REDO Redesigning Logical Syntax Action de Recherche Collaborative 2009 INRIA Saclay - Île-de-France Lutz Straßburger Dale Miller Ivan Gazeau Nicolas Guenot Anne-Laure Poupon François Wirion University of Bath Alessio Guglielmi Guy McCusker Jim Laird Tom Gundersen Ana Carolina Martins Abbud Martin Churchill INRIA Nancy - Grand Est François Lamarche Alessio Guglielmi Paola Bruscoli Novak 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 id a ( a a ) , ( a a ) a id a ,a a ( a a ) , (( a a ) a ) a ,a exch a ( a a ) , a, (( a a ) a ) a id a ,a id a ,a id a, a id a ,a a, a a ,a exch a, a, a a a ,a a, a, a a cut a ,a a, a, a a a ( a a ) , a, a a exch a, a ( a a ) ,a a a ( a ( a a )) ,a a cut 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 id a a id a ( a ( a a )) s a a ( a a ) id (( a a ) a ) a ( a a ) s a ( a a ) a ( a a ) id a a id a ( a ( a a )) s a ( a a ) a id a ( a a ) ( a ( a a )) s a ( a a ) a ( a a ) id a a id a (( a a ) a ) s a ( a a ) a id a ( a a ) (( a a ) a ) s a ( a a ) a ( a a ) a a a a a a a a a a a a a a a a a a id a a id a ( a ( a a )) s a a ( a a ) id (( a a ) a ) a ( a a ) s a ( a a ) a ( a a ) id a a id a ( a ( a a )) s a ( a a ) a id a ( a a ) ( a ( a a )) s a ( a a ) a ( a a ) id a a id a (( a a ) a ) s a ( a a ) a id a ( a a ) (( a a ) a ) s a ( 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 δ 1 ,...,δ k A δ k +1 ,...,δ n B δ 1 ,...,δ n A B A B ,δ 1 ,...,δ n A B ,δ 1 ,...,δ n ( A B ) δ 1 ... δ n . . . . . . d 1 d k a b d k +1 d n . . . a × b d 1 d n . . . a × b d 1 d n . . . ( a × b) d 1 d n ( a × b) d 1 × ... d n × × A , B A B A B δ 1 δ 1 ... δ n δ n ( A B ) δ 1 ... δ n From Deep Inference to Atomic Flows t ai 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 ai f t = t a [ ¯ a t ] ¯ a ai 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 ) ai a f [a b] c ac [( a a ) b] c ac [( a a ) ( b b )] c ac [( 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 b Γ − b → I r b Γ c c I l b Γ b ⊃ c c ⊃ L b Γ c Lf Γ b c [] l Γ b c R l Γ a ⊃ b [c ⊃ L → → → → → → a, a ⊃ b, b ⊃ c here, Γ is the multiset Γ − a → I r Γ b b I l Γ a ⊃ b b ⊃ L Γ b Lf Γ b [] r Γ− b → R r Γ c c I l Γ b ⊃ c c ⊃ L → → → → → →