[PPT] - No-counterexample Interpretations of Logic and the Geometry of PowerPoint Presentation

SLIDE 1

No-counterexample Interpretations of Logic and the Geometry of Interaction

Masaru Shirahata Keio University, Hiyoshi Campus

SLIDE 2

The Plan of the Talk

I will first explain the motivation.
Then I will mostly explain the no-

counterexample interpretation (NCI) according to Tait’s work.

Finally I will add a small observation of

mine and present NCI in a trace-like graphical representation.

SLIDE 3

Introduction

The functional interpretations of logic

have a flavor of game.

The values for existential quantiers are

positive and those for universal quantifiers are negative.

In the negation the positive and the

negative change the roles.

I want to relate them to GoI.

SLIDE 4

GoI and Cut-elimination

GoI is supposed to model the dynamics
f cut-elimination.
For the consistency proof the cut-

elimination of propositional logic is not so interesting….

All techniques of the consistency proof

is to handle the alternating quantifiers.

SLIDE 5

The Consistency Proof of PA

The epsilon substitution method by

Hilbert and Ackermann.

The Cut-elimination method by Gentzen
The Dialectica interpretation by Goedel
No-counterexample interpretation by

Kreisel.

SLIDE 6

The Pre-history

Gentzen’s first version of the

consistency proof is in terms of “reduction”.

Goedel described Gentzen’s idea in his

Zilsel lecture, essentially as a no- counter example interpretation.

It can be stated in terms of game,

recently revived by Coquand

SLIDE 7

Our Convention

Consider the sentences in a classical

first-order logic.

Quantified sentences are regarded as

infinitary disjunctions and conjunctions.

Negations are pushed inside by the De

Morgan duality.

In the games, the player’s moves are

blue and the opponents’ are red.

SLIDE 8

The Henkin Hintikka Game

Start with a sentence.
The Player and the Opponent form a

new sentence from the sentence in the previous stage.

Ends with an atomic sentence.
The Player wins if the atomic sentence

is true. The opponent wins otherwise.

SLIDE 9

A

⇑

k

φ ∑

j

φ j φ

k

φ ∏

p

Done

The Moves in the Henkin Hintikka Game

SLIDE 10

The Gentzen Game

Start with a list of sentences in the

prenex normal form.

The Player and the Opponent form a

new list of sentences from the list in the previous stage.

The player wins if the list contains a true

j

φ , (

k

φ ∑

), .....

p , .....

1

p ,

2

p , .....

The Moves in the Gentzen Game

SLIDE 12

Some Restriction

The Player does not repeat the same

instantiation, in other words, always chooses a different disjunct from the given disjunction.

We regard quantifier free sentences as

prime (atomic).

This restriction is not crucial with

respect to the expressive power.

SLIDE 13

The counter-strategy as a function

The counter-strategy (trying to falsify) of

the Opponent may be seen as a function of the previous moves of the Player.

The re-instantiation of the existential

sentence may be seen as “the change

f mind”.

SLIDE 14

The counter-strategy as a tree

1 2 1

1

p

..... 1 1 .....

1

′

1

′ p

1

2

p

..... ..... .....

2

.....

..... .....

3

p .....

SLIDE 15

The winning strategy as a path finder

1 2 1

1

p

..... 1 1 .....

1

′

1

′ p

1

2

p

..... ..... .....

2

.....

..... .....

3

p .....

True!

SLIDE 16

The no-counterexample Interpretation (NCI)

The universally quantified (negative)

variables are replaced by the functions

f the preceding existentially quantified

(positive) variables.

For a provable sentence one can find

the functionals of those negative functions, yielding the witnesses for the positive variables.

SLIDE 17

A Brief Histroy of NCI

NCI was introduced by G. Kreisel,using

Herbrand’s theorem for FOL and the epsilon substitution for PA.

The direct proofs are given by

Kohlenbach and Tait.

Tait’s work dates back to early 1960’s,

which had been unpublished since then.

