CERES for Propositional Proof Schemata Mikheil Rukhaia joint work - PowerPoint PPT Presentation

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Motivation Example ◮ ψ ( 0 ) : p 0 ⊢ p 0 ¬ : l ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 ∨ : l p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 ◮ ψ ( k + 1 ) : p k + 1 ⊢ p k + 1 ¬ : l ( ψ ( k )) ¬ p k + 1 , p k + 1 ⊢ p k + 2 ⊢ p k + 2 ∨ : l p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 1 p k + 1 , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 cut p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 ∧ : l p 0 , � k + 1 i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 2 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 15 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Motivation Example ◮ ψ ( 0 ) : p 0 ⊢ p 0 ¬ : l ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 ∨ : l p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 ◮ ψ ( k + 1 ) : p k + 1 ⊢ p k + 1 ¬ : l ( ψ ( k )) ¬ p k + 1 , p k + 1 ⊢ p k + 2 ⊢ p k + 2 ∨ : l p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 1 p k + 1 , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 cut p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 ∧ : l p 0 , � k + 1 i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 2 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 15 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Basic Notions ◮ Cut-configuration Ω of ψ is a set of formula occurrences from the end-sequent of ψ . ◮ cl Ω ,ψ is an unique indexed proposition symbol for all proof sym- bols ψ and cut-configurations Ω . ◮ The intended semantics of cl Ω ,ψ will be “the characteristic clause a set of ψ ( a ) , with the cut-configuration Ω ”. CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 16 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Characteristic Clause Term Θ ρ ( π, Ω) is defined inductively: ◮ if ρ is an axiom of the form Γ Ω , Γ C , Γ ⊢ ∆ Ω , ∆ C , ∆ , then Θ ρ ( π, Ω) = Γ Ω , Γ C ⊢ ∆ Ω , ∆ C . ◮ if ρ is a proof link of the form ( ψ ( a )) Γ Ω , Γ C , Γ ⊢ ∆ Ω , ∆ C , ∆ then Θ ρ ( π, Ω) = ⊢ cl Ω ′ ,ψ a where Ω ′ is a set of formula occurrences from Γ Ω , Γ C ⊢ ∆ Ω , ∆ C . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 17 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Characteristic Clause Term (ctd.) ◮ if ρ is an unary rule with immediate predecessor ρ ′ , then Θ ρ ( π, Ω) = Θ ρ ′ ( π, Ω) . ◮ if ρ is a binary rule with immediate predecessors ρ 1 , ρ 2 , then ei- ther Θ ρ ( π, Ω) = Θ ρ 1 ( π, Ω) ⊕ Θ ρ 2 ( π, Ω) or Θ ρ ( π, Ω) = Θ ρ 1 ( π, Ω) ⊗ Θ ρ 2 ( π, Ω) . ◮ Θ( π, Ω) = Θ ρ 0 ( π, Ω) , where ρ 0 is the last inference of π . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 18 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example Ψ = ( ψ ( 0 ) , ψ ( k + 1 )) of p 0 , � n i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p n + 1 , where: ◮ ψ ( 0 ) : p 0 ⊢ p 0 ¬ : l ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 ∨ : l p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 ◮ ψ ( k + 1 ) : p k + 1 ⊢ p k + 1 ¬ : l ( ψ ( k )) ¬ p k + 1 , p k + 1 ⊢ p k + 2 ⊢ p k + 2 ∨ : l p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 1 p k + 1 , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 cut p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 ∧ : l p 0 , � k + 1 i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 2 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 19 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ ψ ( 0 ) : p 0 ⊢ p 0 ¬ : l ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 ∨ : l p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 20 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ ψ ( 0 ) : p 0 ⊢ p 0 ¬ : l ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 ∨ : l p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 Θ( ψ ( 0 ) , ∅ ) = ⊢ CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 20 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ ψ ( 0 ) : p 0 ⊢ p 0 ¬ : l ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 ∨ : l p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 20 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ ψ ( 0 ) : p 0 ⊢ p 0 ¬ : l ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 ∨ : l p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 Θ( ψ ( 0 ) , { p n + 1 } ) = ⊢ p 1 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 