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The Lambda representation of a program is shown by +calling the \texttt{ocamlc} compiler with \texttt{-drawlambda} flag. \begin{lstlisting} (if scrutinee (let (field_1 =a (field 1 scrutinee)) @@ -115,12 +118,6 @@ into simple control flow structures such as \texttt{if, switch}, for example: (makeblock 0 1 (makeblock 0 y))))) [0: 0 0a]) \end{lstlisting} -\begin{comment} -%% TODO: side by side -\end{comment} -The code on 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 bugs typically would result in wrong code @@ -151,6 +148,7 @@ execution. Here are the decision trees for the source and target example program. \begin{minipage}{0.5\linewidth} + \scriptsize \begin{verbatim} Switch(Root) / \ @@ -166,21 +164,22 @@ Here are the decision trees for the source and target example program. \end{verbatim} \end{minipage} \hfill -\begin{minipage}{0.5\linewidth} +\begin{minipage}{0.4\linewidth} + \scriptsize \begin{verbatim} Switch(Root) / \ (= int 0) (!= int 0) / \ Leaf Switch(Root.1) -(makeblock 0 / \ +(mkblock 0 / \ 0 0a) / \ (= int 0) (!= int 0) / \ Leaf Leaf [x = Root.0] [y = Root.1.0] -(makeblock 0 (makeblock 0 - 1 (makeblock 0 x)) 2 (makeblock 0 y)) +(mkblock 0 (mkblock 0 + 1 (mkblock 0 x)) 2 (mkblock 0 y)) \end{verbatim} \end{minipage} @@ -207,7 +206,7 @@ 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 +Or-patterns such as $(p_1 | p_2)$ and pattern variables. We also support \texttt{when} guards, which are interesting as they introduce the evaluation of expressions during matching. Decision trees have nodes of the form: @@ -742,21 +741,17 @@ respectively the /precondition/ and the /postcondition/ that constitute the assertions of the program. C is the directive being executed. The language of the assertions P is: -#+BEGIN_SRC -P ::= true | false | a < b | P_{1} \wedge P_{2} | P_{1} \lor P_{2} | \notP -#+END_SRC +| P ::= true \vert false \vert a < b \vert P_{1} \wedge P_{2} \vert P_{1} \lor P_{2} \vert \not P where a and b are numbers. In the Hoare rules assertions could also take the form of -#+BEGIN_SRC -P ::= \forall i. P | \exists i. P | P_{1} \Rightarrow P_{2} -#+END_SRC +| P ::= \forall i. P \vert \exists i. P \vert P_{1} \Rightarrow P_{2} where i is a logical variable, but assertions of these kinds increases the complexity of the symbolic engine. Execution follows the rules of Hoare logic: - Empty statement : \begin{mathpar} \infer{ } -{ \{P\} skip \{P\} } +{ \{P\}skip\{P\} } \end{mathpar} - Assignment statement : The truthness of P[a/x] is equivalent to the truth of {P} after the assignment. @@ -775,14 +770,14 @@ Execution follows the rules of Hoare logic: - Conditional : \begin{mathpar} \infer { \{P \wedge b \} c_1 \{Q\}, \{P\wedge\not b\}c_2\{Q\} } -{ \{P\}if b then c_1 else c_2\{Q\} } +{ \{P\}\text{if b then $c_1$ else $c_2$}\{Q\} } \end{mathpar} - Loop : {P} is the loop invariant. After the loop is finished /P/ holds and ¬̸b caused the loop to end. \begin{mathpar} \infer { \{P \wedge b \}c\{P\} } -{ \{P\} while b do c \{P \wedge \neg b\} } +{ \{P\}\text{while b do c} \{P \wedge \neg b\} } \end{mathpar} Even if the semantics of symbolic execution engines are well defined, @@ -816,7 +811,7 @@ existing translators that consists of taking the source and the target ** Source program The algorithm takes as its input a source program and translates it -into an algebraic data structure called /decision_tree/. +into an algebraic data structure which type we call /decision_tree/. #+BEGIN_SRC type decision_tree = @@ -824,7 +819,7 @@ type decision_tree = | Failure | Leaf of source_expr | Guard of source_blackbox * decision_tree * decision_tree - | Node of accessor * (constructor * decision_tree) list * 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. @@ -842,18 +837,16 @@ Let's consider some trivial examples: function true -> 1 #+END_SRC -[ ] Converti a disegni - -Is translated to -|Node ([(true, Leaf 1)], Failure) +is translated to +|Switch ([(true, Leaf 1)], Failure) while #+BEGIN_SRC function true -> 1 | false -> 2 #+END_SRC -will give -|Node ([(true, Leaf 1); (false, Leaf 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) @@ -864,7 +857,7 @@ true -> 1 | _ -> . #+END_SRC that gets translated into -Node ([(true, Leaf 1); (false, Leaf 2)], Unreachable) +Switch ([(true, Leaf 1); (false, Leaf 2)], Unreachable) We trust this annotation, which is reasonable as the type-checker verifies that it indeed holds. @@ -875,20 +868,21 @@ 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 +\begin{comment} +[ ] Finisci con Switch [ ] 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 +\end{comment} 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 pattern₁ \to expr₁ +|\vert pattern₂ when guard \to expr₂ +|\vert pattern₃ as var \to expr₃ |⋮ -|\vert pₙ -> exprₙ +|\vert pₙ \to exprₙ and can include any expression that is legal for the OCaml compiler, such as /when/ guards and assignments. Patterns could or could not @@ -896,45 +890,34 @@ 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 pattern₁ \to expr₁ +|\vert pattern₂ when guard \to expr₂ +|\vert pattern₃ as var \to expr₃ |⋮ -|\vert pₙ -> exprₙ +|\vert pₙ \to 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 +| start ::= "match" id "with" patterns \vert{} "function" patterns +| patterns ::= (pattern0\vert{}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 ::= "\vert" clause +| clause ::= lexpr "->" rexpr +| lexpr ::= rule (ε\vert{}condition) +| rexpr ::= _code ;; arbitrary code +| rule ::= wildcard\vert{}variable\vert{}constructor_pattern\vert{}or_pattern ;; +| ;; rules +| wildcard ::= "_" +| variable ::= identifier +| constructor_pattern ::= constructor (rule\vert{}ε) (assignment\vert{}ε) +| constructor ::= int\vert{}float\vert{}char\vert{}string\vert{}bool \vert{}unit\vert{}record\vert{}exn\vert{}objects\vert{}ref \vert{}list\vert{}tuple\vert{}array\vert{}variant\vert{}parameterized_variant ;; data types +| or_pattern ::= rule ("\vert{}" wildcard\vert{}variable\vert{}constructor_pattern)+ +| condition ::= "when" bexpr +| assignment ::= "as" id +| bexpr ::= _code ;; arbitrary code \begin{comment} Check that it is still coherent to this bnf @@ -948,7 +931,7 @@ Patterns are of the form | c(p₁,p₂,...,pₙ) | constructor pattern | | (p₁\vert p₂) | or-pattern | -During compilation by the translators expressions are compiled into +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 @@ -970,7 +953,7 @@ following rules apply |--------------------+---+--------------------+-------------------------------------------| | _ | ≼ | v | ∀v | | x | ≼ | v | ∀v | -| (p₁ \vert\ p₂) | ≼ | v | iff p₁ ≼ v or p₂ ≼ 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] | |--------------------+---+--------------------+-------------------------------------------| @@ -1038,14 +1021,14 @@ 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 +type t = K1 | K2 of t (* declaration of an algebraic and recursive datatype 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 + | K1 -> observe 0 + | K2 K1 -> observe 1 + | K2 x when guard x -> observe 2 + | K2 (K2 x) as y when guard x y -> observe 3 + | K2 _ -> observe 4 #+END_SRC Once parsed, the type declarations and the list of clauses are encoded in the form of a matrix @@ -1069,7 +1052,7 @@ p_{m,1} & p_{m,2} & \cdots & p_{m,n} → lₘ) \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 +is empty and the result of the compilation C₀ is l₁ \begin{equation*} C₀((), @@ -1102,7 +1085,7 @@ following four rules: \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 + 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 @@ -1344,7 +1327,7 @@ 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 | Node(a, (Kᵢ → Cᵢ)^{i∈S} , C?) +| Cₛ ::= Leaf bb | Switch(a, (Kᵢ → Cᵢ)^{i∈S} , C?) | a ::= Here | n.a | vₛ ::= K(vᵢ)^{i∈I} | n ∈ ℕ @@ -1492,11 +1475,11 @@ We prove this by induction on Cₜ: That means that the result of trimming on a Leaf is the Leaf itself b. The same applies to Failure terminal | Failure_{/a→π}(v) = Failure(v) - c. When Cₜ = Node(b, (π→Cᵢ)ⁱ)_{/a→π} then + c. When Cₜ = Switch(b, (π→Cᵢ)ⁱ)_{/a→π} then we look at the accessor /a/ of the subtree Cᵢ and we define πᵢ' = πᵢ if a≠b else πᵢ∩π Trimming a switch node yields the following result: - | Node(b, (π→Cᵢ)ⁱ)_{/a→π} := Node(b, (π'ᵢ→C_{i/a→π})ⁱ) + | Switch(b, (π→Cᵢ)ⁱ)_{/a→π} := Switch(b, (π'ᵢ→C_{i/a→π})ⁱ) \begin{comment} Actually in the proof.org file I transcribed: @@ -1507,10 +1490,10 @@ This is not correct because you don't have Unreachable nodes in target decision For the trimming lemma we have to prove that running the value vₜ against the decistion tree Cₜ is the same as running vₜ against the tree C_{trim} that is the result of the trimming operation on Cₜ -| Cₜ(vₜ) = C_{trim}(vₜ) = Node(b, (πᵢ'→C_{i/a→π})ⁱ)(vₜ) +| Cₜ(vₜ) = C_{trim}(vₜ) = Switch(b, (πᵢ'→C_{i/a→π})ⁱ)(vₜ) We can reason by first noting that when vₜ∉(b→πᵢ)ⁱ the node must be a Failure node. In the case where ∃k| vₜ∈(b→πₖ) then we can prove that -| C_{k/a→π}(vₜ) = Node(b, (πᵢ'→C_{i/a→π})ⁱ)(vₜ) +| C_{k/a→π}(vₜ) = Switch(b, (πᵢ'→C_{i/a→π})ⁱ)(vₜ) because when a ≠ b then πₖ'= πₖ and this means that vₜ∈πₖ' while when a = b then πₖ'=(πₖ∩π) and vt∈πₖ' because: - by the hypothesis, vₜ∈π @@ -1580,8 +1563,8 @@ I think the unreachable case should go at the end. |equiv(S, Failure, Failure) := Yes Given that ∀v, Failure(v) = Failure, the statement holds. 3. When we have a Leaf or a Failure at the left side: - | equiv(S, Failure as Cₛ, Node(a, (πᵢ → Cₜᵢ)ⁱ)) := Forall(equiv( S∩a→π(kᵢ)), Cₛ, Cₜᵢ)ⁱ) - | equiv(S, Leaf bbₛ as Cₛ, Node(a, (πᵢ → Cₜᵢ)ⁱ)) := Forall(equiv( S∩a→π(kᵢ)), Cₛ, Cₜᵢ)ⁱ) + | equiv(S, Failure as Cₛ, Switch(a, (πᵢ → Cₜᵢ)ⁱ)) := Forall(equiv( S∩a→π(kᵢ)), Cₛ, Cₜᵢ)ⁱ) + | equiv(S, Leaf bbₛ as Cₛ, Switch(a, (πᵢ → Cₜᵢ)ⁱ)) := Forall(equiv( S∩a→π(kᵢ)), Cₛ, Cₜᵢ)ⁱ) The algorithm either returns Yes for every sub-input space Sᵢ := S∩(a→π(kᵢ)) and subtree Cₜᵢ | equiv(Sᵢ, Cₛ, Cₜᵢ) = Yes ∀i @@ -1592,11 +1575,11 @@ I think the unreachable case should go at the end. | vₛ≃vₜ∈S ∧ Cₛ(vₛ)≠Cₜ(vₜ) we can say that | equiv(Sᵢ, Cₛ, Cₜᵢ) = No(vₛ, vₜ) for some minimal k∈I -4. When we have a Node on the right we define πₙ as the domain of +4. When we have a Switch on the right we define πₙ as the domain of values not covered but the union of the constructors kᵢ | πₙ = ¬(⋃π(kᵢ)ⁱ) The algorithm proceeds by trimming - | equiv(S, Node(a, (kᵢ → Cₛᵢ)ⁱ, C_{sf}), Cₜ) := + | equiv(S, Switch(a, (kᵢ → Cₛᵢ)ⁱ, C_{sf}), Cₜ) := | Forall(equiv( S∩(a→π(kᵢ)ⁱ), Cₛᵢ, C_{t/a→π(kᵢ)})ⁱ +++ equiv(S∩(a→π(kᵢ)), Cₛ, C_{a→πₙ})) The statement still holds and we show this by first analyzing the /Yes/ case: From 1ec9393e388005eff43bd54f95b170d8c0f857bc Mon Sep 17 00:00:00 2001 From: Francesco Mecca Date: Fri, 10 Apr 2020 00:37:36 +0200 Subject: [PATCH 2/7] correzioni Gabriel piu` apici --- tesi/tesi.pdf | Bin 501049 -> 501126 bytes tesi/tesi_unicode.org | 89 ++++++++++++++++++++++-------------------- 2 files changed, 46 insertions(+), 43 deletions(-) diff --git a/tesi/tesi.pdf b/tesi/tesi.pdf index 683ed0586a633551eb7f871e267f739b37e11e99..0e971de7799d1f906f0dfa8731152548fb01647f 100644 GIT binary patch delta 22638 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zJJWh@Pyf4;FWjCsD`j4Vt+0zy@Seu8?+mQug6Q1Tle_(<(qLui(;5nUArl;UOL<^onr`(@exwXKIgFGfQYw9!?%iM(EH~)l{Wo&}bn?5RPi~klJRB2|G zAqlu|{)gbx+0GiN==Xt1-xC2_SMBkvoGm|kxLcV!{ZnxJLTnN|2}v Date: Fri, 10 Apr 2020 00:46:42 +0200 Subject: [PATCH 3/7] todo coppo --- tesi/tesi_unicode.org | 14 ++++++++++++++ 1 file changed, 14 insertions(+) diff --git a/tesi/tesi_unicode.org b/tesi/tesi_unicode.org index cff2ea3..5a43f89 100644 --- a/tesi/tesi_unicode.org +++ b/tesi/tesi_unicode.org @@ -25,6 +25,20 @@ Magari prima pattern matching poi compilatore? \end{comment} +\begin{comment} +MAIL COPPO: +Dovrei usare il termine tool invece di algoritmo? +Quanto devo scrivere della mia esperienza all'inria? Per ora non ho detto nulla +Descrivigli le parti che salto + +Perche` ha segnato delle note su: +→ l +P → L +clause matrix +[|tₛ|] + +\end{comment} + #+TITLE: Translation Verification of the pattern matching compiler #+AUTHOR: Francesco Mecca #+EMAIL: me@francescomecca.eu From 13b8bb01f24364e88ac33edcf01bbc1809196f2c Mon Sep 17 00:00:00 2001 From: Francesco Mecca Date: Sat, 11 Apr 2020 00:26:05 +0200 Subject: [PATCH 4/7] mail gabriel e coppo --- tesi/.gitignore | 1 + tesi/conv.py | 13 +- tesi/gabriel/Makefile | 22 + tesi/gabriel/part4.org | 354 -------------- tesi/gabriel/part4.pdf | Bin 226968 -> 308599 bytes tesi/gabriel/part4.tex | 817 +++++++++++++++++++++++---------- tesi/gabriel/part4_unicode.org | 701 ++++++++++++++++++++++++++++ tesi/tesi.pdf | Bin 501126 -> 525229 bytes tesi/tesi_unicode.org | 647 ++++++++++++++------------ 9 files changed, 1646 insertions(+), 909 deletions(-) create mode 100644 tesi/gabriel/Makefile delete mode 100644 tesi/gabriel/part4.org create mode 100644 tesi/gabriel/part4_unicode.org diff --git a/tesi/.gitignore b/tesi/.gitignore index 2ea43bc..bbd865c 100644 --- a/tesi/.gitignore +++ b/tesi/.gitignore @@ -2,6 +2,7 @@ tesi.tex .