Merge branch 'master' of http://francescomecca.eu/git/git/pesceWanda/UNITO
This commit is contained in:
Francesco Mecca
commit
f682171afd
10 changed files with 1750 additions and 1012 deletions
1
tesi/.gitignore
vendored
1
tesi/.gitignore
vendored
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@ -2,6 +2,7 @@ tesi.tex
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.#tesi.tex
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.#tesi.tex
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tesi_unicode.tex
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tesi_unicode.tex
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part4_unicode.tex
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## Core latex/pdflatex auxiliary files:
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## Core latex/pdflatex auxiliary files:
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|
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13
tesi/conv.py
13
tesi/conv.py
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@ -2,8 +2,12 @@ import json
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import re
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import re
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from sys import argv
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from sys import argv
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allsymbols = json.load(open('./unicode-latex.json'))
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try:
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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', '≈' ]
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allsymbols = json.load(open('./unicode-latex.json'))
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except:
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allsymbols = json.load(open('../unicode-latex.json'))
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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', '≈', '⊆' ]
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extrasymbols = {'〚': '\llbracket', '〛': r'\rrbracket', '̸': '\neg', '¬̸': '\neg', '∈': '\in ', 'ₛ': '_S', 'ₜ': '_T'}
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extrasymbols = {'〚': '\llbracket', '〛': r'\rrbracket', '̸': '\neg', '¬̸': '\neg', '∈': '\in ', 'ₛ': '_S', 'ₜ': '_T'}
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symbols = {s: allsymbols[s] for s in mysymbols}
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symbols = {s: allsymbols[s] for s in mysymbols}
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@ -39,8 +43,9 @@ def convert(ch, mathmode):
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def latex_errors_replacements(charlist):
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def latex_errors_replacements(charlist):
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text = ''.join(charlist).split(' ')
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text = ''.join(charlist).split(' ')
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replacements = {'\n\end{comment}\n\end{enumerate}\n\end{enumerate}\n\n\subsection{Symbolic':
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replacements = {
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'\n\end{comment}\n\n\subsection{Symbolic'}
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'\n\end{comment}\n\end{enumerate}\n\end{enumerate}\n\n\subsection{Symbolic': '\n\end{comment}\n\n\subsection{Symbolic',
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}
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r_set = set(replacements.keys())
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r_set = set(replacements.keys())
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for word in text:
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for word in text:
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it = r_set.intersection(set([word]))
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it = r_set.intersection(set([word]))
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22
tesi/gabriel/Makefile
Normal file
22
tesi/gabriel/Makefile
Normal file
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@ -0,0 +1,22 @@
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SRC = part4.tex
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AUX = part4.aux
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DEL = part4.aux part4.bbl part4.blg part4.log part4.out part4_unicode.tex part4.pdf part4.tex texput.log
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NAME = part4
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part4:
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@echo
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@echo "=== Removing temporary files ==="
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rm -f $(DEL)
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@echo
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@echo "=== Building from scratch ==="
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emacs -batch part4_unicode.org -f org-latex-export-to-latex --kill
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python3 ../conv.py part4_unicode.tex part4.tex
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pdflatex $(SRC)
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bibtex $(AUX)
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pdflatex $(SRC)
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pdflatex $(SRC)
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@echo
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@echo "=== All done. Generated $(NAME).pdf ==="
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clean:
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rm -f $(DEL)
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1
tesi/gabriel/notations.sty
Symbolic link
1
tesi/gabriel/notations.sty
Symbolic link
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@ -0,0 +1 @@
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../notations.sty
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@ -1,354 +0,0 @@
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# Add headers to export only this section
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* Correctness of the algorithm
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Running a program tₛ or its translation 〚tₛ〛 against an input vₛ
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produces as a result /r/ in the following way:
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| ( 〚tₛ〛ₛ(vₛ) = Cₛ(vₛ) ) → r
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| tₛ(vₛ) → r
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Likewise
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| ( 〚tₜ〛ₜ(vₜ) = Cₜ(vₜ) ) → r
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| tₜ(vₜ) → r
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where result r ::= guard list * (Match blackbox | NoMatch | Absurd)
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and guard ::= blackbox.