20 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ ψ ( k + 1 ) : p k + 1 ⊢ p k + 1 ¬ : l ( ψ ( k )) ¬ p k + 1 , p k + 1 ⊢ p k + 2 ⊢ p k + 2 ∨ : l p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 1 p k + 1 , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 cut p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 ∧ : l p 0 , � k + 1 i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 2 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 20 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ ψ ( k + 1 ) : p k + 1 ⊢ p k + 1 ¬ : l ( ψ ( k )) ¬ p k + 1 , p k + 1 ⊢ p k + 2 ⊢ p k + 2 ∨ : l p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 1 p k + 1 , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 cut p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 ∧ : l p 0 , � k + 1 i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 2 Θ( ψ ( k + 1 ) , ∅ ) = ⊢ cl { p n + 1 } ,ψ ⊕ p k + 1 ⊢ k CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 20 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ ψ ( k + 1 ) : p k + 1 ⊢ p k + 1 ¬ : l ( ψ ( k )) ¬ p k + 1 , p k + 1 ⊢ p k + 2 ⊢ p k + 2 ∨ : l p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 1 p k + 1 , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 cut p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 ∧ : l p 0 , � k + 1 i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 2 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 20 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ ψ ( k + 1 ) : p k + 1 ⊢ p k + 1 ¬ : l ( ψ ( k )) ¬ p k + 1 , p k + 1 ⊢ p k + 2 ⊢ p k + 2 ∨ : l p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 1 p k + 1 , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 cut p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 ∧ : l p 0 , � k + 1 i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 2 Θ( ψ ( k + 1 ) , { p n + 1 } ) = ⊢ cl { p n + 1 } ,ψ ⊕ ( p k + 1 ⊢ ⊗ ⊢ p k + 2 ) k CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 20 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Evaluation of Clause Term ◮ The rewrite rules for clause term symbols: ⊢ cl Ω ,ψ β → Θ( π β ( 0 ) , Ω) , and 0 ⊢ cl Ω ,ψ β → Θ( ν β ( k + 1 ) , Ω) , for all β = 1 , . . . , α . k + 1 ◮ Θ( Ψ, Ω) = Θ( ψ 1 , Ω) , and ◮ Θ( Ψ ) = Θ( Ψ, ∅ ) . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 21 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Evaluation of Clause Term (ctd.) Proposition (Soundness) Let γ ∈ N and Ω be a cut-configuration, then Θ( ψ β , Ω) ↓ γ is a ground clause term for all 1 ≤ β ≤ α . Hence Θ( Ψ ) ↓ γ is a ground clause term. Proof. By induction. CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 22 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Evaluation of Clause Term (ctd.) Proposition (Commutativity) Let Ω be a cut-configuration and γ ∈ N . Then Θ( Ψ ↓ γ , Ω) = Θ( Ψ, Ω) ↓ γ . Proof. By double induction. CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 23 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Term to Set Transformation ◮ Let Γ ⊢ ∆ and Π ⊢ Λ be sequents, then Γ ⊢ ∆ × Π ⊢ Λ = Γ , Π ⊢ ∆ , Λ and P × Q = { S P × S Q | S P ∈ P , S Q ∈ Q } . ◮ Let Θ be a clause term, then we define | Θ | as: | ⊢ cl Ω ′ ,ψ | = C Θ( ψ, Ω ′ ) ( a ) , where C Θ( ψ, Ω ′ ) is a clause set symbol a assigned to Θ( ψ, Ω ′ ) , | Γ ⊢ ∆ | = { Γ ⊢ ∆ } , | Θ 1 ⊗ Θ 2 | = | Θ 1 | × | Θ 2 | , | Θ 1 ⊕ Θ 2 | = | Θ 1 | ∪ | Θ 2 | . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 24 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Characteristic Clause Set Schemata ◮ Let Ψ = � ( π 1 ( 0 ) , ν 1 ( k + 1 )) , . . . , ( π α ( 0 ) , ν α ( k + 1 )) � , then assign each pair of terms, Θ( π β , Ω) and Θ( ν β , Ω) , a unique symbol C γ and define: C γ ( 0 ) = | Θ( π β , Ω) | , C γ ( k + 1 ) = | Θ( ν β , Ω) | . ◮ The characteristic clause set schema CL ( Ψ ) = � ( C 1 ( 0 ) , C 1 ( k + 1 )) , . . . � where C 1 is assigned to the pair of terms Θ( π 1 , ∅ ) and Θ( ν 1 , ∅ ) . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 25 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ CL ( Ψ ) = � ( C ( 0 ) , C ( k + 1 )) , ( D ( 0 ) , D ( k + 1 )) � , where: C ( 0 ) = | Θ( ψ ( 0 ) , ∅ ) | = {⊢} C ( k + 1 ) = | Θ( ψ ( k + 1 ) , ∅ ) | = D ( k ) ∪ { p k + 1 ⊢} D ( 0 ) = | Θ( ψ ( 0 ) , { p n + 1 } ) | = {⊢ p 1 } D ( k + 1 ) = | Θ( ψ ( k + 1 ) , { p n + 1 } ) | = D ( k ) ∪ { p k + 1 ⊢ p k + 2 } CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 26 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ CL ( Ψ ) ↓ 0 : ◮ CL ( Ψ ) ↓ 3 : (1) ⊢ (1) ⊢ p 1 (2) p 1 ⊢ p 2 ◮ CL ( Ψ ) ↓ 1 : (3) p 2 ⊢ p 3 (4) p 3 ⊢ (1) ⊢ p 1 (2) p 1 ⊢ ◮ CL ( Ψ ) ↓ 4 : ◮ CL ( Ψ ) ↓ 2 : (1) ⊢ p 1 (2) p 1 ⊢ p 2 (1) ⊢ p 1 (3) p 2 ⊢ p 3 (2) p 1 ⊢ p 2 (4) p 3 ⊢ p 4 (3) p 2 ⊢ (5) p 4 ⊢ CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 27 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Schematic Projections CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 28 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Basic Notions ◮ pr Ω ,ψ is an unique proof symbol, called projection symbol. ◮ The intended semantics of pr Ω ,ψ ( a ) will be “the set of character- istic projections of ψ ( a ) , with the cut-configuration Ω ”. CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 29 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Characteristic Projection Term Ξ ρ ( π, Ω) is defined inductively: ◮ if ρ is an axiom S , then Ξ ρ ( π, Ω) = S . ◮ if ρ is a proof link of the form ( ψ ( a )) Γ Ω , Γ C , Γ ⊢ ∆ Ω , ∆ C , ∆ then Ξ ρ ( π, Ω) = pr Ω ′ ,ψ ( a ) where Ω ′ is a set of formula occurrences from Γ Ω , Γ C ⊢ ∆ Ω , ∆ C . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 30 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Characteristic Projection Term (ctd.) ◮ If ρ is an unary inference with immediate predecessor ρ ′ , then either Ξ ρ ( π, Ω) = Ξ ρ ′ ( π, Ω) or Ξ ρ ( π, Ω) = ρ (Ξ ρ ′ ( π, Ω)) . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 31 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Characteristic Projection Term (ctd.) ◮ If ρ is a binary inference with immediate predecessors ρ 1 and ρ 2 , then either Ξ ρ ( π, Ω) = w Γ 2 ⊢ ∆ 2 (Ξ ρ 1 ( π, Ω)) ⊕ w Γ 1 ⊢ ∆ 1 (Ξ ρ 2 ( π, Ω)) or Ξ ρ ( π, Ω) = Ξ ρ 1 ( π, Ω) ⊗ ρ Ξ ρ 2 ( π, Ω) ◮ Ξ( π, Ω) = Ξ ρ 0 ( π, Ω) , where ρ 0 is the last inference of π . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 32 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example Ψ = ( ψ ( 0 ) , ψ ( k + 1 )) of p 0 , � n i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p n + 1 , where: ◮ ψ ( 0 ) : p 0 ⊢ p 0 ¬ : l ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 ∨ : l p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 ◮ ψ ( k + 1 ) : p k + 1 ⊢ p k + 1 ¬ : l ( ψ ( k )) ¬ p k + 1 , p k + 1 ⊢ p k + 2 ⊢ p k + 2 ∨ : l p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 1 p k + 1 , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 cut p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 ∧ : l p 0 , � k + 1 i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 2 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 33 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ ψ ( 0 ) : p 0 ⊢ p 0 ¬ : l ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 ∨ : l p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 34 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ ψ ( 0 ) : p 0 ⊢ p 0 ¬ : l ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 ∨ : l p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 Ξ( ψ ( 0 ) , ∅ ) = ¬ l ( p 0 ⊢ p 0 ) ⊗ ∨ l p 1 ⊢ p 1 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 34 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ ψ ( 0 ) : p 0 ⊢ p 0 ¬ : l ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 ∨ : l p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 34 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ ψ ( 0 ) : p 0 ⊢ p 0 ¬ : l ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 ∨ : l p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 Ξ( ψ ( 0 ) , { p n + 1 } ) = ¬ l ( p 0 ⊢ p 0 ) ⊗ ∨ l p 1 ⊢ p 1 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 34 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ ψ ( k + 1 ) : p k + 1 ⊢ p k + 1 ¬ : l ( ψ ( k )) ¬ p k + 1 , p k + 1 ⊢ p k + 2 ⊢ p k + 2 ∨ : l p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 1 p k + 1 , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 cut p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 ∧ : l p 0 , � k + 1 i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 2 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 34 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ ψ ( k + 1 ) : p k + 1 ⊢ p k + 1 ¬ : l ( ψ ( k )) ¬ p k + 1 , p k + 1 ⊢ p k + 2 ⊢ p k + 2 ∨ : l p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 