} false -> 2 +will be translated to + +|Switch ([(true, Leaf 1); (false, Leaf 2)]) +It is possible to produce Unreachable examples by using + refutation clauses (a "dot" in the right-hand-side) +|function +|\vert{} true -> 1 +|\vert{} false -> 2 +|\vert{} _ -> . +that gets translated into +| Switch ([(true, Leaf 1); (false, Leaf 2)], Unreachable) + +We trust this annotation, which is reasonable as the type-checker +verifies that it indeed holds. +We'll see that while we can build Unreachable nodes from source +programs, in the lambda target there isn't a construct equivalent to the refutation clause. + +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. Are K and Here clear here? \end{comment} We say that a translation of a source program to a decision tree @@ -1384,55 +1556,20 @@ In the second case (Kᵥ is distinct from Kₖ for all Kₖ ∈ constrs(pₒⱼ) we have to prove that | (pᵢⱼ)ⁱ(vᵢ)ⁱ = rⱼ_{wild} tail(vᵢ)ⁱ -TODO +# TODO - -** Target translation - -The target 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 a subset -of the Lambda intermediate language, down to OCaml code. -This is the grammar of the target language (TODO: check menhir grammar) -| start ::= sexpr -| sexpr ::= variable \vert{} string \vert{} "(" special_form ")" -| string ::= "\"" identifier "\"" ;; string between doublequotes -| variable ::= identifier -| special_form ::= let\vert{}catch\vert{}if\vert{}switch\vert{}switch-star\vert{}field\vert{}apply\vert{}isout -| let ::= "let" assignment sexpr ;; (assignment sexpr)+ outside of pattern match code -| assignment ::= "function" variable variable+ ;; the first variable is the identifier of the function -| field ::= "field" digit variable -| apply ::= ocaml_lambda_code ;; arbitrary code -| catch ::= "catch" sexpr with sexpr -| with ::= "with" "(" label ")" -| exit ::= "exit" label -| switch-star ::= "switch*" variable case* -| switch::= "switch" variable case* "default:" sexpr -| case ::= "case" casevar ":" sexpr -| casevar ::= ("tag"\vert{}"int") integer -| if ::= "if" bexpr sexpr sexpr -| bexpr ::= "(" ("!="\vert{}"=="\vert{}">="\vert{}"<="\vert{}">"\vert{}"<") sexpr digit \vert{} field ")" -| label ::= integer -The prototype doesn't support strings. - -The AST built by the parser is traversed and evaluated by the symbolic -execution engine. -Given that the target language supports jumps in the form of "catch - exit" -blocks the engine tries to evaluate the instructions inside the blocks -and stores the result of the partial evaluation into a record. -When a jump is encountered, the information at the point allows to -finalize the evaluation of the jump block. -In the environment the engine also stores bindings to values and -functions. -Integer additions and subtractions are simplified in a second pass. +\subsection{From target programs to target decision trees} +# TODO: replace C with D +Symbolic Values during symbolic evaluation have the following form +| vₜ ::= Cell(tag ∈ ℕ, (vᵢ)^{i∈I}) \vert n ∈ ℕ The result of the symbolic evaluation is a target decision tree Cₜ | Cₜ ::= Leaf bb \vert Switch(a, (πᵢ → Cᵢ)^{i∈S} , C?) \vert Failure -| vₜ ::= Cell(tag ∈ ℕ, (vᵢ)^{i∈I}) \vert n ∈ ℕ -Every branch of the decision tree is "constrained" by a domain -| Domain π = { n\vert{}n∈ℕ x n\vert{}n∈Tag⊆ℕ } -Intuitively, the π condition at every branch tells us the set of +Every branch of the decision tree is "constrained" by a domain πₜ that +intuitively tells us the set of possible values that can "flow" through that path. -π conditions are refined by the engine during the evaluation; at the +| Domain π = { n\vert{}n∈ℕ x n\vert{}n∈Tag⊆ℕ } +# TODO: copy the definition from above +πₜ conditions are refined by the engine during the evaluation; at the root of the decision tree the domain corresponds to the set of possible values that the type of the function can hold. 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za{M~oui9)B*W!gtGaR`(zL2Ev6Kw-?b{M`|wyxF-)r23fg!LVXHH=gL=ktWr|8da1Ax%%i z1d$E-5_+Fypl5`^hJYX$2|$7p1F(R=NvQCCaJ1gP@`qjVb9NX4|9_{pvx}jVi-*(C UTxMcr Controlla sia finito - [X] Analisi [27/27] - [X] clearpage su ultime immagini @@ -98,6 +108,9 @@ * Tesi [3/9] - [ ] Rivedere inference rules di Gabriel e aggiustarle con le mie +- [ ] Definizione di First(x_i): serve? +- [ ] Equivalenza R_s R_T (run) +- [ ] TODO t_t - [X] segreteria: tesi inglese - [X] Gatti: inglese - [X] Gatti: Coppo mio relatore @@ -122,5 +135,10 @@ - [ ] Trimming e resto: mi sa che usi left e right male - [ ] Spiega perche` trimming non simmetrico - [ ] Spiega meglio le guards on equivalence checking +- [ ] t_T e t_S o t_t e t_s??? +- [ ] TODO eq_muovi : si parla di eq checking, forse non li` +- [ ] Gabriel: quello che penso sulle equivalenze omesse e` giusto? +- [ ] Cambia le C di constraint tree in D +- [ ] TODO on the org file HALP HALP!