Absorbing the Structural Rules in the Sequent Calculus with - PowerPoint PPT Presentation

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules Franco Parlamento University of Udine Flavio Previale University of Turin Logic Colloquium 2018 Udine July 27th 2018 Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 1 / 34

The G3[mic] systems Initial sequents P , Γ ⇒ ∆ , P ( ⊥ , Γ ⇒ ∆ , ⊥ in G3m ) Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 2 / 34

The G3[mic] systems Initial sequents P , Γ ⇒ ∆ , P ( ⊥ , Γ ⇒ ∆ , ⊥ in G3m ) Logical Rules ⊥ , Γ ⇒ ∆ ( not in G3m ) Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 2 / 34

The G3[mic] systems Initial sequents P , Γ ⇒ ∆ , P ( ⊥ , Γ ⇒ ∆ , ⊥ in G3m ) Logical Rules ⊥ , Γ ⇒ ∆ ( not in G3m ) A , B , Γ ⇒ ∆ Γ ⇒ ∆ , A Γ ⇒ ∆ , B L ∧ R ∧ A ∧ B , Γ ⇒ ∆ Γ ⇒ ∆ , A ∧ B Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 2 / 34

The G3[mic] systems Initial sequents P , Γ ⇒ ∆ , P ( ⊥ , Γ ⇒ ∆ , ⊥ in G3m ) Logical Rules ⊥ , Γ ⇒ ∆ ( not in G3m ) A , B , Γ ⇒ ∆ Γ ⇒ ∆ , A Γ ⇒ ∆ , B L ∧ R ∧ A ∧ B , Γ ⇒ ∆ Γ ⇒ ∆ , A ∧ B Γ ⇒ ∆ , A B , Γ ⇒ ∆ A , Γ ⇒ ∆ , B L → R → A → B , Γ ⇒ ∆ Γ ⇒ ∆ , A → B Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 2 / 34

A [ x / t ] , ∀ xA , Γ ⇒ ∆ Γ ⇒ ∆ , A [ x / a ] L ∀ R ∀ ∀ xA , Γ ⇒ ∆ Γ ⇒ ∆ , ∀ xA Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 3 / 34

A [ x / t ] , ∀ xA , Γ ⇒ ∆ Γ ⇒ ∆ , A [ x / a ] L ∀ R ∀ ∀ xA , Γ ⇒ ∆ Γ ⇒ ∆ , ∀ xA A [ x / a ] , Γ ⇒ ∆ Γ ⇒ ∆ , ∃ xA , A [ x / t ] L ∃ R ∃ ∃ xA , Γ ⇒ ∆ Γ ⇒ ∆ , ∃ xA Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 3 / 34

In G3mi the rules L → , R → and R ∀ are replaced by: A → B , Γ ⇒ ∆ , A B , Γ ⇒ ∆ A , Γ ⇒ B L i → R i → A → B , Γ ⇒ ∆ Γ ⇒ ∆ , A → B Γ ⇒ A [ x / a ] R i ∀ Γ ⇒ ∆ , ∀ xA Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 4 / 34

GK3 [ mic ] Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 5 / 34

GK3 [ mic ] A ∧ B , A , B , Γ ⇒ ∆ A ∧ B , Γ ⇒ ∆ Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 5 / 34

Atomic Rules Q 1 , Γ 1 ⇒ ∆ 1 , � � Q n , Γ n ⇒ ∆ n , � � Q ′ Q ′ 1 . . . n � P , Γ 1 , . . . , Γ n ⇒ ∆ 1 , . . . , ∆ n , � P ′ Q 1 , � Q n , � where � 1 , . . . , � n , � P , � P ′ are sequences (possibly empty) of Q ′ Q ′ atomic formulae and Γ 1 , . . . , Γ n , ∆ 1 , . . . ∆ n are finite sequences (possibly empty) of formulae that are not active in t Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 6 / 34

Equality Rules t = t , Γ ⇒ ∆ Ref Ref Γ ⇒ ∆ Γ ⇒ ∆ , t = t Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 7 / 34

Equality Rules t = t , Γ ⇒ ∆ Ref Ref Γ ⇒ ∆ Γ ⇒ ∆ , t = t s = r , P [ x / s ] , P [ x / r ] , Γ ⇒ ∆ Repl s = r , P [ x / s ] , Γ ⇒ ∆ Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 7 / 34

Equality Rules t = t , Γ ⇒ ∆ Ref Ref Γ ⇒ ∆ Γ ⇒ ∆ , t = t s = r , P [ x / s ] , P [ x / r ] , Γ ⇒ ∆ Repl s = r , P [ x / s ] , Γ ⇒ ∆ r = s , P [ x / r ] , Γ ⇒ ∆ r = s , Γ ⇒ ∆ , P [ x / r ] Repl r Repl l 1 r = s , P [ x / s ] , Γ ⇒ ∆ r = s , Γ ⇒ ∆ , P [ x / s ] 1 Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 7 / 34

