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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

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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

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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

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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

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The G3[mic] systems

Initial sequents P, Γ ⇒ ∆, P (⊥, Γ ⇒ ∆, ⊥ in G3m) Logical Rules ⊥, Γ ⇒ ∆ ( not in G3m) A, B, Γ ⇒ ∆ L∧ Γ ⇒ ∆, A Γ ⇒ ∆, B R∧ A ∧ B, Γ ⇒ ∆ Γ ⇒ ∆, A ∧ B

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 2 / 34

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The G3[mic] systems

Initial sequents P, Γ ⇒ ∆, P (⊥, Γ ⇒ ∆, ⊥ in G3m) Logical Rules ⊥, Γ ⇒ ∆ ( not in G3m) A, B, Γ ⇒ ∆ L∧ Γ ⇒ ∆, A Γ ⇒ ∆, B R∧ A ∧ B, Γ ⇒ ∆ Γ ⇒ ∆, A ∧ B Γ ⇒ ∆, A B, Γ ⇒ ∆ L → A, Γ ⇒ ∆, B R → A → B, Γ ⇒ ∆ Γ ⇒ ∆, A → B

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 2 / 34

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A[x/t], ∀xA, Γ ⇒ ∆ L∀ Γ ⇒ ∆, A[x/a] R∀ ∀xA, Γ ⇒ ∆ Γ ⇒ ∆, ∀xA

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 3 / 34

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A[x/t], ∀xA, Γ ⇒ ∆ L∀ Γ ⇒ ∆, A[x/a] R∀ ∀xA, Γ ⇒ ∆ Γ ⇒ ∆, ∀xA A[x/a], Γ ⇒ ∆ L∃ Γ ⇒ ∆, ∃xA, A[x/t] R∃ ∃xA, Γ ⇒ ∆ Γ ⇒ ∆, ∃xA

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 3 / 34

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In G3mi the rules L →, R → and R∀ are replaced by: A → B, Γ ⇒ ∆, A B, Γ ⇒ ∆ Li → A, Γ ⇒ B Ri → A → B, Γ ⇒ ∆ Γ ⇒ ∆, A → B Γ ⇒ A[x/a] Ri∀ Γ ⇒ ∆, ∀xA

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 4 / 34

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GK3[mic]

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 5 / 34

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GK3[mic] A ∧ B, A, B, Γ ⇒ ∆ A ∧ B, Γ ⇒ ∆

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 5 / 34

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Atomic Rules

  • Q1, Γ1 ⇒ ∆1,

Q′

1 . . .

  • Qn, Γn ⇒ ∆n,

Q′

n

  • P, Γ1, . . . , Γn ⇒ ∆1, . . . , ∆n,

P′ where Q1, Q′

1, . . . ,

Qn, Q′

n,

P, P′ are sequences (possibly empty) of atomic formulae and Γ1, . . . , Γn, ∆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

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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

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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

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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 ⇒ ∆ Repll

1

r = s, Γ ⇒ ∆, P[x/r] Replr

1

r = s, P[x/s], Γ ⇒ ∆ r = s, Γ ⇒ ∆, P[x/s]

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 7 / 34

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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 ⇒ ∆ Repll

1

r = s, Γ ⇒ ∆, P[x/r] Replr

1

r = s, P[x/s], Γ ⇒ ∆ r = s, Γ ⇒ ∆, P[x/s] s = r, P[x/r], Γ ⇒ ∆ Repll

2

s = r, Γ ⇒ ∆, P[x/r] Replr

2

s = r, P[x/s], Γ ⇒ ∆ s = r, Γ ⇒ ∆, P[x/s]

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 7 / 34

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Γ1 ⇒ ∆1, r = s Γ2 ⇒ ∆2, P[x/r] CNG Γ1, Γ2 ⇒ ∆1, ∆2, P[x/s]

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 8 / 34

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Γ1 ⇒ ∆1, r = s Γ2 ⇒ ∆2, P[x/r] CNG Γ1, Γ2 ⇒ ∆1, ∆2, P[x/s]

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 8 / 34

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Admissibilty of the structural rules in G3[mic]

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 9 / 34

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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

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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 Ri → and Ri∀

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 9 / 34

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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 Ri → and Ri∀ iii) Admissibility of the contraction rules via hight preserving admissibility

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 9 / 34

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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 Ri → and Ri∀ 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

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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

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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

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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

  • r equal 1 in such a system.

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 10 / 34

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The Separation Theorem

Definition A derivation in G3[mic]R + Cutcs is said to be separated if no logical inference precedes an R or Cutcs-inference.

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 11 / 34

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The Separation Theorem

Definition A derivation in G3[mic]R + Cutcs is said to be separated if no logical inference precedes an R or Cutcs-inference. Theorem Every derivation in G3[mic]R + Cutcs can be transformed into a separated derivation of its endsequent.

