778 lines
32 KiB
Org Mode
Executable file
778 lines
32 KiB
Org Mode
Executable file
#+LANGUAGE: en
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#+LaTeX_CLASS: article
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#+LaTeX_HEADER: \linespread{1.25}
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#+LaTeX_HEADER: \usepackage{algorithm}
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#+LaTeX_HEADER: \usepackage{comment}
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#+LaTeX_HEADER: \usepackage{algpseudocode}
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#+LaTeX_HEADER: \newtheorem{definition}{Definition}
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#+LaTeX_HEADER: \newcommand{\YesTEX}{\mathsf{Yes}}
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#+LaTeX_HEADER: \newcommand{\NoTEX}[2]{\mathsf{No}(#1, #2)}
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#+LaTeX_HEADER: moredelim=[is][\color{red}]{/[}{]/},
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#+LaTeX_HEADER: \lstset{aboveskip=0.4ex,belowskip=0.4ex}
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#+EXPORT_SELECT_TAGS: export
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#+EXPORT_EXCLUDE_TAGS: noexport
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#+OPTIONS: H:2 toc:nil \n:nil @:t ::t |:t ^:{} _:{} *:t TeX:t LaTeX:t
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#+STARTUP: showall
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* Translation validation of the Pattern Matching Compiler
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** Source program
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Our algorithm takes as its input a source program and translates it
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into an algebraic data structure which type we call /decision_tree/.
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#+BEGIN_SRC
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type decision_tree =
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| Unreachable
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| Failure
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| Leaf of source_expr
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| Guard of source_blackbox * decision_tree * decision_tree
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| Switch of accessor * (constructor * decision_tree) list * decision_tree
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#+END_SRC
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Unreachable, Leaf of source_expr and Failure are the terminals of the three.
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We distinguish
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- Unreachable: statically it is known that no value can go there
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- Failure: a value matching this part results in an error
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- Leaf: a value matching this part results into the evaluation of a
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source black box of code
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Our algorithm doesn't support type-declaration-based analysis
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to know the list of constructors at a given type.
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Let's consider some trivial examples:
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| function true -> 1
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is translated to
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|Switch ([(true, Leaf 1)], Failure)
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while
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| function
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| \vert{} true -> 1
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| \vert{} false -> 2
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will be translated to
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|Switch ([(true, Leaf 1); (false, Leaf 2)])
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It is possible to produce Unreachable examples by using
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refutation clauses (a "dot" in the right-hand-side)
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|function
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|\vert{} true -> 1
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|\vert{} false -> 2
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|\vert{} _ -> .
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that gets translated into
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| Switch ([(true, Leaf 1); (false, Leaf 2)], Unreachable)
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We trust this annotation, which is reasonable as the type-checker
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verifies that it indeed holds.
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Guard nodes of the tree are emitted whenever a guard is found. Guards
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node contains a blackbox of code that is never evaluated and two
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branches, one that is taken in case the guard evaluates to true and
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the other one that contains the path taken when the guard evaluates to
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true.
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\begin{comment}
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[ ] Finisci con Switch
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[ ] Spiega del fallback
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\end{comment}
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The source code of a pattern matching function has the
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following form:
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|match variable with
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|\vert pattern₁ \to expr₁
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|\vert pattern₂ when guard \to expr₂
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|\vert pattern₃ as var \to expr₃
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|⋮
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|\vert pₙ \to exprₙ
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Patterns could or could not be exhaustive.
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Pattern matching code could also be written using the more compact form:
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|function
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|\vert pattern₁ \to expr₁
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|\vert pattern₂ when guard \to expr₂
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|\vert pattern₃ as var \to expr₃
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|⋮
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|\vert pₙ \to exprₙ
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This BNF grammar describes formally the grammar of the source program:
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| start ::= "match" id "with" patterns \vert{} "function" patterns
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| patterns ::= (pattern0\vert{}pattern1) pattern1+
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| ;; pattern0 and pattern1 are needed to distinguish the first case in which
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| ;; we can avoid writing the optional vertical line
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| pattern0 ::= clause
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| pattern1 ::= "\vert" clause
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| clause ::= lexpr "->" rexpr
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| lexpr ::= rule (ε\vert{}condition)
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| rexpr ::= _code ;; arbitrary code
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| rule ::= wildcard\vert{}variable\vert{}constructor_pattern\vert or_pattern ;;
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| wildcard ::= "_"
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| variable ::= identifier
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| constructor_pattern ::= constructor (rule\vert{}ε) (assignment\vert{}ε)
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| 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
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| or_pattern ::= rule ("\vert{}" wildcard\vert{}variable\vert{}constructor_pattern)+
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| condition ::= "when" bexpr
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| assignment ::= "as" id
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| bexpr ::= _code ;; arbitrary code
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The source program is parsed using the ocaml-compiler-libs library.