Equality Rules t = t , Γ ⇒ ∆ Ref Ref Γ ⇒ ∆ Γ ⇒ ∆ , t = t s = r , P [ x / s ] , P [ x / r ] , Γ ⇒ ∆ Repl s = r , P [ x / s ] , Γ ⇒ ∆ r = s , P [ x / r ] , Γ ⇒ ∆ r = s , Γ ⇒ ∆ , P [ x / r ] Repl r Repl l 1 r = s , P [ x / s ] , Γ ⇒ ∆ r = s , Γ ⇒ ∆ , P [ x / s ] 1 s = r , P [ x / r ] , Γ ⇒ ∆ s = r , Γ ⇒ ∆ , P [ x / r ] Repl r Repl l 2 s = r , P [ x / s ] , Γ ⇒ ∆ s = r , Γ ⇒ ∆ , P [ x / s ] 2 Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 7 / 34

Γ 1 ⇒ ∆ 1 , r = s Γ 2 ⇒ ∆ 2 , P [ x / r ] CNG Γ 1 , Γ 2 ⇒ ∆ 1 , ∆ 2 , P [ x / s ] Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 8 / 34

Γ 1 ⇒ ∆ 1 , r = s Γ 2 ⇒ ∆ 2 , P [ x / r ] CNG Γ 1 , Γ 2 ⇒ ∆ 1 , ∆ 2 , P [ x / s ] Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 8 / 34

Admissibilty of the structural rules in G3 [ mic ] Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 9 / 34

Admissibilty of the structural rules in G3 [ mic ] i) Hight preserving admissibility of the weakening rules Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 9 / 34

Admissibilty of the structural rules in G3 [ mic ] i) Hight preserving admissibility of the weakening rules ii) Height preserving invertibility of the logical rules different from R i → and R i ∀ Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 9 / 34

Admissibilty of the structural rules in G3 [ mic ] i) Hight preserving admissibility of the weakening rules ii) Height preserving invertibility of the logical rules different from R i → and R i ∀ iii) Admissibility of the contraction rules via hight preserving admissibility Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 9 / 34

Admissibilty of the structural rules in G3 [ mic ] i) Hight preserving admissibility of the weakening rules ii) Height preserving invertibility of the logical rules different from R i → and R i ∀ iii) Admissibility of the contraction rules via hight preserving admissibility iv) Admissibility of the Cut rule, using i ) and iii ) “all the time" Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 9 / 34

a = f ( a ) , a = f ( a ) ⇒ a = f ( f ( a )) has derivations of height equal 1 in the systems obtained by adding Ref and Repl to G3 [ mic ] : Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 10 / 34

a = f ( a ) , a = f ( a ) ⇒ a = f ( f ( a )) has derivations of height equal 1 in the systems obtained by adding Ref and Repl to G3 [ mic ] : a = f ( a ) , a = f ( a ) , a = f ( f ( a )) ⇒ a = f ( f ( a )) a = f ( a ) , a = f ( a ) ⇒ a = f ( f ( a )) Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 10 / 34

a = f ( a ) , a = f ( a ) ⇒ a = f ( f ( a )) has derivations of height equal 1 in the systems obtained by adding Ref and Repl to G3 [ mic ] : a = f ( a ) , a = f ( a ) , a = f ( f ( a )) ⇒ a = f ( f ( a )) a = f ( a ) , a = f ( a ) ⇒ a = f ( f ( a )) but a = f ( a ) ⇒ a = f ( f ( a )) cannot have a derivation of height less than or equal 1 in such a system. Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 10 / 34

The Separation Theorem Definition A derivation in G3 [ mic ] R + Cut cs is said to be separated if no logical inference precedes an R or Cut cs -inference. Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 11 / 34

The Separation Theorem Definition A derivation in G3 [ mic ] R + Cut cs is said to be separated if no logical inference precedes an R or Cut cs -inference. Theorem Every derivation in G3 [ mic ] R + Cut cs can be transformed into a separated derivation of its endsequent. Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 11 / 34

The R -Admissibility Theorem Theorem If the structural rules are admissible in R , then they are admissible in G3 [ mic ] R as well. Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 12 / 34

The R -Admissibility Theorem Theorem If the structural rules are admissible in R , then they are admissible in G3 [ mic ] R as well. E F , F , Γ ⇒ ∆ F , Γ ⇒ ∆ Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 12 / 34

The R -Admissibility Theorem Theorem If the structural rules are admissible in R , then they are admissible in G3 [ mic ] R as well. E F , F , Γ ⇒ ∆ F , Γ ⇒ ∆ can be replaced by: I E F , Γ ⇒ F F , F , Γ ⇒ ∆ F , Γ ⇒ ∆ where, in case F is not atomic, I is a derivation in G3m or in G3i . Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 12 / 34

Proof of the Separation Theorem Lemma If the premisses of an R -inference R have a separated derivation, then its conclusion has a separated derivation. Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 13 / 34

Proof of the Separation Theorem Lemma If Γ ⇒ ∆ , A and A , Γ ⇒ ∆ have separated derivation in G3c [ mic ] R + Cut cs , then Γ ⇒ ∆ has a separated derivation in the same system. Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 14 / 34