#tesi.tex tesi_unicode.tex +part4_unicode.tex ## Core latex/pdflatex auxiliary files: diff --git a/tesi/conv.py b/tesi/conv.py index 15f2be9..7f79a5e 100644 --- a/tesi/conv.py +++ b/tesi/conv.py @@ -2,8 +2,12 @@ import json import re from sys import argv -allsymbols = json.load(open('./unicode-latex.json')) -mysymbols = ['≡', '≠', '≼', '→', '←', '⊀', '⋠', '≺', '∀', 'ε', '₀', '₂', '₁', '₃', 'ₐ', 'ₖ', 'ₘ', 'ₙ', 'ᵢ', 'ⁱ', '⋮', 'ₛ', 'ₜ', '≃', '⇔', '∧', '∅', 'ℕ', 'ⱼ', 'ʲ', '⊥', 'π', 'α', 'β', '∞', 'σ', '≤', '⊈', '∧', '∨', '∃', '⇒', '∩', '∉', '⋃', 'ᵏ', 'ₗ', 'ˡ', 'ₒ', 'ᵣ', 'ᴵ', '≈' ] +try: + allsymbols = json.load(open('./unicode-latex.json')) +except: + allsymbols = json.load(open('../unicode-latex.json')) + +mysymbols = ['≡', '≠', '≼', '→', '←', '⊀', '⋠', '≺', '∀', 'ε', '₀', '₂', '₁', '₃', 'ₐ', 'ₖ', 'ₘ', 'ₙ', 'ᵢ', 'ⁱ', '⋮', 'ₛ', 'ₜ', '≃', '⇔', '∧', '∅', 'ℕ', 'ⱼ', 'ʲ', '⊥', 'π', 'α', 'β', '∞', 'σ', '≤', '⊈', '∧', '∨', '∃', '⇒', '∩', '∉', '⋃', 'ᵏ', 'ₗ', 'ˡ', 'ₒ', 'ᵣ', 'ᴵ', '≈', '⊆' ] extrasymbols = {'〚': '\llbracket', '〛': r'\rrbracket', '̸': '\neg', '¬̸': '\neg', '∈': '\in ', 'ₛ': '_S', 'ₜ': '_T'} symbols = {s: allsymbols[s] for s in mysymbols} @@ -39,8 +43,9 @@ def convert(ch, mathmode): def latex_errors_replacements(charlist): text = ''.join(charlist).split(' ') - replacements = {'\n\end{comment}\n\end{enumerate}\n\end{enumerate}\n\n\subsection{Symbolic': - '\n\end{comment}\n\n\subsection{Symbolic'} + replacements = { + '\n\end{comment}\n\end{enumerate}\n\end{enumerate}\n\n\subsection{Symbolic': '\n\end{comment}\n\n\subsection{Symbolic', + } r_set = set(replacements.keys()) for word in text: it = r_set.intersection(set([word])) diff --git a/tesi/gabriel/Makefile b/tesi/gabriel/Makefile new file mode 100644 index 0000000..efd61b7 --- /dev/null +++ b/tesi/gabriel/Makefile @@ -0,0 +1,22 @@ +SRC = part4.tex +AUX = part4.aux +DEL = part4.aux part4.bbl part4.blg part4.log part4.out part4_unicode.tex part4.pdf part4.tex texput.log +NAME = part4 + +part4: + @echo + @echo "=== Removing temporary files ===" + rm -f $(DEL) + @echo + @echo "=== Building from scratch ===" + emacs -batch part4_unicode.org -f org-latex-export-to-latex --kill + python3 ../conv.py part4_unicode.tex part4.tex + pdflatex $(SRC) + bibtex $(AUX) + pdflatex $(SRC) + pdflatex $(SRC) + @echo + @echo "=== All done. Generated $(NAME).pdf ===" + +clean: + rm -f $(DEL) diff --git a/tesi/gabriel/part4.org b/tesi/gabriel/part4.org deleted file mode 100644 index 5cb1a98..0000000 --- a/tesi/gabriel/part4.org +++ /dev/null @@ -1,354 +0,0 @@ -# Add headers to export only this section -* Correctness of the algorithm -Running a program tₛ or its translation 〚tₛ〛 against an input vₛ -produces as 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 -of decision 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 -decision 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 decision tree -is correct when for every possible input, the source program and its -respective decision 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 -decision tree Cₜ. (TODO: rephrase) - -Theorem: Given an input space /S/ and a couple of decision trees, where -the target decision tree Cₜ covers the input space /S/, we say that -the two decision 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 decision 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 - - -*** Proof of the correctness of the translation from source programs to source decision trees - -We define a source term tₛ as a collection of patterns pointing to blackboxes -| tₛ ::= (p → bb)^{i∈I} - -A pattern is defined as either a constructor pattern, an or pattern or -a constant pattern -| p ::= | K(pᵢ)ⁱ, i ∈ I | (p|q) | n ∈ ℕ - -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 | Node(a, (Kᵢ → Cᵢ)^{i∈S} , C?) -| a ::= Here | n.a -| vₛ ::= K(vᵢ)^{i∈I} | n ∈ ℕ - -\begin{comment} -Are K and Here clear here? -\end{comment} - -We define the decomposition matrix /mₛ/ as -| SMatrix mₛ := (aⱼ)^{j∈J}, ((p_{ij})^{j∈J} → bbᵢ)^{i∈I} -\begin{comment} -Correggi prendendo in considerazione l'accessor -\end{comment} - -We define the decision tree of source programs -〚tₛ〛 -in terms of the decision tree of pattern matrices -〚mₛ〛 -by the following: -〚((pᵢ → bbᵢ)^{i∈I}〛 := 〚(Root), (pᵢ → bbᵢ)^{i∈I} 〛 - -decision tree computed from pattern matrices respect the following invariant: -| ∀v (vᵢ)^{i∈I} = v(aᵢ)^{i∈I} → 〚m〛(v) = m(vᵢ)^{i∈I} for m = ((aᵢ)^{i∈I}, (rᵢ)^{i∈I}) -where -| v(Here) = v -| K(vᵢ)ⁱ(k.a) = vₖ(a) if k ∈ [0;n[ -\begin{comment} -TODO: EXPLAIN -\end{comment} - -We proceed to show the correctness of the invariant by a case -analysys. - -Base cases: -1. [| ∅, (∅ → bbᵢ)ⁱ |] := Leaf bbᵢ where i := min(I), that is a - decision tree [|ms|] defined by an empty accessor and empty - patterns pointing to blackboxes /bbᵢ/. This respects the invariant - because a decomposition matrix in the case of empty rows returns - the first expression and we known that (Leaf bb)(v) := Match bb -2. [| (aⱼ)ʲ, ∅ |] := Failure - -Regarding non base cases: -Let's first define -| let Idx(k) := [0; arity(k)[ -| let First(∅) := ⊥ -| let First((aⱼ)ʲ) := a_{min(j∈J≠∅)} -\[ -m := ((a_i)^i ((p_{ij})^i \to e_j)^{ij}) -\] -\[ - (k_k)^k := headconstructor(p_{i0})^i -\] -\begin{equation} - Groups(m) := ( k_k \to ((a)_{0.l})^{l \in Idx(k_k)} +++ (a_i)^{i \in I\backslash \{0\} }), \\ - ( if p_{0j} is k(q_l) then \\ - (qₗ)^{l \in Idx(k_k)} +++ (p_{ij})^{i \in I\backslash \{0\}} \to e_j \\ - if p_{0j} is \_ then \\ - (\_)^{l \in Idx(k_k)} +++ (p_{ij})^{i \in I\backslash \{0\}} \to e_j \\ - else \bot )^j ), \\ - ((a_i)^{i \in I\backslash \{0\}}, ((p_{ij})^{i \in I\backslash \{0\}} \to eⱼ if p_{0j} is \_ else \bot)^{j \in J}) -\end{equation} - -Groups(m) is an auxiliary function that decomposes a matrix m into -submatrices, according to the head constructor of their first pattern. -Groups(m) returns one submatrix m_r for each head constructor k that -occurs on the first row of m, plus one "wildcard submatrix" m_{wild} -that matches on all values that do not start with one of those head -constructors. - -Intuitively, m is equivalent to its decompositionin the following -sense: if the first pattern of an input vector (v_i)^i starts with one -of the head constructors k, then running (v_i)^i against m is the same -as running it against the submatrix m_k; otherwise (its head -constructor is none of the k) it is equivalent to running it against -the wildcard submatrix. - -We formalize this intuition as follows: -Lemma (Groups): -Let \[m\] be a matrix with \[Groups(m) = (k_r \to m_r)^k, m_{wild}\]. -For any value vector \[(v_i)^l\] such that \[v_0 = k(v'_l)^l\] for some -constructor k, -we have: -\[ - if k = kₖ for some k then - m(vᵢ)ⁱ = mₖ((v'ₗ)ˡ +++ (vᵢ)^{i∈I\backslash \{0\}}) - else - m(vᵢ)ⁱ = m_{wild}(vᵢ)^{i∈I\backslash \{0\}} -\] - -*** Proof: -Let \[m\] be a matrix with \[Group(m) = (k_r \to m_r)^k, m_{wild}\]. -Let \[(v_i)^i\] be an input matrix with \[v_0 = k(v'_l)^l\] for some k. -We proceed by case analysis: -- either k is one of the kₖ for some k -- or k is none of the (kₖ)ᵏ - -Both m(vᵢ)ⁱ and mₖ(vₖ)ᵏ are defined as the first matching result of -a family over each row rⱼ of a matrix - -We know, from the definition of -Groups(m), that mₖ is -\[ - ((a){0.l})^{l∈Idx(kₖ)} +++ (aᵢ)^{i∈I\backslash \{0\}}), - ( - if p_{0j} is k(qₗ) then - (qₗ)ˡ +++ (p_{ij})^{i∈I\backslash \{0\}} → eⱼ - if p_{0j} is _ then - (_)ˡ +++ (p_{ij})^{i∈I\backslash \{0\}} → eⱼ - else ⊥ - )ʲ -\] - -By definition, m(vᵢ)ⁱ is -m(vᵢ)ⁱ = First(rⱼ(vᵢ)ⁱ)ʲ for m = ((aᵢ)ⁱ, (rⱼ)ʲ) -(pᵢ)ⁱ (vᵢ)ⁱ = { - if p₀ = k(qₗ)ˡ, v₀ = k'(v'ₖ)ᵏ, k=Idx(k') and l=Idx(k) - if k ≠ k' then ⊥ - if k = k' then ((qₗ)ˡ +++ (pᵢ)^{i∈I\backslash \{0\}}) ((v'ₖ)ᵏ +++ (vᵢ)^{i∈I\backslash \{0\}}) - if p₀ = (q₁|q₂) then - First( (q₁pᵢ^{i∈I \backslash \{0\}}) vᵢ^{i∈I \backslash \{0\}}, (q₂pᵢ^{i∈I \backslash \{0\}}) vᵢ^{i∈I \backslash \{0\}} ) -} - -For this reason, if we can prove that -| ∀j, rⱼ(vᵢ)ⁱ = r'ⱼ((v'ₖ)ᵏ ++ (vᵢ)ⁱ) -it follows that -| m(vᵢ)ⁱ = mₖ((v'ₖ)ᵏ ++ (vᵢ)ⁱ) -from the above definition. - -We can also show that aᵢ = a_{0.l}ˡ +++ a_{i∈I\backslash \{0\}} because v(a₀) = K(v(a){0.l})ˡ) - - - - -** Proof of equivalence checking -\begin{comment} -TODO: put ^i∈I where needed -\end{comment} -\subsubsection{The trimming lemma} -The trimming lemma allows to reduce the size of a decision tree given -an accessor → π relation (TODO: expand) -| ∀vₜ ∈ (a→π), Cₜ(vₜ) = C_{t/a→π(kᵢ)}(vₜ) -We prove this by induction on Cₜ: - a. Cₜ = Leaf_{bb}: when the decision tree is a leaf terminal, we - know that - | Leaf_{bb/a→π}(v) = Leaf_{bb}(v) - That means that the result of trimming on a Leaf is the Leaf itself - b. The same applies to Failure terminal - | Failure_{/a→π}(v) = Failure(v) - c. When Cₜ = Node(b, (π→Cᵢ)ⁱ)_{/a→π} then - we look at the accessor /a/ of the subtree Cᵢ and - we define πᵢ' = πᵢ if a≠b else πᵢ∩π - Trimming a switch node yields the following result: - | Node(b, (π→Cᵢ)ⁱ)_{/a→π} := Node(b, (π'ᵢ→C_{i/a→π})ⁱ) - -\begin{comment} -Actually in the proof.org file I transcribed: - e. Unreachabe → ⊥ -This is not correct because you don't have Unreachable nodes in target decision trees -\end{comment} - -For the trimming lemma we have to prove that running the value vₜ against -the decistion tree Cₜ is the same as running vₜ against the tree -C_{trim} that is the result of the trimming operation on Cₜ -| Cₜ(vₜ) = C_{trim}(vₜ) = Node(b, (πᵢ'→C_{i/a→π})ⁱ)(vₜ) -We can reason by first noting that when vₜ∉(b→πᵢ)ⁱ the node must be a Failure node. -In the case where ∃k| vₜ∈(b→πₖ) then we can prove that -| C_{k/a→π}(vₜ) = Node(b, (πᵢ'→C_{i/a→π})ⁱ)(vₜ) -because when a ≠ b then πₖ'= πₖ and this means that vₜ∈πₖ' -while when a = b then πₖ'=(πₖ∩π) and vt∈πₖ' because: -- by the hypothesis, vₜ∈π -- we are in the case where vₜ∈πₖ -So vₜ ∈ πₖ' and by induction -| Cₖ(vₜ) = C_{k/a→π}(vₜ) -We also know that ∀vₜ∈(b→πₖ) → Cₜ(vₜ) = Cₖ(vₜ) -By putting together the last two steps, we have proven the trimming -lemma. - -\begin{comment} -TODO: what should I say about covering??? I swap π and π' -Covering lemma: - ∀a,π covers(Cₜ,S) → covers(C_{t/a→π}, (S∩a→π)) - Uᵢπⁱ ≈ Uᵢπ'∩(a→π) ≈ (Uᵢπ')∩(a→π) %% - - -%%%%%%% Also: Should I swap π and π' ? -\end{comment} - -\subsubsection{Equivalence checking} -The equivalence checking algorithm takes as parameters an input space -/S/, a source decision tree /Cₛ/ and a target decision tree /Cₜ/: -| equiv(S, Cₛ, Cₜ) → Yes | No(vₛ, vₜ) - -When the algorithm returns Yes and the input space is covered by /Cₛ/ -we can say that the couple of decision trees are the same for -every couple of source value /vₛ/ and target value /vₜ/ that are equivalent. -\begin{comment} -Define "covered" -Is it correct to say the same? How to correctly distinguish in words ≃ and = ? -\end{comment} -| equiv(S, Cₛ, Cₜ) = Yes and cover(Cₜ, S) → ∀ vₛ ≃ vₜ∈S ∧ Cₛ(vₛ) = Cₜ(vₜ) -In the case where the algorithm returns No we have at least a couple -of counter example values vₛ and vₜ for which the two decision trees -outputs a different result. -| equiv(S, Cₛ, Cₜ) = No(vₛ,vₜ) and cover(Cₜ, S) → ∀ vₛ ≃ vₜ∈S ∧ Cₛ(vₛ) ≠ Cₜ(vₜ) - -We define the following -| Forall(Yes) = Yes -| Forall(Yes::l) = Forall(l) -| Forall(No(vₛ,vₜ)::_) = No(vₛ,vₜ) -There exists and are injective: -| int(k) ∈ ℕ (arity(k) = 0) -| tag(k) ∈ ℕ (arity(k) > 0) -| π(k) = {n\vert int(k) = n} x {n\vert tag(k) = n} -where k is a constructor. - -\begin{comment} -TODO: explain: -∀v∈a→π, C_{/a→π}(v) = C(v) -\end{comment} - -We proceed by case analysis: -\begin{comment} -I start numbering from zero to leave the numbers as they were on the blackboard, were we skipped some things -I think the unreachable case should go at the end. -\end{comment} -0. in case of unreachable: -| Cₛ(vₛ) = Absurd(Unreachable) ≠ Cₜ(vₜ) ∀vₛ,vₜ -1. In the case of an empty input space - | equiv(∅, Cₛ, Cₜ) := Yes - and that is trivial to prove because there is no pair of values (vₛ, vₜ) that could be - tested against the decision trees. - In the other subcases S is always non-empty. -2. When there are /Failure/ nodes at both sides the result is /Yes/: - |equiv(S, Failure, Failure) := Yes - Given that ∀v, Failure(v) = Failure, the statement holds. -3. When we have a Leaf or a Failure at the left side: - | equiv(S, Failure as Cₛ, Node(a, (πᵢ → Cₜᵢ)ⁱ)) := Forall(equiv( S∩a→π(kᵢ)), Cₛ, Cₜᵢ)ⁱ) - | equiv(S, Leaf bbₛ as Cₛ, Node(a, (πᵢ → Cₜᵢ)ⁱ)) := Forall(equiv( S∩a→π(kᵢ)), Cₛ, Cₜᵢ)ⁱ) - The algorithm either returns Yes for every sub-input space Sᵢ := S∩(a→π(kᵢ)) and - subtree Cₜᵢ - | equiv(Sᵢ, Cₛ, Cₜᵢ) = Yes ∀i - or we have a counter example vₛ, vₜ for which - | vₛ≃vₜ∈Sₖ ∧ cₛ(vₛ) ≠ Cₜₖ(vₜ) - then because - | vₜ∈(a→πₖ) → Cₜ(vₜ) = Cₜₖ(vₜ) , - | vₛ≃vₜ∈S ∧ Cₛ(vₛ)≠Cₜ(vₜ) - we can say that - | equiv(Sᵢ, Cₛ, Cₜᵢ) = No(vₛ, vₜ) for some minimal k∈I -4. When we have a Node on the right we define πₙ as the domain of - values not covered but the union of the constructors kᵢ - | πₙ = ¬(⋃π(kᵢ)ⁱ) - The algorithm proceeds by trimming - | equiv(S, Node(a, (kᵢ → Cₛᵢ)ⁱ, C_{sf}), Cₜ) := - | Forall(equiv( S∩(a→π(kᵢ)ⁱ), Cₛᵢ, C_{t/a→π(kᵢ)})ⁱ +++ equiv(S∩(a→π(kᵢ)), Cₛ, C_{a→πₙ})) - The statement still holds and we show this by first analyzing the - /Yes/ case: - | Forall(equiv( S∩(a→π(kᵢ)ⁱ), Cₛᵢ, C_{t/a→π(kᵢ)})ⁱ = Yes - The constructor k is either included in the set of constructors kᵢ: - | k \vert k∈(kᵢ)ⁱ ∧ Cₛ(vₛ) = Cₛᵢ(vₛ) - We also know that - | (1) Cₛᵢ(vₛ) = C_{t/a→πᵢ}(vₜ) - | (2) C_{T/a→πᵢ}(vₜ) = Cₜ(vₜ) - (1) is true by induction and (2) is a consequence of the trimming lemma. - Putting everything together: - | Cₛ(vₛ) = Cₛᵢ(vₛ) = C_{T/a→πᵢ}(vₜ) = Cₜ(vₜ) - - When the k∉(kᵢ)ⁱ [TODO] - - The auxiliary Forall function returns /No(vₛ, vₜ)/ when, for a minimum k, - | equiv(Sₖ, Cₛₖ, C_{T/a→πₖ} = No(vₛ, vₜ) - Then we can say that - | Cₛₖ(vₛ) ≠ 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zYiNs79&9!