Lecture 4: Cut-elimination for MLL

Teacher: Michele Pagani

Reminder:

Consistency ⟺ $\not ⊢ 0$, and not $\not ⊢ ⊥$

If you have a proof $⊢ 0$, then the cut-elimination equational theory collapses.

On top of that, you can prove any $Γ$:

\[\infer{⊢ Γ}{ ⊢ 0 & ⊢ ⊤, Γ}\]

But not with $\not⊢ ⊥$ (and in fact there are theories with $mix_0$ where $⊢ ⊥$ is provable)

How to show that $Γ \not ⊢ Δ$: cut-elimination theorem ⟹ subformula property, and then proof search to show that no rule can be applied to prove $Γ ⊢ Δ$

MLL Proof nets

Semi-distributivity of tensor over parr

\[A ⊗ (B ⅋ C) ⊢ (A ⊗ B) ⅋ C\]

NB: we don’t have the converse

cf. picture

Now, you have to check if this proof structure is a proof net:

  • 4 possible switching graph (erase first left, second left / first left, second right / etc…)

  • in each case: no cycle + connected graph ⟹ we do have a proof net

You have several possible permutation (here: only the identity) to determine axioms. Cut-elimination = composition of permutations

NB: with additives or exponentials: much more challenging, as you can introduce erase some fomulas (ex: as with $\&$) or duplicate other (ex: contraction duplicates $!A$)

Link:

a node, and all its premises and conclusions

cf. picture

NB: the correctness criterion for units is in NPTIME, that’s why we don’t consider units

Cut-elimination for MLL

Th: If there exists a MLL proof nets $π ⊢ Γ$, then there exists a cut-free proof net $π’ ⊢ Γ$

cf. picture

Cut-elimination rewriting: reduction of ax-redexes and $⊗$/$⅋$-redexes.

For ax-redexes: note that we’re only considering proof nets, so there’s no cycle (therefore $α ≠ α’$)

cf. picture

Proof net rewriting preserve interfaces (input/output formulas).

Example: cf. picture

NB:

  • this rewriting is confluent, so the cut-free proof obtained is independent of the rewriting path used

  • if you associate a permutation to each proof net, cut-elimination corresponds to composition of permutations (correctness corresponds to properties of the permutations ⟶ geometry of interaction ($λ$-terms are over spaces) (Olivier Laurent)) cf. picture

  • normal forms: cycle ax-cut, or cut-free proof nets

Reminder on rewriting systems

$(A, \underbrace{⟶}_{⊆ A^2})$

Weak normalization:

$∀a ∈ A, ∃ k ∈ ℕ, ∃ \lbrace a_i \rbrace_{i ≤ k} ⊆ A$ st

  • $a_0 = a$
  • $a_k$ is a normal form: $∀ a’, a_k \not⟶ a’$
  • $∀i < k, a_i ⟶ a_{i+1}$
Strong normalization:

$∀a ∈ A, \not ∃ \lbrace a_i \rbrace_{i ∈ ℕ} ⊆ A$ st $a_0 = a$ and $∀i, a_i ⟶ a_{i+1}$

NB: this amounts to show that the order induced by $⟶$ is well-founded.

Ex:

  digraph {
    rankdir=LR;
    a -> b -> c;
    b -> a;
  }

is weakly normalizing but not strongly normalizing.

Lemma: the cut-elimination rewriting over MLL proof structure is strongly normalizing

NB:

  • the problem is open for MELL
  • weakly normalizing for MELL: way harder

Idea: By setting $μ: PS ⟶ ℕ$ to be the number of nodes in the proof net, we can show that:

\[π ⟶ π' ⟹ μ(π) > μ(π')\]

which will yield the result, as $<$ in $ℕ$ is well-founded. Indeed, as long as the reduction preserve proof nets, the normal form has to be a proof net (so not ax-cut cycle), thus it is a cut-free proof (otherwise we would have an ax-redex or a $⊗$/$⅋$-redex, and it wouldn’t be a normal form).

Warning: we have to prove that

Lemma: Given a reduction step $π ⟶ π’$, if $π$ is a proof net, then $π’$ is a proof net as well.

Sketch: show that if there is a switching cycle with the contractum, there is one as well in the proof net with the redex. And then use the following lemma:

Lemma: if $G$ is an acyclic undirected graph, the the number of connected components if $G$ equals $nodes(G)-edges(G)$

Confluence (Church-Rosser property): the cut-elimination rewriting is confluent.

Because if you have two cuts:

  • for $⊗$/$⅋$-redexes: reducing one cut doesn’t change/touch the other one
  • pay attention to ax-redexes that overlap: cf. picture (the final result yields the same graph)

Leave a comment