From 13b8bb01f24364e88ac33edcf01bbc1809196f2c Mon Sep 17 00:00:00 2001 From: Francesco Mecca Date: Sat, 11 Apr 2020 00:26:05 +0200 Subject: [PATCH] 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