{tcha9wb*Arj4efDNQP@{OJapq70RZlBswGms0s0UFC(8JpW0UJ_=_^O zua&>{z(#p~%-81yeMz`x-*ER>4P@7dlgPzN?75}(qrCZ_akrDq6;ND6u>E+r= zMCa$;O4$ZDb&!1dox5s?xwI}6MTX#)QlSYPXI+;wE^3s1P>&e!@F~8T4yI0UjI4Bm zbABB@CM_ty^S2Xfg=CCN08+n`Edcuh3d547sq)QWxNywO?EkTBk&%fhh(rGmAOKvT diff --git a/tesi/gabriel/part4.tex b/tesi/gabriel/part4.tex index 1e8c80d..05c321e 100644 --- a/tesi/gabriel/part4.tex +++ b/tesi/gabriel/part4.tex @@ -1,4 +1,4 @@ -% Created 2020-04-09 Thu 01:36 +% Created 2020-04-11 Sat 00:25 % Intended LaTeX compiler: pdflatex \documentclass[11pt]{article} \usepackage[utf8]{inputenc} @@ -29,6 +29,7 @@ \newcommand{\EquivTEX}[3]{\mathsf{equiv}(#1, #2, #3)} % \equiv is already taken \newcommand{\coversTEX}[2]{#1 \mathrel{\mathsf{covers}} #2} \newcommand{\YesTEX}{\mathsf{Yes}} +\newcommand{\DZ}{\backslash\text{\{0\}}} \newcommand{\NoTEX}[2]{\mathsf{No}(#1, #2)} \usepackage{comment} \usepackage{mathpartir} @@ -51,94 +52,262 @@ moredelim=[is][\color{red}]{/[}{]/}, xleftmargin=1em, } \lstset{aboveskip=0.4ex,belowskip=0.4ex} -\author{Francesco Mecca} -\date{} +\date{\today} +\title{} \hypersetup{ + pdfauthor={}, + pdftitle={}, pdfkeywords={}, pdfsubject={}, pdfcreator={Emacs 26.3 (Org mode 9.1.9)}, pdflang={English}} \begin{document} -\maketitle -\begin{comment} -TODO: neg is parsed incorrectly -TODO: chiedi a Gabriel se T e S vanno bene, ma prima controlla che siano coerenti -\section{{\bfseries\sffamily TODO} Scaletta [1/6]} -\label{sec:org530707f} +\section{Translation validation of the Pattern Matching Compiler} +\label{sec:org45bb046} + +\subsection{Source program} +\label{sec:orgf4d7ca7} +The algorithm takes as its input a source program and translates it +into an algebraic data structure which type we call \emph{decision\_tree}. + +\begin{verbatim} +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{verbatim} + +Unreachable, Leaf of source\_expr and Failure are the terminals of the three. +We distinguish \begin{itemize} -\item[{$\boxtimes$}] Introduction -\item[{$\boxminus$}] Background [80\%] -\begin{itemize} -\item[{$\boxtimes$}] Low level representation -\item[{$\boxtimes$}] Lambda code [0\%] -\item[{$\boxtimes$}] Pattern matching -\item[{$\boxtimes$}] Symbolic execution -\item[{$\square$}] Translation Validation -\end{itemize} -\item[{$\square$}] Translation validation of the Pattern Matching Compiler -\begin{itemize} -\item[{$\square$}] Source translation -\begin{itemize} -\item[{$\square$}] Formal Grammar -\item[{$\square$}] Compilation of source patterns -\item[{$\square$}] Rest? -\end{itemize} -\item[{$\square$}] Target translation -\begin{itemize} -\item[{$\square$}] Formal Grammar -\item[{$\square$}] Symbolic execution of the Lambda target -\end{itemize} -\item[{$\square$}] Equivalence between source and target -\end{itemize} -\item[{$\square$}] Proof of correctness -\item[{$\square$}] Practical results +\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 black box of code \end{itemize} -Magari prima pattern matching poi compilatore? +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: -\end{comment} -\section{Correctness of the algorithm} -\label{sec:org4188d38} -Running a program t$_S$ or its translation $\llbracket$t$_S$$\rrbracket$ against an input v$_S$ -produces as a result \emph{r} in the following way: +\begin{verbatim} +function true -> 1 +\end{verbatim} + +is translated to \begin{center} \begin{tabular}{l} -( $\llbracket$t$_S$$\rrbracket$$_S$(v$_S$) = C$_S$(v$_S$) ) $\to$ r\\ +Switch ([(true, Leaf 1)], Failure)\\ +\end{tabular} +\end{center} +while +\begin{verbatim} +function +true -> 1 +| false -> 2 +\end{verbatim} +will be translated to +\begin{center} +\begin{tabular}{l} +Switch ([(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 +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 +[ ] rivedi compilazione per tenere in considerazione il tuo albero invece che le Lambda +\end{comment} + +The source code of a pattern matching function has the +following form: + +\begin{center} +\begin{tabular}{l} +match variable with\\ +\(\vert{}\) pattern$_1$ \(\to\) expr$_1$\\ +\(\vert{}\) pattern$_2$ when guard \(\to\) expr$_2$\\ +\(\vert{}\) pattern$_3$ as var \(\to\) expr$_3$\\ +$\vdots$\\ +\(\vert{}\) p$_n$ \(\to\) expr$_n$\\ +\end{tabular} +\end{center} + +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\\ +\(\vert{}\) pattern$_1$ \(\to\) expr$_1$\\ +\(\vert{}\) pattern$_2$ when guard \(\to\) expr$_2$\\ +\(\vert{}\) pattern$_3$ as var \(\to\) expr$_3$\\ +$\vdots$\\ +\(\vert{}\) p$_n$ \(\to\) expr$_n$\\ +\end{tabular} +\end{center} + + +This BNF grammar describes formally the grammar of the source program: + +\begin{center} +\begin{tabular}{l} +start ::= "match" id "with" patterns \(\vert{}\) "function" patterns\\ +patterns ::= (pattern0\(\vert{}\)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 ::= "\(\vert{}\)" clause\\ +clause ::= lexpr "->" rexpr\\ +lexpr ::= rule ($\varepsilon$\(\vert{}\)condition)\\ +rexpr ::= \_code ;; arbitrary code\\ +rule ::= wildcard\(\vert{}\)variable\(\vert{}\)constructor\_pattern\(\vert{}\)\{\}or\_pattern ;;\\ +;; rules\\ +wildcard ::= "\_"\\ +variable ::= identifier\\ +constructor\_pattern ::= constructor (rule\(\vert{}\)$\varepsilon$) (assignment\(\vert{}\)$\varepsilon$)\\ +constructor ::= int\(\vert{}\)float\(\vert{}\)char\(\vert{}\)string\(\vert{}\)bool \(\vert{}\)unit\(\vert{}\)record\(\vert{}\)exn\(\vert{}\)objects\(\vert{}\)ref \(\vert{}\)list\(\vert{}\)tuple\(\vert{}\)array\(\vert{}\)variant\(\vert{}\)parameterized\_variant ;; data types\\ +or\_pattern ::= rule ("\(\vert{}\)" wildcard\(\vert{}\)variable\(\vert{}\)constructor\_pattern)+\\ +condition ::= "when" bexpr\\ +assignment ::= "as" id\\ +bexpr ::= \_code ;; arbitrary code\\ +\end{tabular} +\end{center} + +A source program t$_S$ is a collection of pattern clauses pointing to +\emph{bb} terms. Running a program t$_S$ against an input value v$_S$ produces as +a result \emph{r}: +\begin{center} +\begin{tabular}{l} +t$_S$ ::= (p $\to$ bb)\(^{\text{i$\in $I}}\)\\ +p ::= \(\vert{}\) K(p$_i$)$^i$, i $\in $ I \(\vert{}\) (p\(\vert{}\)q) \(\vert{}\) n $\in $ $\mathbb{N}$\\ +r ::= guard list * (Match bb \(\vert{}\) NoMatch \(\vert{}\) Absurd)\\ t$_S$(v$_S$) $\to$ r\\ \end{tabular} \end{center} -Likewise + +TODO: argument on what it means to run a source program + +\emph{guard} and \emph{bb} expressions are treated as blackboxes of OCaml code. +A sound approach for treating these blackboxes 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{verbatim} +external guard : 'a -> 'b = "guard" +external observe : 'a -> 'b = "observe" +\end{verbatim} + +We assume these two external functions \emph{guard} and \emph{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{verbatim} +type t = K1 | K2 of t (* declaration of an algebraic and recursive datatype t *) + +let _ = function + | K1 -> observe 0 + | K2 K1 -> observe 1 + | K2 x when guard x -> observe 2 + | K2 (K2 x) as y when guard x y -> observe 3 + | K2 _ -> observe 4 +\end{verbatim} + + + +In our prototype we make use of accessors to encode stored values. +\begin{minipage}{0.2\linewidth} +\begin{verbatim} +let value = 1 :: 2 :: 3 :: [] +(* that can also be written *) +let value = [] + |> List.cons 3 + |> List.cons 2 + |> List.cons 1 +\end{verbatim} +\end{minipage} +\hfill +\begin{minipage}{0.5\linewidth} +\begin{verbatim} + + + (field 0 x) = 1 + (field 0 (field 1 x)) = 2 + (field 0 (field 1 (field 1 x)) = 3 + (field 0 (field 1 (field 1 (field 1 x)) = [] +\end{verbatim} +\end{minipage} +An \emph{accessor} \emph{a} represents the +access path to a value that can be reached by deconstructing the +scrutinee. \begin{center} \begin{tabular}{l} -( $\llbracket$t$_T$$\rrbracket$$_T$(v$_T$) = C$_T$(v$_T$) ) $\to$ r\\ -t$_T$(v$_T$) $\to$ r\\ +a ::= Here \(\vert{}\) n.a\\ \end{tabular} \end{center} -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 - v$_S$ $\simeq$ v$_T$ (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 -of decision trees +The above example, in encoded form: +\begin{verbatim} +Here = 1 +1.Here = 2 +1.1.Here = 3 +1.1.1.Here = [] +\end{verbatim} +In our prototype the source matrix m$_S$ is defined as follows \begin{center} \begin{tabular}{l} -t$_S$ $\simeq$ t$_T$ := $\forall$v$_S$$\simeq$v$_T$, t$_S$(v$_S$) = t$_T$(v$_T$)\\ -C$_S$ $\simeq$ C$_T$ := $\forall$v$_S$$\simeq$v$_T$, C$_S$(v$_S$) = C$_T$(v$_T$)\\ +SMatrix m$_S$ := (a$_j$)\(^{\text{j$\in $J}}\), ((p\(_{\text{ij}}\))\(^{\text{j$\in $J}}\) $\to$ bb$_i$)\(^{\text{i$\in $I}}\)\\ \end{tabular} \end{center} -The proposed equivalence algorithm that works on a couple of -decision trees is returns either \emph{Yes} or \emph{No(v$_S$, v$_T$)} where v$_S$ and -v$_T$ are a couple of possible counter examples for which the constraint -trees produce a different result. - -\subsection{Statements} -\label{sec:orgccf94c1} -Theorem. We say that a translation of a source program to a decision tree +Source matrices are used to build source decision trees C$_S$. +A decision tree is defined as either a Leaf, a Failure terminal or +an intermediate node with different children sharing the same accessor \emph{a} +and an optional fallback. +Failure is emitted only when the patterns don't cover the whole set of +possible input values \emph{S}. The fallback is not needed when the user +doesn't use a wildcard pattern. +\%\%\% Give example of thing +\begin{center} +\begin{tabular}{l} +C$_S$ ::= Leaf bb \(\vert{}\) Switch(a, (K$_i$ $\to$ C$_i$)\(^{\text{i$\in $S}}\) , C?) \(\vert{}\) Failure \(\vert{}\) Unreachable\\ +v$_S$ ::= K(v$_i$)\(^{\text{i$\in $I}}\) \(\vert{}\) n $\in $ $\mathbb{N}$\\ +\end{tabular} +\end{center} +\begin{comment} +Are K and Here clear here? +\end{comment} +We say that a translation of a source program to a decision tree is correct when for every possible input, the source program and its respective decision tree produces the same result @@ -148,121 +317,20 @@ $\forall$v$_S$, t$_S$(v$_S$) = $\llbracket$t$_S$$\rrbracket$$_S$(v$_S$)\\ \end{tabular} \end{center} - -Likewise, for the target language: - -\begin{center} -\begin{tabular}{l} -$\forall$v$_T$, t$_T$(v$_T$) = $\llbracket$t$_T$$\rrbracket$$_T$(v$_T$)\\ -\end{tabular} -\end{center} - -Definition: in the presence of guards we can say that two results are -equivalent modulo the guards queue, written \emph{r$_1$ $\simeq$gs r$_2$}, when: -\begin{center} -\begin{tabular}{l} -(gs$_1$, r$_1$) $\simeq$gs (gs$_2$, r$_2$) $\Leftrightarrow$ (gs$_1$, r$_1$) = (gs$_2$ ++ gs, r$_2$)\\ -\end{tabular} -\end{center} - -Definition: we say that C$_T$ covers the input space \emph{S}, written -/covers(C$_T$, S) when every value v$_S$$\in $S is a valid input to the -decision tree C$_T$. (TODO: rephrase) - -Theorem: Given an input space \emph{S} and a couple of decision trees, where -the target decision tree C$_T$ covers the input space \emph{S}, we say that -the two decision trees are equivalent when: - -\begin{center} -\begin{tabular}{l} -equiv(S, C$_S$, C$_T$, gs) = Yes $\wedge$ covers(C$_T$, S) $\to$ $\forall$v$_S$$\simeq$v$_T$ $\in $ S, C$_S$(v$_S$) $\simeq$gs C$_T$(v$_T$)\\ -\end{tabular} -\end{center} - -Similarly we say that a couple of decision trees in the presence of -an input space \emph{S} are \emph{not} equivalent when: - -\begin{center} -\begin{tabular}{l} -equiv(S, C$_S$, C$_T$, gs) = No(v$_S$,v$_T$) $\wedge$ covers(C$_T$, S) $\to$ v$_S$$\simeq$v$_T$ $\in $ S $\wedge$ C$_S$(v$_S$) $\ne$gs C$_T$(v$_T$)\\ -\end{tabular} -\end{center} - -Corollary: For a full input space \emph{S}, that is the universe of the -target program we say: - -\begin{center} -\begin{tabular}{l} -equiv(S, $\llbracket$t$_S$$\rrbracket$$_S$, $\llbracket$t$_T$$\rrbracket$$_T$, $\varnothing$) = Yes $\Leftrightarrow$ t$_S$ $\simeq$ t$_T$\\ -\end{tabular} -\end{center} - - -\begin{enumerate} -\item Proof of the correctness of the translation from source programs to source decision trees -\label{sec:orgfa2a152} - -We define a source term t$_S$ as a collection of patterns pointing to blackboxes -\begin{center} -\begin{tabular}{l} -t$_S$ ::= (p $\to$ bb)\(^{\text{i$\in $I}}\)\\ -\end{tabular} -\end{center} - -A pattern is defined as either a constructor pattern, an or pattern or -a constant pattern -\begin{center} -\begin{tabular}{lllll} -p ::= & K(p$_i$)$^i$, i $\in $ I & (p & q) & n $\in $ $\mathbb{N}$\\ -\end{tabular} -\end{center} - -A decision tree is defined as either a Leaf, a Failure terminal or -an intermediate node with different children sharing the same accessor \emph{a} -and an optional fallback. -Failure is emitted only when the patterns don't cover the whole set of -possible input values \emph{S}. The fallback is not needed when the user -doesn't use a wildcard pattern. -\%\%\% Give example of thing - -\begin{center} -\begin{tabular}{ll} -C$_S$ ::= Leaf bb & Node(a, (K$_i$ $\to$ C$_i$)\(^{\text{i$\in $S}}\) , C?)