Sequent Calculus Cut Elimination Akim Demaille [email protected] EPITA — École Pour l’Informatique et les Techniques Avancées June 10, 2016 Sequent Calculus Cut Elimination 1 Problems of Natural Deduction 2 Sequent Calculus 3 Natural Deduction in Sequent Calculus A. Demaille Sequent Calculus, Cut Elimination 2 / 40 Preamble The following slides are implicitly dedicated to classical logic. A. Demaille Sequent Calculus, Cut Elimination 3 / 40 Problems of Natural Deduction 1 Problems of Natural Deduction 2 Sequent Calculus 3 Natural Deduction in Sequent Calculus A. Demaille Sequent Calculus, Cut Elimination 4 / 40 Normalization A. Demaille Sequent Calculus, Cut Elimination 5 / 40 Proofs hard to find Some elimination rules used formulas coming out of the blue. [A] [B] · · · · · · · · · A _ B C C _E C A. Demaille Sequent Calculus, Cut Elimination 6 / 40 Negation is awkward A. Demaille Sequent Calculus, Cut Elimination 7 / 40 Sequent Calculus 1 Problems of Natural Deduction 2 Sequent Calculus Syntax LK — Classical Sequent Calculus Cut Elimination 3 Natural Deduction in Sequent Calculus A. Demaille Sequent Calculus, Cut Elimination 8 / 40 Gerhard Karl Erich Gentzen (1909–1945) German logician and mathematician. SA (1933) An assistant of David Hilbert in Göttingen (1935–1939). Joined the Nazi Party (1937). Worked on the V2. Died of malnutrition in a prison camp (August 4, 1945). A. Demaille Sequent Calculus, Cut Elimination 9 / 40 Syntax 1 Problems of Natural Deduction 2 Sequent Calculus Syntax LK — Classical Sequent Calculus Cut Elimination 3 Natural Deduction in Sequent Calculus A. Demaille Sequent Calculus, Cut Elimination 10 / 40 Sequents Sequent A sequent is an expression Γ ` ∆, where Γ; ∆ are (finite) sequences of formulas. Variants: Sets Γ; ∆ can be taken as (finite) sets. Simplifies the structural group Prevents close examination of. the structural rules Single Sided Sequents can be forced to be ` Γ Half the number of rules is needed Not for intuitionistic systems A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents Sequent A sequent is an expression Γ ` ∆, where Γ; ∆ are (finite) sequences of formulas. Variants: Sets Γ; ∆ can be taken as (finite) sets. Simplifies the structural group Prevents close examination of. the structural rules Single Sided Sequents can be forced to be ` Γ Half the number of rules is needed Not for intuitionistic systems A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents Sequent A sequent is an expression Γ ` ∆, where Γ; ∆ are (finite) sequences of formulas. Variants: Sets Γ; ∆ can be taken as (finite) sets. Simplifies the structural group Prevents close examination of. the structural rules Single Sided Sequents can be forced to be ` Γ Half the number of rules is needed Not for intuitionistic systems A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents Sequent A sequent is an expression Γ ` ∆, where Γ; ∆ are (finite) sequences of formulas. Variants: Sets Γ; ∆ can be taken as (finite) sets. Simplifies the structural group Prevents close examination of. the structural rules Single Sided Sequents can be forced to be ` Γ Half the number of rules is needed Not for intuitionistic systems A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents