REDORedesigning Logical Syntax

Action de Recherche Collaborative 2009INRIA Saclay - Île-de-France

Lutz StraßburgerDale Miller

Ivan GazeauNicolas Guenot

Anne-Laure PouponFrançois Wirion

University of Bath

Alessio GuglielmiGuy McCusker

Jim LairdTom Gundersen

Ana Carolina Martins AbbudMartin Churchill

INRIA Nancy - Grand Est

François LamarcheAlessio Guglielmi

Paola BruscoliNovak Novakovic

Yves Guiraud

This project is a grouping of three teams (2 in France, 1 in the UK) through their common belief in the need for a new way of looking at syntax in proof theory. On one side, syntax is a blessing, because it is the handle that algorithms can work on. On the other side it is a curse because it comes with bureaucracy that disguises the essence of proofs and very often causes unnecessary exponential blow-up in the complexity of proof search. We intend to tackle the problem of bureaucracy by using approaches based on proof nets, which are intrinsically bureaucracy-free; on deep inference, which allows to design deductive systems with reduced bureaucracy; on focussing, which tells us how to reduce the search space during proof search; and on games semantics, which provides new computational models for proof search. The close relation between these fields has been revealed only within the last few years, and we plan to further unify them.

From Sequent Calculus to Proof Nets

ida (a a) , (a a ) a

ida , a

a (a a) , (( a a ) a) a , aexch

a (a a) , a, (( a a ) a) a

ida , a

ida , a

ida, a

ida , a

a, a a , aexch

a, a, a a

a , a a, a, a acut

a , a a, a, a a

a (a a) , a, a aexch

a, a (a a) , a a

a (a (a a)) , a acut

a (a a) , a, a a

a (a a) , a (a a )

id

id id id id id

cut

cut

a (a a)

(a a ) a a

a

(( a a ) a) a a (a (a a))

a

a

a

a a a a a

a a

a (a a)

a a

a (a a )

How can we represent proofs ?

From Deep Inference to Proof Nets

ida a

ida (a (a a ))

sa a (a a )

id(( a a) a) a (a a )

sa (a a) a (a a )

ida a

ida (a (a a ))

sa (a a ) a

ida (a a) (a (a a ))

sa (a a) a (a a )

ida a

ida (( a a ) a)

sa (a a ) a

ida (a a ) (( a a ) a )

sa (a a) a (a a )

a a a a a a a a a a a a a a a a a a

ida a

ida (a (a a ))

sa a (a a )

id(( a a) a) a (a a )

sa (a a) a (a a )

ida a

ida (a (a a ))

sa (a a ) a

ida (a a) (a (a a ))

sa (a a) a (a a )

ida a

ida (( a a ) a)

sa (a a ) a

ida (a a ) (( a a ) a )

sa (a a) a (a a )

( ) ( )a (a a ) a (a a ) ( ) ( )a (a a) a (a a ) ( ) ( )a (a a) a (a a )

From Sequent Calculus to Neutral Games

δ1, . . . , δ k A δk+1 , . . . , δ n Bδ1, . . . , δ n A BA B , δ1, . . . , δ n

A B , δ1, . . . , δ n

(A B ) δ1 . . . δn

...

...

d1dka

bdk+1dn

...

a × bd1

dn

...

a × bd1

dn

...

(a × b)d1

dn

(a × b) d1 × . . . dn× ×

A , BA BA B

δ1δ1 . . .

δnδn

(A B ) δ1 . . . δn

From Deep Inference to Atomic Flows

tai

a a=(a t) ( t a )

m[a t] [t a ]

=[a t] [ a t]

s([a t] a ) t

=(a [a t]) t

s[(a a ) t] t

=(a a ) t

aif t

=t

a [a t] aai

a [a [ a a ]] a=(a [[a a ] a ]) a

s[(a [a a ]) a ] a

ac[(a a) a ] a

ai[f a ] a

=a a

ac(a a ) a

=a (a a )

aia f

[a b] cac

[(a a ) b] cac

[(a a ) ( b b )] cac

[(a a ) (b b)] (c c )m([a b ] [a b]) (c c )

=([a b ] c ) ([a b ] c )

Forward Chaining versus Backward Chaining

Γ −a →I r

bΓ − b→Ir bΓ c c

I l

b Γ b⊃ c c⊃ L

b Γ cLf

Γ b c[]l

Γ b cR l

Γ a ⊃ b [c⊃ L

→

→

→→→

→

a, a ⊃ b, b ⊃ chere, Γ is the multiset

Γ −a →I r Γ b b

I l

Γ a⊃ b b⊃ L

Γ bLf

Γ b[]r

Γ − b→R r Γ c c

I l

Γ b⊃ c c⊃ L

→

→→

→

→

→