\\ -a ::= Here & n.a\\ -v$_S$ ::= K(v$_i$)\(^{\text{i$\in $I}}\) & n $\in $ $\mathbb{N}$\\ -\end{tabular} -\end{center} - -\begin{comment} -Are K and Here clear here? -\end{comment} - -We define the decomposition matrix \emph{m$_S$} as -\begin{center} -\begin{tabular}{l} -SMatrix m$_S$ := (a$_j$)\(^{\text{j$\in $J}}\), ((p\(_{\text{ij}}\))\(^{\text{j$\in $J}}\) $\to$ bb$_i$)\(^{\text{i$\in $I}}\)\\ -\end{tabular} -\end{center} -\begin{comment} -Correggi prendendo in considerazione l'accessor -\end{comment} - We define the decision tree of source programs $\llbracket$t$_S$$\rrbracket$ in terms of the decision tree of pattern matrices $\llbracket$m$_S$$\rrbracket$ by the following: -$\llbracket$((p$_i$ $\to$ bb$_i$)\(^{\text{i$\in $I}}\)$\rrbracket$ := $\llbracket$(Root), (p$_i$ $\to$ bb$_i$)\(^{\text{i$\in $I}}\) $\rrbracket$ - -decision tree computed from pattern matrices respect the following invariant: +\begin{center} +\begin{tabular}{l} +$\llbracket$((p$_i$ $\to$ bb$_i$)\(^{\text{i$\in $I}}\)$\rrbracket$ := $\llbracket$(Here), (p$_i$ $\to$ bb$_i$)\(^{\text{i$\in $I}}\) $\rrbracket$\\ +\end{tabular} +\end{center} +Decision tree computed from pattern matrices respect the following invariant: \begin{center} \begin{tabular}{l} $\forall$v (v$_i$)\(^{\text{i$\in $I}}\) = v(a$_i$)\(^{\text{i$\in $I}}\) $\to$ $\llbracket$m$\rrbracket$(v) = m(v$_i$)\(^{\text{i$\in $I}}\) for m = ((a$_i$)\(^{\text{i$\in $I}}\), (r$_i$)\(^{\text{i$\in $I}}\))\\ -\end{tabular} -\end{center} -where -\begin{center} -\begin{tabular}{l} v(Here) = v\\ K(v$_i$)$^i$(k.a) = v$_k$(a) if k $\in $ [0;n[\\ \end{tabular} @@ -276,12 +344,12 @@ analysys. Base cases: \begin{enumerate} -\item\relax [| $\varnothing$, ($\varnothing$ $\to$ bb$_i$)$^i$ |] := Leaf bb$_i$ where i := min(I), that is a +\item\relax [| $\varnothing$, ($\varnothing$ $\to$ bb$_i$)$^i$ |] $\equiv$ Leaf bb$_i$ where i := min(I), that is a decision tree [|ms|] defined by an empty accessor and empty patterns pointing to blackboxes \emph{bb$_i$}. This respects the invariant -because a decomposition matrix in the case of empty rows returns -the first expression and we known that (Leaf bb)(v) := Match bb -\item\relax [| (a$_j$)$^j$, $\varnothing$ |] := Failure +because a source matrix in the case of empty rows returns +the first expression and (Leaf bb)(v) := Match bb +\item\relax [| (a$_j$)$^j$, $\varnothing$ |] $\equiv$ Failure \end{enumerate} Regarding non base cases: @@ -300,46 +368,53 @@ m := ((a_i)^i ((p_{ij})^i \to e_j)^{ij}) (k_k)^k := headconstructor(p_{i0})^i \] \begin{equation} - Groups(m) := ( k_k \to ((a)_{0.l})^{l \in Idx(k_k)} +++ (a_i)^{i \in I\backslash \{0\} }), \\ + Groups(m) := ( k_k \to ((a)_{0.l})^{l \in Idx(k_k)} +++ (a_i)^{i \in I\DZ}), \\ ( if p_{0j} is k(q_l) then \\ - (q_l)^{l \in Idx(k_k)} +++ (p_{ij})^{i \in I\backslash \{0\}} \to e_j \\ + (q_l)^{l \in Idx(k_k)} +++ (p_{ij})^{i \in I\DZ} \to e_j \\ if p_{0j} is \_ then \\ - (\_)^{l \in Idx(k_k)} +++ (p_{ij})^{i \in I\backslash \{0\}} \to e_j \\ + (\_)^{l \in Idx(k_k)} +++ (p_{ij})^{i \in I\DZ} \to e_j \\ else \bot )^j ), \\ - ((a_i)^{i \in I\backslash \{0\}}, ((p_{ij})^{i \in I\backslash \{0\}} \to e_j if p_{0j} is \_ else \bot)^{j \in J}) + ((a_i)^{i \in I\DZ}, ((p_{ij})^{i \in I\DZ} \to e_j if p_{0j} is \_ else \bot)^{j \in J}) \end{equation} -Groups(m) is an auxiliary function that decomposes a matrix m into +Groups(m) is an auxiliary function that source a matrix m into submatrices, according to the head constructor of their first pattern. Groups(m) returns one submatrix m\_r for each head constructor k that occurs on the first row of m, plus one "wildcard submatrix" m\(_{\text{wild}}\) that matches on all values that do not start with one of those head constructors. -Intuitively, m is equivalent to its decompositionin the following -sense: if the first pattern of an input vector (v\_i)\^{}i starts with one -of the head constructors k, then running (v\_i)\^{}i against m is the same -as running it against the submatrix m\_k; otherwise (its head +Intuitively, m is equivalent to its decomposition in the following +sense: if the first pattern of an input vector (v$_i$)$^i$ starts with one +of the head constructors k, then running (v$_i$)$^i$ against m is the same +as running it against the submatrix m$_k$; otherwise (its head constructor is none of the k) it is equivalent to running it against the wildcard submatrix. We formalize this intuition as follows: Lemma (Groups): Let \[m\] be a matrix with \[Groups(m) = (k_r \to m_r)^k, m_{wild}\]. -For any value vector \[(v_i)^l\] such that \[v_0 = k(v'_l)^l\] for some +For any value vector \((v_i)^l\) such that \(v_0 = k(v'_l)^l\) for some constructor k, we have: -\[ - if k = k_k for some k then - m(v_i)^i = m_k((v'_l)^l +++ (v_i)^{i\in I\backslash \{0\}}) - else - m(v_i)^i = m_{wild}(v_i)^{i\in I\backslash \{0\}} -\] +\begin{center} +\begin{tabular}{l} +if k = k$_k$ \text{ for some k then}\\ +\quad m(v$_i$)$^i$ = m$_k$((v\(_{\text{l}}\)')$^l$ \sout{+} (v\(_{\text{i}}\))\(^{\text{i$\in $I}\DZ}\))\\ +\text{else}\\ +\quad m(v$_i$)$^i$ = m\(_{\text{wild}}\)(v$_i$)\(^{\text{i$\in $I}\DZ}\)\\ +\end{tabular} +\end{center} +\begin{comment} +TODO: fix \{0} +\end{comment} + +\begin{enumerate} \item Proof: -\label{sec:org8dad6fc} -Let \[m\] be a matrix with \[Group(m) = (k_r \to m_r)^k, m_{wild}\]. -Let \[(v_i)^i\] be an input matrix with \[v_0 = k(v'_l)^l\] for some k. +\label{sec:org9868f7d} +Let \(m\) be a matrix with \[Group(m) = (k_r \to m_r)^k, m_{wild}\] +Let \((v_i)^i\) be an input matrix with \(v_0 = k(v'_l)^l\) for some k. We proceed by case analysis: \begin{itemize} \item either k is one of the k$_k$ for some k @@ -351,26 +426,31 @@ a family over each row r$_j$ of a matrix We know, from the definition of Groups(m), that m$_k$ is -\[ - ((a){0.l})^{l\in Idx(k_k)} +++ (a_i)^{i\in I\backslash \{0\}}), - ( - if p_{0j} is k(q_l) then - (q_l)^l +++ (p_{ij})^{i\in I\backslash \{0\}} \to e_j - if p_{0j} is _ then - (_)^l +++ (p_{ij})^{i\in I\backslash \{0\}} \to e_j - else \bot - )^j -\] +\begin{center} +\begin{tabular}{l} +((a)\{0.l\})\(^{\text{l$\in $Idx(k$_k$)}}\) \sout{+} (a$_i$)\(^{\text{i$\in $I}\DZ}\)),\\ +(\\ +\quad if p\(_{\text{0j}}\) is k(q$_l$) then\\ +\quad \quad (q$_l$)$^l$ \sout{+} (p\(_{\text{ij}}\))\(^{\text{i$\in $I}\DZ\ }\) $\to$ e$_j$\\ +\quad if p\(_{\text{0j}}\) is \_ then\\ +\quad \quad (\_)$^l$ \sout{+} (p\(_{\text{ij}}\))\(^{\text{i$\in $I}\DZ}\) $\to$ e$_j$\\ +\quad else $\bot$\\ +)\(^{\text{j$\in $J}}\)\\ +\end{tabular} +\end{center} By definition, m(v$_i$)$^i$ is -m(v$_i$)$^i$ = First(r$_j$(v$_i$)$^i$)$^j$ for m = ((a$_i$)$^i$, (r$_j$)$^j$) -(p$_i$)$^i$ (v$_i$)$^i$ = \{ - if p$_0$ = k(q$_l$)$^l$, v$_0$ = k'(v'$_k$)$^k$, k=Idx(k') and l=Idx(k) - if k $\ne$ k' then $\bot$ - if k = k' then ((q$_l$)$^l$ \sout{+} (p$_i$)\(^{\text{i$\in $I}\backslash\ \text{$\backslash${0$\backslash$}}}\)) ((v'$_k$)$^k$ \sout{+} (v$_i$)\(^{\text{i$\in $I}\backslash\ \text{$\backslash${0$\backslash$}}}\)) - if p$_0$ = (q$_1$|q$_2$) then - First( (q$_1$p$_i$\(^{\text{i$\in $I }\backslash\ \text{$\backslash${0$\backslash$}}}\)) v$_i$\(^{\text{i$\in $I }\backslash\ \text{$\backslash${0$\backslash$}}}\), (q$_2$p$_i$\(^{\text{i$\in $I }\backslash\ \text{$\backslash${0$\backslash$}}}\)) v$_i$\(^{\text{i$\in $I }\backslash\ \text{$\backslash${0$\backslash$}}}\) ) -\} +\begin{center} +\begin{tabular}{l} +m(v$_i$)$^i$ = First(r$_j$(v$_i$)$^i$)$^j$ for m = ((a$_i$)$^i$, (r$_j$)$^j$)\\ +(p$_i$)$^i$ (v$_i$)$^i$ = \{\\ +\quad if p$_0$ = k(q$_l$)$^l$, v$_0$ = k'(v'$_k$)$^k$, k=Idx(k') and l=Idx(k)\\ +\quad \quad if k $\ne$ k' then $\bot$\\ +if k = k' then ((q$_l$)$^l$ \sout{+} (p$_i$)\(^{\text{i$\in $I}\DZ}\)) ((v'$_k$)$^k$ \sout{+} (v$_i$)\(^{\text{i$\in $I}\DZ}\) )\\ +if p$_0$ = (q$_1$\(\vert{}\)q$_2$) then\\ +First( (q$_1$p$_i$\(^{\text{i$\in $I}\DZ}\)) v$_i$\(^{\text{i$\in $I}\DZ}\), (q$_2$p$_i$\(^{\text{i$\in $I }\DZ}\)) v$_i$\(^{\text{i$\in $I}\DZ}\))\}\\ +\end{tabular} +\end{center} For this reason, if we can prove that \begin{center} @@ -386,73 +466,318 @@ m(v$_i$)$^i$ = m$_k$((v'$_k$)$^k$ ++ (v$_i$)$^i$)\\ \end{center} from the above definition. -We can also show that a$_i$ = a\(_{\text{0.l}}\)$^l$ \sout{+} a\(_{\text{i$\in $I}\backslash\ \text{$\backslash${0$\backslash$}}}\) because v(a$_0$) = K(v(a)\{0.l\})$^l$) +We can also show that a$_i$ = (a\(_{\text{0.l}}\))$^l$ \sout{+} a\(_{\text{i$\in $I}\DZ}\) because v(a$_0$) = K(v(a)\{0.l\})$^l$) \end{enumerate} +\subsection{Target translation} +\label{sec:org5f1edb6} + +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) +\begin{center} +\begin{tabular}{l} +start ::= sexpr\\ +sexpr ::= variable \(\vert{}\) string \(\vert{}\) "(" special\_form ")"\\ +string ::= "$\backslash$"" identifier "$\backslash$"" ;; 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 ::= "(" ("!="\(\vert{}\)"\texttt{="\textbackslash{}vert\{\}">}"\(\vert{}\)"<="\(\vert{}\)">"\(\vert{}\)"<") sexpr digit \(\vert{}\) field ")"\\ +label ::= integer\\ +\end{tabular} +\end{center} +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. +The result of the symbolic evaluation is a target decision tree C$_T$ +\begin{center} +\begin{tabular}{l} +C$_T$ ::= Leaf bb \(\vert{}\) Switch(a, ($\pi$$_i$ $\to$ C$_i$)\(^{\text{i$\in $S}}\) , C?) \(\vert{}\) Failure\\ +v$_T$ ::= Cell(tag $\in $ $\mathbb{N}$, (v$_i$)\(^{\text{i$\in $I}}\)) \(\vert{}\) n $\in $ $\mathbb{N}$\\ +\end{tabular} +\end{center} +Every branch of the decision tree is "constrained" by a domain +\begin{center} +\begin{tabular}{l} +Domain $\pi$ = \{ n\(\vert{}\)n$\in $$\mathbb{N}$ x n\(\vert{}\)n$\in $Tag$\subseteq$$\mathbb{N}$ \}\\ +\end{tabular} +\end{center} +Intuitively, the $\pi$ condition at every branch tells us the set of +possible values that can "flow" through that path. +$\pi$ 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. +C? is the fallback node of the tree that is taken whenever the value +at that point of the execution can't flow to any other subbranch. +Intuitively, the $\pi$\(_{\text{fallback}}\) condition of the C? fallback node is +\begin{center} +\begin{tabular}{l} +$\pi$\(_{\text{fallback}}\) = ¬\bigcup\limits\(_{\text{i$\in $I}}\)$\pi$$_i$\\ +\end{tabular} +\end{center} +The fallback node can be omitted in the case where the domain of the +children nodes correspond to set of possible values pointed by the +accessor at that point of the execution +\begin{center} +\begin{tabular}{l} +domain(v$_S$(a)) = \bigcup\limits\(_{\text{i$\in $I}}\)$\pi$$_i$\\ +\end{tabular} +\end{center} +We say that a translation of a target program to a decision tree +is correct when for every possible input, the target program and its +respective decision tree produces the same result +\begin{center} +\begin{tabular}{l} +$\forall$v$_T$, t$_T$(v$_T$) = $\llbracket$t$_T$$\rrbracket$$_T$(v$_T$)\\ +\end{tabular} +\end{center} + + + +\subsection{Equivalence checking} +\label{sec:org4d9383d} + +The equivalence checking algorithm takes as input a domain of +possible values \emph{S} and a pair of source and target decision trees and +in case the two trees are not equivalent it returns a counter example. +The algorithm respects the following correctness statement: + +\begin{comment} +TODO: we have to define what \coversTEX mean for readers to understand the specifications +(or we use a simplifying lie by hiding \coversTEX in the statements). +\end{comment} + +\begin{align*} + \EquivTEX S {C_S} {C_T} \emptyqueue = \YesTEX \;\land\; \coversTEX {C_T} S + & \implies + \forall v_S \approx v_T \in S,\; C_S(v_S) = C_T(v_T) +\\ + \EquivTEX S {C_S} {C_T} \emptyqueue = \NoTEX {v_S} {v_T} \;\land\; \coversTEX {C_T} S + & \implies + v_S \approx v_T \in S \;\land\; C_S(v_S) \neq C_T(v_T) +\end{align*} +Our equivalence-checking algorithm \(\EquivTEX S {C_S} {C_T} G\) is +a exactly decision procedure for the provability of the judgment +\((\EquivTEX S {C_S} {C_T} G)\), defined below. +\begin{mathpar} + \begin{array}{l@{~}r@{~}l} + & & \text{\emph{constraint trees}} \\ + C & \bnfeq & \Leaf t \\ + & \bnfor & \Failure \\ + & \bnfor & \Switch a {\Fam i {\pi_i, C_i}} \Cfb \\ + & \bnfor & \Guard t {C_0} {C_1} \\ + \end{array} + + \begin{array}{l@{~}r@{~}l} + & & \text{\emph{boolean result}} \\ + b & \in & \{ 0, 1 \} \\[0.5em] + & & \text{\emph{guard queues}} \\ + G & \bnfeq & (t_1 = b_1), \dots, (t_n = b_n) \\ + \end{array} + + \begin{array}{l@{~}r@{~}l} + & & \text{\emph{input space}} \\ + S & \subseteq & \{ (v_S, v_T) \mid \vrel {v_S} {v_T} \} \\ + \end{array} + \\ + \infer{ } + {\EquivTEX \emptyset {C_S} {C_T} G} + + \infer{ } + {\EquivTEX S \Failure \Failure \emptyqueue} + + \infer + {\trel {t_S} {t_T}} + {\EquivTEX S {\Leaf {t_S}} {\Leaf {t_T}} \emptyqueue} + + \infer + {\forall i,\; + \EquivTEX + {(S \land a = K_i)} + {C_i} {\trim {C_T} {a = K_i}} G + \\ + \EquivTEX + {(S \land a \notin \Fam i {K_i})} + \Cfb {\trim {C_T} {a \notin \Fam i {K_i}}} G + } + {\EquivTEX S + {\Switch a {\Fam i {K_i, C_i}} \Cfb} {C_T} G} + +\begin{comment} + % TODO explain somewhere why the judgment is not symmetric: + % we avoid defining trimming on source trees, which would + % require more expressive conditions than just constructors +\end{comment} + \infer + {C_S \in {\Leaf t, \Failure} + \\ + \forall i,\; \EquivTEX {(S \land a \in D_i)} {C_S} {C_i} G + \\ + \EquivTEX {(S \land a \notin \Fam i {D_i})} {C_S} \Cfb G} + {\EquivTEX S + {C_S} {\Switch a {\Fam i {D_i} {C_i}} \Cfb} G} + + \infer + {\EquivTEX S {C_0} {C_T} {G, (t_S = 0)} + \\ + \EquivTEX S {C_1} {C_T} {G, (t_S = 1)}} + {\EquivTEX S + {\Guard {t_S} {C_0} {C_1}} {C_T} G} + + \infer + {\trel {t_S} {t_T} + \\ + \EquivTEX S {C_S} {C_b} G} + {\EquivTEX S + {C_S} {\Guard {t_T} {C_0} {C_1}} {(t_S = b), G}} +\end{mathpar} +Running a program t$_S$ or its translation $\llbracket$t$_S$$\rrbracket$ against an input v$_S$ +produces as a result \emph{r} in the following way: +\begin{center} +\begin{tabular}{l} +( $\llbracket$t$_S$$\rrbracket$$_S$(v$_S$) $\equiv$ C$_S$(v$_S$) ) $\to$ r\\ +t$_S$(v$_S$) $\to$ r\\ +\end{tabular} +\end{center} +Likewise +\begin{center} +\begin{tabular}{l} +( $\llbracket$t$_T$$\rrbracket$$_T$(v$_T$) $\equiv$ C$_T$(v$_T$) ) $\to$ r\\ +t$_T$(v$_T$) $\to$ r\\ +result r ::= guard list * (Match blackbox \(\vert{}\) NoMatch \(\vert{}\) Absurd)\\ +guard ::= blackbox.