1 p k + 1 , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 cut p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 ∧ : l p 0 , � k + 1 i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 2 Ξ( ψ ( k + 1 ) , ∅ ) = ∧ l ( w ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 ( pr { p n + 1 } ,ψ ( k )) ⊕ w p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ ( ¬ l ( p k + 1 ⊢ p k + 1 ) ⊗ ∨ l p k + 2 ⊢ p k + 2 )) CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 34 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ ψ ( k + 1 ) : p k + 1 ⊢ p k + 1 ¬ : l ( ψ ( k )) ¬ p k + 1 , p k + 1 ⊢ p k + 2 ⊢ p k + 2 ∨ : l p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 1 p k + 1 , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 cut p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 ∧ : l p 0 , � k + 1 i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 2 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 34 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ ψ ( k + 1 ) : p k + 1 ⊢ p k + 1 ¬ : l ( ψ ( k )) ¬ p k + 1 , p k + 1 ⊢ p k + 2 ⊢ p k + 2 ∨ : l p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 1 p k + 1 , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 cut p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) , ¬ p k + 1 ∨ p k + 2 ⊢ p k + 2 ∧ : l p 0 , � k + 1 i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p k + 2 Ξ( ψ ( k + 1 ) , { p n + 1 } ) = ∧ l ( w ¬ p k + 1 ∨ p k + 2 ⊢ ( pr { p n + 1 } ,ψ ( k )) ⊕ w p 0 , � k i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ ( ¬ l ( p k + 1 ⊢ p k + 1 ) ⊗ ∨ l p k + 2 ⊢ p k + 2 )) CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 34 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Evaluation of Clause Term ◮ The rewrite rules for clause term symbols: pr Ω ,ψ β ( 0 ) → Ξ( π β ( 0 ) , Ω) , and pr Ω ,ψ β ( k + 1 ) → Ξ( ν β ( k + 1 ) , Ω) , for all β = 1 , . . . , α . ◮ Ξ( Ψ, Ω) = Ξ( ψ 1 , Ω) , and ◮ Ξ( Ψ ) = Ξ( Ψ, ∅ ) . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 35 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Evaluation of Clause Term (ctd.) Proposition (Soundness) Let γ ∈ N and Ω be a cut-configuration, then Ξ( ψ β , Ω) ↓ γ is a ground projection term for all 1 ≤ β ≤ α . Hence Ξ( Ψ ) ↓ γ is a ground projection term. Proof. By induction. CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 36 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Evaluation of Clause Term (ctd.) Proposition (Commutativity) Let Ω be a cut-configuration and γ ∈ N . Then Ξ( Ψ ↓ γ , Ω) = Ξ( Ψ, Ω) ↓ γ . Proof. By double induction. CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 37 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Term to Set Transformation ◮ Let ρ be an unary and σ a binary rule. Let φ, ψ be LKS -proofs, then ρ ( φ ) is the LKS -proof obtained from the φ by applying ρ , and σ ( φ, ψ ) is the proof obtained from the proofs φ and ψ by ap- plying σ . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 38 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Term to Set Transformation ◮ Let ρ be an unary and σ a binary rule. Let φ, ψ be LKS -proofs, then ρ ( φ ) is the LKS -proof obtained from the φ by applying ρ , and σ ( φ, ψ ) is the proof obtained from the proofs φ and ψ by ap- plying σ . φ = p 0 ⊢ p 0 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 38 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Term to Set Transformation ◮ Let ρ be an unary and σ a binary rule. Let φ, ψ be LKS -proofs, then ρ ( φ ) is the LKS -proof obtained from the φ by applying ρ , and σ ( φ, ψ ) is the proof obtained from the proofs φ and ψ by ap- plying σ . p 0 ⊢ p 0 ¬ l ( φ ) = ¬ : l ¬ p 0 , p 0 ⊢ CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 38 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Term to Set Transformation ◮ Let ρ be an unary and σ a binary rule. Let φ, ψ be LKS -proofs, then ρ ( φ ) is the LKS -proof obtained from the φ by applying ρ , and σ ( φ, ψ ) is the proof obtained from the proofs φ and ψ by ap- plying σ . p 0 ⊢ p 0 ψ = p 1 ⊢ p 1 ¬ l ( φ ) = ¬ : l ¬ p 0 , p 0 ⊢ CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 38 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Term to Set Transformation ◮ Let ρ be an unary and σ a binary rule. Let φ, ψ be LKS -proofs, then ρ ( φ ) is the LKS -proof obtained from the φ by applying ρ , and σ ( φ, ψ ) is the proof obtained from the proofs φ and ψ by ap- plying σ . p 0 ⊢ p 0 ¬ : l ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 ∨ l ( ¬ l ( φ ) , ψ ) = ∨ : l p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 38 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Term to Set Transformation (ctd.) ◮ P Γ ⊢ ∆ = { ψ Γ ⊢ ∆ | ψ ∈ P } , where ψ Γ ⊢ ∆ is ψ followed by weak- enings adding Γ ⊢ ∆ . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 39 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Term to Set Transformation (ctd.) ◮ P Γ ⊢ ∆ = { ψ Γ ⊢ ∆ | ψ ∈ P } , where ψ Γ ⊢ ∆ is ψ followed by weak- enings adding Γ ⊢ ∆ . p 0 ⊢ p 0 ¬ : l ψ = ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 ∨ : l p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 39 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Term to Set Transformation (ctd.) ◮ P Γ ⊢ ∆ = { ψ Γ ⊢ ∆ | ψ ∈ P } , where ψ Γ ⊢ ∆ is ψ followed by weak- enings adding Γ ⊢ ∆ . p 0 ⊢ p 0 ¬ : l ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 ∨ : l ψ Γ ⊢ ∆ = p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 w : l ∗ p 0 , ¬ p 0 ∨ p 1 , Γ ⊢ p 1 w : r ∗ p 0 , ¬ p 0 ∨ p 1 , Γ ⊢ ∆ , p 1 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 39 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Term to Set Transformation (ctd.) ◮ P , Q : sets of LKS -proofs. ◮ P × σ Q = { σ ( φ, ψ ) | φ ∈ P , ψ ∈ Q } . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 40 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Term to Set Transformation (ctd.) ◮ P , Q : sets of LKS -proofs. ◮ P × σ Q = { σ ( φ, ψ ) | φ ∈ P , ψ ∈ Q } . � � p 0 ⊢ p 0 q 0 ⊢ q 0 P = , w : l ¬ : l ¬ p 0 , p 0 ⊢ ¬ p 0 , q 0 ⊢ q 0 � q 1 ⊢ q 1 � Q = , p 1 ⊢ p 1 w : l p 1 , q 1 ⊢ q 1 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 40 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Term to Set Transformation (ctd.) � p 0 ⊢ p 0 ¬ : l ◮ P × ∨ l Q = , ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 ∨ : l p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 q 0 ⊢ q 0 w : l ¬ p 0 , q 0 ⊢ q 0 p 1 ⊢ p 1 , ∨ : l q 0 , ¬ p 0 ∨ p 1 ⊢ q 0 , p 1 p 0 ⊢ p 0 q 1 ⊢ q 1 ¬ : l w : l ¬ p 0 , p 0 ⊢ p 1 , q 1 ⊢ q 1 , ∨ : l p 0 , q 1 , ¬ p 0 ∨ p 1 ⊢ q 1 � q 0 ⊢ q 0 q 1 ⊢ q 1 w : l w : l ¬ p 0 , q 0 ⊢ q 0 p 1 , q 1 ⊢ q 1 ∨ : l q 0 , q 1 , ¬ p 0 ∨ p 1 ⊢ q 0 , q 1 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 40 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Term to Set Transformation (ctd.) ◮ Let Ξ be a ground projection term, then we define | Ξ | as: | Γ ⊢ ∆ | = Γ ⊢ ∆ , | ρ (Ξ) | = ρ ( | Ξ | ) for unary rule symbols ρ , | w Γ ⊢ ∆ (Ξ) | = | Ξ | Γ ⊢ ∆ , | Ξ 1 ⊕ Ξ 2 | = | Ξ 1 | ∪ | Ξ 2 | , | Ξ 1 ⊗ σ Ξ 2 | = | Ξ 1 | × σ | Ξ 2 | for binary rule symbols σ . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 41 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Term to Set Transformation (ctd.) ◮ For ground LKS -proofs π and cut-configurations Ω , define PR ( π, Ω) = | Ξ( π, Ω) | and PR ( π ) = PR ( π, ∅ ) . ◮ PR ( Ψ ) ↓ γ = | Ξ( Ψ ) ↓ γ | . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 42 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ PR ( Ψ ) ↓ 0 : � � p 0 ⊢ p 0 ¬ : l ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 ∨ : l p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 43 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ PR ( Ψ ) ↓ 1 : p 0 ⊢ p 0 ¬ : l ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 � ∨ : l p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 w : l , r p 0 , ¬ p 0 ∨ p 1 , ¬ p 1 ∨ p 2 ⊢ p 2 , p 1 ∧ : l p 0 , � 1 i = 0 ¬ p i ∨ p i + 1 ⊢ p 2 , p 1 p 1 ⊢ p 1 ¬ : l ¬ p 1 , p 1 ⊢ p 2 ⊢ p 2 � ∨ : l p 1 , ¬ p 1 ∨ p 2 ⊢ p 2 w : l p 1 , p 0 , � 0 i = 0 ( ¬ p i ∨ p i + 1 ) , ¬ p 1 ∨ p 2 ⊢ p 2 ∧ : l p 1 , p 0 , � 1 i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p 2 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 43 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ PR ( Ψ ) ↓ 2 : � p 0 ⊢ p 0 ¬ : l ¬ p 0 , p 0 ⊢ p 1 ⊢ p 1 ∨ : l p 0 , ¬ p 0 ∨ p 1 ⊢ p 1 w : l p 0 , ¬ p 0 ∨ p 1 , ¬ p 1 ∨ p 2 ⊢ p 1 ∧ : l p 0 , � 1 i = 0 ¬ p i ∨ p i + 1 ⊢ p 1 w : l , r p 0 , � 1 i = 0 ¬ p i ∨ p i + 1 , ¬ p 2 ∨ p 3 ⊢ p 3 , p 1 ∧ : l p 0 , � 2 i = 0 ¬ p i ∨ p i + 1 ⊢ p 3 , p 1 � . . . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 43 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ PR ( Ψ ) ↓ 2 : � . . . p 1 ⊢ p 1 ¬ : l ¬ p 1 , p 1 ⊢ p 2 ⊢ p 2 ∨ : l p 1 , ¬ p 1 ∨ p 2 ⊢ p 2 w : l p 1 , p 0 , � 0 i = 0 ( ¬ p i ∨ p i + 1 ) , ¬ p 1 ∨ p 2 ⊢ p 2 ∧ : l p 1 , p 0 , � 1 i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p 2 w : l , r p 1 , p 0 , � 1 i = 0 ( ¬ p i ∨ p i + 1 ) , ¬ p 2 ∨ p 3 ⊢ p 3 , p 2 ∧ : l p 1 , p 0 , � 2 i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p 3 , p 2 � . . . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 43 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin An Example (ctd.) ◮ PR ( Ψ ) ↓ 2 : � . . . p 2 ⊢ p 2 ¬ : l ¬ p 2 , p 2 ⊢ p 3 ⊢ p 3 ∨ : l p 2 , ¬ p 2 ∨ p 3 ⊢ p 3 w : l p 2 , p 0 , � 1 i = 0 ( ¬ p i ∨ p i + 1 ) , ¬ p 2 ∨ p 3 ⊢ p 3 ∧ : l p 2 , p 0 , � 2 i = 0 ( ¬ p i ∨ p i + 1 ) ⊢ p 3 � CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 43 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Term to Set Transformation (ctd.) Proposition (Soundness) Let π be a ground LKS -proof with end-sequent S, then for all clauses C ∈ CL ( π ) , there exists a ground LKS -proof π ∈ PR ( π ) with end-sequent S ◦ C. Proposition (Commutativity) Let γ ∈ N , then PR ( Ψ ↓ γ ) = PR ( Ψ ) ↓ γ . Proposition (Correctness) Let γ ∈ N , then for every clause C ∈ CL ( Ψ ) ↓ γ there exists a ground LKS -proof π ∈ PR ( Ψ ) ↓ γ with end-sequent C ◦ S ( γ ) . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 44 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Resolution Schemata CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 45 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Clause Schemata ◮ We define s-clause as: clause variables, denoted with X , Y , . . . , are s-clauses, clauses are s-clauses, if s 1 , s 2 are s-clauses, then s 1 ◦ s 2 is an s-clause. ◮ A clause schema is a term t ( a , X 1 , . . . , X α ) w.r.t a rewrite system R : t ( 0 , X 1 , . . . , X α ) → s 0 , t ( k + 1 , X 1 , . . . , X α ) → t ( k , s 1 , . . . , s α ) , for s 0 , . . . , s α being s- clauses with clause variables in { X 1 , . . . , X α } . ◮ Example: consider t ( n , X ) w.r.t t ( 0 , X ) → ( ⊢ p 0 ) ◦ X , t ( k + 1 , X ) → t ( k , ( ⊢ p k + 1 ) ◦ X ) , then t ( α, ⊢ q 0 ) ↓ are ⊢ q 0 , p 0 , . . . , p α for all α ≥ 0. CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 46 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Resolution Term ◮ We define resolution terms inductively: s-clauses are resolution terms, clause schemata are resolution terms, if r 1 , r 2 are resolution terms w.r.t. R 1 and R 2 , then r ( r 1 ; r 2 ; p a ) is a resolution term w.r.t. R = R 1 ∪ R 2 . ◮ A resolution term r based on a set of clause schemata C is a reso- lution term s.t. all s-clauses and clause schemata in r are also in C . ◮ Example: r ( r ( t ( n , X ); p n ⊢ ; p n ); q 0 ⊢ ; q 0 ) is a resolution term. CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 47 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Resolution Deduction ◮ Let Γ ⊢ ∆ and Π ⊢ Λ be clauses. If p a occurs in ∆ and Π , then res (Γ ⊢ ∆ , Π ⊢ Λ , p a ) = Γ , Π \ p a ⊢ ∆ \ p a , Λ is called resolvent. ◮ We define resolution deduction inductively: if C is a clause, then C is a resolution deduction and ES ( C ) = C , if δ 1 and δ 2 are resolution deductions, ES ( δ 1 ) = C 1 , ES ( δ 2 ) = C 2 and res ( C 1 , C 2 , p a ) = D , then r ( δ 1 , δ 2 , p a ) is a resolution deduc- tion and ES ( r ( δ 1 , δ 2 , p a )) = D . ◮ δ is called resolution refutation, if ES ( δ ) = ⊢ . ◮ Examples: r ( r ( ⊢ q 0 , p 0 , p 1 ; p 1 ⊢ ; p 1 ); q 0 ⊢ ; q 0 ) is a resolution deduction. r ( r ( ⊢ q 0 , p 0 ; p 0 ⊢ ; p 0 ); q 0 ⊢ ; q 0 ) is a resolution refutation. CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 48 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Tree Transformation ◮ Let δ be a resolution deduction. If: δ = C , then T ( δ ) = C , δ = r ( δ 1 ; δ 2 ; p a ) , ES ( δ 1 ) = C 1 , ES ( δ 2 ) = C 2 and res ( C 1 , C 2 , p a ) = C , then T ( δ ) = ( T ( δ 1 )) ( T ( δ 2 )) C 1 C 2 C ◮ Example: T ( r ( r ( ⊢ q 0 , p 0 , p 1 ; p 1 ⊢ ; p 1 ); q 0 ⊢ ; q 0 )) is: ⊢ q 0 , p 0 , p 1 p 1 ⊢ ⊢ q 0 , p 0 q 0 ⊢ ⊢ p 0 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 49 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Resolution Refutation Schema ◮ A resolution proof schema with clause variables X 1 , . . . , X β is a structure R = (( ̺ 1 , . . . , ̺ α ) , R , D , R ′ ) where the ̺ i denote res- olution terms, D is a finite set of clause schemata w.r.t. R ′ and R = R 1 ∪ . . . ∪ R α , where the R i (for 0 ≤ i ≤ α ) are defined as follows: ̺ i ( 0 , X 1 , . . . , X β ) → s i , s i 0 ) , ̺ l 1 ( a i s i 1 ) , . . . , ̺ l j ( i ) ( a i s i ̺ i ( k + 1 , X 1 , . . . , X β ) → t i [ ̺ i ( k , ¯ 1 , ¯ j ( i ) , ¯ j ( i ) )] , where s i is a resolution term containing some of X 1 , . . . , X β , a i 1 , . . . , a i j ( i ) are arithmetic terms, s i s i ¯ 0 , . . . , ¯ j ( i ) are vectors of clause schemata over X 1 , . . . , X β , s i 0 ) , ̺ l 1 ( a i s i 1 ) , . . . , ̺ l j ( i ) ( a i s i the t i [ ̺ i ( k , ¯ 1 , ¯ j ( i ) , ¯ j ( i ) )] are resolution terms based on D after replacement of some clause schemata by the terms s i 0 ) , ̺ l 1 ( a i s i 1 ) , . . . , ̺ l r ( a i s i ̺ i ( k , ¯ 1 , ¯ j ( i ) , ¯ j ( i ) ) where i < min { l 1 , . . . , l j ( i ) } and max { l 1 , . . . , l j ( i ) } ≤ α . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 50 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Resolution Refutation Schema (ctd.) ◮ A resolution proof schema is called a resolution refutation schema of a clause schema C ( n ) if there exist clauses C 1 , . . . , C α s.t. ̺ 1 ( β, C 1 , . . . , C α ) ↓ is a resolution refutation of C ( β ) ↓ . ◮ Example: We define the resolution refutation schema R = (( ̺, δ ) , R , ∅ , ∅ ) where R is: ̺ ( 0 ) → ⊢ ̺ ( k + 1 ) → r ( δ ( k ); p k + 1 ⊢ ; p k + 1 ) , δ ( 0 ) → ⊢ p 1 , δ ( k + 1 ) → r ( δ ( k ); p k + 1 ⊢ p k + 2 ; p k + 1 ) . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 51 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Atomic Cut Normal Form Theorem (ACNF) Let Ψ be a proof schema with end-sequent S ( n ) , and let R be a reso- lution refutation schema of CL ( Ψ ) . Then for all α ∈ N there exists a ground LKS -proof π of S ( α ) with at most atomic cuts such that its size l ( π ) is polynomial in l ( R ↓ α ) · l ( PR ( Ψ ) ↓ α ) . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 52 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin The Adder Example CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 53 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Formula Definitions ◮ We introduce the following “shortcuts” for formulas: A ⊕ B ( A ∧ ¬ B ) ∨ ( ¬ A ∧ B ) = def A ⇔ B = def ( ¬ A ∨ B ) ∧ ( ¬ B ∨ A ) ˆ = def S i ⇔ ( A i ⊕ B i ) ⊕ C i S i S ′ ˆ S ′ i ⇔ ( B i ⊕ A i ) ⊕ C ′ = def i i ˆ C i = def C i + 1 ⇔ ( A i ∧ B i ) ∨ ( C i ∧ A i ) ∨ ( C i ∧ B i ) C ′ ˆ C ′ i + 1 ⇔ ( B i ∧ A i ) ∨ ( C ′ i ∧ B i ) ∨ ( C ′ = def i ∧ A i ) i � n i = 0 ˆ S i ∧ � n i = 0 ˆ = def C i ∧ ¬ C 0 Adder n � n i = 0 ˆ i ∧ � n i = 0 ˆ Adder ′ S ′ C ′ i ∧ ¬ C ′ = def n 0 � n i = 0 ( C i ⇔ C ′ = def i ) EqC n � n i = 0 ( S i ⇔ S ′ = def i ) EqS n CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 54 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin The Adder Proof ◮ The proof schema Ψ is: � ( ψ ( 0 ) , ψ ( k + 1 )) , ( ϕ ( 0 ) , ϕ ( k + 1 )) , ( φ ( 0 ) , φ ( k + 1 )) , ( χ ( 0 ) , χ ( k + 1 )) � , where ψ ( k ) is: ( ϕ ( k )) ( χ ( k )) 0 , � k i = 0 ˆ C i , � k i = 0 ˆ EqC k , � k i = 0 ˆ S i , � k i = 0 ˆ ¬ C 0 , ¬ C ′ C ′ S ′ i ⊢ EqC k i ⊢ EqS k cut ¬ C 0 , ¬ C ′ 0 , � k i = 0 ˆ C i , � k i = 0 ˆ C ′ i , � k i = 0 ˆ S i , � k i = 0 ˆ S ′ i ⊢ EqS k ∧ : l ∗ Adder k , Adder ′ k ⊢ EqS k CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 55 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin The Adder Proof (ctd.) ◮ ϕ ( k + 1 ) is: ( ϕ ( k )) ( φ ( k )) ¬ C 0 , ¬ C ′ 0 , � k i = 0 ˆ C i , � k i = 0 ˆ C ′ ¬ C 0 , ¬ C ′ 0 , � k i = 0 ˆ C i , � k i = 0 ˆ C ′ i ⊢ C k + 1 ⇔ C ′ i ⊢ EqC k k + 1 ∧ : r , c : l ∗ ¬ C 0 , ¬ C ′ 0 , � k i = 0 ˆ C i , � k i = 0 ˆ C ′ i ⊢ EqC k + 1 ∧ : l ∗ 0 , � k + 1 C i , � k + 1 ¬ C 0 , ¬ C ′ i = 0 ˆ i = 0 ˆ C ′ i ⊢ EqC k + 1 ◮ φ ( k + 1 ) is: . . ( φ ( k )) . ¬ C 0 , ¬ C ′ 0 , � k i = 0 ˆ C i , � k i = 0 ˆ C ′ i ⊢ C k + 1 ⇔ C ′ C k + 1 ⇔ C ′ k + 1 , ˆ C k + 1 , ˆ C ′ k + 1 ⊢ C k + 2 ⇔ C ′ k + 1 k + 2 cut ¬ C 0 , ¬ C ′ 0 , � k i = 0 ˆ C i , � k i = 0 ˆ C ′ i , ˆ C k + 1 , ˆ C ′ k + 1 ⊢ C k + 2 ⇔ C ′ k + 2 ∧ : l ∗ 0 , � k + 1 C i , � k + 1 ¬ C 0 , ¬ C ′ i = 0 ˆ i = 0 ˆ C ′ i ⊢ C k + 2 ⇔ C ′ k + 2 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 56 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin The Adder Proof (ctd.) ◮ Finally, χ ( k + 1 ) is: . . ( χ ( k )) . EqC k , � k i = 0 ˆ S i , � k i = 0 ˆ S ′ C k + 1 ⇔ C ′ k + 1 , ˆ S k + 1 , ˆ S ′ k + 1 ⊢ S k + 1 ⇔ S ′ i ⊢ EqS k k + 