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The result of parsing, when successful, results in a list of clauses
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and a list of type declarations.
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Every clause consists of three objects: a left-hand-side that is the
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kind of pattern expressed, an option guard and a right-hand-side expression.
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Patterns are encoded in the following way:
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| pattern | type |
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|-----------------+-------------|
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| _ | Wildcard |
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| p₁ as x | Assignment |
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| c(p₁,p₂,...,pₙ) | Constructor |
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| (p₁\vert p₂) | Orpat |
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Once parsed, the type declarations and the list of clauses are encoded in the form of a matrix
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that is later evaluated using a matrix decomposition algorithm.
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Patterns are of the form
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| pattern | type of pattern |
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|-----------------+---------------------|
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| _ | wildcard |
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| x | variable |
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| c(p₁,p₂,...,pₙ) | constructor pattern |
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| (p₁\vert p₂) | or-pattern |
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The pattern /p/ matches a value /v/, written as p ≼ v, when one of the
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following rules apply
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|--------------------+---+--------------------+-------------------------------------------|
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| _ | ≼ | v | ∀v |
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| x | ≼ | v | ∀v |
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| (p₁ \vert p₂) | ≼ | v | iff p₁ ≼ v or p₂ ≼ v |
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| c(p₁, p₂, ..., pₐ) | ≼ | c(v₁, v₂, ..., vₐ) | iff (p₁, p₂, ..., pₐ) ≼ (v₁, v₂, ..., vₐ) |
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| (p₁, p₂, ..., pₐ) | ≼ | (v₁, v₂, ..., vₐ) | iff pᵢ ≼ vᵢ ∀i ∈ [1..a] |
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|--------------------+---+--------------------+-------------------------------------------|
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When a value /v/ matches pattern /p/ we say that /v/ is an /instance/ of /p/.
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During compilation by the translators, expressions at the
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right-hand-side are compiled into
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Lambda code and are referred as lambda code actions lᵢ.
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We define the /pattern matrix/ P as the matrix |m x n| where m bigger
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or equal than the number of clauses in the source program and n is
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equal to the arity of the constructor with the gratest arity.
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\begin{equation*}
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P =
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\begin{pmatrix}
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p_{1,1} & p_{1,2} & \cdots & p_{1,n} \\
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p_{2,1} & p_{2,2} & \cdots & p_{2,n} \\
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\vdots & \vdots & \ddots & \vdots \\
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p_{m,1} & p_{m,2} & \cdots & p_{m,n} )
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\end{pmatrix}
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\end{equation*}
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every row of /P/ is called a pattern vector
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$\vec{p_i}$ = (p₁, p₂, ..., pₙ); In every instance of P pattern
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vectors appear normalized on the length of the longest pattern vector.