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 11 / 34

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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

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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

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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

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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

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Proof of the Separation Theorem

Lemma If Γ ⇒ ∆, A and A, Γ ⇒ ∆ have separated derivation in G3c[mic]R + Cutcs, then Γ ⇒ ∆ has a separated derivation in the same system.

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 14 / 34

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Proof of the Separation Theorem

Lemma a) Hight-preserving separated invertibility of the logical rules in G3cR + Cutcs If the conclusion of a logical rule has a separated derivation of height bounded by h, then also its premisses have separated derivations of height bounded by h. b) The same holds for G3[mi]R + Cutcs, except for the rules Ri → and Ri∀.

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 15 / 34

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Proof of the Separation Theorem

Both D and E end with a logical rule:

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 16 / 34

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Proof of the Separation Theorem

Both D and E end with a logical rule: D E Γ ⇒ ∆, A A, Γ ⇒ ∆ Γ ⇒ ∆

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 16 / 34

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Proof of the Separation Theorem

Both D and E end with a logical rule: D E Γ ⇒ ∆, A A, Γ ⇒ ∆ Γ ⇒ ∆ Principal induction on h(A), secondary induction on h(D) + h(E)

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 16 / 34

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Proof of the Separation Theorem

Classical case A not principal in the last inference of D or in the last inference of E A principal in both the last inference of D and the last inference of E

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 17 / 34

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Proof of the Separation Theorem

Classical case A not principal in the last inference of D or in the last inference of E A principal in both the last inference of D and the last inference of E Minimal and intuitionistic case A not principal in the last inference of D A principal in the last inference of D but not in the last inference of E A principal in both the last inference of D and the last inference of E

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 17 / 34

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Proof of the Separation Theorem

D0 E0 Γ ⇒ ∆′, E → F, ∃xB, B[x/t] ∃xB, E, Γ ⇒ F Γ ⇒ ∆′, E → F, ∃xB ∃xB, Γ ⇒ ∆′, E → F Γ ⇒ ∆′, E → F

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 18 / 34

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Proof of the Separation Theorem

D0 E0 Γ ⇒ ∆′, E → F, ∃xB, B[x/t] ∃xB, E, Γ ⇒ F Γ ⇒ ∆′, E → F, ∃xB ∃xB, Γ ⇒ ∆′, E → F Γ ⇒ ∆′, E → F is transformed into: E0 D0 ∃xB, E, Γ ⇒ F Γ ⇒ ∆′, E → F, ∃xB, B[x/t] ∃xB, Γ ⇒ ∆′, E → F, B[x/t] ind Γ ⇒ ∆′, E → F, B[x/t]

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 18 / 34

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Proof of the Separation Theorem

E0 D0 ∃xB, E, Γ ⇒ F Γ ⇒ ∆′, E → F, ∃xB, B[x/t] ∃xB, Γ ⇒ ∆′, E → F, B[x/t] ind Γ ⇒ ∆′, E → F, B[x/t]

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 19 / 34

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Proof of the Separation Theorem

E0 D0 ∃xB, E, Γ ⇒ F Γ ⇒ ∆′, E → F, ∃xB, B[x/t] ∃xB, Γ ⇒ ∆′, E → F, B[x/t] ind Γ ⇒ ∆′, E → F, B[x/t] E0 ∃xB, E, Γ ⇒ F ∃xB, Γ ⇒ ∆′, E → F inv B[x/a], Γ ⇒ ∆′, E → F Sub[a/t] B[x/t], Γ ⇒ ∆′, E → F

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 19 / 34

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Proof of the Separation Theorem

E0 D0 ∃xB, E, Γ ⇒ F Γ ⇒ ∆′, E → F, ∃xB, B[x/t] ∃xB, Γ ⇒ ∆′, E → F, B[x/t] ind Γ ⇒ ∆′, E → F, B[x/t] E0 ∃xB, E, Γ ⇒ F ∃xB, Γ ⇒ ∆′, E → F inv B[x/a], Γ ⇒ ∆′, E → F Sub[a/t] B[x/t], Γ ⇒ ∆′, E → F Γ ⇒ ∆′, E → F, B[x/t] B[x/t], Γ ⇒ ∆′, E → F ind Γ ⇒ ∆′, E → F

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 19 / 34

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Logical applications

For R = ∅: Admissibility of the structural rules in G3[mic], without using height-preserving admissibility of the contraction rules

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 20 / 34

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Logical applications

If the structural rules are admissible in R, then they are admissible in GK3[mic]R as well.