Sequent A sequent is an expression Γ ` ∆, where Γ; ∆ are (finite) sequences of formulas. Variants: Sets Γ; ∆ can be taken as (finite) sets. Simplifies the structural group Prevents close examination of. the structural rules Single Sided Sequents can be forced to be ` Γ Half the number of rules is needed Not for intuitionistic systems A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents Sequent A sequent is an expression Γ ` ∆, where Γ; ∆ are (finite) sequences of formulas. Variants: Sets Γ; ∆ can be taken as (finite) sets. Simplifies the structural group Prevents close examination of. the structural rules Single Sided Sequents can be forced to be ` Γ Half the number of rules is needed Not for intuitionistic systems A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents Sequent A sequent is an expression Γ ` ∆, where Γ; ∆ are (finite) sequences of formulas. Variants: Sets Γ; ∆ can be taken as (finite) sets. Simplifies the structural group Prevents close examination of. the structural rules Single Sided Sequents can be forced to be ` Γ Half the number of rules is needed Not for intuitionistic systems A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Sequents Sequent A sequent is an expression Γ ` ∆, where Γ; ∆ are (finite) sequences of formulas. Variants: Sets Γ; ∆ can be taken as (finite) sets. Simplifies the structural group Prevents close examination of. the structural rules Single Sided Sequents can be forced to be ` Γ Half the number of rules is needed Not for intuitionistic systems A. Demaille Sequent Calculus, Cut Elimination 11 / 40 Reading a Sequent If all the formulas of Γ are true, Γ ` ∆ then one of the formulas of ∆ is true. Commas on the left hand side stand for “and” Turnstile, `, stands for “implies” Commas on the right hand side stand for “or” A. Demaille Sequent Calculus, Cut Elimination 12 / 40 Reading a Sequent If all the formulas of Γ are true, Γ ` ∆ then one of the formulas of ∆ is true. Commas on the left hand side stand for “and” Turnstile, `, stands for “implies” Commas on the right hand side stand for “or” A. Demaille Sequent Calculus, Cut Elimination 12 / 40 Reading a Sequent If all the formulas of Γ are true, Γ ` ∆ then one of the formulas of ∆ is true. Commas on the left hand side stand for “and” Turnstile, `, stands for “implies” Commas on the right hand side stand for “or” A. Demaille Sequent Calculus, Cut Elimination 12 / 40 Reading a Sequent If all the formulas of Γ are true, Γ ` ∆ then one of the formulas of ∆ is true. Commas on the left hand side stand for “and” Turnstile, `, stands for “implies” Commas on the right hand side stand for “or” A. Demaille Sequent Calculus, Cut Elimination 12 / 40 ` AA is true Γ ` Γ is in contradiction A ` :A ` contradiction Some Special Sequents Γ ` AA is true under the hypotheses Γ A. Demaille Sequent Calculus, Cut Elimination 13 / 40 Γ ` Γ is in contradiction A ` :A ` contradiction Some Special Sequents Γ ` AA is true under the hypotheses Γ ` AA is true A. Demaille Sequent Calculus, Cut Elimination 13 / 40 A ` :A ` contradiction Some Special Sequents Γ ` AA is true under the hypotheses Γ ` AA is true Γ ` Γ is in contradiction A. Demaille Sequent Calculus, Cut Elimination 13 / 40 ` contradiction Some Special Sequents Γ ` AA is true under the hypotheses Γ ` AA is true Γ ` Γ is in contradiction A ` :A A. Demaille Sequent Calculus, Cut Elimination 13 / 40 Some Special Sequents Γ ` AA is true under the hypotheses Γ ` AA is true Γ ` Γ is in contradiction A ` :A ` contradiction A. Demaille Sequent Calculus, Cut Elimination 13 / 40 LK — Classical Sequent Calculus 1 Problems of Natural Deduction 2 Sequent Calculus Syntax LK — Classical Sequent Calculus Cut Elimination 3 Natural Deduction in Sequent Calculus A. Demaille Sequent Calculus, Cut Elimination 14 / 40 LK — Classical Sequent Calculus LK — Gentzen 1934 logistischerklassischer Kalkül. There are several possible exposures Different sets of inference rules We follow [Girard, 2011] A. Demaille Sequent Calculus, Cut Elimination 15 / 40 Identity Group Γ ` A; ∆ Γ0; A ` ∆0 Id Cut A ` A Γ; Γ0 ` ∆; ∆0 Gentzen’s Hauptsatz The cut rule is redundant, i.e., any sequent provable with cuts is provable without. A. Demaille Sequent Calculus, Cut Elimination 16 / 40 Identity Group Γ ` A; ∆ Γ0; A ` ∆0 Id Cut A ` A Γ; Γ0 ` ∆; ∆0 Gentzen’s Hauptsatz The cut rule is redundant, i.e., any sequent provable with cuts is provable without. A. Demaille Sequent Calculus, Cut Elimination 16 / 40 Γ ` ∆ Γ ` ∆ `W W` Γ ` A; ∆ Γ; A ` ∆ Γ ` A; A; ∆ Γ; A; A ` ∆ `C C` Γ ` A; ∆ Γ; A ` ∆ Structural Group Γ ` ∆ Γ ` ∆ `X X` Γ ` τ(∆) σ(Γ) ` ∆ A. Demaille Sequent Calculus, Cut Elimination 17 / 40 Γ ` A; A; ∆ Γ; A; A ` ∆ `C C` Γ ` A; ∆ Γ; A ` ∆ Structural Group Γ ` ∆ Γ ` ∆ `X X` Γ ` τ(∆) σ(Γ) ` ∆ Γ ` ∆ Γ ` ∆ `W W` Γ ` A; ∆ Γ; A ` ∆ A. Demaille Sequent Calculus, Cut Elimination 17 / 40 Structural Group Γ ` ∆ Γ ` ∆ `X X` Γ ` τ(∆) σ(Γ) ` ∆ Γ ` ∆ Γ ` ∆ `W W` Γ ` A; ∆ Γ; A ` ∆ Γ ` A; A; ∆ Γ; A; A ` ∆ `C C` Γ ` A; ∆ Γ; A ` ∆ A. Demaille Sequent Calculus, Cut Elimination 17 / 40 Logical Group: Negation Γ; A ` ∆ Γ ` A; ∆ `: :` Γ ` :A; ∆ Γ; :A ` ∆ A. Demaille Sequent Calculus, Cut Elimination 18 / 40 Additive ˘ Multiplicative ⊗ Γ; A ` ∆ l^` Γ; A ^ B ` ∆ Γ; B ` ∆ r^` Γ; A ^ B ` ∆ Γ ` A; ∆ Γ0 ` B; ∆0 Γ; A; B ` ∆ `^ ^` Γ; Γ0 ` A ^ B; ∆; ∆0 Γ; A ^ B ` ∆ Logical Group: Conjunction Γ ` A; ∆ Γ ` B; ∆ `^ Γ ` A ^ B; ∆ A. Demaille Sequent Calculus, Cut Elimination 19 / 40 Additive ˘ Multiplicative ⊗ Γ; B ` ∆ r^` Γ; A ^ B ` ∆ Γ ` A; ∆ Γ0 ` B; ∆0 Γ; A; B ` ∆ `^ ^` Γ; Γ0 ` A ^ B; ∆; ∆0 Γ; A ^ B ` ∆ Logical Group: Conjunction Γ; A ` ∆ l^` Γ ` A; ∆ Γ ` B; ∆ Γ; A ^ B ` ∆ `^ Γ ` A ^ B; ∆ A. Demaille Sequent Calculus, Cut Elimination 19 / 40 Multiplicative ⊗ Additive ˘ Γ ` A; ∆ Γ0 ` B; ∆0 Γ; A; B ` ∆ `^ ^` Γ; Γ0 ` A ^ B; ∆; ∆0 Γ; A ^ B ` ∆ Logical Group: Conjunction Γ; A ` ∆ l^` Γ ` A; ∆ Γ ` B; ∆ Γ; A ^ B ` ∆ `^ Γ ` A ^ B; ∆ Γ; B ` ∆ r^` Γ; A ^ B ` ∆ A. Demaille Sequent Calculus, Cut Elimination 19 / 40 Additive ˘ Multiplicative ⊗ Logical Group: Conjunction Γ; A ` ∆ l^` Γ ` A; ∆ Γ ` B; ∆ Γ; A ^ B ` ∆ `^ Γ ` A ^ B; ∆ Γ; B ` ∆ r^` Γ; A ^ B ` ∆ Γ ` A; ∆ Γ0 ` B; ∆0 Γ; A; B ` ∆ `^ ^` Γ; Γ0 ` A ^ B; ∆; ∆0 Γ; A ^ B ` ∆ A. Demaille Sequent Calculus, Cut Elimination 19 / 40 ˘ ⊗ Logical Group: Conjunction Γ; A ` ∆ l^` Γ ` A; ∆ Γ ` B; ∆ Γ; A ^ B ` ∆ Additive `^ Γ ` A ^ B; ∆ Γ; B ` ∆ r^` Γ; A ^ B ` ∆ Γ ` A; ∆ Γ0 ` B; ∆0 Γ; A; B ` ∆ Multiplicative `^ ^` Γ; Γ0 ` A ^ B; ∆; ∆0 Γ; A ^ B ` ∆ A.