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Considering the pattern matrix P we say that the value vector
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$\vec{v}$ = (v₁, v₂, ..., vᵢ) matches the pattern vector pᵢ in P if and only if the following two
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conditions are satisfied:
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- p_{i,1}, p_{i,2}, \cdots, p_{i,n} ≼ (v₁, v₂, ..., vᵢ)
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- ∀j < i p_{j,1}, p_{j,2}, \cdots, p_{j,n} ⋠ (v₁, v₂, ..., vᵢ)
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We can define the following three relations with respect to patterns:
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- Pattern p is less precise than pattern q, written p ≼ q, when all
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instances of q are instances of p
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- Pattern p and q are equivalent, written p ≡ q, when their instances
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are the same
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- Patterns p and q are compatible when they share a common instance
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Wit the support of two auxiliary functions
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- tail of an ordered family
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| tail((xᵢ)^{i ∈ I}) := (xᵢ)^{i ≠ min(I)}
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- first non-⊥ element of an ordered family
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| First((xᵢ)ⁱ) := ⊥ \quad \quad \quad if ∀i, xᵢ = ⊥
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| First((xᵢ)ⁱ) := x_min{i \vert{} xᵢ ≠ ⊥} \quad if ∃i, xᵢ ≠ ⊥
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we now define what it means to run a pattern row against a value
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vector of the same length, that is (pᵢ)ⁱ(vᵢ)ⁱ
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| pᵢ | vᵢ | result_{pat} |
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|--------------------------+----------------------+-------------------------------------------------|
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| ∅ | (∅) | [] |
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| (_, tail(pᵢ)ⁱ) | (vᵢ) | tail(pᵢ)ⁱ(tail(vᵢ)ⁱ) |
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| (x, tail(pᵢ)ⁱ) | (vᵢ) | σ[x↦v₀] if tail(pᵢ)ⁱ(tail(vᵢ)ⁱ) = σ |
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| (K(qⱼ)ʲ, tail(pᵢ)ⁱ) | (K(v'ⱼ)ʲ,tail(vⱼ)ʲ) | ((qⱼ)ʲ +++ tail(pᵢ)ⁱ)((v'ⱼ)ʲ +++ tail(vᵢ)ⁱ) |
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| (K(qⱼ)ʲ, tail(pᵢ)ⁱ) | (K'(v'ₗ)ˡ,tail(vⱼ)ʲ) | ⊥ if K ≠ K' |
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| (q₁\vert{}q₂, tail(pᵢ)ⁱ) | (vᵢ)ⁱ | First((q₁,tail(pᵢ)ⁱ)(vᵢ)ⁱ, (q₂,tail(pᵢ)ⁱ)(vᵢ)ⁱ) |
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A source program tₛ is a collection of pattern clauses pointing to
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/bb/ terms. Running a program tₛ against an input value vₛ, written
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tₛ(vₛ) produces a result /r/ belonging to the following grammar:
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| tₛ ::= (p → bb)^{i∈I}
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| tₛ(vₛ) → r
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| r ::= guard list * (Match bb \vert{} NoMatch \vert{} Absurd)
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\begin{comment}
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TODO: understand how to explain guard
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\end{comment}
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We can define what it means to run an input value vₛ against a source
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program tₛ:
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| tₛ(vₛ) := ⊥ \quad \quad \quad if ∀i, pᵢ(vₛ) = ⊥
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| First((xᵢ)ⁱ) := x_min{i \vert{} xᵢ ≠ ⊥} \quad if ∃i, xᵢ ≠ ⊥
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| tₛ(vₛ) = Absurd if bb_min{pᵢ → bbᵢ \vert{} pᵢ(vₛ) ≠ ⊥} = /refutation clause/
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| tₛ(vₛ) = Match bb_min{pᵢ → bbᵢ \vert{} pᵢ(vₛ) ≠ ⊥}
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[ ... ] Big part that I think doesn't need revision from you [ ... ]
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In our prototype we make use of accessors to encode stored values.
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\begin{minipage}{0.6\linewidth}
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\begin{verbatim}
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let value = 1 :: 2 :: 3 :: []
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(* that can also be written *)
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let value = []
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|> List.cons 3
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|> List.cons 2
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|> List.cons 1
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\end{verbatim}
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\end{minipage}
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\hfill
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\begin{minipage}{0.5\linewidth}
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\begin{verbatim}
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(field 0 x) = 1
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(field 0 (field 1 x)) = 2
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(field 0 (field 1 (field 1 x)) = 3
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(field 0 (field 1 (field 1 (field 1 x)) = []
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\end{verbatim}
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\end{minipage}
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An \emph{accessor} /a/ represents the
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access path to a value that can be reached by deconstructing the
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scrutinee.
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| a ::= Here \vert n.a
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The above example, in encoded form:
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\begin{verbatim}
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Here = 1
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1.Here = 2
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1.1.Here = 3
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1.1.1.Here = []
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\end{verbatim}
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In our prototype the source matrix mₛ is defined as follows
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| SMatrix mₛ := (aⱼ)^{j∈J}, ((p_{ij})^{j∈J} → bbᵢ)^{i∈I}
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Source matrices are used to build source decision trees Cₛ.
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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 \vert Switch(a, (Kᵢ → Cᵢ)^{i∈S} , C?) \vert Failure \vert Unreachable
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| vₛ ::= K(vᵢ)^{i∈I} \vert 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 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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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}〛 := 〚(Here), (pᵢ → bbᵢ)^{i∈I} 〛
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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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| 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 source matrix in the case of empty rows returns
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the first expression and (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 some auxiliary functions
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- The index family of a constructor
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| Idx(K) := [0; arity(K)[
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- head of an ordered family (we write x for any object here, value, pattern etc.)
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| head((xᵢ)^{i ∈ I}) = x_min(I)
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- tail of an ordered family
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| tail((xᵢ)^{i ∈ I}) := (xᵢ)^{i ≠ min(I)}
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- head constructor of a value or pattern
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| constr(K(xᵢ)ⁱ) = K
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| constr(_) = ⊥
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| constr(x) = ⊥
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- first non-⊥ element of an ordered family
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| First((xᵢ)ⁱ) := ⊥ \quad \quad \quad if ∀i, xᵢ = ⊥
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| First((xᵢ)ⁱ) := x_min{i \vert{} xᵢ ≠ ⊥} \quad if ∃i, xᵢ ≠ ⊥
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- definition of group decomposition:
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| let constrs((pᵢ)^{i ∈ I}) = { K \vert{} K = constr(pᵢ), i ∈ I }
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| let Groups(m) where m = ((aᵢ)ⁱ ((pᵢⱼ)ⁱ → eⱼ)ⁱʲ) =
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| \quad \quad let (Kₖ)ᵏ = constrs(pᵢ₀)ⁱ in
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| \quad \quad ( Kₖ →
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| \quad \quad \quad \quad ((a₀.ₗ)ˡ +++ tail(aᵢ)ⁱ)
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| \quad \quad \quad \quad (
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| \quad \quad \quad \quad if pₒⱼ is Kₖ(qₗ) then
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| \quad \quad \quad \quad \quad \quad (qₗ)ˡ +++ tail(pᵢⱼ)ⁱ → eⱼ
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| \quad \quad \quad \quad if pₒⱼ is _ then
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| \quad \quad \quad \quad \quad \quad (_)ˡ +++ tail(pᵢⱼ)ⁱ → eⱼ
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| \quad \quad \quad \quad else ⊥
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| \quad \quad \quad \quad )ʲ
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| \quad \quad ), (
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| \quad \quad \quad \quad tail(aᵢ)ⁱ, (tail(pᵢⱼ)ⁱ → eⱼ if p₀ⱼ is _ else ⊥)ʲ
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| \quad \quad )
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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 decomposition in 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 Kₖ, then running (v_i)^i against m is the same
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as running it against the submatrix m_{Kₖ}; otherwise (its head
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constructor ∉ (Kₖ)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
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| 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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| 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 ((aᵢ)ⁱ, ((pᵢⱼ)ⁱ → eⱼ)ʲ) with
|
||
| Groups(m) = (Kₖ → mₖ)ᵏ, m_{wild}
|
||
Below we are going to assume that m is a simplified matrix such that
|
||
the first row does not contain an or-pattern or a binding to a
|
||
variable.
|
||
|
||
Let (vᵢ)ⁱ be an input matrix with v₀ = Kᵥ(v'_{l})ˡ for some constructor Kᵥ.
|
||
We have to show that:
|
||
- if Kₖ = Kᵥ for some Kₖ ∈ constrs(p₀ⱼ)ʲ, then
|
||
m(vᵢ)ⁱ = mₖ((v'ₗ)ˡ +++ tail(vᵢ)ⁱ)
|
||
- otherwise
|
||
m(vᵢ)ⁱ = m_{wild}(tail(vᵢ)ⁱ)
|
||
Let us call (rₖⱼ) the j-th row of the submatrix mₖ, and rⱼ_{wild}
|
||
the j-th row of the wildcard submatrix m_{wild}.