SLIDE 18

      ,

2

F f ,

2

f

1

F f

     

2

F f

            where

f ≡

1

f

2

f

⇓ ⇓

Consider a counter-strategy. The winning strategy finds a path.

SLIDE 19

Modus Ponens

∃

1

x ∀

1

u ∃

2

x ∀

2

u A

1

x ,

1

u ,

2

x ,

2

u

( )

∀

1

x ∃

1

u ∀

2

x ∃

2

u ∃ v ∀ y ¬ A

1

x ,

1

u ,

2

x ,

2

u

( ) ∨ B v , y

           

∀

1

x ∃

1

u ∀

2

x ∃

2

u ¬ A

1

x ,

1

u ,

2

x ,

2

u

( ) ∨ ∃ v ∀ y B v , y

     

⇓

∃ v ∀ y B v , y

     

SLIDE 20

Modus Ponens in NCI

A

1

F f ,

1

f

1

F f

      ,

2

F f ,

2

f

1

F f

     

2

F f

           

¬ A

1

g,

1

G g ,

2

g

1

G g

      ,

2

G g

      ∨ B H g ,

3

g

1

G g

     

2

G g

      H g            

⇓

B J h , h J h

( )

¬ A

1

g ,

1

G g ,

2

g

1

G g

      ,

2

G g

      ∨ B H g ,

3

g H g

           

B H

1

g

2

g h , h H

1

g

2

g h

( )

with

1

g ,

2

g such that A

1

g ,

1

G

1

g

2

g h ,

2

g

1

G

1

g

2

g h

( ),

2

G

1

g

2

g h

( )

SLIDE 21

Finding the Counter Strategies

A

1

F

1

f

2

f ,

1

f

1

F

1

f

2

f

( ),

2

F

1

f

2

f ,

2

f

1

F

1

f

2

f

( )

2

F

1

f

2

f

( )

     

A

1

g ,

1

G

1

g

2

g h ,

2

g

1

G

1

g

2

g h

( ),

2

G

1

g

2

g h

     

1

ˆ g =

1

F

1

f

2

f

1

ˆ f

1

F

1

ˆ f

2

f

      =

1

G

1

ˆ g

2

g h

2

ˆ g

1

G

1

ˆ g

2

ˆ g h

( ) =

2

F

1

ˆ f

2

f

2

ˆ f

1

F

1

ˆ f

2

ˆ f

     

2

F

1

ˆ f

2

ˆ f

      =

2

G

1

ˆ g

2

ˆ g h

SLIDE 22

The General Pattern

Find the solution h for h L h

( )

( ) = K h

( )

u x h K h

( )

L h

( )

h L h

( )

Positive variable Preceding negative variable Solved with h

SLIDE 23

The Approximation

h m

( ) =0

n+1

h m

( )= K

n

h

( ) if m = L

n

h

( )

n

h m

( ) otherwise

    

n

h

n+1

h

L

n

h

( ), K

n

h

( )

update

SLIDE 24

The System of α-recursive Functionals

Tait introduced the system of recursively

definable functionals, allowing the recursion along a primitive recursively definable well-founded partial order α.

One can keep track of how much of the

initial segment of input functions is necessary to compute the value of the functional, along α.

SLIDE 25

The Solution in the System of α-recursive Functionals

Let ˆ h be

n+1

h such that L

n+1

h

( )= L

n

h

( )

We have L h

( )= L

′ h

( )⇒ K h ( )= K

′ h

( )

Hence

n+1

h L

n+1

h

( )

( )=

n+1

h L

n

h

( )

( )= K

n

h

( )= K

n+1

h

( )

SLIDE 26

The Solution as the Fixpoint of the Update Operator

Assume L

n+1

h

( )= L

n

h

( )

For m = L

n+1

h

( )

n+2

h L

n+1

h

( )

( ) = K

n+1

h

( ) = K

n

h

( ) =

n+1

h L

n

h

( )

Otherwise

n+2

h m

( ) =

n+1

h m

( )

n+1

h L

n+1

h

( )

( )=

n+2

h L

n+1

h

( )

( )= K

n+1

h

( )

SLIDE 27

The Solution is the Fixpoint of the Update Operator

Assume h = U h

( )

Then h L h

( )

( ) =U h

( ) L h ( )

( ) = K h

( )

Assume h L h

( )

( )= K h

( )

For m = L h

( )

U h

( ) m ( ) = K h ( )= h m ( )

Otherwise U h

( ) m ( )= h m ( )

SLIDE 28

The Similarity between NCI and GoI

The morphisms in GoI and the

interpretations in NCI are functions from “negatives” to “positives”.