\\ +\end{tabular} +\end{center} +Having defined equivalence between two inputs of which one is +expressed in the source language and the other in the target language, +v$_S$ $\simeq$ v$_T$, we can define the equivalence between a couple of programs or +a couple of decision trees +\begin{center} +\begin{tabular}{l} +t$_S$ $\simeq$ t$_T$ := $\forall$v$_S$$\simeq$v$_T$, t$_S$(v$_S$) = t$_T$(v$_T$)\\ +C$_S$ $\simeq$ C$_T$ := $\forall$v$_S$$\simeq$v$_T$, C$_S$(v$_S$) = C$_T$(v$_T$)\\ +\end{tabular} +\end{center} +The result of the proposed equivalence algorithm is \emph{Yes} or \emph{No(v$_S$, +v$_T$)}. In particular, in the negative case, v$_S$ and v$_T$ are a couple of +possible counter examples for which the decision trees produce a +different result. + +In the presence of guards we can say that two results are +equivalent modulo the guards queue, written \emph{r$_1$ $\simeq$gs r$_2$}, when: +\begin{center} +\begin{tabular}{l} +(gs$_1$, r$_1$) $\simeq$gs (gs$_2$, r$_2$) $\Leftrightarrow$ (gs$_1$, r$_1$) = (gs$_2$ ++ gs, r$_2$)\\ +\end{tabular} +\end{center} +We say that C$_T$ covers the input space \emph{S}, written +\emph{covers(C$_T$, S)} when every value v$_S$$\in $S is a valid input to the +decision tree C$_T$. (TODO: rephrase) +Given an input space \emph{S} and a couple of decision trees, where +the target decision tree C$_T$ covers the input space \emph{S}, we say that +the two decision trees are equivalent when: +\begin{center} +\begin{tabular}{l} +equiv(S, C$_S$, C$_T$, gs) = Yes $\wedge$ covers(C$_T$, S) $\to$ $\forall$v$_S$$\simeq$v$_T$ $\in $ S, C$_S$(v$_S$) $\simeq$gs C$_T$(v$_T$)\\ +\end{tabular} +\end{center} +Similarly we say that a couple of decision trees in the presence of +an input space \emph{S} are \emph{not} equivalent when: +\begin{center} +\begin{tabular}{l} +equiv(S, C$_S$, C$_T$, gs) = No(v$_S$,v$_T$) $\wedge$ covers(C$_T$, S) $\to$ v$_S$$\simeq$v$_T$ $\in $ S $\wedge$ C$_S$(v$_S$) $\ne$gs C$_T$(v$_T$)\\ +\end{tabular} +\end{center} +Corollary: For a full input space \emph{S}, that is the universe of the +target program we say: +\begin{center} +\begin{tabular}{l} +equiv(S, $\llbracket$t$_S$$\rrbracket$$_S$, $\llbracket$t$_T$$\rrbracket$$_T$, $\varnothing$) = Yes $\Leftrightarrow$ t$_S$ $\simeq$ t$_T$\\ +\end{tabular} +\end{center} + -\subsection{Proof of equivalence checking} -\label{sec:orgffce3ea} \begin{comment} TODO: put ^i$\in $I where needed \end{comment} \subsubsection{The trimming lemma} The trimming lemma allows to reduce the size of a decision tree given -an accessor $\to$ $\pi$ relation (TODO: expand) +an accessor \emph{a} $\to$ $\pi$ relation (TODO: expand) \begin{center} \begin{tabular}{l} -$\forall$v$_T$ $\in $ (a$\to$$\pi$), C$_T$(v$_T$) = C\(_{\text{t/a$\to$$\pi$(k$_i$)}}\)(v$_T$)\\ +$\forall$v$_T$ $\in $ (a$\to$$\pi$), C$_T$(v$_T$) = C\(_{\text{t/a$\to$$\pi$}}\)(v$_T$)\\ \end{tabular} \end{center} We prove this by induction on C$_T$: - a. C$_T$ = Leaf\(_{\text{bb}}\): when the decision tree is a leaf terminal, we - know that + +\begin{itemize} +\item C$_T$ = Leaf\(_{\text{bb}}\): when the decision tree is a leaf terminal, the result of trimming on a Leaf is the Leaf itself \begin{center} \begin{tabular}{l} Leaf\(_{\text{bb/a$\to$$\pi$}}\)(v) = Leaf\(_{\text{bb}}\)(v)\\ \end{tabular} \end{center} -That means that the result of trimming on a Leaf is the Leaf itself -b. The same applies to Failure terminal +\item The same applies to Failure terminal \begin{center} \begin{tabular}{l} Failure\(_{\text{/a$\to$$\pi$}}\)(v) = Failure(v)\\ \end{tabular} \end{center} -c. When C$_T$ = Node(b, ($\pi$$\to$C$_i$)$^i$)\(_{\text{/a$\to$$\pi$}}\) then -we look at the accessor \emph{a} of the subtree C$_i$ and -we define $\pi$$_i$' = $\pi$$_i$ if a$\ne$b else $\pi$$_i$$\cap$$\pi$ -Trimming a switch node yields the following result: +\item When C$_T$ = Switch(b, ($\pi$$\to$C$_i$)$^i$)\(_{\text{/a$\to$$\pi$}}\) then we look at the accessor +\emph{a} of the subtree C$_i$ and we define $\pi$$_i$' = $\pi$$_i$ if a$\ne$b else $\pi$$_i$$\cap$$\pi$ Trimming +a switch node yields the following result: \begin{center} \begin{tabular}{l} -Node(b, ($\pi$$\to$C$_i$)$^i$)\(_{\text{/a$\to$$\pi$}}\) := Node(b, ($\pi$'$_i$$\to$C\(_{\text{i/a$\to$$\pi$}}\))$^i$)\\ +Switch(b, ($\pi$$\to$C$_i$)\(^{\text{i$\in $I}}\))\(_{\text{/a$\to$$\pi$}}\) := Switch(b, ($\pi$'$_i$$\to$C\(_{\text{i/a$\to$$\pi$}}\))\(^{\text{i$\in $I}}\))\\ \end{tabular} \end{center} - +\end{itemize} \begin{comment} -Actually in the proof.org file I transcribed: - e. Unreachabe $\to$ $\bot$ -This is not correct because you don't have Unreachable nodes in target decision trees +TODO: understand how to properly align lists +check that every list is aligned \end{comment} - For the trimming lemma we have to prove that running the value v$_T$ against -the decistion tree C$_T$ is the same as running v$_T$ against the tree +the decision tree C$_T$ is the same as running v$_T$ against the tree C\(_{\text{trim}}\) that is the result of the trimming operation on C$_T$ \begin{center} \begin{tabular}{l} -C$_T$(v$_T$) = C\(_{\text{trim}}\)(v$_T$) = Node(b, ($\pi$$_i$'$\to$C\(_{\text{i/a$\to$$\pi$}}\))$^i$)(v$_T$)\\ +C$_T$(v$_T$) = C\(_{\text{trim}}\)(v$_T$) = Switch(b, ($\pi$$_i$'$\to$C\(_{\text{i/a$\to$$\pi$}}\))\(^{\text{i$\in $I}}\))(v$_T$)\\ \end{tabular} \end{center} We can reason by first noting that when v$_T$$\notin$(b$\to$$\pi$$_i$)$^i$ the node must be a Failure node. -In the case where $\exists$k| v$_T$$\in $(b$\to$$\pi$$_k$) then we can prove that +In the case where $\exists$k \(\vert{}\) v$_T$$\in $(b$\to$$\pi$$_k$) then we can prove that \begin{center} \begin{tabular}{l} -C\(_{\text{k/a$\to$$\pi$}}\)(v$_T$) = Node(b, ($\pi$$_i$'$\to$C\(_{\text{i/a$\to$$\pi$}}\))$^i$)(v$_T$)\\ +C\(_{\text{k/a$\to$$\pi$}}\)(v$_T$) = Switch(b, ($\pi$$_i$'$\to$C\(_{\text{i/a$\to$$\pi$}}\))\(^{\text{i$\in $I}}\))(v$_T$)\\ \end{tabular} \end{center} because when a $\ne$ b then $\pi$$_k$'= $\pi$$_k$ and this means that v$_T$$\in $$\pi$$_k$' -while when a = b then $\pi$$_k$'=($\pi$$_k$$\cap$$\pi$) and vt$\in $$\pi$$_k$' because: +while when a = b then $\pi$$_k$'=($\pi$$_k$$\cap$$\pi$) and v$_T$$\in $$\pi$$_k$' because: \begin{itemize} \item by the hypothesis, v$_T$$\in $$\pi$ \item we are in the case where v$_T$$\in $$\pi$$_k$ @@ -481,8 +806,8 @@ Covering lemma: The equivalence checking algorithm takes as parameters an input space \emph{S}, a source decision tree \emph{C$_S$} and a target decision tree \emph{C$_T$}: \begin{center} -\begin{tabular}{ll} -equiv(S, C$_S$, C$_T$) $\to$ Yes & No(v$_S$, v$_T$)\\ +\begin{tabular}{l} +equiv(S, C$_S$, C$_T$) $\to$ Yes \(\vert{}\) No(v$_S$, v$_T$)\\ \end{tabular} \end{center} @@ -506,7 +831,6 @@ outputs a different result. equiv(S, C$_S$, C$_T$) = No(v$_S$,v$_T$) and cover(C$_T$, S) $\to$ $\forall$ v$_S$ $\simeq$ v$_T$$\in $S $\wedge$ C$_S$(v$_S$) $\ne$ C$_T$(v$_T$)\\ \end{tabular} \end{center} - We define the following \begin{center} \begin{tabular}{l} @@ -563,8 +887,8 @@ Given that $\forall$v, Failure(v) = Failure, the statement holds. \item When we have a Leaf or a Failure at the left side: \begin{center} \begin{tabular}{l} -equiv(S, Failure as C$_S$, Node(a, ($\pi$$_i$ $\to$ C$_T$$_i$)$^i$)) := Forall(equiv( S$\cap$a$\to$$\pi$(k$_i$)), C$_S$, C$_T$$_i$)$^i$)\\ -equiv(S, Leaf bb$_S$ as C$_S$, Node(a, ($\pi$$_i$ $\to$ C$_T$$_i$)$^i$)) := Forall(equiv( S$\cap$a$\to$$\pi$(k$_i$)), C$_S$, C$_T$$_i$)$^i$)\\ +equiv(S, Failure as C$_S$, Switch(a, ($\pi$$_i$ $\to$ C$_T$$_i$)\(^{\text{i$\in $I}}\))) := Forall(equiv( S$\cap$a$\to$$\pi$(k$_i$)), C$_S$, C$_T$$_i$)\(^{\text{i$\in $I}}\))\\ +equiv(S, Leaf bb$_S$ as C$_S$, Switch(a, ($\pi$$_i$ $\to$ C$_T$$_i$)\(^{\text{i$\in $I}}\))) := Forall(equiv( S$\cap$a$\to$$\pi$(k$_i$)), C$_S$, C$_T$$_i$)\(^{\text{i$\in $I}}\))\\ \end{tabular} \end{center} The algorithm either returns Yes for every sub-input space S$_i$ := S$\cap$(a$\to$$\pi$(k$_i$)) and @@ -593,25 +917,25 @@ we can say that equiv(S$_i$, C$_S$, C$_T$$_i$) = No(v$_S$, v$_T$) for some minimal k$\in $I\\ \end{tabular} \end{center} -\item When we have a Node on the right we define $\pi$$_n$ as the domain of +\item When we have a Switch on the right we define $\pi$\(_{\text{fallback}}\) as the domain of values not covered but the union of the constructors k$_i$ \begin{center} \begin{tabular}{l} -$\pi$$_n$ = ¬($\bigcup$$\pi$(k$_i$)$^i$)\\ +$\pi$\(_{\text{fallback}}\) = ¬\bigcup\limits\(_{\text{i$\in $I}}\)$\pi$(k$_i$)\\ \end{tabular} \end{center} The algorithm proceeds by trimming \begin{center} \begin{tabular}{l} -equiv(S, Node(a, (k$_i$ $\to$ C$_S$$_i$)$^i$, C\(_{\text{sf}}\)), C$_T$) :=\\ -Forall(equiv( S$\cap$(a$\to$$\pi$(k$_i$)$^i$), C$_S$$_i$, C\(_{\text{t/a$\to$$\pi$(k$_i$)}}\))$^i$ \sout{+} equiv(S$\cap$(a$\to$$\pi$(k$_i$)), C$_S$, C\(_{\text{a$\to$$\pi$$_n$}}\)))\\ +equiv(S, Switch(a, (k$_i$ $\to$ C$_S$$_i$)\(^{\text{i$\in $I}}\), C\(_{\text{sf}}\)), C$_T$) :=\\ +Forall(equiv( S$\cap$(a$\to$$\pi$(k$_i$)\(^{\text{i$\in $I}}\)), C$_S$$_i$, C\(_{\text{t/a$\to$$\pi$(k$_i$)}}\))\(^{\text{i$\in $I}}\) \sout{+} equiv(S$\cap$(a$\to$$\pi$$_n$), C$_S$, C\(_{\text{a$\to$$\pi$}_{\text{fallback}}}\)))\\ \end{tabular} \end{center} The statement still holds and we show this by first analyzing the \emph{Yes} case: \begin{center} \begin{tabular}{l} -Forall(equiv( S$\cap$(a$\to$$\pi$(k$_i$)$^i$), C$_S$$_i$, C\(_{\text{t/a$\to$$\pi$(k$_i$)}}\))$^i$ = Yes\\ +Forall(equiv( S$\cap$(a$\to$$\pi$(k$_i$)\(^{\text{i$\in $I}}\)), C$_S$$_i$, C\(_{\text{t/a$\to$$\pi$(k$_i$)}}\))\(^{\text{i$\in $I}}\) = Yes\\ \end{tabular} \end{center} The constructor k is either included in the set of constructors k$_i$: @@ -657,3 +981,4 @@ C$_S$$_k$(v$_S$) $\ne$ (C\(_{\text{t/a$\to$$\pi$$_k$}}\)(v$_T$) = C$_T$(v$_T$))\ \end{center} \end{enumerate} \end{document} + \ No newline at end of file diff --git a/tesi/gabriel/part4_unicode.org b/tesi/gabriel/part4_unicode.org new file mode 100644 index 0000000..5083f82 --- /dev/null +++ b/tesi/gabriel/part4_unicode.org @@ -0,0 +1,701 @@ +#+LANGUAGE: en +#+LaTeX_CLASS: article +#+LaTeX_HEADER: \linespread{1.25} +#+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{mathpartir} +#+LaTeX_HEADER: \usepackage{graphicx} +#+LaTeX_HEADER: \usepackage{listings} +#+LaTeX_HEADER: \usepackage{color} +#+LaTeX_HEADER: \usepackage{stmaryrd} +#+LaTeX_HEADER: \newcommand{\semTEX}[1]{{\llbracket{#1}\rrbracket}} +#+LaTeX_HEADER: \newcommand{\EquivTEX}[3]{\mathsf{equiv}(#1, #2, #3)} % \equiv is already taken +#+LaTeX_HEADER: \newcommand{\coversTEX}[2]{#1 \mathrel{\mathsf{covers}} #2} +#+LaTeX_HEADER: \newcommand{\YesTEX}{\mathsf{Yes}} +#+LaTeX_HEADER: \newcommand{\DZ}{\backslash\text{\{0\}}} +#+LaTeX_HEADER: \newcommand{\NoTEX}[2]{\mathsf{No}(#1, #2)} +#+LaTeX_HEADER: \usepackage{comment} +#+LaTeX_HEADER: \usepackage{mathpartir} +#+LaTeX_HEADER: \usepackage{stmaryrd} % llbracket, rrbracket +#+LaTeX_HEADER: \usepackage{listings} +#+LaTeX_HEADER: \usepackage{notations} +#+LaTeX_HEADER: \lstset{ +#+LaTeX_HEADER: mathescape=true, +#+LaTeX_HEADER: language=[Objective]{Caml}, +#+LaTeX_HEADER: basicstyle=\ttfamily, +#+LaTeX_HEADER: extendedchars=true, +#+LaTeX_HEADER: showstringspaces=false, +#+LaTeX_HEADER: aboveskip=\smallskipamount, +#+LaTeX_HEADER: % belowskip=\smallskipamount, +#+LaTeX_HEADER: columns=fullflexible, +#+LaTeX_HEADER: moredelim=**[is][\color{blue}]{/*}{*/}, +#+LaTeX_HEADER: moredelim=**[is][\color{green!60!black}]{/!