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Having defined equivalence between two inputs of which one is
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expressed in the source language and the other in the target language
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vₛ ≃ vₜ (TODO define, this talks about the representation of source values in the target)
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we can define the equivalence between a couple of programs or a couple
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of decision trees
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| tₛ ≃ tₜ := ∀vₛ≃vₜ, tₛ(vₛ) = tₜ(vₜ)
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| Cₛ ≃ Cₜ := ∀vₛ≃vₜ, Cₛ(vₛ) = Cₜ(vₜ)
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The proposed equivalence algorithm that works on a couple of
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decision trees is returns either /Yes/ or /No(vₛ, vₜ)/ where vₛ and
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vₜ are a couple of possible counter examples for which the constraint
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trees produce a different result.
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** Statements
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Theorem. We say that a translation of a source program to a decision tree
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is correct when for every possible input, the source program and its
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respective decision tree produces the same result
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| ∀vₛ, tₛ(vₛ) = 〚tₛ〛ₛ(vₛ)
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Likewise, for the target language:
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| ∀vₜ, tₜ(vₜ) = 〚tₜ〛ₜ(vₜ)
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Definition: in the presence of guards we can say that two results are
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equivalent modulo the guards queue, written /r₁ ≃gs r₂/, when:
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| (gs₁, r₁) ≃gs (gs₂, r₂) ⇔ (gs₁, r₁) = (gs₂ ++ gs, r₂)
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Definition: we say that Cₜ covers the input space /S/, written
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/covers(Cₜ, S) when every value vₛ∈S is a valid input to the
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decision tree Cₜ. (TODO: rephrase)
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Theorem: Given an input space /S/ and a couple of decision trees, where
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the target decision tree Cₜ covers the input space /S/, we say that
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the two decision trees are equivalent when:
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| equiv(S, Cₛ, Cₜ, gs) = Yes ∧ covers(Cₜ, S) → ∀vₛ≃vₜ ∈ S, Cₛ(vₛ) ≃gs Cₜ(vₜ)
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Similarly we say that a couple of decision trees in the presence of
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an input space /S/ are /not/ equivalent when:
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| equiv(S, Cₛ, Cₜ, gs) = No(vₛ,vₜ) ∧ covers(Cₜ, S) → vₛ≃vₜ ∈ S ∧ Cₛ(vₛ) ≠gs Cₜ(vₜ)
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Corollary: For a full input space /S/, that is the universe of the
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target program we say:
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| equiv(S, 〚tₛ〛ₛ, 〚tₜ〛ₜ, ∅) = Yes ⇔ tₛ ≃ tₜ
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*** Proof of the correctness of the translation from source programs to source decision trees
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We define a source term tₛ as a collection of patterns pointing to blackboxes
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| tₛ ::= (p → bb)^{i∈I}
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A pattern is defined as either a constructor pattern, an or pattern or
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a constant pattern
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| p ::= | K(pᵢ)ⁱ, i ∈ I | (p|q) | n ∈ ℕ
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A decision tree is defined as either a Leaf, a Failure terminal or
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an intermediate node with different children sharing the same accessor /a/
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and an optional fallback.
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Failure is emitted only when the patterns don't cover the whole set of
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possible input values /S/. The fallback is not needed when the user
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doesn't use a wildcard pattern.
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%%% Give example of thing
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| Cₛ ::= Leaf bb | Node(a, (Kᵢ → Cᵢ)^{i∈S} , C?)
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| a ::= Here | n.a
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| vₛ ::= K(vᵢ)^{i∈I} | n ∈ ℕ
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\begin{comment}
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Are K and Here clear here?
