Normal forms in sequent calculus Lu´ ıs Pinto Centro de Matem´ atica, Univ. Minho, Portugal Computational Logic Workshop in honour of Roy Dyckhoff 18-19 November 2011 Univ. St Andrews, Scotland Joint work with Jos´ e Esp´ ırito Santo and Maria Jo˜ ao Frade
Plan Part I: Revisiting permutative conversions in sequent calculus Part II: A calculus of multiary sequent terms Part III: β -normal λ -terms in sequent calculus Part IV: Refinements
PART I Revisiting permutative conversions in sequent calculus
Permutations in intuitionistic sequent calculus ◮ One of Kleene’s permutation for intuitionistic implication: D 2 D ′ D 2 1 Γ , y : B , z : C ⊢ D Γ , z : C ⊢ A Γ , y : B , z : C ⊢ D D 1 Γ , y : B ⊢ C ⊃ D ⊃ R ⊃ L � Γ ⊢ A Γ , x : A ⊃ B , z : C ⊢ D ⊃ L Γ , x : A ⊃ B ⊢ C ⊃ D ⊃ R Γ , x : A ⊃ B ⊢ C ⊃ D ◮ Permutability Thm: D 1 , D 2 are inter-permutable iff ϕ D 1 = ϕ D 2 , for ϕ Prawitz’s mapping of sequent calculus into nat. deduction. ◮ Zucker and Pottinger: cuts are present and permutations involve cut or contraction. ◮ Mints, Dyckhoff&P. and Schwichtenberg: cut-free fragments and permutations involve typically logical inferences.
Permutations in intuitionistic sequent calculus ◮ One of Kleene’s permutation for intuitionistic implication: D 2 D ′ D 2 1 Γ , y : B , z : C ⊢ D Γ , z : C ⊢ A Γ , y : B , z : C ⊢ D D 1 Γ , y : B ⊢ C ⊃ D ⊃ R ⊃ L � Γ ⊢ A Γ , x : A ⊃ B , z : C ⊢ D ⊃ L Γ , x : A ⊃ B ⊢ C ⊃ D ⊃ R Γ , x : A ⊃ B ⊢ C ⊃ D ◮ Permutability Thm: D 1 , D 2 are inter-permutable iff ϕ D 1 = ϕ D 2 , for ϕ Prawitz’s mapping of sequent calculus into nat. deduction. ◮ Zucker and Pottinger: cuts are present and permutations involve cut or contraction. ◮ Mints, Dyckhoff&P. and Schwichtenberg: cut-free fragments and permutations involve typically logical inferences.
Permutations in intuitionistic sequent calculus ◮ One of Kleene’s permutation for intuitionistic implication: D 2 D ′ D 2 1 Γ , y : B , z : C ⊢ D Γ , z : C ⊢ A Γ , y : B , z : C ⊢ D D 1 Γ , y : B ⊢ C ⊃ D ⊃ R ⊃ L � Γ ⊢ A Γ , x : A ⊃ B , z : C ⊢ D ⊃ L Γ , x : A ⊃ B ⊢ C ⊃ D ⊃ R Γ , x : A ⊃ B ⊢ C ⊃ D ◮ Permutability Thm: D 1 , D 2 are inter-permutable iff ϕ D 1 = ϕ D 2 , for ϕ Prawitz’s mapping of sequent calculus into nat. deduction. ◮ Zucker and Pottinger: cuts are present and permutations involve cut or contraction. ◮ Mints, Dyckhoff&P. and Schwichtenberg: cut-free fragments and permutations involve typically logical inferences.
