Redesigning Logical Syntax - École Polytechniquelutz/orgs/REDO-poster.pdf · REDO Redesigning...
Transcript of Redesigning Logical Syntax - École Polytechniquelutz/orgs/REDO-poster.pdf · REDO Redesigning...
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
→
→→
→
→
→