Lecture 7: Static translation

Teacher: Delia Kesner

cf picture

Theorem:

  1. If you have $t =_c t’$, then $T(t)$ (which means: the translation of a type derivation of $t$) $≃_A T(t’)$
  2. If $t ⟶_{App3, Lamb} t'$, then $T(t) ≃_{A,B} T(t')$
  3. If $t ⟶_{λs\backslash \lbrace App3, Lamb\rbrace} t'$, $T(t) ⟶^+_{R/E} W[T(t')]$

Notations:

  • $R$: all the reduction rules
  • $E$: all the equations for Proof Nets
  • $R/E$: “$R$ modulo $E$”

Observation: $⟶_{App3} ∪ ⟶_{Lamb}$: is (strongly) terminating

NB: $⟶_B$ and $⟶_S$ are terminating, but what about the union?

Corollary: From a theorem coming from abstract rewriting theory, we can conclude that the relation $⟶_{λs}$ is strongly normalizing by using the fact that $⟶_{R/E}$ is strongly normalizing.

NB: we are showing normalization of $λs$ by using normalization of proof nets.

\[t[x/u][y/v] =_c t[y/v][x/u] \qquad x ∉ fv(v) ∧ y ∉ fv(u)\]

Reminder: recall that there are two kinds of “boxes” in proof nets: the substitution boxes and the application ones.

\[\infer{⊢ t[x/u][y/v]}{ ⊢ v & \infer{⊢ t[x/u]:A}{ ⊢ u & ⊢ t} }\]

  • Fist (easy) case: $x, y ∈ fv(t)$

    \[\infer{⊢ t[x/u][y/v]:A}{ ⊢ v:C & \infer{⊢ t[x/u]:A}{ ⊢ u:B & y: C, x:B ⊢ t:A} }\]

    Notation: $Γ_{t’, t’’}$: variables of $t’$ and $t’’$

    cf picture : write the complete derivation tree

  • $(λx. u)[y/v] ⟶_{Lamb} λx.u[y/v]$

    First case: $x ∈ fv(u), y ∈ fv(u)$

    cf picture

  • Other cases: cf. pictures

Inductive Cases:

For example:

  • \[t = uv ⟶ u'v = t' \text{ where } u ⟶ u'\]

    Trivial case, since the weakenings arising from $u ⟶ u’$ are outside: they are also weakenings for the whole proof net.

  • $t= u[x/v] ⟶ u[x/v’] = t$ where $v ⟶ v’$

    IH: \(T(v) ⟶^+ W[T(v')]\)

    To prove:

    \[T(u[x/v]) ⟶^+ W[T[u[x/v']]]\]
  • $U$ rule: used for \((zxy)[x'/y] ⟶ zxy\)

Reminder:

$λ$-calculus ⟶ $λs$-terms ⟶ Pnets

But what about going the other way round? ⟹ $λlxr$-calculus

$λlxr$-calculus

Specificity:

  • Weakening: $W_x(y)$ allowed, but not $W_x(x)$

  • Contraction: $C_x^{y,z}(t)$, where $x$ is free, $y,z$ are bound:

    • $C_x^{yz}(x_1 y z)$ allowed
    • $C_x^{yz}(x_1 x_1 x_2)$ not allowed
\[fv(W_x(t)) = fv(t) ∪ \lbrace x\rbrace\\ fv(C_x^{yz}(t)) = (fv(t) \backslash \lbrace y, z\rbrace) ∪ \lbrace x \rbrace\]

Complusory presence

For all the binders, the variables need to be there to be bound

\[λx.y \leadsto λx. W_x(y)\\ x[x'/y] \leadsto W_{x'}(x)[x'/y]\]

Linearity

A term is linear:

iff each variable occurs at most once

$λx.xx$ not allowed:

\[λx.xx \leadsto λx. C_x^{x_1, x_2}(x_1 x_2)\]

NB: In proof nets, it’s the same: to bound/contract a variable, you need wires to be there in the first place!

Recap examples:

\[λx.y \quad \leadsto \quad λx. W_x(y)\\ x[z/w] \quad \leadsto \quad W_z(x) [z/w]\\ C_x^{y,z}(wy) \quad \leadsto \quad C_x^{y,z}(W_z(wy))\\\]

Barendregt convention: $λx. x[x/z]$ is ambiguous ⟶ rename

Typing rules

Weakening and contraction are now built in: there is an explicit notation to denote them

\[\cfrac{Γ ⊢ t:A}{Γ, x:B ⊢ W_x(t):A}\]

and

\[\cfrac{Γ, x:A, y:A ⊢ t:B}{Γ, z:A ⊢ C_z^{x,y}(t):B}\]

Linearity: $Γ, Δ$ is well-defined only if $Γ$ and $Δ$ do not share variables.

No contractions anymore in the proof nets, as there are no shared variables in the contexts.

Equations

Contrary to proof nets, where many thing happen in parallel, now we need to sequentialize.

Rule $A$ for proof nets becomes:

\[C_w^{x,v}(C^{z,y}_x(t)) ≡ C_w^{x,y}(C_x^{z,v}(t)) \qquad \text{ if } x≠y, v\]

And equality of proof nets is made algebraic:

The fact that wires commute:

\[C_x^{y,z}(t) ≡ C_x^{z,y}(t)\]

Concurrency of parallel contractions:

\[C_{x'}^{y',z'}(C^{y,z}_x(t)) ≡ C^{y,z}_x(C_{x'}^{y',z'}(t)) \qquad \text{ if } x≠y', z' \quad ∧ \quad x' ≠ y, z\]

etc…

Commutativity of substitution:

\[t[x/u][y/v] ≡ t[y/v][x/u] \qquad x ∉ fv(v) ∧ y ∉ fv(u)\]

etc…

SubSystem $x$

Rewriting rules to deal with explicit substitutions.

As terms are linear, we only have two $App$ rules now.

Example:

\[W_x(y_1 y_2)[x/z_1 z_2] ⟶ W_{z_2}(W_{z_1}(y_1 y_2)) \qquad \text{(the order of the } z_i \text{'s don't matter}\\\]

NB: in these rules,

  • all the substitutions go down
  • all the weakenings go up
  • all the contractions go down

In $Cont1$: $Δ$ and $\Pi$ are fresh variables. $u_1 = R^Φ_Δ(u)$ where $Φ = fv(u)$ is a meta-operation meaning: “make a copy of $u$, where the free variables of $u$ have been renamed as $Δ$”

Notations: $x$: system pushing down substitutions, $t$: system handling weakening and contractions

Only rule “erasing” variables: $Weak1$, but free variables of $u$ are actually not discarded ⟹ free variables are preserved (weakening keep track of “erased” variables)

$xt = x ∪ t$ is convergent (terminating and confluent) ⟶ it defines a function: a term in normal form is a term where all the weakenings are up and all the contractions are down.

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