}{!/}, +#+LaTeX_HEADER: moredelim=**[is][\color{orange}]{/(}{)/}, +#+LaTeX_HEADER: moredelim=[is][\color{red}]{/[}{]/}, +#+LaTeX_HEADER: xleftmargin=1em, +#+LaTeX_HEADER: } +#+LaTeX_HEADER: \lstset{aboveskip=0.4ex,belowskip=0.4ex} + +#+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 +* 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 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 + +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 + +is translated to +|Switch ([(true, Leaf 1)], Failure) +while +#+BEGIN_SRC +function +true -> 1 +| false -> 2 +#+END_SRC +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) +#+BEGIN_SRC +function +true -> 1 +| false -> 2 +| _ -> . +#+END_SRC +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 +[ ] rivedi compilazione per tenere in considerazione il tuo albero invece che le Lambda +\end{comment} + +The source code of a pattern matching function has the +following form: + +|match variable with +|\vert pattern₁ \to expr₁ +|\vert pattern₂ when guard \to expr₂ +|\vert pattern₃ as var \to expr₃ +|⋮ +|\vert pₙ \to exprₙ + +Patterns could or could not be exhaustive. + +Pattern matching code could also be written using the more compact form: +|function +|\vert pattern₁ \to expr₁ +|\vert pattern₂ when guard \to expr₂ +|\vert pattern₃ as var \to expr₃ +|⋮ +|\vert pₙ \to exprₙ + + +This BNF grammar describes formally the grammar of the source program: + +| start ::= "match" id "with" patterns \vert{} "function" patterns +| patterns ::= (pattern0\vert{}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 ::= "\vert" clause +| clause ::= lexpr "->" rexpr +| lexpr ::= rule (ε\vert{}condition) +| rexpr ::= _code ;; arbitrary code +| rule ::= wildcard\vert{}variable\vert{}constructor_pattern\vert{}or_pattern ;; +| ;; rules +| wildcard ::= "_" +| variable ::= identifier +| constructor_pattern ::= constructor (rule\vert{}ε) (assignment\vert{}ε) +| constructor ::= int\vert{}float\vert{}char\vert{}string\vert{}bool \vert{}unit\vert{}record\vert{}exn\vert{}objects\vert{}ref \vert{}list\vert{}tuple\vert{}array\vert{}variant\vert{}parameterized_variant ;; data types +| or_pattern ::= rule ("\vert{}" wildcard\vert{}variable\vert{}constructor_pattern)+ +| condition ::= "when" bexpr +| assignment ::= "as" id +| bexpr ::= _code ;; arbitrary code + +A source program tₛ is a collection of pattern clauses pointing to +/bb/ terms. Running a program tₛ against an input value vₛ produces as +a result /r/: +| tₛ ::= (p → bb)^{i∈I} +| p ::= \vert K(pᵢ)ⁱ, i ∈ I \vert (p\vert{}q) \vert n ∈ ℕ +| r ::= guard list * (Match bb \vert{} NoMatch \vert{} Absurd) +| tₛ(vₛ) → r + +TODO: argument on what it means to run a source program + +/guard/ and /bb/ expressions are treated as blackboxes of OCaml code. +A sound approach for treating these blackboxes 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 = K1 | K2 of t (* declaration of an algebraic and recursive datatype t *) + +let _ = function + | K1 -> observe 0 + | K2 K1 -> observe 1 + | K2 x when guard x -> observe 2 + | K2 (K2 x) as y when guard x y -> observe 3 + | K2 _ -> observe 4 +#+END_SRC + + + +In our prototype we make use of accessors to encode stored values. +\begin{minipage}{0.2\linewidth} +\begin{verbatim} +let value = 1 :: 2 :: 3 :: [] +(* that can also be written *) +let value = [] + |> List.cons 3 + |> List.cons 2 + |> List.cons 1 +\end{verbatim} +\end{minipage} +\hfill +\begin{minipage}{0.5\linewidth} +\begin{verbatim} + + + (field 0 x) = 1 + (field 0 (field 1 x)) = 2 + (field 0 (field 1 (field 1 x)) = 3 + (field 0 (field 1 (field 1 (field 1 x)) = [] +\end{verbatim} +\end{minipage} +An \emph{accessor} /a/ represents the +access path to a value that can be reached by deconstructing the +scrutinee. +| a ::= Here \vert n.a +The above example, in encoded form: +\begin{verbatim} +Here = 1 +1.Here = 2 +1.1.Here = 3 +1.1.1.Here = [] +\end{verbatim} +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. +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} +Are K and Here clear here? +\end{comment} +We say that a translation of a source program to a decision tree +is correct when for every possible input, the source program and its +respective decision tree produces the same result + +| ∀vₛ, tₛ(vₛ) = 〚tₛ〛ₛ(vₛ) + +We define the decision tree of source programs +〚tₛ〛 +in terms of the decision tree of pattern matrices +〚mₛ〛 +by the following: +| 〚((pᵢ → bbᵢ)^{i∈I}〛 := 〚(Here), (pᵢ → bbᵢ)^{i∈I} 〛 +Decision tree computed from pattern matrices respect the following invariant: +| ∀v (vᵢ)^{i∈I} = v(aᵢ)^{i∈I} → 〚m〛(v) = m(vᵢ)^{i∈I} for m = ((aᵢ)^{i∈I}, (rᵢ)^{i∈I}) +| v(Here) = v +| K(vᵢ)ⁱ(k.a) = vₖ(a) if k ∈ [0;n[ +\begin{comment} +TODO: EXPLAIN +\end{comment} + +We proceed to show the correctness of the invariant by a case +analysys. + +Base cases: +1. [| ∅, (∅ → bbᵢ)ⁱ |] ≡ Leaf bbᵢ where i := min(I), that is a + decision tree [|ms|] defined by an empty accessor and empty + patterns pointing to blackboxes /bbᵢ/. This respects the invariant + because a source matrix in the case of empty rows returns + the first expression and (Leaf bb)(v) := Match bb +2. [| (aⱼ)ʲ, ∅ |] ≡ Failure + +Regarding non base cases: +Let's first define +| let Idx(k) := [0; arity(k)[ +| let First(∅) := ⊥ +| let First((aⱼ)ʲ) := a_{min(j∈J≠∅)} +\[ +m := ((a_i)^i ((p_{ij})^i \to e_j)^{ij}) +\] +\[ + (k_k)^k := headconstructor(p_{i0})^i +\] +\begin{equation} + Groups(m) := ( k_k \to ((a)_{0.l})^{l \in Idx(k_k)} +++ (a_i)^{i \in I\DZ}), \\ + ( if p_{0j} is k(q_l) then \\ + (qₗ)^{l \in Idx(k_k)} +++ (p_{ij})^{i \in I\DZ} \to e_j \\ + if p_{0j} is \_ then \\ + (\_)^{l \in Idx(k_k)} +++ (p_{ij})^{i \in I\DZ} \to e_j \\ + else \bot )^j ), \\ + ((a_i)^{i \in I\DZ}, ((p_{ij})^{i \in I\DZ} \to eⱼ if p_{0j} is \_ else \bot)^{j \in J}) +\end{equation} + +Groups(m) is an auxiliary function that source a matrix m into +submatrices, according to the head constructor of their first pattern. +Groups(m) returns one submatrix m_r for each head constructor k that +occurs on the first row of m, plus one "wildcard submatrix" m_{wild} +that matches on all values that do not start with one of those head +constructors. + +Intuitively, m is equivalent to its decomposition in the following +sense: if the first pattern of an input vector (vᵢ)ⁱ starts with one +of the head constructors k, then running (vᵢ)ⁱ against m is the same +as running it against the submatrix mₖ; otherwise (its head +constructor is none of the k) it is equivalent to running it against +the wildcard submatrix. + +We formalize this intuition as follows: +Lemma (Groups): +Let \[m\] be a matrix with \[Groups(m) = (k_r \to m_r)^k, m_{wild}\]. +For any value vector $(v_i)^l$ such that $v_0 = k(v'_l)^l$ for some +constructor k, +we have: +| if k = kₖ \text{ for some k then} +| \quad m(vᵢ)ⁱ = mₖ((v_{l}')ˡ +++ (v_{i})^{i∈I\DZ}) +| \text{else} +| \quad m(vᵢ)ⁱ = m_{wild}(vᵢ)^{i∈I\DZ} + +\begin{comment} +TODO: fix \{0} +\end{comment} + +*** Proof: +Let $m$ be a matrix with \[Group(m) = (k_r \to m_r)^k, m_{wild}\] +Let $(v_i)^i$ be an input matrix with $v_0 = k(v'_l)^l$ for some k. +We proceed by case analysis: +- either k is one of the kₖ for some k +- or k is none of the (kₖ)ᵏ + +Both m(vᵢ)ⁱ and mₖ(vₖ)ᵏ are defined as the first matching result of +a family over each row rⱼ of a matrix + +We know, from the definition of +Groups(m), that mₖ is +| ((a){0.l})^{l∈Idx(kₖ)} +++ (aᵢ)^{i∈I\DZ}), +| ( +| \quad if p_{0j} is k(qₗ) then +| \quad \quad (qₗ)ˡ +++ (p_{ij})^{i∈I\DZ } → eⱼ +| \quad if p_{0j} is _ then +| \quad \quad (_)ˡ +++ (p_{ij})^{i∈I\DZ} → eⱼ +| \quad else ⊥ +| )^{j∈J} + +By definition, m(vᵢ)ⁱ is +| m(vᵢ)ⁱ = First(rⱼ(vᵢ)ⁱ)ʲ for m = ((aᵢ)ⁱ, (rⱼ)ʲ) +| (pᵢ)ⁱ (vᵢ)ⁱ = { +| \quad if p₀ = k(qₗ)ˡ, v₀ = k'(v'ₖ)ᵏ, k=Idx(k') and l=Idx(k) +| \quad \quad if k ≠ k' then ⊥ +| if k = k' then ((qₗ)ˡ +++ (pᵢ)^{i∈I\DZ}) ((v'ₖ)ᵏ +++ (vᵢ)^{i∈I\DZ} ) +| if p₀ = (q₁\vert{}q₂) then +| First( (q₁pᵢ^{i∈I\DZ}) vᵢ^{i∈I\DZ}, (q₂pᵢ^{i∈I \DZ}) vᵢ^{i∈I\DZ})} + +For this reason, if we can prove that +| ∀j, rⱼ(vᵢ)ⁱ = r'ⱼ((v'ₖ)ᵏ ++ (vᵢ)ⁱ) +it follows that +| m(vᵢ)ⁱ = mₖ((v'ₖ)ᵏ ++ (vᵢ)ⁱ) +from the above definition. + +We can also show that aᵢ = (a_{0.l})ˡ +++ a_{i∈I\DZ} because v(a₀) = K(v(a){0.l})ˡ) + + + +** 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. +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 +possible values that can "flow" through that path. +π 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. +C? is the fallback node of the tree that is taken whenever the value +at that point of the execution can't flow to any other subbranch. +Intuitively, the π_{fallback} condition of the C? fallback node is +| π_{fallback} = ¬\bigcup\limits_{i∈I}πᵢ +The fallback node can be omitted in the case where the domain of the +children nodes correspond to set of possible values pointed by the +accessor at that point of the execution +| domain(vₛ(a)) = \bigcup\limits_{i∈I}πᵢ +We say that a translation of a target program to a decision tree +is correct when for every possible input, the target program and its +respective decision tree produces the same result +| ∀vₜ, tₜ(vₜ) = 〚tₜ〛ₜ(vₜ) + + + +** Equivalence checking + +The equivalence checking algorithm takes as input a domain of +possible values \emph{S} and a pair of source and target decision trees and +in case the two trees are not equivalent it returns a counter example. +The algorithm respects the following correctness statement: + +\begin{comment} +TODO: we have to define what \coversTEX mean for readers to understand the specifications +(or we use a simplifying lie by hiding \coversTEX in the statements). +\end{comment} + +\begin{align*} + \EquivTEX S {C_S} {C_T} \emptyqueue = \YesTEX \;\land\; \coversTEX {C_T} S + & \implies + \forall v_S \approx v_T \in S,\; C_S(v_S) = C_T(v_T) +\\ + \EquivTEX S {C_S} {C_T} \emptyqueue = \NoTEX {v_S} {v_T} \;\land\; \coversTEX {C_T} S + & \implies + v_S \approx v_T \in S \;\land\; C_S(v_S) \neq C_T(v_T) +\end{align*} +Our equivalence-checking algorithm $\EquivTEX S {C_S} {C_T} G$ is +a exactly decision procedure for the provability of the judgment +$(\EquivTEX S {C_S} {C_T} G)$, defined below. +\begin{mathpar} + \begin{array}{l@{~}r@{~}l} + & & \text{\emph{constraint trees}} \\ + C & \bnfeq & \Leaf t \\ + & \bnfor & \Failure \\ + & \bnfor & \Switch a {\Fam i {\pi_i, C_i}} \Cfb \\ + & \bnfor & \Guard t {C_0} {C_1} \\ + \end{array} + + \begin{array}{l@{~}r@{~}l} + & & \text{\emph{boolean result}} \\ + b & \in & \{ 0, 1 \} \\[0.5em] + & & \text{\emph{guard queues}} \\ + G & \bnfeq & (t_1 = b_1), \dots, (t_n = b_n) \\ + \end{array} + + \begin{array}{l@{~}r@{~}l} + & & \text{\emph{input space}} \\ + S & \subseteq & \{ (v_S, v_T) \mid \vrel {v_S} {v_T} \} \\ + \end{array} + \\ + \infer{ } + {\EquivTEX \emptyset {C_S} {C_T} G} + + \infer{ } + {\EquivTEX S \Failure \Failure \emptyqueue} + + \infer + {\trel {t_S} {t_T}} + {\EquivTEX S {\Leaf {t_S}} {\Leaf {t_T}} \emptyqueue} + + \infer + {\forall i,\; + \EquivTEX + {(S \land a = K_i)} + {C_i} {\trim {C_T} {a = K_i}} G + \\ + \EquivTEX + {(S \land a \notin \Fam i {K_i})} + \Cfb {\trim {C_T} {a \notin \Fam i {K_i}}} G + } + {\EquivTEX S + {\Switch a {\Fam i {K_i, C_i}} \Cfb} {C_T} G} + +\begin{comment} + % TODO explain somewhere why the judgment is not symmetric: + % we avoid defining trimming on source trees, which would + % require more expressive conditions than just constructors +\end{comment} + \infer + {C_S \in {\Leaf t, \Failure} + \\ + \forall i,\; \EquivTEX {(S \land a \in D_i)} {C_S} {C_i} G + \\ + \EquivTEX {(S \land a \notin \Fam i {D_i})} {C_S} \Cfb G} + {\EquivTEX S + {C_S} {\Switch a {\Fam i {D_i} {C_i}} \Cfb} G} + + \infer + {\EquivTEX S {C_0} {C_T} {G, (t_S = 0)} + \\ + \EquivTEX S {C_1} {C_T} {G, (t_S = 1)}} + {\EquivTEX S + {\Guard {t_S} {C_0} {C_1}} {C_T} G} + + \infer + {\trel {t_S} {t_T} + \\ + \EquivTEX S {C_S} {C_b} G} + {\EquivTEX S + {C_S} {\Guard {t_T} {C_0} {C_1}} {(t_S = b), G}} +\end{mathpar} +Running a program tₛ or its translation 〚tₛ〛 against an input vₛ +produces as 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 +| result r ::= guard list * (Match blackbox \vert{} NoMatch \vert{} Absurd) +| 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, we can define the equivalence between a couple of programs or +a couple of decision trees +| tₛ ≃ tₜ := ∀vₛ≃vₜ, tₛ(vₛ) = tₜ(vₜ) +| Cₛ ≃ Cₜ := ∀vₛ≃vₜ, Cₛ(vₛ) = Cₜ(vₜ) +The result of the proposed equivalence algorithm is /Yes/ or /No(vₛ, +vₜ)/. In particular, in the negative case, vₛ and vₜ are a couple of +possible counter examples for which the decision trees produce a +different result. + +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) +We say that Cₜ covers the input space /S/, written +/covers(Cₜ, S)/ when every value vₛ∈S is a valid input to the +decision tree Cₜ. (TODO: rephrase) +Given an input space /S/ and a couple of decision trees, where +the target decision tree Cₜ covers the input space /S/, we say that +the two decision 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 decision 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 + + +\begin{comment} +TODO: put ^i∈I where needed +\end{comment} +\subsubsection{The trimming lemma} +The trimming lemma allows to reduce the size of a decision tree given +an accessor /a/ → π relation (TODO: expand) +| ∀vₜ ∈ (a→π), Cₜ(vₜ) = C_{t/a→π}(vₜ) +We prove this by induction on Cₜ: + +- Cₜ = Leaf_{bb}: when the decision tree is a leaf terminal, the result of trimming on a Leaf is the Leaf itself + | Leaf_{bb/a→π}(v) = Leaf_{bb}(v) +- The same applies to Failure terminal + | Failure_{/a→π}(v) = Failure(v) +- When Cₜ = Switch(b, (π→Cᵢ)ⁱ)_{/a→π} then we look at the accessor + /a/ of the subtree Cᵢ and we define πᵢ' = πᵢ if a≠b else πᵢ∩π Trimming + a switch node yields the following result: + | Switch(b, (π→Cᵢ)^{i∈I})_{/a→π} := Switch(b, (π'ᵢ→C_{i/a→π})^{i∈I}) +\begin{comment} +TODO: understand how to properly align lists +check that every list is aligned +\end{comment} +For the trimming lemma we have to prove that running the value vₜ against +the decision tree Cₜ is the same as running vₜ against the tree +C_{trim} that is the result of the trimming operation on Cₜ +| Cₜ(vₜ) = C_{trim}(vₜ) = Switch(b, (πᵢ'→C_{i/a→π})^{i∈I})(vₜ) +We can reason by first noting that when vₜ∉(b→πᵢ)ⁱ the node must be a Failure node. +In the case where ∃k \vert{} vₜ∈(b→πₖ) then we can prove that +| C_{k/a→π}(vₜ) = Switch(b, (πᵢ'→C_{i/a→π})^{i∈I})(vₜ) +because when a ≠ b then πₖ'= πₖ and this means that vₜ∈πₖ' +while when a = b then πₖ'=(πₖ∩π) and vₜ∈πₖ' because: +- by the hypothesis, vₜ∈π +- we are in the case where vₜ∈πₖ +So vₜ ∈ πₖ' and by induction +| Cₖ(vₜ) = C_{k/a→π}(vₜ) +We also know that ∀vₜ∈(b→πₖ) → Cₜ(vₜ) = Cₖ(vₜ) +By putting together the last two steps, we have proven the trimming +lemma. + +\begin{comment} +TODO: what should I say about covering??? I swap π and π' +Covering lemma: + ∀a,π covers(Cₜ,S) → covers(C_{t/a→π}, (S∩a→π)) + Uᵢπⁱ ≈ Uᵢπ'∩(a→π) ≈ (Uᵢπ')∩(a→π) %% + + +%%%%%%% Also: Should I swap π and π' ? +\end{comment} + +\subsubsection{Equivalence checking} +The equivalence checking algorithm takes as parameters an input space +/S/, a source decision tree /Cₛ/ and a target decision tree /Cₜ/: +| equiv(S, Cₛ, Cₜ) → Yes \vert{} No(vₛ, vₜ) + +When the algorithm returns Yes and the input space is covered by /Cₛ/ +we can say that the couple of decision trees are the same for +every couple of source value /vₛ/ and target value /vₜ/ that are equivalent. +\begin{comment} +Define "covered" +Is it correct to say the same? How to correctly distinguish in words ≃ and = ? +\end{comment} +| equiv(S, Cₛ, Cₜ) = Yes and cover(Cₜ, S) → ∀ vₛ ≃ vₜ∈S ∧ Cₛ(vₛ) = Cₜ(vₜ) +In the case where the algorithm returns No we have at least a couple +of counter example values vₛ and vₜ for which the two decision trees +outputs a different result. +| equiv(S, Cₛ, Cₜ) = No(vₛ,vₜ) and cover(Cₜ, S) → ∀ vₛ ≃ vₜ∈S ∧ Cₛ(vₛ) ≠ Cₜ(vₜ) +We define the following +| Forall(Yes) = Yes +| Forall(Yes::l) = Forall(l) +| Forall(No(vₛ,vₜ)::_) = No(vₛ,vₜ) +There exists and are injective: +| int(k) ∈ ℕ (arity(k) = 0) +| tag(k) ∈ ℕ (arity(k) > 0) +| π(k) = {n\vert int(k) = n} x {n\vert tag(k) = n} +where k is a constructor. + +\begin{comment} +TODO: explain: +∀v∈a→π, C_{/a→π}(v) = C(v) +\end{comment} + +We proceed by case analysis: +\begin{comment} +I start numbering from zero to leave the numbers as they were on the blackboard, were we skipped some things +I think the unreachable case should go at the end. +\end{comment} +0. in case of unreachable: +| Cₛ(vₛ) = Absurd(Unreachable) ≠ Cₜ(vₜ) ∀vₛ,vₜ +1. In the case of an empty input space + | equiv(∅, Cₛ, Cₜ) := Yes + and that is trivial to prove because there is no pair of values (vₛ, vₜ) that could be + tested against the decision trees. + In the other subcases S is always non-empty. +2. When there are /Failure/ nodes at both sides the result is /Yes/: + |equiv(S, Failure, Failure) := Yes + Given that ∀v, Failure(v) = Failure, the statement holds. +3. When we have a Leaf or a Failure at the left side: + | equiv(S, Failure as Cₛ, Switch(a, (πᵢ → Cₜᵢ)^{i∈I})) := Forall(equiv( S∩a→π(kᵢ)), Cₛ, Cₜᵢ)^{i∈I}) + | equiv(S, Leaf bbₛ as Cₛ, Switch(a, (πᵢ → Cₜᵢ)^{i∈I})) := Forall(equiv( S∩a→π(kᵢ)), Cₛ, Cₜᵢ)^{i∈I}) + The algorithm either returns Yes for every sub-input space Sᵢ := S∩(a→π(kᵢ)) and + subtree Cₜᵢ + | equiv(Sᵢ, Cₛ, Cₜᵢ) = Yes ∀i + or we have a counter example vₛ, vₜ for which + | vₛ≃vₜ∈Sₖ ∧ cₛ(vₛ) ≠ Cₜₖ(vₜ) + then because + | vₜ∈(a→πₖ) → Cₜ(vₜ) = Cₜₖ(vₜ) , + | vₛ≃vₜ∈S ∧ Cₛ(vₛ)≠Cₜ(vₜ) + we can say that + | equiv(Sᵢ, Cₛ, Cₜᵢ) = No(vₛ, vₜ) for some minimal k∈I +4. When we have a Switch on the right we define π_{fallback} as the domain of + values not covered but the union of the constructors kᵢ + | π_{fallback} = ¬\bigcup\limits_{i∈I}π(kᵢ) + The algorithm proceeds by trimming + | equiv(S, Switch(a, (kᵢ → Cₛᵢ)^{i∈I}, C_{sf}), Cₜ) := + | Forall(equiv( S∩(a→π(kᵢ)^{i∈I}), Cₛᵢ, C_{t/a→π(kᵢ)})^{i∈I} +++ equiv(S∩(a→πₙ), Cₛ, C_{a→π_{fallback}})) + The statement still holds and we show this by first analyzing the + /Yes/ case: + | Forall(equiv( S∩(a→π(kᵢ)^{i∈I}), Cₛᵢ, C_{t/a→π(kᵢ)})^{i∈I} = Yes + The constructor k is either included in the set of constructors kᵢ: + | k \vert k∈(kᵢ)ⁱ ∧ Cₛ(vₛ) = Cₛᵢ(vₛ) + We also know that + | (1) Cₛᵢ(vₛ) = C_{t/a→πᵢ}(vₜ) + | (2) C_{T/a→πᵢ}(vₜ) = Cₜ(vₜ) + (1) is true by induction and (2) is a consequence of the trimming lemma. + Putting everything together: + | Cₛ(vₛ) = Cₛᵢ(vₛ) = C_{T/a→πᵢ}(vₜ) = Cₜ(vₜ) + + When the k∉(kᵢ)ⁱ [TODO] + + The auxiliary Forall function returns /No(vₛ, vₜ)/ when, for a minimum k, + | equiv(Sₖ, 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This would allow to test for equality with the respective blackbox at @@ -1046,9 +974,74 @@ let _ = function | K2 _ -> observe 4 #+END_SRC +The source program is parsed using the ocaml-compiler-libs library. +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 | + + 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. +Patterns are of the form +| pattern | type of pattern | +|-----------------+---------------------| +| _ | wildcard | +| x | variable | +| c(p₁,p₂,...,pₙ) | constructor pattern | +| (p₁\vert p₂) | or-pattern | + +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/. + + +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*} + +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: +- Pattern 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{Matrix decomposition of pattern clauses} The initial input of the decomposition algorithm C consists of a vector of variables @@ -1076,8 +1069,7 @@ C₀((), → l₂ \\ → \vdots \\ → lₘ -\end{pmatrix}) -) = l₁ +\end{pmatrix}) = l₁ \end{equation*} When $\vec{x}$ ≠ () then the compilation advances using one of the @@ -1118,25 +1110,25 @@ following four rules: 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} + \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} + \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. @@ -1161,9 +1153,239 @@ following four rules: apply, and P₂ → L₂ containing the remaining rows. The algorithm is applied to both matrices. + +In our prototype we make use of accessors to encode stored values. +\begin{minipage}{0.2\linewidth} +\begin{verbatim} +let value = 1 :: 2 :: 3 :: [] +(* that can also be written *) +let value = [] + |> List.cons 3 + |> List.cons 2 + |> List.cons 1 +\end{verbatim} +\end{minipage} +\hfill +\begin{minipage}{0.5\linewidth} +\begin{verbatim} + + + (field 0 x) = 1 + (field 0 (field 1 x)) = 2 + (field 0 (field 1 (field 1 x)) = 3 + (field 0 (field 1 (field 1 (field 1 x)) = [] +\end{verbatim} +\end{minipage} +An \emph{accessor} /a/ represents the +access path to a value that can be reached by deconstructing the +scrutinee. +| a ::= Here \vert n.a +The above example, in encoded form: +\begin{verbatim} +Here = 1 +1.Here = 2 +1.1.Here = 3 +1.1.1.Here = [] +\end{verbatim} +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. +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} +Are K and Here clear here? +\end{comment} +We say that a translation of a source program to a decision tree +is correct when for every possible input, the source program and its +respective decision tree produces the same result + +| ∀vₛ, tₛ(vₛ) = 〚tₛ〛ₛ(vₛ) + +We define the decision tree of source programs +〚tₛ〛 +in terms of the decision tree of pattern matrices +〚mₛ〛 +by the following: +| 〚((pᵢ → bbᵢ)^{i∈I}〛 := 〚(Here), (pᵢ → bbᵢ)^{i∈I} 〛 +Decision tree computed from pattern matrices respect the following invariant: +| ∀v (vᵢ)^{i∈I} = v(aᵢ)^{i∈I} → 〚m〛(v) = m(vᵢ)^{i∈I} for m = ((aᵢ)^{i∈I}, (rᵢ)^{i∈I}) +| v(Here) = v +| K(vᵢ)ⁱ(k.a) = vₖ(a) if k ∈ [0;n[ +\begin{comment} +TODO: EXPLAIN +\end{comment} + +We proceed to show the correctness of the invariant by a case +analysys. + +Base cases: +1. [| ∅, (∅ → bbᵢ)ⁱ |] ≡ Leaf bbᵢ where i := min(I), that is a + decision tree [|ms|] defined by an empty accessor and empty + patterns pointing to blackboxes /bbᵢ/. This respects the invariant + because a source matrix in the case of empty rows returns + the first expression and (Leaf bb)(v) := Match bb +2. [| (aⱼ)ʲ, ∅ |] ≡ Failure + +Regarding non base cases: +Let's first define +| let Idx(k) := [0; arity(k)[ +| let First(∅) := ⊥ +| let First((aⱼ)ʲ) := a_{min(j∈J≠∅)} +\[ +m := ((a_i)^i ((p_{ij})^i \to e_j)^{ij}) +\] +\[ + (k_k)^k := headconstructor(p_{i0})^i +\] +\begin{equation} + Groups(m) := ( k_k \to ((a)_{0.l})^{l \in Idx(k_k)} +++ (a_i)^{i \in I\DZ}), \\ + ( if p_{0j} is k(q_l) then \\ + (qₗ)^{l \in Idx(k_k)} +++ (p_{ij})^{i \in I\DZ} \to e_j \\ + if p_{0j} is \_ then \\ + (\_)^{l \in Idx(k_k)} +++ (p_{ij})^{i \in I\DZ} \to e_j \\ + else \bot )^j ), \\ + ((a_i)^{i \in I\DZ}, ((p_{ij})^{i \in I\DZ} \to eⱼ if p_{0j} is \_ else \bot)^{j \in J}) +\end{equation} + +Groups(m) is an auxiliary function that source a matrix m into +submatrices, according to the head constructor of their first pattern. +Groups(m) returns one submatrix m_r for each head constructor k that +occurs on the first row of m, plus one "wildcard submatrix" m_{wild} +that matches on all values that do not start with one of those head +constructors. + +Intuitively, m is equivalent to its decomposition in the following +sense: if the first pattern of an input vector (vᵢ)ⁱ starts with one +of the head constructors k, then running (vᵢ)ⁱ against m is the same +as running it against the submatrix mₖ; otherwise (its head +constructor is none of the k) it is equivalent to running it against +the wildcard submatrix. + +We formalize this intuition as follows: +Lemma (Groups): +Let \[m\] be a matrix with \[Groups(m) = (k_r \to m_r)^k, m_{wild}\]. +For any value vector $(v_i)^l$ such that $v_0 = k(v'_l)^l$ for some +constructor k, +we have: +| if k = kₖ \text{ for some k then} +| \quad m(vᵢ)ⁱ = mₖ((v_{l}')ˡ +++ (v_{i})^{i∈I\DZ}) +| \text{else} +| \quad m(vᵢ)ⁱ = m_{wild}(vᵢ)^{i∈I\DZ} + +\begin{comment} +TODO: fix \{0} +\end{comment} + +*** Proof: +Let $m$ be a matrix with \[Group(m) = (k_r \to m_r)^k, m_{wild}\] +Let $(v_i)^i$ be an input matrix with $v_0 = k(v'_l)^l$ for some k. +We proceed by case analysis: +- either k is one of the kₖ for some k +- or k is none of the (kₖ)ᵏ + +Both m(vᵢ)ⁱ and mₖ(vₖ)ᵏ are defined as the first matching result of +a family over each row rⱼ of a matrix + +We know, from the definition of +Groups(m), that mₖ is +| ((a){0.l})^{l∈Idx(kₖ)} +++ (aᵢ)^{i∈I\DZ}), +| ( +| \quad if p_{0j} is k(qₗ) then +| \quad \quad (qₗ)ˡ +++ (p_{ij})^{i∈I\DZ } → eⱼ +| \quad if p_{0j} is _ then +| \quad \quad (_)ˡ +++ (p_{ij})^{i∈I\DZ} → eⱼ +| \quad else ⊥ +| )^{j∈J} + +By definition, m(vᵢ)ⁱ is +| m(vᵢ)ⁱ = First(rⱼ(vᵢ)ⁱ)ʲ for m = ((aᵢ)ⁱ, (rⱼ)ʲ) +| (pᵢ)ⁱ (vᵢ)ⁱ = { +| \quad if p₀ = k(qₗ)ˡ, v₀ = k'(v'ₖ)ᵏ, k=Idx(k') and l=Idx(k) +| \quad \quad if k ≠ k' then ⊥ +| if k = k' then ((qₗ)ˡ +++ (pᵢ)^{i∈I\DZ}) ((v'ₖ)ᵏ +++ (vᵢ)^{i∈I\DZ} ) +| if p₀ = (q₁\vert{}q₂) then +| First( (q₁pᵢ^{i∈I\DZ}) vᵢ^{i∈I\DZ}, (q₂pᵢ^{i∈I \DZ}) vᵢ^{i∈I\DZ})} + +For this reason, if we can prove that +| ∀j, rⱼ(vᵢ)ⁱ = r'ⱼ((v'ₖ)ᵏ ++ (vᵢ)ⁱ) +it follows that +| m(vᵢ)ⁱ = mₖ((v'ₖ)ᵏ ++ (vᵢ)ⁱ) +from the above definition. + +We can also show that aᵢ = (a_{0.l})ˡ +++ a_{i∈I\DZ} because v(a₀) = K(v(a){0.l})ˡ) + + + ** Target translation -TODO +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. +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 +possible values that can "flow" through that path. +π 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. +C? is the fallback node of the tree that is taken whenever the value +at that point of the execution can't flow to any other subbranch. +Intuitively, the π_{fallback} condition of the C? fallback node is +| π_{fallback} = ¬\bigcup\limits_{i∈I}πᵢ +The fallback node can be omitted in the case where the domain of the +children nodes correspond to set of