From 048affd516fdb8e1c4f7b54a9f61891844afdea0 Mon Sep 17 00:00:00 2001 From: Francesco Mecca Date: Thu, 9 Apr 2020 01:44:19 +0200 Subject: [PATCH] gabriel draft sent --- tesi/gabriel/part4.org | 354 ++++++++++++++++++++++ tesi/gabriel/part4.pdf | Bin 0 -> 226968 bytes tesi/gabriel/part4.tex | 659 +++++++++++++++++++++++++++++++++++++++++ tesi/tesi.pdf | Bin 468498 -> 469378 bytes tesi/tesi_unicode.org | 79 +++-- 5 files changed, 1060 insertions(+), 32 deletions(-) create mode 100644 tesi/gabriel/part4.org create mode 100644 tesi/gabriel/part4.pdf create mode 100644 tesi/gabriel/part4.tex diff --git a/tesi/gabriel/part4.org b/tesi/gabriel/part4.org new file mode 100644 index 0000000..5cb1a98 --- /dev/null +++ b/tesi/gabriel/part4.org @@ -0,0 +1,354 @@ +# 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) ≠ C_{t/a→πₖ}(vₜ) + that is enough for proving that + | Cₛₖ(vₛ) ≠ (C_{t/a→πₖ}(vₜ) = Cₜ(vₜ)) diff --git a/tesi/gabriel/part4.pdf b/tesi/gabriel/part4.pdf new file mode 100644 index 0000000000000000000000000000000000000000..3c85d5a0426711955aa7e6318b7494b5f4c4f86b GIT binary patch literal 226968 zcma(2LzFN;vbBwt{g!Rpwr$(CZQHhO+qP}ncK!X`b!UABoht`1&OtnT=Z+we7Z#;q zpk;v~nO`1Rhhk*Gr^mN5w1nd3hN6=;u{Cox$7f(+X2t*i9Vj|c3u|W+M|?U_YXfH! zVG|=eV-qM|UMMGLM-u}ZDEG}74JoHBR`}j8wfZvo90-Eld^1P$rOupU2~$lSP8qmn zVX18LVo5~@&Lq4&P}F1z$Y$da3onLn(0hXhxNC5Zh*Sl{ALoSV=Dj)ET%85x!sKGb zMsrsrh7kxQ%ESs{(MyL;%0ftq zq-4!|;!H*v=C$z8t5)hLM@Zi4{pdZvt zY7&a*H&YdsLR6chg8|YRPOGTO=Tdqf03wJJm}J)~z0g;Hu9Ctmghf?&Fj%N_&TYvA z5PQ!i-4Iu%;DPo%8ZA?ijpkkV5reRMSY0=>fUi<;yRg0PfdMLkl+h>j7Yp;cl|y&MxCru*MK)Lj|?R^>~8 zsXV&m$o(US!rZ-KVhh`I3q*84`>YcfJa9i!Ehq*yJ7}1C&uQ`!$)`DJeUJr%H{=h^4U=F|jPgIiaq0eTlNj{%{irsA&X?v(LxU>9F zEs`TG;1ws7&~E<5=N+ zL7CnQ9W`a`y+&-(UZ^w8ky(~nTrd3Xvy$~Qi*f2#FK3tSfpwVRZ}$5{RcZG0Z0lo? 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unreachable case should go at the end. +\end{comment} +\begin{enumerate} +\item in case of unreachable: +\end{enumerate} +\begin{center} +\begin{tabular}{l} +C$_S$(v$_S$) = Absurd(Unreachable) $\ne$ C$_T$(v$_T$) $\forall$v$_S$,v$_T$\\ +\end{tabular} +\end{center} +\begin{enumerate} +\item In the case of an empty input space +\begin{center} +\begin{tabular}{l} +equiv($\varnothing$, C$_S$, C$_T$) := Yes\\ +\end{tabular} +\end{center} +and that is trivial to prove because there is no pair of values (v$_S$, v$_T$) that could be +tested against the decision trees. +In the other subcases S is always non-empty. +\item When there are \emph{Failure} nodes at both sides the result is \emph{Yes}: +\begin{center} +\begin{tabular}{l} +equiv(S, Failure, Failure) := Yes\\ +\end{tabular} +\end{center} +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$)\\ +\end{tabular} +\end{center} +The algorithm either returns Yes for every sub-input