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 21 / 34

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Logical applications

If the structural rules are admissible in R, then they are admissible in GK3[mic]R as well. A ⇒ A

LW

B ⇒ B

LW

A, B ⇒ A A, B ⇒ B

R∧

A, B ⇒ A ∧ B A ∧ B, A, B, Γ ⇒ ∆

Cut

A, B, A, B, Γ ⇒ ∆

LC

A, B, Γ ⇒ ∆

L∧

A ∧ B, Γ ⇒ ∆

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 21 / 34

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Applications to logic with equality

The systems that contain Ref or Ref and at least one of the equality rules for which the structural rules are admissible are all equivalent.

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 22 / 34

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Applications to logic with equality

G3[mic]= = G3[mic] + {Ref, Repl} t = t, Γ ⇒ ∆ Ref s = r, P[x/s], P[x/r], Γ ⇒ ∆ Repl Γ ⇒ ∆ s = r, P[x/s], Γ ⇒ ∆

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 23 / 34

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Applications to logic with equality

G3[mic]= = G3[mic] + {Ref, Repl} t = t, Γ ⇒ ∆ Ref s = r, P[x/s], P[x/r], Γ ⇒ ∆ Repl Γ ⇒ ∆ s = r, P[x/s], Γ ⇒ ∆ Theorem The structural rules are admissible in G3[mic]=

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 23 / 34

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Applications to logic with equality

G3[mic]= = G3[mic] + {Ref, Repl} t = t, Γ ⇒ ∆ Ref s = r, P[x/s], P[x/r], Γ ⇒ ∆ Repl Γ ⇒ ∆ s = r, P[x/s], Γ ⇒ ∆ Theorem The structural rules are admissible in G3[mic]=

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 23 / 34

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Applications to logic with equality

t = t, Γ ⇒ ∆ Ref s = r, P[x/r], Γ ⇒ ∆ Repll

2

Γ ⇒ ∆ s = r, P[x/s], Γ ⇒ ∆

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 24 / 34

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SLIDE 53

Applications to logic with equality

t = t, Γ ⇒ ∆ Ref s = r, P[x/r], Γ ⇒ ∆ Repll

2

Γ ⇒ ∆ s = r, P[x/s], Γ ⇒ ∆ Theorem The structural rules are admissible in G3[mic] + Ref + Repl l

2

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 24 / 34

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Applications to logic with equality

Ref Γ ⇒ ∆, t = t r = s, Γ ⇒ ∆, P[x/r] Replr

1

s = r, Γ ⇒ ∆, P[x/r] Replr

2

r = s, Γ ⇒ ∆, P[x/s] s = r, Γ ⇒ ∆, P[x/s]

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 25 / 34

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Applications to logic with equality

Ref Γ ⇒ ∆, t = t r = s, Γ ⇒ ∆, P[x/r] Replr

1

s = r, Γ ⇒ ∆, P[x/r] Replr

2

r = s, Γ ⇒ ∆, P[x/s] s = r, Γ ⇒ ∆, P[x/s] Theorem The structural rules are admissible in G3[mic] + Ref + Repl r

1 + Repl r 2

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 25 / 34

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Applications to logic with equality

P not an equality, E an equality: r = s, P[x/r], Γ ⇒ ∆ Repll=

1

r = s, Γ ⇒ ∆, E[x/r] Replr=

1

r = s, P[x/s], Γ ⇒ ∆ r = s, Γ ⇒ ∆, E[x/s]

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 26 / 34

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Applications to logic with equality

P not an equality, E an equality: r = s, P[x/r], Γ ⇒ ∆ Repll=

1

r = s, Γ ⇒ ∆, E[x/r] Replr=

1

r = s, P[x/s], Γ ⇒ ∆ r = s, Γ ⇒ ∆, E[x/s] s = r, P[x/r], Γ ⇒ ∆ Repll =

2

s = r, Γ ⇒ ∆, E[x/r] Replr=

2

s = r, P[x/s], Γ ⇒ ∆ s = r, Γ ⇒ ∆, E[x/s] Theorem The structural rules are admissible in G3[mic] + Ref + Repl l=

1

+ Repl l=

2 Repl r= 1

+ Repl r=

2

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 26 / 34

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SLIDE 58

Applications to logic with equality

Repll+

1 and Repll+ 2 same as Repll 1 and Repll 2 except that their

instances concerning an equality E are strengthened into: s = r, E[x/s], E[x/r], Γ ⇒ ∆ and r = s, E[x/s], E[x/r], Γ ⇒ ∆ s = r, E[x/s], Γ ⇒ ∆ r = s, E[x/s], Γ ⇒ ∆ respectively.