possible values pointed by the +accessor at that point of the execution +| domain(vₛ(a)) = \bigcup\limits_{i∈I}πᵢ +We say that a translation of a target program to a decision tree +is correct when for every possible input, the target program and its +respective decision tree produces the same result +| ∀vₜ, tₜ(vₜ) = 〚tₜ〛ₜ(vₜ) + + ** Equivalence checking @@ -1186,7 +1408,6 @@ TODO: we have to define what \coversTEX mean for readers to understand the speci & \implies v_S \approx v_T \in S \;\land\; C_S(v_S) \neq C_T(v_T) \end{align*} - Our equivalence-checking algorithm $\EquivTEX S {C_S} {C_T} G$ is a exactly decision procedure for the provability of the judgment $(\EquivTEX S {C_S} {C_T} G)$, defined below. @@ -1262,8 +1483,6 @@ $(\EquivTEX S {C_S} {C_T} G)$, defined below. {\EquivTEX S {C_S} {\Guard {t_T} {C_0} {C_1}} {(t_S = b), G}} \end{mathpar} - -* Correctness of the algorithm Running a program tₛ or its translation 〚tₛ〛 against an input vₛ produces as a result /r/ in the following way: | ( 〚tₛ〛ₛ(vₛ) ≡ Cₛ(vₛ) ) → r @@ -1271,211 +1490,37 @@ produces as a result /r/ in the following way: Likewise | ( 〚tₜ〛ₜ(vₜ) ≡ Cₜ(vₜ) ) → r | tₜ(vₜ) → r -where result r ::= guard list * (Match blackbox | NoMatch | Absurd) -and guard ::= blackbox. - +| result r ::= guard list * (Match blackbox \vert{} NoMatch \vert{} Absurd) +| 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 -of decision trees +expressed in the source language and the other in the target language, +vₛ ≃ vₜ, we can define the equivalence between a couple of programs or +a couple of decision trees | tₛ ≃ tₜ := ∀vₛ≃vₜ, tₛ(vₛ) = tₜ(vₜ) | Cₛ ≃ Cₜ := ∀vₛ≃vₜ, Cₛ(vₛ) = Cₜ(vₜ) - The result of the proposed equivalence algorithm is /Yes/ or /No(vₛ, vₜ)/. In particular, in the negative case, vₛ and vₜ are a couple of possible counter examples for which the decision trees produce a different result. -** Statements -Theorem. We say that a translation of a source program to a decision tree -is correct when for every possible input, the source program and its -respective decision 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 +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 +We say that Cₜ covers the input space /S/, written +/covers(Cₜ, S)/ when every value vₛ∈S is a valid input to the decision tree Cₜ. (TODO: rephrase) - -Theorem: Given an input space /S/ and a couple of decision trees, where +Given an input space /S/ and a couple of decision trees, where the target decision tree Cₜ covers the input space /S/, we say that the two decision 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 decision 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 -*** Proof of the correctness of the translation from source programs to source decision trees - -We define the source term tₛ as a collection of patterns pointing to blackboxes -| tₛ ::= (p → bb)^{i∈I} - -A pattern is defined as either a constructor pattern, an or pattern or -a constant pattern -| p ::= \vert K(pᵢ)ⁱ, i ∈ I \vert (p\vert{}q) \vert n ∈ ℕ - -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?) -| a ::= Here \vert n.a -| vₛ ::= K(vᵢ)^{i∈I} \vert n ∈ ℕ - -\begin{comment} -Are K and Here clear here? -\end{comment} - -We define the decomposition matrix /mₛ/ as -| SMatrix mₛ := (aⱼ)^{j∈J}, ((p_{ij})^{j∈J} → bbᵢ)^{i∈I} -\begin{comment} -Correggi prendendo in considerazione l'accessor -\end{comment} - -We define the decision tree of source programs -〚tₛ〛 -in terms of the decision tree of pattern matrices -〚mₛ〛 -by the following: -〚((pᵢ → bbᵢ)^{i∈I}〛 := 〚(Root), (pᵢ → bbᵢ)^{i∈I} 〛 - -decision tree computed from pattern matrices respect the following invariant: -| ∀v (vᵢ)^{i∈I} = v(aᵢ)^{i∈I} → 〚m〛(v) = m(vᵢ)^{i∈I} for m = ((aᵢ)^{i∈I}, (rᵢ)^{i∈I}) -where -| v(Here) = v -| K(vᵢ)ⁱ(k.a) = vₖ(a) if k ∈ [0;n[ -\begin{comment} -TODO: EXPLAIN -\end{comment} - -We proceed to show the correctness of the invariant by a case -analysys. - -Base cases: -1. [| ∅, (∅ → bbᵢ)ⁱ |] ≡ Leaf bbᵢ where i := min(I), that is a - decision tree [|ms|] defined by an empty accessor and empty - patterns pointing to blackboxes /bbᵢ/. This respects the invariant - because a decomposition matrix in the case of empty rows returns - the first expression and (Leaf bb)(v) := Match bb -2. [| (aⱼ)ʲ, ∅ |] ≡ Failure - -Regarding non base cases: -Let's first define -| let Idx(k) := [0; arity(k)[ -| let First(∅) := ⊥ -| let First((aⱼ)ʲ) := a_{min(j∈J≠∅)} -\[ -m := ((a_i)^i ((p_{ij})^i \to e_j)^{ij}) -\] -\[ - (k_k)^k := headconstructor(p_{i0})^i -\] -\begin{equation} - Groups(m) := ( k_k \to ((a)_{0.l})^{l \in Idx(k_k)} +++ (a_i)^{i \in I\backslash \{0\} }), \\ - ( if p_{0j} is k(q_l) then \\ - (qₗ)^{l \in Idx(k_k)} +++ (p_{ij})^{i \in I\backslash \{0\}} \to e_j \\ - if p_{0j} is \_ then \\ - (\_)^{l \in Idx(k_k)} +++ (p_{ij})^{i \in I\backslash \{0\}} \to e_j \\ - else \bot )^j ), \\ - ((a_i)^{i \in I\backslash \{0\}}, ((p_{ij})^{i \in I\backslash \{0\}} \to eⱼ if p_{0j} is \_ else \bot)^{j \in J}) -\end{equation} - -Groups(m) is an auxiliary function that decomposes a matrix m into -submatrices, according to the head constructor of their first pattern. -Groups(m) returns one submatrix m_r for each head constructor k that -occurs on the first row of m, plus one "wildcard submatrix" m_{wild} -that matches on all values that do not start with one of those head -constructors. - -Intuitively, m is equivalent to its decomposition in the following -sense: if the first pattern of an input vector (vᵢ)ⁱ starts with one -of the head constructors k, then running (vᵢ)ⁱ against m is the same -as running it against the submatrix mₖ; otherwise (its head -constructor is none of the k) it is equivalent to running it against -the wildcard submatrix. - -We formalize this intuition as follows: -Lemma (Groups): -Let \[m\] be a matrix with \[Groups(m) = (k_r \to m_r)^k, m_{wild}\]. -For any value vector \[(v_i)^l\] such that \[v_0 = k(v'_l)^l\] for some -constructor k, -we have: -\[ - if k = kₖ for some k then - m(vᵢ)ⁱ = mₖ((v'ₗ)ˡ +++ (vᵢ)^{i∈I\backslash \{0\}}) - else - m(vᵢ)ⁱ = m_{wild}(vᵢ)^{i∈I\backslash \{0\}} -\] - -*** Proof: -Let \[m\] be a matrix with \[Group(m) = (k_r \to m_r)^k, m_{wild}\]. -Let \[(v_i)^i\] be an input matrix with \[v_0 = k(v'_l)^l\] for some k. -We proceed by case analysis: -- either k is one of the kₖ for some k -- or k is none of the (kₖ)ᵏ - -Both m(vᵢ)ⁱ and mₖ(vₖ)ᵏ are defined as the first matching result of -a family over each row rⱼ of a matrix - -We know, from the definition of -Groups(m), that mₖ is -\[ - ((a){0.l})^{l∈Idx(kₖ)} +++ (aᵢ)^{i∈I\backslash \{0\}}), - ( - if p_{0j} is k(qₗ) then - (qₗ)ˡ +++ (p_{ij})^{i∈I\backslash \{0\}} → eⱼ - if p_{0j} is _ then - (_)ˡ +++ (p_{ij})^{i∈I\backslash \{0\}} → eⱼ - else ⊥ - )ʲ -\] - -By definition, m(vᵢ)ⁱ is -m(vᵢ)ⁱ = First(rⱼ(vᵢ)ⁱ)ʲ for m = ((aᵢ)ⁱ, (rⱼ)ʲ) -(pᵢ)ⁱ (vᵢ)ⁱ = { - if p₀ = k(qₗ)ˡ, v₀ = k'(v'ₖ)ᵏ, k=Idx(k') and l=Idx(k) - if k ≠ k' then ⊥ - if k = k' then ((qₗ)ˡ +++ (pᵢ)^{i∈I\backslash \{0\}}) ((v'ₖ)ᵏ +++ (vᵢ)^{i∈I\backslash \{0\}}) - if p₀ = (q₁|q₂) then - First( (q₁pᵢ^{i∈I \backslash \{0\}}) vᵢ^{i∈I \backslash \{0\}}, (q₂pᵢ^{i∈I \backslash \{0\}}) vᵢ^{i∈I \backslash \{0\}} ) -} - -For this reason, if we can prove that -| ∀j, rⱼ(vᵢ)ⁱ = r'ⱼ((v'ₖ)ᵏ ++ (vᵢ)ⁱ) -it follows that -| m(vᵢ)ⁱ = mₖ((v'ₖ)ᵏ ++ (vᵢ)ⁱ) -from the above definition. - -We can also show that aᵢ = a_{0.l}ˡ +++ a_{i∈I\backslash \{0\}} because v(a₀) = K(v(a){0.l})ˡ) - - - - -** Proof of equivalence checking \begin{comment} TODO: put ^i∈I where needed \end{comment} @@ -1493,17 +1538,10 @@ We prove this by induction on Cₜ: /a/ of the subtree Cᵢ and we define πᵢ' = πᵢ if a≠b else πᵢ∩π Trimming a switch node yields the following result: | Switch(b, (π→Cᵢ)^{i∈I})_{/a→π} := Switch(b, (π'ᵢ→C_{i/a→π})^{i∈I}) - \begin{comment} TODO: understand how to properly align lists check that every list is aligned \end{comment} -\begin{comment} -Actually in the proof.org file I transcribed: - e. Unreachabe → ⊥ -This is not correct because you don't have Unreachable nodes in target decision trees -\end{comment} - For the trimming lemma we have to prove that running the value vₜ against the decision tree Cₜ is the same as running vₜ against the tree C_{trim} that is the result of the trimming operation on Cₜ @@ -1548,7 +1586,6 @@ In the case where the algorithm returns No we have at least a couple of counter example values vₛ and vₜ for which the two decision trees outputs a different result. | equiv(S, Cₛ, Cₜ) = No(vₛ,vₜ) and cover(Cₜ, S) → ∀ vₛ ≃ vₜ∈S ∧ Cₛ(vₛ) ≠ Cₜ(vₜ) - We define the following | Forall(Yes) = Yes | Forall(Yes::l) = Forall(l) @@ -1592,12 +1629,12 @@ I think the unreachable case should go at the end. | vₛ≃vₜ∈S ∧ Cₛ(vₛ)≠Cₜ(vₜ) we can say that | equiv(Sᵢ, Cₛ, Cₜᵢ) = No(vₛ, vₜ) for some minimal k∈I -4. When we have a Switch on the right we define πₙ as the domain of +4. When we have a Switch on the right we define π_{fallback} as the domain of values not covered but the union of the constructors kᵢ - | πₙ = ¬(⋃π(kᵢ)^{i∈I}) + | π_{fallback} = ¬\bigcup\limits_{i∈I}π(kᵢ) The algorithm proceeds by trimming | equiv(S, Switch(a, (kᵢ → Cₛᵢ)^{i∈I}, C_{sf}), Cₜ) := - | Forall(equiv( S∩(a→π(kᵢ)^{i∈I}), Cₛᵢ, C_{t/a→π(kᵢ)})^{i∈I} +++ equiv(S∩(a→πₙ), Cₛ, C_{a→πₙ})) + | Forall(equiv( S∩(a→π(kᵢ)^{i∈I}), Cₛᵢ, C_{t/a→π(kᵢ)})^{i∈I} +++ equiv(S∩(a→πₙ), Cₛ, C_{a→π_{fallback}})) The statement still holds and we show this by first analyzing the /Yes/ case: | Forall(equiv( S∩(a→π(kᵢ)^{i∈I}), Cₛᵢ, C_{t/a→π(kᵢ)})^{i∈I} = Yes From a4328b0b1408ba6abc066cfa52fcae020912f46f Mon Sep 17 00:00:00 2001 From: Francesco Mecca Date: Sat, 11 Apr 2020 00:26:52 +0200 Subject: [PATCH 5/7] mail gabriel e coppo --- tesi/.gitignore | 1 + tesi/gabriel/part4.pdf | Bin 308599 -> 308599 bytes tesi/gabriel/part4.tex | 12 ++++++------ 3 files changed, 7 insertions(+), 6 deletions(-) diff --git a/tesi/.gitignore b/tesi/.gitignore index bbd865c..03ea693 100644 --- a/tesi/.gitignore +++ b/tesi/.gitignore @@ -3,6 +3,7 @@ tesi.tex tesi_unicode.tex part4_unicode.tex +part4.tex ## Core latex/pdflatex auxiliary files: diff --git a/tesi/gabriel/part4.pdf b/tesi/gabriel/part4.pdf index c744289c541b708de5baabc0268da379dcce2ac4..4d09d10c6684cd526c482ec7f380a3f3d43bee25 100644 GIT binary patch delta 110 zcmezVO6dD5p@tU57N!>FEi5c6n9Ym~rn9eLQ2;TvYpr0}$>r>3;$rD+VCZUSZ0c-g iVdCuU=;&l&VCrOSVPtOL=3;24U_(gBcD2ncflL4r0v(+I delta 110 zcmezVO6dD5p@tU57N!>FEi5c6m`#mLrn9eLQ2;TvYpr0}$>nTl;$r4v>FQ?g;^yRH iXlm|e>SkzS;^O9LU}$1) Date: Sat, 11 Apr 2020 00:30:01 +0200 Subject: [PATCH 6/7] mail gabriel e coppo --- tesi/tesi.pdf | Bin 525229 -> 524613 bytes 1 file changed, 0 insertions(+), 0 deletions(-) diff --git a/tesi/tesi.pdf b/tesi/tesi.pdf index 0f108e9c7137973b0d23d858a1ea3d818710a0db..da8201403bc4b84263f03fcdf7a6985255104c07 100644 GIT binary patch delta 51394 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auxiliary files: diff --git a/tesi/gabriel/notations.sty b/tesi/gabriel/notations.sty new file mode 120000 index 0000000..24ae4a2 --- /dev/null +++ b/tesi/gabriel/notations.sty @@ -0,0 +1 @@ +../notations.sty \ No newline at end of file diff --git a/tesi/gabriel/part4.pdf b/tesi/gabriel/part4.pdf index 4d09d10c6684cd526c482ec7f380a3f3d43bee25..a91659ee326c9966e5952bec3a39572da4649d3e 100644 GIT binary patch delta 112 zcmezVO6dD5p@tU57N!>FEiB9{Sd0w~OsBK2V37y2wrj0m*~#T>ZeZ-};%sK(WNGH&>ga4~r(i=!$#%8PEP+e_KNKDV delta 112 zcmezVO6dD5p@tU57N!>FEiB9{Sd7e!45qWMV37y2wrj0m*~#VXX5wP$Y+&eWXl&|i jW?|y&?C9uZVPNWHY++<>;O1gzr(i=!$#%8PEP+e_MTi~& diff --git a/tesi/gabriel/part4.tex b/tesi/gabriel/part4.tex index a0d244c..017c251 100644 --- a/tesi/gabriel/part4.tex +++ b/tesi/gabriel/part4.tex @@ -1,4 +1,4 @@ -% Created 2020-04-11 Sat 00:26 +% Created 2020-04-11 Sat 00:31 % Intended LaTeX compiler: pdflatex \documentclass[11pt]{article} \usepackage[utf8]{inputenc} @@ -64,10 +64,10 @@ xleftmargin=1em, \begin{document} \section{Translation validation of the Pattern Matching Compiler} -\label{sec:org3d79c1b} +\label{sec:org9d32dd4} \subsection{Source program} -\label{sec:orgc2c278d} +\label{sec:org0032637} The algorithm takes as its input a source program and translates it into an algebraic data structure which type we call \emph{decision\_tree}. @@ -412,7 +412,7 @@ TODO: fix \{0} \begin{enumerate} \item Proof: -\label{sec:org160a47b} +\label{sec:orgb73fb4f} Let \(m\) be a matrix with \[Group(m) = (k_r \to m_r)^k, m_{wild}\] Let \((v_i)^i\) be an input matrix with \(v_0 = k(v'_l)^l\) for some k. We proceed by case analysis: @@ -472,7 +472,7 @@ We can also show that a$_i$ = (a\(_{\text{0.l}}\))$^l$ \sout{+} a\(_{\text{i$\in \subsection{Target translation} -\label{sec:orgd2ea1d0} +\label{sec:org0337fa8} 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. @@ -560,7 +560,7 @@ $\forall$v$_T$, t$_T$(v$_T$) = $\llbracket$t$_T$$\rrbracket$$_T$(v$_T$)\\ \subsection{Equivalence checking} -\label{sec:org33b978b} +\label{sec:org7268798} The equivalence checking algorithm takes as input a domain of possible values \emph{S} and a pair of source and target decision trees and