space S$_i$ := S$\cap$(a$\to$$\pi$(k$_i$)) and +subtree C$_T$$_i$ +\begin{center} +\begin{tabular}{l} +equiv(S$_i$, C$_S$, C$_T$$_i$) = Yes $\forall$i\\ +\end{tabular} +\end{center} +or we have a counter example v$_S$, v$_T$ for which +\begin{center} +\begin{tabular}{l} +v$_S$$\simeq$v$_T$$\in $S$_k$ $\wedge$ c$_S$(v$_S$) $\ne$ C$_T$$_k$(v$_T$)\\ +\end{tabular} +\end{center} +then because +\begin{center} +\begin{tabular}{l} +v$_T$$\in $(a$\to$$\pi$$_k$) $\to$ C$_T$(v$_T$) = C$_T$$_k$(v$_T$) ,\\ +v$_S$$\simeq$v$_T$$\in $S $\wedge$ C$_S$(v$_S$)$\ne$C$_T$(v$_T$)\\ +\end{tabular} +\end{center} +we can say that +\begin{center} +\begin{tabular}{l} +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 +values not covered but the union of the constructors k$_i$ +\begin{center} +\begin{tabular}{l} +$\pi$$_n$ = ¬($\bigcup$$\pi$(k$_i$)$^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$}}\)))\\ +\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\\ +\end{tabular} +\end{center} +The constructor k is either included in the set of constructors k$_i$: +\begin{center} +\begin{tabular}{l} +k \(\vert{}\) k$\in $(k$_i$)$^i$ $\wedge$ C$_S$(v$_S$) = C$_S$$_i$(v$_S$)\\ +\end{tabular} +\end{center} +We also know that +\begin{center} +\begin{tabular}{l} +(1) C$_S$$_i$(v$_S$) = C\(_{\text{t/a$\to$$\pi$$_i$}}\)(v$_T$)\\ +(2) C\(_{\text{T/a$\to$$\pi$$_i$}}\)(v$_T$) = C$_T$(v$_T$)\\ +\end{tabular} +\end{center} +(1) is true by induction and (2) is a consequence of the trimming lemma. +Putting everything together: +\begin{center} +\begin{tabular}{l} +C$_S$(v$_S$) = C$_S$$_i$(v$_S$) = C\(_{\text{T/a$\to$$\pi$$_i$}}\)(v$_T$) = C$_T$(v$_T$)\\ +\end{tabular} +\end{center} + +When the k$\notin$(k$_i$)$^i$ [TODO] + +The auxiliary Forall function returns \emph{No(v$_S$, v$_T$)} when, for a minimum k, +\begin{center} +\begin{tabular}{l} +equiv(S$_k$, C$_S$$_k$, C\(_{\text{T/a$\to$$\pi$$_k$}}\) = No(v$_S$, v$_T$)\\ +\end{tabular} +\end{center} +Then we can say that +\begin{center} +\begin{tabular}{l} +C$_S$$_k$(v$_S$) $\ne$ C\(_{\text{t/a$\to$$\pi$$_k$}}\)(v$_T$)\\ +\end{tabular} +\end{center} +that is enough for proving that +\begin{center} +\begin{tabular}{l} +C$_S$$_k$(v$_S$) $\ne$ (C\(_{\text{t/a$\to$$\pi$$_k$}}\)(v$_T$) = C$_T$(v$_T$))\\ +\end{tabular} +\end{center} +\end{enumerate} +\end{document} diff --git a/tesi/tesi.pdf b/tesi/tesi.pdf index 6644301d884301683b5843896fda6433561519e1..51f413b2cddc6c331f0f2109976744dc730ec3fe 100644 GIT binary patch delta 52552 zcmV)ZK&!u!%p8K#9DuX|&SL^MIg=3#D1Yr+OK;mo5Wf3Yu%}Q+wAuF~g;5|t6SU}| zMGf?j_MqBQtk{wqNv;#L{qLQ9kmAxqiBcRTDRPo54rgcQ`OWN_7Ox9hygZ@l^ZD7y z(-%w^j1os3E6yebQ-V;#3auF7$`)rA#jA34dp_W{yc`UL;N|*qv>yC%_T$qRQh#Ny zN=k$>#gJ*@go&^I6)8C@3~`1ALPfZ>Kt{CCk%~qtmhm&hXi+q%PSI5vQScs(Bv%+C zO(+)?