|
||
|
||
Our goal contains same-behavior equalities between matrices, for
|
||
a fixed input vector (vᵢ)ⁱ. It suffices to show same-behavior
|
||
equalities between each row of the matrices for this input
|
||
vector. We show that for any j,
|
||
- if Kₖ = Kᵥ for some Kₖ ∈ constrs(p₀ⱼ)ʲ, then
|
||
| (pᵢⱼ)ⁱ(vᵢ)ⁱ = rₖⱼ((v'ₗ)ˡ +++ tail(vᵢ)ⁱ
|
||
- otherwise
|
||
| (pᵢⱼ)ⁱ(vᵢ)ⁱ = rⱼ_{wild} tail(vᵢ)ⁱ
|
||
In the first case (Kᵥ is Kₖ for some Kₖ ∈ constrs(p₀ⱼ)ʲ), we
|
||
have to prove that
|
||
| (pᵢⱼ)ⁱ(vᵢ)ⁱ = rₖⱼ((v'ₗ)ˡ +++ tail(vᵢ)ⁱ
|
||
By definition of mₖ we know that rₖⱼ is equal to
|
||
| if pₒⱼ is Kₖ(qₗ) then
|
||
| \quad (qₗ)ˡ +++ tail(pᵢⱼ)ⁱ → eⱼ
|
||
| if pₒⱼ is _ then
|
||
| \quad (_)ˡ +++ tail(pᵢⱼ)ⁱ → eⱼ
|
||
| else ⊥
|
||
\begin{comment}
|
||
Maybe better as a table?
|
||
| pₒⱼ | rₖⱼ |
|
||
|--------+---------------------------|
|
||
| Kₖ(qₗ) | (qₗ)ˡ +++ tail(pᵢⱼ)ⁱ → eⱼ |
|
||
| _ | (_)ˡ +++ tail(pᵢⱼ)ⁱ → eⱼ |
|
||
| else | ⊥ |
|
||
\end{comment}
|
||
By definition of (pᵢ)ⁱ(vᵢ)ⁱ we know that (pᵢⱼ)ⁱ(vᵢ) is equal to
|
||
| (K(qⱼ)ʲ, tail(pᵢⱼ)ⁱ) (K(v'ₗ)ˡ,tail(vᵢ)ⁱ) := ((qⱼ)ʲ +++ tail(pᵢⱼ)ⁱ)((v'ₗ)ˡ +++ tail(vᵢ)ⁱ)
|
||
| (_, tail(pᵢⱼ)ⁱ) (vᵢ) := tail(pᵢⱼ)ⁱ(tail(vᵢ)ⁱ)
|
||
| (K(qⱼ)ʲ, tail(pᵢⱼ)ⁱ) (K'(v'ₗ)ˡ,tail(vⱼ)ʲ) := ⊥ if K ≠ K'
|
||
|
||
We prove this first case by a second case analysis on p₀ⱼ.
|
||
|
||
TODO
|
||
|
||
In the second case (Kᵥ is distinct from Kₖ for all Kₖ ∈ constrs(pₒⱼ)ʲ),
|
||
we have to prove that
|
||
| (pᵢⱼ)ⁱ(vᵢ)ⁱ = rⱼ_{wild} tail(vᵢ)ⁱ
|
||
|
||
TODO
|
||
|
||
|
||
** Target translation
|
||
|
||
The target program of the following general form is parsed using a parser
|
||
generated by Menhir, a LR(1) parser generator for the OCaml programming language.
|
||
Menhir compiles LR(1) a grammar specification, in this case a subset
|
||
of the Lambda intermediate language, down to OCaml code.
|
||
This is the grammar of the target language (TODO: check menhir grammar)
|
||
| start ::= sexpr
|
||
| sexpr ::= variable \vert{} string \vert{} "(" special_form ")"
|
||
| string ::= "\"" identifier "\"" ;; string between doublequotes
|
||
| variable ::= identifier
|
||
| special_form ::= let\vert{}catch\vert{}if\vert{}switch\vert{}switch-star\vert{}field\vert{}apply\vert{}isout
|
||
| let ::= "let" assignment sexpr ;; (assignment sexpr)+ outside of pattern match code
|
||
| assignment ::= "function" variable variable+ ;; the first variable is the identifier of the function
|
||
| field ::= "field" digit variable
|
||
| apply ::= ocaml_lambda_code ;; arbitrary code
|
||
| catch ::= "catch" sexpr with sexpr
|
||
| with ::= "with" "(" label ")"
|
||
| exit ::= "exit" label
|
||
| switch-star ::= "switch*" variable case*
|
||
| switch::= "switch" variable case* "default:" sexpr
|
||
| case ::= "case" casevar ":" sexpr
|
||
| casevar ::= ("tag"\vert{}"int") integer
|
||
| if ::= "if" bexpr sexpr sexpr
|
||
| bexpr ::= "(" ("!="\vert{}"=="\vert{}">="\vert{}"<="\vert{}">"\vert{}"<") sexpr digit \vert{} field ")"
|
||
| label ::= integer
|
||
The prototype doesn't support strings.