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\end{comment}
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We define the decomposition matrix /mₛ/ as
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| SMatrix mₛ := (aⱼ)^{j∈J}, ((p_{ij})^{j∈J} → bbᵢ)^{i∈I}
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\begin{comment}
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Correggi prendendo in considerazione l'accessor
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\end{comment}
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We define the decision tree of source programs
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〚tₛ〛
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in terms of the decision tree of pattern matrices
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〚mₛ〛
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by the following:
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〚((pᵢ → bbᵢ)^{i∈I}〛 := 〚(Root), (pᵢ → bbᵢ)^{i∈I} 〛
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|
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decision tree computed from pattern matrices respect the following invariant:
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| ∀v (vᵢ)^{i∈I} = v(aᵢ)^{i∈I} → 〚m〛(v) = m(vᵢ)^{i∈I} for m = ((aᵢ)^{i∈I}, (rᵢ)^{i∈I})
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where
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| v(Here) = v
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| K(vᵢ)ⁱ(k.a) = vₖ(a) if k ∈ [0;n[
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\begin{comment}
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TODO: EXPLAIN
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\end{comment}
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We proceed to show the correctness of the invariant by a case
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analysys.
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Base cases:
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1. [| ∅, (∅ → bbᵢ)ⁱ |] := Leaf bbᵢ where i := min(I), that is a
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decision tree [|ms|] defined by an empty accessor and empty
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patterns pointing to blackboxes /bbᵢ/. This respects the invariant
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because a decomposition matrix in the case of empty rows returns
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the first expression and we known that (Leaf bb)(v) := Match bb
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2. [| (aⱼ)ʲ, ∅ |] := Failure
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Regarding non base cases:
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Let's first define
|
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| let Idx(k) := [0; arity(k)[
|
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| let First(∅) := ⊥
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| let First((aⱼ)ʲ) := a_{min(j∈J≠∅)}
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\[
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m := ((a_i)^i ((p_{ij})^i \to e_j)^{ij})
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\]
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\[
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(k_k)^k := headconstructor(p_{i0})^i
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\]
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\begin{equation}
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Groups(m) := ( k_k \to ((a)_{0.l})^{l \in Idx(k_k)} +++ (a_i)^{i \in I\backslash \{0\} }), \\
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( if p_{0j} is k(q_l) then \\
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(qₗ)^{l \in Idx(k_k)} +++ (p_{ij})^{i \in I\backslash \{0\}} \to e_j \\
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if p_{0j} is \_ then \\
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(\_)^{l \in Idx(k_k)} +++ (p_{ij})^{i \in I\backslash \{0\}} \to e_j \\
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else \bot )^j ), \\
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((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})
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\end{equation}
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Groups(m) is an auxiliary function that decomposes a matrix m into
|
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submatrices, according to the head constructor of their first pattern.
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Groups(m) returns one submatrix m_r for each head constructor k that
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occurs on the first row of m, plus one "wildcard submatrix" m_{wild}
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that matches on all values that do not start with one of those head
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constructors.
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Intuitively, m is equivalent to its decompositionin the following
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sense: if the first pattern of an input vector (v_i)^i starts with one
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of the head constructors k, then running (v_i)^i against m is the same
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as running it against the submatrix m_k; otherwise (its head
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constructor is none of the k) it is equivalent to running it against
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the wildcard submatrix.
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We formalize this intuition as follows:
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Lemma (Groups):
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Let \[m\] be a matrix with \[Groups(m) = (k_r \to m_r)^k, m_{wild}\].
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For any value vector \[(v_i)^l\] such that \[v_0 = k(v'_l)^l\] for some
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|
||||||
constructor k,
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we have:
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\[
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if k = kₖ for some k then
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m(vᵢ)ⁱ = mₖ((v'ₗ)ˡ +++ (vᵢ)^{i∈I\backslash \{0\}})
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else
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m(vᵢ)ⁱ = m_{wild}(vᵢ)^{i∈I\backslash \{0\}}
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|
||||||
\]
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|
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*** Proof:
|
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Let \[m\] be a matrix with \[Group(m) = (k_r \to m_r)^k, m_{wild}\].
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Let \[(v_i)^i\] be an input matrix with \[v_0 = k(v'_l)^l\] for some k.