Dyckhoff&P.’s approach to the Permutability Thm. ◮ Terms are used to represent derivations: x : A , Γ ⊢ x : A Ax x : A , Γ ⊢ t : B Γ , x ⊢ u : A y : B , Γ , x ⊢ v : C Γ ⊢ λ x . t : A ⊃ B ⊃ R ⊃ L Γ , x : A ⊃ B ⊢ x ( u ( y ) v ): C ◮ Normal cut-free forms: ◮ at x ( u ( y ) v ) impose v is y -normal (ie v = y or v = y ( u ′ ( z ) v ′ ) , y �∈ u ′ , v ′ , and v ′ is z -normal); ◮ in bijection with β -nfs of λ -calculus, via Herbelin’s nfs of λ : ✲ x ( u , ( y ) y ( u ′ ( z ) z ) ( x ; [ u , u ′ ]) ✛ ) ✛ ✲ � �� � y / ✲ ∈ ✛ xuu ′
Dyckhoff&P.’s approach to the Permutability Thm. ◮ Terms are used to represent derivations: x : A , Γ ⊢ x : A Ax x : A , Γ ⊢ t : B Γ , x ⊢ u : A y : B , Γ , x ⊢ v : C Γ ⊢ λ x . t : A ⊃ B ⊃ R ⊃ L Γ , x : A ⊃ B ⊢ x ( u ( y ) v ): C ◮ Normal cut-free forms: ◮ at x ( u ( y ) v ) impose v is y -normal (ie v = y or v = y ( u ′ ( z ) v ′ ) , y �∈ u ′ , v ′ , and v ′ is z -normal); ◮ in bijection with β -nfs of λ -calculus, via Herbelin’s nfs of λ : ✲ x ( u , ( y ) y ( u ′ ( z ) z ) ( x ; [ u , u ′ ]) ✛ ) ✛ ✲ � �� � y / ✲ ∈ ✛ xuu ′
Dyckhoff&P.’s approach to the Permutability Thm. ◮ Terms are used to represent derivations: x : A , Γ ⊢ x : A Ax x : A , Γ ⊢ t : B Γ , x ⊢ u : A y : B , Γ , x ⊢ v : C Γ ⊢ λ x . t : A ⊃ B ⊃ R ⊃ L Γ , x : A ⊃ B ⊢ x ( u ( y ) v ): C ◮ Normal cut-free forms: ◮ at x ( u ( y ) v ) impose v is y -normal (ie v = y or v = y ( u ′ ( z ) v ′ ) , y �∈ u ′ , v ′ , and v ′ is z -normal); ◮ in bijection with β -nfs of λ -calculus, via Herbelin’s nfs of λ : ✲ x ( u , ( y ) y ( u ′ ( z ) z ) ( x ; [ u , u ′ ]) ✛ ) ✛ ✲ � �� � y / ✲ ∈ ✛ xuu ′
Dyckhoff&P.’s approach to the Permutability Thm. ◮ Permutations are oriented and induce a rewriting system whose nfs are the normal cut-free forms: ( i ) x ( u ( y ) v ) → y if y �∈ v ( ii ) x ( u ( y ) z ( v ( w ) t )) → z ( x ( u ( y ) v ) ( w ) x ( u ( y ) t )) if y � = z and y ∈ v or t ( ii ′ ) x ( u ( y ) y ( v ( w ) t )) → x ( u ( y ) y ( x ( u ( y ) v ) ( w ) x ( u ( y ) t ))) if y ∈ v or t ( iii ) x ( u ( y ) λ z . v ) → λ z . x ( u ( y ) v ) ◮ x ( u ( y ) v ) is approx. the explicit substitution of xu for y in v . ◮ The induced rewriting system is confluent and WN.
Dyckhoff&P.’s approach to the Permutability Thm. ◮ Permutations are oriented and induce a rewriting system whose nfs are the normal cut-free forms: ( i ) x ( u ( y ) v ) → y if y �∈ v ( ii ) x ( u ( y ) z ( v ( w ) t )) → z ( x ( u ( y ) v ) ( w ) x ( u ( y ) t )) if y � = z and y ∈ v or t ( ii ′ ) x ( u ( y ) y ( v ( w ) t )) → x ( u ( y ) y ( x ( u ( y ) v ) ( w ) x ( u ( y ) t ))) if y ∈ v or t ( iii ) x ( u ( y ) λ z . v ) → λ z . x ( u ( y ) v ) ◮ x ( u ( y ) v ) is approx. the explicit substitution of xu for y in v . ◮ The induced rewriting system is confluent and WN.