The composition in GoI and Modus

Ponens of NCI are formed by taking trace and fixpoint, connecting the corresponding negatives and positives.

The simple duality is lost in NCI.

SLIDE 29

Trace-like Operation

Given

i

Φ ,

j

F with

i

Φ ,

j

F :

1

X ,.....

l

X ,.....

1

X →

l

X Take the fixpoint

f

k

Φ and substitute it in

i

Φ ,

j

F

l

X

l

X

k

Φ

SLIDE 30

The Counter Strategies in Cyclic Graphs

A

1

F

1

f

2

f ,

1

f

1

F

1

f

2

f

( ),

2

F

1

f

2

f ,

2

f

1

F

1

f

2

f

( )

2

F

1

f

2

f

( )

     

A

1

x ,

1

u ,

2

x ,

2

u

      ∨ B

v , y

     

A

1

g ,

1

G

1

g

2

g h ,

2

g

1

G

1

g

2

g h

( ),

2

G

1

g

2

g h

     

1

x

1

x y

1

u

1

u

2

u

2

u

2

x v

2

x

1

F

2

F

1

G

2

G

3

G

SLIDE 31

Stage 1

1

ˆ g =

1

F

1

f

2

f

1

x

1

x y

1

u

1

u

2

u

2

u

2

x v

2

x

1

F

2

F

1

G

2

G

3

G

1

F

1

x

SLIDE 32

Stage 2

1

ˆ f

1

F

1

ˆ f

2

f

      =

1

G

1

ˆ g

2

g h

1

x

1

x y

1

u

1

u

2

u

2

u

2

x v

2

x

1

F

2

F

1

G

2

G

3

G

1

F

1

x

1

G

1

u

1

x

1

u

U

Pr

1

u

1

F

SLIDE 33

Stage 3

2

ˆ g

1

G

1

ˆ g

2

ˆ g h

( ) =

2

F

1

ˆ f

2

f

1

x

1

x y

1

u

1

u

2

u

2

u

2

x v

2

x

1

F

2

F

1

G

2

G

3

G

1

F

1

x

1

G

1

u

1

x

1

u

U

Pr

1

u

1

G

1

u

1

F Pr

2

x

2

F

2

x

U

2

x

SLIDE 34

Stage 4

1

x

1

x y

1

u

1

u

2

u

2

u

2

x v

2

x

1

F

2

F

1

G

2

G

3

G

1

F

1

x

1

G

1

u

1

x

1

u

U

Pr

1

u

1

G

1

u

1

F Pr

2

x

2

F

2

x

U

2

x

2

ˆ f

1

F

1

ˆ f

2

ˆ f

     

2

F

1

ˆ f

2

ˆ f

      =

2

G

1

ˆ g

2

ˆ g h

2

G

2

u

2

F

1

F Pr

U

2

x

1

x

2

u

2

u

SLIDE 35

The General Pattern

K

1

u x

n

u x x L Pr

U

x

SLIDE 36

The Categorical NCI?

Adding the propositional structure is just

straightforward.

The “trace” here is partial. We need to

find a suitable framework.

Study the formal properties of our

“trace”.

SLIDE 37

The Dialectica Interpretation and NCI

In Dialectica we have the duality at the

cost of higher-types.

In Dialectica the cut is composition

while in NCI the cut is taking the “trace”.

NCI is a germ of Dialectica?

SLIDE 38

Conclusion

We have seen the similarity between

NCI and GoI.

The predicate logic is quite relevant.
The unified framework?