#qU&6jE^K-@`|XE+GJ8|!8WTwnW_7`dF}fCu zA^f~lqPb|rEi>AXS$G9VhTIUzd4D9GESBD2tXwT-<7m3`#eB71-kz_kM^{hcgFA2E zas2IaG~fnQ;D33BsfxlnU}Bm?pH2otMlG&;=7<0_(tRzMX(6f}9L5-vQRR#jVZ3M% zCAs`wVMAUESFl6xz~lK0^*BeA?Bgh+?I;p=?7tv-JIxNDo-5$(*8JwqU# 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zgsV-`ohMywR#tbz)fQ#LueoYfHut8hHf7JwyV|Pko!?!xEBpL@wN2U2AG_MFLUpyP z4i!55k5ukd;o1+bx>We~X;(W`xc7#uohnw}bk(im%vD#rRDAmjS3N3z{f(<$m3pUL z?N;f%MOS?)Eskfp+N086Kf3By#pCa~8c@aEpIi-spFS607W|~~R4>C7V4(TZKwE0L zEEiP*!!1RDF+@4H!?Ih3MC~)PeuxVQyMS; diff --git a/tesi/tesi_unicode.org b/tesi/tesi_unicode.org index c4409c2..5ddbbbc 100644 --- a/tesi/tesi_unicode.org +++ b/tesi/tesi_unicode.org @@ -18,7 +18,6 @@ TODO: chiedi a Gabriel se T e S vanno bene, ma prima controlla che siano coerent - [ ] Formal Grammar - [ ] Symbolic execution of the Lambda target - [ ] Equivalence between source and target - - [ ] Statement of correctness - [ ] Proof of correctness - [ ] Practical results @@ -1362,9 +1361,9 @@ Correggi prendendo in considerazione l'accessor We define the decision tree of source programs 〚tₛ〛 in terms of the decision tree of pattern matrices - 〚mₛ〛 +〚mₛ〛 by the following: - 〚((pᵢ → bbᵢ)^{i∈I}〛 := 〚(Root), (pᵢ → bbᵢ)^{i∈I} 〛 +〚((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}) @@ -1467,13 +1466,13 @@ m(vᵢ)ⁱ = First(rⱼ(vᵢ)ⁱ)ʲ for m = ((aᵢ)ⁱ, (rⱼ)ʲ) 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 prova that +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) +We can also show that aᵢ = a_{0.l}ˡ +++ a_{i∈I\backslash \{0\}} because v(a₀) = K(v(a){0.l})ˡ) @@ -1484,36 +1483,52 @@ 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 (In other words???) +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. Case where Cₜ = Leaf_{bb}: - Leaf_{bb/a→π}(v) = Leaf_{bb}(v) - That means that the result of trimming on a Leaf is - the Leaf itself - b. Same for failure terminal - e. Unreachabe → ⊥ - c. When Cₜ = Node(b, (π→Cᵢ)ⁱ)_{/a→π} then - we define πᵢ' = πᵢ if a≠b else πᵢ∩π ; - Node(b, (π→Cᵢ)ⁱ)_{/a→π} := Node(b, (π'ᵢ→C_{i/a→π})ⁱ) +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) - Goal: prove that Cₜ(vₜ) = Node(b, (πᵢ'→C_{i/a→π})ⁱ)(vₜ) - Either vₜ∉(b→πᵢ)ⁱ and that means that the node is a Failure node - Or vₜ∈(b→πₖ) for some k, then - C_{k/a→π}(vₜ) = Node(b, (πᵢ'→C_{i/a→π})ⁱ)(vₜ) - because when a ≠ b then πₖ'= πₖ => vₜ∈πₖ' - while when a = b then πₖ'=(πₖ∩π) - - vₜ∈π because of hypothesis - - we already know that vₜ∈πₖ - So vₜ ∈ πₖ' and by induction Cₖ(vₜ) = C_{k/a→π}(vₜ) - and Cₜ(vₜ) = Cₖ(vₜ) when vₜ∈(b→πₖ) - This proves that Node(b, (πᵢ'→C_{i/a→π})ⁱ)(vₜ) = Cₜ(vₜ) - - Covering lemma: - ∀a,π covers(Cₜ,S) => covers(C_{t/a→π}, (S∩a→π)) - Uᵢπⁱ ≈ Uᵢπ'∩(a→π) ≈ (Uᵢπ')∩(a→π) %% \begin{comment} -Should I swap π and π' +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}