Teacher: Michele Pagani

Reminder:

Consistency ⟺ $⊬0$, and not $⊬\perp$

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

On top of that, you can prove any $\Gamma$:

But not with $⊬\perp$ (and in fact there are theories with $mi{x}_0$ where $⊢\perp$ is provable)

How to show that $\Gamma ⊬\Delta$: cut-elimination theorem ⟹ subformula property, and then proof search to show that no rule can be applied to prove $\Gamma ⊢\Delta$

MLL Proof nets

Semi-distributivity of tensor over parr

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 $\mathrm{\&}$) 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 $\pi ⊢\Gamma$, then there exists a cut-free proof net ${\pi }^{\prime }⊢\Gamma$

cf. picture

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

For ax-redexes: note that we’re only considering proof nets, so there’s no cycle (therefore $\alpha \ne {\alpha }^{\prime }$)

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 ($\lambda$-terms are over spaces) (Olivier Laurent)) cf. picture

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

Reminder on rewriting systems

$(A,\underset{\subseteq A^2}{\underbrace{⟶}})$

Weak normalization:

$\forall a\in A,\exists k\in \mathbb{N},\exists \{a_i{\}}_{i\le k}\subseteq A$ st

• $a_0=a$
• $a_k$ is a normal form: $\forall a^{\prime },a_k⟶̸a^{\prime }$
• $\forall i<k,a_i⟶a_{i+1}$

Strong normalization:

$\forall a\in A,\nexists \{a_i{\}}_{i\in \mathbb{N}}\subseteq A$ st $a_0=a$ and $\forall i,a_i⟶a_{i+1}$

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

Ex:

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 $\mu :PS⟶\mathbb{N}$ to be the number of nodes in the proof net, we can show that:

which will yield the result, as $<$ in $\mathbb{N}$ 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 $\otimes$/$⅋$-redex, and it wouldn’t be a normal form).

Warning: we have to prove that

Lemma: Given a reduction step $\pi ⟶{\pi }^{\prime }$, if $\pi$ is a proof net, then ${\pi }^{\prime }$ 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 $\otimes$/$⅋$-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)