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||||||
We proceed by case analysis:
|
|
||||||
- either k is one of the kₖ for some k
|
|
||||||
- or k is none of the (kₖ)ᵏ
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|
||||||
|
|
||||||
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
|
|
||||||
|
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||||||
We know, from the definition of
|
|
||||||
Groups(m), that mₖ is
|
|
||||||
\[
|
|
||||||
((a){0.l})^{l∈Idx(kₖ)} +++ (aᵢ)^{i∈I\backslash \{0\}}),
|
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||||||
(
|
|
||||||
if p_{0j} is k(qₗ) then
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||||||
(qₗ)ˡ +++ (p_{ij})^{i∈I\backslash \{0\}} → eⱼ
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|
||||||
if p_{0j} is _ then
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||||||
(_)ˡ +++ (p_{ij})^{i∈I\backslash \{0\}} → eⱼ
|
|
||||||
else ⊥
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|
||||||
)ʲ
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|
||||||
\]
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|
||||||
|
|
||||||
By definition, m(vᵢ)ⁱ is
|
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||||||
m(vᵢ)ⁱ = First(rⱼ(vᵢ)ⁱ)ʲ for m = ((aᵢ)ⁱ, (rⱼ)ʲ)
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(pᵢ)ⁱ (vᵢ)ⁱ = {
|
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||||||
if p₀ = k(qₗ)ˡ, v₀ = k'(v'ₖ)ᵏ, k=Idx(k') and l=Idx(k)
|
|
||||||
if k ≠ k' then ⊥
|
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||||||
if k = k' then ((qₗ)ˡ +++ (pᵢ)^{i∈I\backslash \{0\}}) ((v'ₖ)ᵏ +++ (vᵢ)^{i∈I\backslash \{0\}})
|
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||||||
if p₀ = (q₁|q₂) then
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First( (q₁pᵢ^{i∈I \backslash \{0\}}) vᵢ^{i∈I \backslash \{0\}}, (q₂pᵢ^{i∈I \backslash \{0\}}) vᵢ^{i∈I \backslash \{0\}} )
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}
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|
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For this reason, if we can prove that
|
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||||||
| ∀j, rⱼ(vᵢ)ⁱ = r'ⱼ((v'ₖ)ᵏ ++ (vᵢ)ⁱ)
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||||||
it follows that
|
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| m(vᵢ)ⁱ = mₖ((v'ₖ)ᵏ ++ (vᵢ)ⁱ)
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||||||
from the above definition.
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||||||
|
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||||||
We can also show that aᵢ = a_{0.l}ˡ +++ a_{i∈I\backslash \{0\}} because v(a₀) = K(v(a){0.l})ˡ)
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|
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||||||
|
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||||||
|
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||||||
|
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||||||
** Proof of equivalence checking
|
|
||||||
\begin{comment}
|
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||||||
TODO: put ^i∈I where needed
|
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||||||
\end{comment}
|
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\subsubsection{The trimming lemma}
|
|
||||||
The trimming lemma allows to reduce the size of a decision tree given
|
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||||||
an accessor → π relation (TODO: expand)
|
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||||||
| ∀vₜ ∈ (a→π), Cₜ(vₜ) = C_{t/a→π(kᵢ)}(vₜ)
|
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||||||
We prove this by induction on Cₜ:
|
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||||||
a. Cₜ = Leaf_{bb}: when the decision tree is a leaf terminal, we
|
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||||||
know that
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||||||
| Leaf_{bb/a→π}(v) = Leaf_{bb}(v)
|
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||||||
That means that the result of trimming on a Leaf is the Leaf itself
|
|
||||||
b. The same applies to Failure terminal
|
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||||||
| Failure_{/a→π}(v) = Failure(v)
|
|
||||||
c. When Cₜ = Node(b, (π→Cᵢ)ⁱ)_{/a→π} then
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|
||||||
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))
|
|
||||||
Binary file not shown.
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Load diff
701
tesi/gabriel/part4_unicode.org
Normal file
701
tesi/gabriel/part4_unicode.org
Normal file
|
|
@ -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 <arg> <arg> <arg>}, where the
|
||||||
|
<arg> are expressed using the OCaml pattern matching language.
|
||||||
|
Similarly, all the right-hand-side expressions are of the form
|
||||||
|
\texttt{observe <arg> <arg> ...} 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, 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))
|
||||||
BIN
tesi/tesi.pdf
BIN
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Reference in a new issue