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 27 / 34

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SLIDE 59

Applications to logic with equality

Repll+

1 and Repll+ 2 same as Repll 1 and Repll 2 except that their

instances concerning an equality E are strengthened into: s = r, E[x/s], E[x/r], Γ ⇒ ∆ and r = s, E[x/s], E[x/r], Γ ⇒ ∆ s = r, E[x/s], Γ ⇒ ∆ r = s, E[x/s], Γ ⇒ ∆ respectively. Theorem The structural rules are admissible in G3[mic] + Ref + Repl l+

2

+ Repl r

2

The same holds for G3[mic] + Ref + Repl l+

1

+ Repl r

1 .

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 27 / 34

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SLIDE 60

Applications to logic with equality

Repll+

1 and Repll+ 2 same as Repll 1 and Repll 2 except that their

instances concerning an equality E are strengthened into: s = r, E[x/s], E[x/r], Γ ⇒ ∆ and r = s, E[x/s], E[x/r], Γ ⇒ ∆ s = r, E[x/s], Γ ⇒ ∆ r = s, E[x/s], Γ ⇒ ∆ respectively. Theorem The structural rules are admissible in G3[mic] + Ref + Repl l+

2

+ Repl r

2

The same holds for G3[mic] + Ref + Repl l+

1

+ Repl r

1 .

Problem Can we do without strengthening Repll

1 and Repll 2 into

Repll+

1 and Repll+ 2 ?

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 27 / 34

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SLIDE 61

Applications to logic with equality

Let ≺ be any antisymmetric relation on terms, i.e. if r ≺ s, then s ≺ r. Problem Are the structural rules admissible over Γ ⇒ ∆, t = t and r = s, P[x/r], Γ ⇒ ∆ r ≺ s r = s, Γ ⇒ ∆, P[x/r] r ≺ s r = s, P[x/s], Γ ⇒ ∆ r = s, Γ ⇒ ∆, P[x/s] s = r, P[x/r], Γ ⇒ ∆ s ≺ r s = r, Γ ⇒ ∆, P[x/r] s ≺ r s = r, P[x/s], Γ ⇒ ∆ s = r, Γ ⇒ ∆, P[x/s] ? Does it depend on ≺?

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 28 / 34

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Applications to Semantic Tableau Systems

Tableau System: Can extend a branch by adding: T.t = t (Tableau Reflexivity Rule) T.P[x/r], if it contains T.s = r and T.P[x/s] (Tableau Replacement Rule)

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 29 / 34

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Applications to Semantic Tableau Systems

Alternate Tableau System Can close a branch if it contains F.t = t and can extend a branch that contains T.r = s or T.s = r by adding: T.P[x/r] if it contains T.P[x/s] where P is any atomic formula F.E[x/r] if it contains F.E[x/s] where E is an equality (Alternate Tableau Replacement Rule)

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 30 / 34

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SLIDE 64

Applications to Semantic Tableau Systems

A closed semantic tableau for T.Γ, F.∆ corresponds to a derivation of Γ ⇒ ∆ in GK3c + Ref + Repl.

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 31 / 34

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Applications to Semantic Tableau Systems

A closed semantic tableau for T.Γ, F.∆ corresponds to a derivation of Γ ⇒ ∆ in GK3c + Ref + Repl. GK3c + Ref + Repl is complete

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 31 / 34

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SLIDE 66

Applications to Semantic Tableau Systems

A closed semantic tableau for T.Γ, F.∆ corresponds to a derivation of Γ ⇒ ∆ in GK3c + Ref + Repl. GK3c + Ref + Repl is complete Every system equivalent to GK3c + Ref + Repl is complete

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 31 / 34

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SLIDE 67

Applications to Semantic Tableau Systems

A closed semantic tableau for T.Γ, F.∆ corresponds to a derivation of Γ ⇒ ∆ in GK3c + Ref + Repl. GK3c + Ref + Repl is complete Every system equivalent to GK3c + Ref + Repl is complete G3c + Ref + Repll

2 is complete

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 31 / 34

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SLIDE 68

Applications to Semantic Tableau Systems

Theorem The Tableau System remains sound and complete even if the strictness condition is imposed on the non γ logical rules, including the Tableau Replacement Rule.

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 32 / 34

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SLIDE 69

Applications to Semantic Tableau Systems

Theorem The Tableau System remains sound and complete even if the strictness condition is imposed on the non γ logical rules, including the Tableau Replacement Rule. Theorem The Alternate Tableau System is complete and remains complete also when the Alternate Tableau Replacement Rule is applied to T.P only when P is not an equality and the full strictness condition is imposed.

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 32 / 34

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SLIDE 70

Applications to Semantic Tableau Systems

Theorem The Alternate Replacement Rules can be made directional as in the Tableau System, i.e. only left-right replacement is allowed, provided it is applied to formulae of the form T.P as well as F.P. Furthermore also the strictness condition can be imposed, except when the Alternate Replacement Rule is applied to signed formulae of the form T.E, where E is an equality.

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 33 / 34

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SLIDE 71

THANK YOU !

Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules 34 / 34