Lecture 6: MELL Proof nets

Teacher: Delia Kesner

Translation of $\lambda s$-calculus to proof nets: negation ${\bullet }^{\perp }$ is crucial, because it appears in

• atoms
• cuts

The operation $A^{\perp }$ (involutive: $A^{\perp \perp }=A$)

Grammar of formulas and negation operation

MELL:

$$A,B\stackrel{\scriptscriptstyle\mathrm{def}}{=}p\mid p^{\perp }\mid A⅋B\mid A\otimes B\mid ?A\mid !A$$

Base formulas: $\{p,\underset{\text{also written }\underset{―}{p}}{\underbrace{p^{\perp }}}\}$

Definition of the negation operation (defined symbol):

NB:

• in negative base formulas (of the form $p^{\perp }$), negation is a constructor
• whereas the negation operation defined above is a defined symbol (operation defined outside)

Proof nets

$\frac{}{⊢A,A^{\perp }}ax$ $⇝$

$\frac{⊢\Gamma ,A,B}{⊢\Gamma ,A⅋B}⅋$ $⇝$

etc… cf picture

Cut-elimination rules

cf. pictures

Confluent, Strongly Normalizing

Theorem: Cut-elimination for MELL Proof-nets is confluent and strongly normalizing

Alternative syntax for proof-nets: using $n$-ary contractions (as contraction is associative and commutative)

Why are $A,B$ not rewriting rules?

• Because in $\lambda s$-calculus, we will need to use in them in both directions

• other possible syntax for proof-nets: contractions are at the boundary of the boxes (neither inside nor outside, no need to worry about that) ⟶ but not the original Girard’s proof-nets

• $V$ goes in one way only: you can’t put the weakening wire inside a box, because this wire may be connected down the way to a wire of the box

Examples: cf. picture

Untyped proof-nets: proof-nets without formula labelling ⟶ can simulate untyped $\lambda$-calculus (and other calculi)

The reduction system constituting of cut-elimination+$U$+$V$ modulo $A,B$ is

• confluent

  $$M{\sim }_{A,B}{M}^{\prime }\ast \ast M_0\ast M_0^{\prime }\ast M_1{\sim }_{A,B}{M}_1^{\prime }$$

• (strongly) normalizing: there is no $\infty$ reduction sequence starting at a typed proof-net where reduction means MELL proof-nets+U,V modulo equations A,B.

Translation of typed $\lambda s$-calculus into MELL PNets

Translation of types: Call-by-name translation

Translating typed $\lambda s$-terms

cf pictures

Example:

Warning: Proof-nets of normal terms are not necessarily in normal form.

Only, but big, advantage of proof-nets: there’s no artificial order between things that shouldn’t be ordered (parallel cuts, whereas in proof trees in SC: everything ordered). LL is independent from that: you can use other logics as well.

Proof nets can be seen as an “operational” semantics, more “concrete” than denotational semantics.