Lecture 4: Cutelimination for MLL
Teacher: Michele Pagani
Reminder:
Consistency ⟺ $\not ⊢ 0$, and not $\not ⊢ ⊥$
If you have a proof $⊢ 0$, then the cutelimination 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 $Γ \not ⊢ Δ$: cutelimination theorem ⟹ subformula property, and then proof search to show that no rule can be applied to prove $Γ ⊢ Δ$
MLL Proof nets
Semidistributivity 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. Cutelimination = 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
Cutelimination for MLL
Th: If there exists a MLL proof nets $π ⊢ Γ$, then there exists a cutfree proof net $π’ ⊢ Γ$
cf. picture
Cutelimination rewriting: reduction of axredexes and $⊗$/$⅋$redexes.
For axredexes: 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 cutfree proof obtained is independent of the rewriting path used

if you associate a permutation to each proof net, cutelimination 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 axcut, or cutfree 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 wellfounded.
Ex:
digraph {
rankdir=LR;
a > b > c;
b > a;
}
is weakly normalizing but not strongly normalizing.
Lemma: the cutelimination 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 wellfounded. Indeed, as long as the reduction preserve proof nets, the normal form has to be a proof net (so not axcut cycle), thus it is a cutfree proof (otherwise we would have an axredex 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 (ChurchRosser property): the cutelimination rewriting is confluent.
Because if you have two cuts:
 for $⊗$/$⅋$redexes: reducing one cut doesn’t change/touch the other one
 pay attention to axredexes that overlap: cf. picture (the final result yields the same graph)
Leave a comment