1 ∧ : r EqC k , � k i = 0 ˆ S i , � k i = 0 ˆ k + 1 , ˆ S k + 1 , ˆ S ′ i , C k + 1 ⇔ C ′ S ′ k + 1 ⊢ EqS k + 1 ∧ : l ∗ EqC k + 1 , � k + 1 S i , � k + 1 i = 0 ˆ i = 0 ˆ S ′ i ⊢ EqS k + 1 CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 57 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Characteristic Clause Set ◮ We get the following schema: CL ( Ψ ) = � ( C 1 ( 0 ) , C 1 ( k + 1 )) , . . . , ( C 4 ( 0 ) , C 4 ( k + 1 )) � where: C 1 ( k ) = C 2 ( k ) ∪ C 4 ( k ) , C 2 ( 0 ) = � C 0 ⊢ ; C ′ 0 ⊢ � , C 2 ( k + 1 ) = C 2 ( k ) ∪ C 3 ( k ) CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 58 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Characteristic Clause Set (ctd.) ◮ C 3 ( 0 ) = � C 1 ⊢ C ′ 1 ; C ′ 1 ⊢ C 1 � , ◮ C 3 ( k + 1 ) = C 3 ( k ) ∪ � C k + 1 ⊢ C ′ k + 1 , C k ; C ′ k + 1 ⊢ C k + 1 , C ′ k ; C ′ k , C k + 1 ⊢ C ′ k + 1 ; C k , C ′ k + 1 ⊢ C k + 1 � CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 59 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Characteristic Clause Set (ctd.) ◮ C 4 ( 0 ) = � ⊢ C 0 , C ′ 0 ; C 0 , C ′ 0 ⊢ � , ◮ C 4 ( k + 1 ) = C 4 ( k ) ◦ {⊢ C k + 1 , C ′ k + 1 } ∪ C 4 ( k ) ◦ { C k + 1 , C ′ k + 1 ⊢} . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 60 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Characteristic Clause Set (ctd.) ◮ CL ( Ψ ) ↓ 2 : ◮ CL ( Ψ ) ↓ 0 : (1) C 0 ⊢ (1) C 0 ⊢ (2) C ′ 0 ⊢ (2) C ′ 0 ⊢ (3) C 1 ⊢ C ′ 1 (3) ⊢ C 0 , C ′ (4) C ′ 1 ⊢ C 1 0 (4) C 0 , C ′ 0 ⊢ (5) C 2 ⊢ C ′ 2 , C 1 (6) C ′ 2 ⊢ C 2 , C ′ 1 ◮ CL ( Ψ ) ↓ 1 : (7) C ′ 1 , C 2 ⊢ C ′ 2 (8) C 1 , C ′ 2 ⊢ C 2 (1) C 0 ⊢ (9) ⊢ C 2 , C 0 , C ′ 0 , C 1 , C ′ 1 , C ′ (2) C ′ 0 ⊢ 2 (10) C ′ 2 , C 2 ⊢ C 1 , C ′ 1 , C 0 , C ′ (3) C 1 ⊢ C ′ 0 1 (11) C ′ 1 , C 1 ⊢ C 2 , C ′ 2 , C 0 , C ′ (4) C ′ 1 ⊢ C 1 0 (12) C ′ 2 , C 2 , C ′ 1 , C 1 ⊢ C 0 , C ′ (5) ⊢ C 0 , C ′ 0 , C 1 , C ′ 0 1 (13) C ′ 0 , C 0 ⊢ C 2 , C ′ 2 , C 1 , C ′ (6) C ′ 0 , C 0 ⊢ C 1 , C ′ 1 1 (14) C ′ 2 , C 2 , C ′ 0 , C 0 ⊢ C 1 , C ′ (7) C ′ 1 , C 1 ⊢ C 0 , C ′ 1 0 (15) C ′ 1 , C 1 , C ′ 0 , C 0 ⊢ C 2 , C ′ (8) C 0 , C ′ 0 , C 1 , C ′ 1 ⊢ 2 (16) C 2 , C 0 , C ′ 0 , C 1 , C ′ 1 , C ′ 2 ⊢ CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 61 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Refutation Schema ◮ A resolution refutation schema of CL ( Ψ ) is R = (( ̺, δ, η ) , R , ∅ , ∅ ) where: ̺ ( 0 , X ) → r ( r (( ⊢ C 0 , C ′ 0 ) ◦ X ; C 0 ⊢ ; C 0 ); C ′ 0 ⊢ ; C ′ 0 ) , ̺ ( k + 1 , X ) → r ( r ( ̺ ( k , ( ⊢ C k + 1 , C ′ k + 1 ) ◦ X ); η ( k ); C ′ k + 1 ) ; r ( δ ( k ); ̺ ( k , ( C k + 1 , C ′ k + 1 ⊢ ) ◦ X ); C ′ k + 1 ) ; C k + 1 ) . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 62 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Refutation Schema (ctd.) ◮ and δ ( 0 ) → C 1 ⊢ C ′ 1 , δ ( k + 1 ) → r ( C k + 2 ⊢ C ′ k + 2 , C k + 1 ; r ( δ ( k ); C ′ k + 1 , C k + 2 ⊢ C ′ k + 2 ; C ′ k + 1 ) ; C k + 1 ) . η ( 0 ) → C ′ 1 ⊢ C 1 , η ( k + 1 ) → r ( C ′ k + 2 ⊢ C k + 2 , C ′ k + 1 ; r ( η ( k ); C k + 1 , C ′ k + 2 ⊢ C k + 2 ; C k + 1 ) ; C ′ k + 1 ) . ◮ Finally, refutation of CL ( Ψ ) ↓ α is defined by ̺ ( α, ⊢ ) ↓ . CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 63 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Refutation Schema (ctd.) ◮ T ( ̺ ( 0 , ⊢ ) ↓ ) is: ⊢ C 0 , C ′ C 0 ⊢ 0 ⊢ C ′ C ′ 0 ⊢ 0 ⊢ ◮ T ( ̺ ( 1 , ⊢ ) ↓ ) is: ( ̺ ( 0 , ⊢ C 1 , C ′ ( ̺ ( 0 , C 1 , C ′ 1 ) ↓ ) 1 ⊢ ) ↓ ) ⊢ C 1 , C ′ C ′ C 1 ⊢ C ′ C 1 , C ′ 1 ⊢ C 1 1 ⊢ 1 1 ⊢ C 1 C 1 ⊢ ⊢ CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 64 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Refutation Schema (ctd.) ◮ T ( ̺ ( 0 , ⊢ C 1 , C ′ 1 ) ↓ ) is: ⊢ C 0 , C ′ 0 , C 1 , C ′ C 0 ⊢ 1 ⊢ C ′ 0 , C 1 , C ′ C ′ 0 ⊢ 1 ⊢ C 1 , C ′ 1 ◮ T ( ̺ ( 1 , ⊢ ) ↓ ) is: ( ̺ ( 0 , ⊢ C 1 , C ′ ( ̺ ( 0 , C 1 , C ′ 1 ) ↓ ) 1 ⊢ ) ↓ ) ⊢ C 1 , C ′ C ′ C 1 ⊢ C ′ C 1 , C ′ 1 ⊢ C 1 1 ⊢ 1 1 ⊢ C 1 C 1 ⊢ ⊢ CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 64 / 65

Intro Schematic LK Schematic Clause Set Schematic Projections Resolution Schemata Adder Example Fin Refutation Schema (ctd.) ◮ T ( ̺ ( 0 , C 1 , C ′ 1 ⊢ ) ↓ ) is: C 1 , C ′ 1 ⊢ C 0 , C ′ C 0 ⊢ 0 C 1 , C ′ 1 ⊢ C ′ C ′ 0 ⊢ 0 C 1 , C ′ 1 ⊢ ◮ T ( ̺ ( 1 , ⊢ ) ↓ ) is: ( ̺ ( 0 , ⊢ C 1 , C ′ ( ̺ ( 0 , C 1 , C ′ 1 ) ↓ ) 1 ⊢ ) ↓ ) ⊢ C 1 , C ′ C ′ C 1 ⊢ C ′ C 1 , C ′ 1 ⊢ C 1 1 ⊢ 1 1 ⊢ C 1 C 1 ⊢ ⊢ CERES for Proof Schemata M. Rukhaia Laboratory of Informatics of Grenoble Mar 29, 2012 64 / 65