Schwichtenberg’s approach via multiary sequent terms ◮ x -normality can be represented with lists, eg µ x ( u , u ′ :: [] , ( z ) z ) ✛ x ( u , ( y ) y ( u ′ ( z ) z ) ) ✲ � �� � y / ✛ ∈ xuu ′ ◮ Schwichtenberg considers a family of left rules (one for each k ∈ N 0 ) ⊢ u : A ⊢ u 1 : B 1 . . . ⊢ u k : B k y : C ⊢ v : D x : A ⊃ B 1 ⊃ . . . ⊃ B k ⊃ C ⊢ x ( u , u 1 :: ... :: u k :: [] , ( y ) v ): D ⊃ L k ◮ The µ -nfs are determined by the µ -rule (where a stands for ”append”): x ( u , l , ( y ) y ( u ′ , l ′ , ( z ) v )) → x ( u , a ( l , u ′ :: l ′ ) , ( z ) v ) if y �∈ u ′ , l ′ , v ◮ Permutative rules aim at trivialising generality to x ( u , l , ( z ) z ). ◮ SN holds and implies SN for a restriction of Dyckhoff&P.’s rules.
Schwichtenberg’s approach via multiary sequent terms ◮ x -normality can be represented with lists, eg µ x ( u , u ′ :: [] , ( z ) z ) ✛ x ( u , ( y ) y ( u ′ ( z ) z ) ) ✲ � �� � y / ✛ ∈ xuu ′ ◮ Schwichtenberg considers a family of left rules (one for each k ∈ N 0 ) ⊢ u : A ⊢ u 1 : B 1 . . . ⊢ u k : B k y : C ⊢ v : D x : A ⊃ B 1 ⊃ . . . ⊃ B k ⊃ C ⊢ x ( u , u 1 :: ... :: u k :: [] , ( y ) v ): D ⊃ L k ◮ The µ -nfs are determined by the µ -rule (where a stands for ”append”): x ( u , l , ( y ) y ( u ′ , l ′ , ( z ) v )) → x ( u , a ( l , u ′ :: l ′ ) , ( z ) v ) if y �∈ u ′ , l ′ , v ◮ Permutative rules aim at trivialising generality to x ( u , l , ( z ) z ). ◮ SN holds and implies SN for a restriction of Dyckhoff&P.’s rules.
Schwichtenberg’s approach via multiary sequent terms ◮ x -normality can be represented with lists, eg µ x ( u , u ′ :: [] , ( z ) z ) ✛ x ( u , ( y ) y ( u ′ ( z ) z ) ) ✲ � �� � y / ✛ ∈ xuu ′ ◮ Schwichtenberg considers a family of left rules (one for each k ∈ N 0 ) ⊢ u : A ⊢ u 1 : B 1 . . . ⊢ u k : B k y : C ⊢ v : D x : A ⊃ B 1 ⊃ . . . ⊃ B k ⊃ C ⊢ x ( u , u 1 :: ... :: u k :: [] , ( y ) v ): D ⊃ L k ◮ The µ -nfs are determined by the µ -rule (where a stands for ”append”): x ( u , l , ( y ) y ( u ′ , l ′ , ( z ) v )) → x ( u , a ( l , u ′ :: l ′ ) , ( z ) v ) if y �∈ u ′ , l ′ , v ◮ Permutative rules aim at trivialising generality to x ( u , l , ( z ) z ). ◮ SN holds and implies SN for a restriction of Dyckhoff&P.’s rules.
Schwichtenberg’s approach via multiary sequent terms ◮ x -normality can be represented with lists, eg µ x ( u , u ′ :: [] , ( z ) z ) ✛ x ( u , ( y ) y ( u ′ ( z ) z ) ) ✲ � �� � y / ✛ ∈ xuu ′ ◮ Schwichtenberg considers a family of left rules (one for each k ∈ N 0 ) ⊢ u : A ⊢ u 1 : B 1 . . . ⊢ u k : B k y : C ⊢ v : D x : A ⊃ B 1 ⊃ . . . ⊃ B k ⊃ C ⊢ x ( u , u 1 :: ... :: u k :: [] , ( y ) v ): D ⊃ L k ◮ The µ -nfs are determined by the µ -rule (where a stands for ”append”): x ( u , l , ( y ) y ( u ′ , l ′ , ( z ) v )) → x ( u , a ( l , u ′ :: l ′ ) , ( z ) v ) if y �∈ u ′ , l ′ , v ◮ Permutative rules aim at trivialising generality to x ( u , l , ( z ) z ). ◮ SN holds and implies SN for a restriction of Dyckhoff&P.’s rules.
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