|
||
|
||
The AST built by the parser is traversed and evaluated by the symbolic
|
||
execution engine.
|
||
Given that the target language supports jumps in the form of "catch - exit"
|
||
blocks the engine tries to evaluate the instructions inside the blocks
|
||
and stores the result of the partial evaluation into a record.
|
||
When a jump is encountered, the information at the point allows to
|
||
finalize the evaluation of the jump block.
|
||
In the environment the engine also stores bindings to values and
|
||
functions.
|
||
Integer additions and subtractions are simplified in a second pass.
|
||
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.
|
||
Our 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 can define equivalence:
|
||
| 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 in the following way:
|
||
| 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:
|
||
| 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})
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| equiv(S, Leaf bbₛ as Cₛ, Switch(a, (πᵢ → Cₜᵢ)^{i∈I})) := Forall(equiv( S∩a→π(kᵢ)), Cₛ, Cₜᵢ)^{i∈I})
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Our algorithm either returns Yes for every sub-input space Sᵢ := S∩(a→π(kᵢ)) and
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subtree Cₜᵢ
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| equiv(Sᵢ, Cₛ, Cₜᵢ) = Yes ∀i
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or we have a counter example vₛ, vₜ for which
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| vₛ≃vₜ∈Sₖ ∧ cₛ(vₛ) ≠ Cₜₖ(vₜ)
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then because
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| vₜ∈(a→πₖ) → Cₜ(vₜ) = Cₜₖ(vₜ) ,
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| vₛ≃vₜ∈S ∧ Cₛ(vₛ)≠Cₜ(vₜ)
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we can say that
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| equiv(Sᵢ, Cₛ, Cₜᵢ) = No(vₛ, vₜ) for some minimal k∈I
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4. When we have a Switch on the right we define π_{fallback} as the domain of
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values not covered but the union of the constructors kᵢ
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| π_{fallback} = ¬\bigcup\limits_{i∈I}π(kᵢ)
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Our algorithm proceeds by trimming
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| equiv(S, Switch(a, (kᵢ → Cₛᵢ)^{i∈I}, C_{sf}), Cₜ) :=
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| 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}}))
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The statement still holds and we show this by first analyzing the
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/Yes/ case:
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| Forall(equiv( S∩(a→π(kᵢ)^{i∈I}), Cₛᵢ, C_{t/a→π(kᵢ)})^{i∈I} = Yes
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The constructor k is either included in the set of constructors kᵢ:
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| k \vert k∈(kᵢ)ⁱ ∧ Cₛ(vₛ) = Cₛᵢ(vₛ)
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We also know that
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| (1) Cₛᵢ(vₛ) = C_{t/a→πᵢ}(vₜ)
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||
| (2) C_{T/a→πᵢ}(vₜ) = Cₜ(vₜ)
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(1) is true by induction and (2) is a consequence of the trimming lemma.
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Putting everything together:
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| Cₛ(vₛ) = Cₛᵢ(vₛ) = C_{T/a→πᵢ}(vₜ) = Cₜ(vₜ)
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||
|
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When the k∉(kᵢ)ⁱ [TODO]
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||
|
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The auxiliary Forall function returns /No(vₛ, vₜ)/ when, for a minimum k,
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| equiv(Sₖ, Cₛₖ, C_{T/a→πₖ} = No(vₛ, vₜ)
|
||
Then we can say that
|
||
| Cₛₖ(vₛ) ≠ C_{t/a→πₖ}(vₜ)
|
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that is enough for proving that
|
||
| Cₛₖ(vₛ) ≠ (C_{t/a→πₖ}(vₜ) = Cₜ(vₜ))
|