Type Systems for Programming Languages, Fall 2008

Lecture 26: A Taste Of Linear Logic

December 3 - Final lecture

We've been taking advantage at every point of this class as a notion of hypothetical judgment that obeys certain structural properties:

WEAKENING:
If I can prove
J₁, ... J_n |- J,
then I can prove
J₁, ... J_n, J' |- J,

CONTRACTION:
If I can prove
J₁, ... J_n, J', J' |- J,
then I can prove
J₁, ... J_n, J' |- J,

EXCHANGE:
If I can prove
J₁, ... J_m, J', J'', J_m+1, ... J_n |- J,
then I can prove
J₁, ... J_m, J'', J', J_m+1, ... J_n |- J,

Turns out, there are a lot of substructural logics that don't obey these properties:

Linear logic - has only exchange
Affine logic - has only exchange and weakening (no contraction)
Relevant logic - has only exchange and contraction (no weakening)
Ordered logic - has none of these!

We will focus on linear logic today.

Remember that in the logic of the class, it is important that all of the hypothesis in the conclusion of of a rule go to each of the rule, which led to some really wide derivations:

      let Γ = A -> C proved, A + C proved
      let Γ' = A -> C proved, A + C proved, A proved

                             ------------------- hyp  -------------- hyp
                             Γ' |- A -> C proved      Γ' |- A proved  
      ----------------- hyp  --------------------------------------- imp-i ----------------------- hyp
      &Gamma |- A + C proved      Γ, A proved |- C proved                       Γ, C proved |- C proved
      -------------------------------------------------------------------------------------------- or-e
      Γ |- C proved

The logic of this lecture doesn't necessarly allow all the conclusions to flow into the premises! First, the hypothesis rule is now exact:

      --------------------- hyp (NEW, linear logic)
      A proved ||- A proved

Instead of allowing multiple copies

      ---------------------------------------------- hyp (OLD)
      A proved, B proved, C proved, .... |- A proved

Rules

Linear implication

      A₁ pr, ..., A_n pr, A pr ||- B pr
      ------------------------------------------------- -oI
      A₁ pr, ..., A_n pr ||- A -o B pr

      A₁ pr, ..., A_n pr ||- A -o B pr
      B₁ pr, ..., B_m pr ||- A pr 
      ------------------------------------------------- -oE
      A₁ pr, ..., A_n pr, B₁ pr, ..., B_m pr ||- B pr

Simultaneous conjunection

      A₁ pr, ..., A_n pr ||- A pr
      B₁ pr, ..., B_m pr ||- B pr 
      ------------------------------------------------- ⊗I
      A₁ pr, ..., A_n pr, B₁ pr, ..., B_m pr ||- A ⊗ B pr

      A₁ pr, ..., A_n pr ||- A ⊗ B pr
      A₁ pr, ..., A_n pr, A pr, B pr ||- C pr
      ------------------------------------------------- ⊗E
      A₁ pr, ..., A_n pr ||- C pr

Alternative conjuction (you decide)

      A₁ pr, ..., A_n pr ||- A 
      A₁ pr, ..., A_n pr ||- B
      ------------------------------------------------- &I
      A₁ pr, ..., A_n pr ||- A & B pr

      A₁ pr, ..., A_n pr ||- A & B pr
      ------------------------------------------------- &E1
      A₁ pr, ..., A_n pr ||- A pr

      A₁ pr, ..., A_n pr ||- A & B pr
      ------------------------------------------------- &E2
      A₁ pr, ..., A_n pr ||- B pr

Disjunction (we report)

      A₁ pr, ..., A_n pr ||- A pr
      ------------------------------------------------- ⊕I1
      A₁ pr, ..., A_n pr ||- A ⊕ B pr

      A₁ pr, ..., A_n pr ||- B pr
      ------------------------------------------------- ⊕I2
      A₁ pr, ..., A_n pr ||- A ⊕ B pr

      A₁ pr, ..., A_n pr ||- A ⊕ B pr 
      B₁ pr, ..., B_m pr, A pr ||- C pr 
      B₁ pr, ..., B_m pr, B pr ||- C pr 
      ------------------------------------------------- ⊕E
      A₁ pr, ..., A_n pr, B₁ pr, ..., B_m pr ||- C pr

Impossibility

Examples

Assumed:

$1 -o cake
$1 -o pizza
cake -o full
pizza -o full

For one dollar, you can have your cake, you can eat your cake, but you can't have your cake and eat it too (additive disjuction)

                                                                -------------- rule ---------- hyp
                                                                ||- $1 -o cake      $1 ||- $1
      -------------- rule ---------- hyp  ---------------- rule ----------------------------- -oE
      ||- $1 -o cake      $1 ||- $1       ||- cake -o full      $1 ||- cake 
      ----------------------------- -oE   --------------------------------------- -oE
      $1 ||- cake                         $1 ||- full 
      -------------------------------------------------- &I
      $1 ||- cake & full
      ----------------------- -oI
      ||- $1 -o (cake & full)

However, it takes two dollars to have my cake and eat it too:

                                                                 -------------- rule ---------- hyp
                                                                 ||- $1 -o cake      $1 ||- $1
      -------------- rule ---------- hyp   ---------------- rule ----------------------------- -oE
      ||- $1 -o cake pr     $1 ||- $1      ||- cake -o full      $1 ||- cake       
      ----------------------------- -oE    ----------------------------- -oE    
      $1 ||- cake pr                       $1 ||- full pr
      --------------------------------------------------- ⊗I
      $1, $1 ||- (cake ⊗ full) pr
      -------------------------------- -oI
      $1 ||- $1 -o (cake ⊗ full) pr
      -------------------------------- -oI
      ||- $1 -o $1 -o (cake ⊗ full) pr

French Menu

Following Frank Pfenning's 2001 notes:

For 20 Euros	`20€ -o`
Onion Soup or Clear Broth	`((OS ⊕ CB) ⊗`
Honey-Glazed Duck	`⊗ HGD`
Peas or Red Cabbage (according to season)	`⊗ (P ⊕ RC)`
New Potatoes	`⊗ NP`
Chocolate Mousse (3 Euros extra)	`⊗ ((3€ -o CM) & 1)`
Coffee (if you want)	`⊗ (C & 1))`

$LastChangedDate: 2008-12-06 19:54:28 -0500 (Sat, 06 Dec 2008) $
$Author: rjsimmon $
$Rev: 1074 $