Cut Elimination – Splitting versus Proof Nets Antrittsvortrag zur Diplomarbeit von Matthias Horbach Betreuer: Lutz Straßburger

Outline What is a proof? What is a proof net? Cut elimination and the splitting lemma Objectives of my thesis

What is a Proof? Intuitionistic Logic: proof normal proof λ-term β-η-normal form cut eliminationnormalization

What is a Proof? (Multiplicative) Linear Logic: proof normal proof MLL-proof net normal proof net cut eliminationnormalization

What is a Proof? Classical Logic: proof ? proof net ? cut eliminationnormalization

Proof Nets ¹ b a ¹ a ¹ bb a ¹ a b £ ¹ b ^ a ; ¹ a ^ ¹ b ; b ^ a ; ¹ a ^ b ¤ £ ¹ b ^ a ; ¹ a ^ ¹ b ; b ^ ¹ b ; b ^ a ; ¹ a ^ b ¤ ai↑ £ ¹ b ^ a ; ¹ a ^ ¹ b ; b ¤ ^ £ ¹ b ; b ^ a ; ¹ a ^ b ¤ 2x s £ ¹ b ^ [ a ; ¹ a ] ^ ¹ b ; b ; b ¤ ^ £ ¹ b ; ¹ b ; b ^ [ a ; ¹ a ] ^ b ¤ 4x s [ a ; ¹ a ] ^ £ ¹ b ; b ¤ ^ £ ¹ b ; b ¤ ^ £ ¹ b ; b ¤ ^ £ ¹ b ; b ¤ ^ [ a ; ¹ a ] £ ¹ b ^ a ; ¹ a ^ ¹ b ; b ; b ¤ ^ £ ¹ b ; ¹ b ; b ^ a ; ¹ a ^ b ¤ 2x c↓ 6x ai↓ ¹ b a ¹ a ¹ bb ¹ bb a ¹ a b ¹ b a ¹ a ¹ bb a ¹ a b

Splitting t k [( R ^ T ) ; K ] t t ^ t k [ R ; K R ] ^ [ T ; K T ] [( R ^ T ) ; K R ; K T ] k [( R ^ T ) ; K ]

Cut Elimination via Splitting Initial proof with exactly one cut: Step 3: Combining the killers t k K A _ K ¹ A t k S f A ^ ¹ A g S f f g t k K A _ K ¹ A k K f _ K k S f f g Step 1: Context reduction fg _ K k S fg Step 2: Splitting t k ¹ A _ K ¹ A t k A _ K A and K A _ K ¹ A k K

Main Goals of the Thesis Find corresponding cut elimination procedures for proof nets and for deep inference, and thus a notion of normal forms. If possible, find a bound on the size of the proofs created by splitting, i.e. of normal forms.

Previous Work Jean-Yves Girard, A new Constructive Logic: Classical Logic Edmund Robinson, Proof Nets for Classical Logic. J. of Logic Computation, vol. 13, no. 5, pp , 2003 Alessio Guglielmi, A System of Interaction and Structure. ACM Transactions on Computational Logic, Straßburger, Linear Logic and Noncommutativity in the Calculus of Structures. PhD Thesis, TU Dresden, 2003 Kai Brünnler, Deep Inference and Symmetry in Classical Proofs. PhD Thesis, TU Dresden, 2003 Lamarche and Straßburger, Naming Proofs in Classical Propositional Logic. In Pawel Urzyczyn, editor, Typed Lambda Calculi and Applications, TLCA 2005 Straßburger, From Deep Inference to Proof Nets. SD/ICALP, Lamarche and Straßburger, Constructing free Boolean Categories. LICS, 2005

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Simulation of the Atomic Sequent Cut Rule with ai↑ [ F ; a ] ^ [ G ; ¹ a ] [ F ; a ^ [ G ; ¹ a ]] [ F ; a ^ ¹ a ; G ] [ F ; G ] s s ai↑ R ^ [ T ; P ] [ R ; P ] a ^ ¹ a f swhere: