On a computational interpretation of sequent calculus for ... › ~yfukuda › slide ›...
Transcript of On a computational interpretation of sequent calculus for ... › ~yfukuda › slide ›...
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On a computational interpretation ofsequent calculus for modal logic S4
Yosuke Fukuda
Graduate School of Informatics, Kyoto University
Second Workshop on Mathematical Logic and Its ApplicationsMarch 8th, 2018
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Modal logic in the theory of programming languages
Some studies [Kobayashi ’97][Benton+ ’98][Pfenning+ ’01][Kimura+ ’11]
discovered that S4 corresponds to various typed λ-calculi for“meta-programming”
In the logical foundation, �-modality plays an essential role:
�A means the set of programs which “encode” programs oftype A
(this is similar to the intuition in Logic of Proof, etc:�A means the proposition of a “proof” of A)
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Modal logic in the theory of programming languages
Some studies [Kobayashi ’97][Benton+ ’98][Pfenning+ ’01][Kimura+ ’11]
discovered that S4 corresponds to various typed λ-calculi for“meta-programming”
In the logical foundation, �-modality plays an essential role:
�A means the set of programs which “encode” programs oftype A
(this is similar to the intuition in Logic of Proof, etc:�A means the proposition of a “proof” of A)
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Modal logic in the theory of programming languages
Some studies [Kobayashi ’97][Benton+ ’98][Pfenning+ ’01][Kimura+ ’11]
discovered that S4 corresponds to various typed λ-calculi for“meta-programming”
In the logical foundation, �-modality plays an essential role:
�A means the set of programs which “encode” programs oftype A
(this is similar to the intuition in Logic of Proof, etc:�A means the proposition of a “proof” of A)
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Problem from a practical viewpoint
All the previous studies only consider natural-deduction-style
λ-calculi, and they use the “one-step” substitution as usual:
(λx .M)N M[x := N]
However, the operation is too rich from a practical viewpoint
Natural-deduction-style λ-calculus is not enough to capturethe structure of computation
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Problem from a practical viewpoint
All the previous studies only consider natural-deduction-style
λ-calculi, and they use the “one-step” substitution as usual:
(λx .M)N M[x := N]
However, the operation is too rich from a practical viewpoint
Natural-deduction-style λ-calculus is not enough to capturethe structure of computation
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Problem from a practical viewpoint
All the previous studies only consider natural-deduction-style
λ-calculi, and they use the “one-step” substitution as usual:
(λx .M)N M[x := N]
However, the operation is too rich from a practical viewpoint
Natural-deduction-style λ-calculus is not enough to capturethe structure of computation
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This talk
Aim of this talk� �To create another computational model for modal logic S4,in terms of sequent calculus� �
To do this, a sequent calculus and its corresponding calculus forintuitionistic S4 are proposed
1 proof-theoretically based on:
a modal sequent calculus and the G3-style system[Troelstra&Schwichtenberg ’96]a higher-arity modal natural deduction [Pfenning&Davies ’01]
2 type-theoretically based on:
the higher-arity modal λ-calculus [Pfenning&Davies ’01]the Curry–Howard correspondence for a G3-style sequent calc.[Ohori ’99]
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This talk
Aim of this talk� �To create another computational model for modal logic S4,in terms of sequent calculus� �
To do this, a sequent calculus and its corresponding calculus forintuitionistic S4 are proposed
1 proof-theoretically based on:
a modal sequent calculus and the G3-style system[Troelstra&Schwichtenberg ’96]a higher-arity modal natural deduction [Pfenning&Davies ’01]
2 type-theoretically based on:
the higher-arity modal λ-calculus [Pfenning&Davies ’01]the Curry–Howard correspondence for a G3-style sequent calc.[Ohori ’99]
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This talk
Aim of this talk� �To create another computational model for modal logic S4,in terms of sequent calculus� �
To do this, a sequent calculus and its corresponding calculus forintuitionistic S4 are proposed
1 proof-theoretically based on:
a modal sequent calculus and the G3-style system[Troelstra&Schwichtenberg ’96]a higher-arity modal natural deduction [Pfenning&Davies ’01]
2 type-theoretically based on:
the higher-arity modal λ-calculus [Pfenning&Davies ’01]the Curry–Howard correspondence for a G3-style sequent calc.[Ohori ’99]
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This talk
Aim of this talk� �To create another computational model for modal logic S4,in terms of sequent calculus� �
To do this, a sequent calculus and its corresponding calculus forintuitionistic S4 are proposed
1 proof-theoretically based on:
a modal sequent calculus and the G3-style system[Troelstra&Schwichtenberg ’96]a higher-arity modal natural deduction [Pfenning&Davies ’01]
2 type-theoretically based on:
the higher-arity modal λ-calculus [Pfenning&Davies ’01]the Curry–Howard correspondence for a G3-style sequent calc.[Ohori ’99]
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Higher-arity Sequent Calculus for intuitionistic S4
We propose a “higher-arity” sequent calc. for (∧,∨,⊃,�)-fragmentof intuitionistic S4, HLJS4, based on [Troelstra&Schwichtenberg ’96]
Definition (Formula)
A,B ::= p | A ∧ B | A ∨ B | A ⊃ B | �A
Definition (Higher-arity judgment [Pfenning+ ’01])
A judgment is defined by the following higher-arity form:
∆; Γ ` A
which intuitively means (∧�∆) ∧ (
∧Γ) ⊃ A
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Inference rules of HLJS4
Ax∅;A ` A�Ax
A; ∅ ` A
∆; Γ ` A ∆; Γ ` B ∧R∆; Γ ` A ∧ B
∆; Γ,Ai ` B∧L
∆; Γ,A1 ∧ A2 ` B
∆; Γ ` Ai ∨R∆; Γ ` A1 ∨ A2
∆; Γ,A ` C ∆; Γ,B ` C∨L
∆; Γ,A ∨ B ` C
∆; Γ,A ` B⊃ R
∆; Γ ` A ⊃ B
∆; Γ ` A ∆; Γ,B ` C⊃ L
∆; Γ,A ⊃ B ` C
∆; ∅ ` A�R
∆; ∅ ` �A∆,A; Γ ` B
�L∆; Γ,�A ` B
∆; Γ ` BW
∆;Γ,A ` B∆; Γ ` B
�W∆,A; Γ ` B
∆; Γ,A,A ` BC
∆; Γ,A ` B
∆,A,A; Γ ` B�C
∆,A; Γ ` B
∆; Γ ` A ∆; Γ,A ` BCut
∆; Γ ` B∆; ∅ ` A ∆,A; Γ ` B
�Cut∆; Γ ` B
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Init rules of HLJS4
Init rule
Ax∅;A ` A�Ax
A; ∅ ` A
Ax∅;A ` A
Intuition of Ax� ��Ax�A ⊃ A� �
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Init rules of HLJS4
Init rule
Ax∅;A ` A�Ax
A; ∅ ` A
Ax∅;A ` A
Intuition of Ax� ��Ax�A ⊃ A� �
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Logical rules of HLJS4
Logical rule
∆; Γ ` A ∆; Γ ` B ∧R∆; Γ ` A ∧ B
∆; Γ,Ai ` B∧L
∆; Γ,A1 ∧ A2 ` B
∆; Γ ` Ai ∨R∆; Γ ` A1 ∨ A2
∆; Γ,A ` C ∆; Γ,B ` C∨L
∆; Γ,A ∨ B ` C
∆; Γ,A ` B⊃ R
∆; Γ ` A ⊃ B
∆; Γ ` A ∆; Γ,B ` C⊃ L
∆; Γ,A ⊃ B ` C
∆; ∅ ` A�R
∆; ∅ ` �A∆,A; Γ ` B
�L∆; Γ,�A ` B
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Logical rules of HLJS4
Logical rule
∆;Γ ` A ∆;Γ ` B ∧R∆;Γ ` A ∧ B
∆;Γ,Ai ` B∧L
∆;Γ,A1 ∧ A2 ` B
∆;Γ ` Ai ∨R∆;Γ ` A1 ∨ A2
∆;Γ,A ` C ∆;Γ,B ` C∨L
∆;Γ,A ∨ B ` C
∆;Γ,A ` B⊃ R
∆;Γ ` A ⊃ B
∆;Γ ` A ∆;Γ,B ` C⊃ L
∆;Γ,A ⊃ B ` C
∆; ∅ ` A�R
∆; ∅ ` �A∆,A; Γ ` B
�L∆; Γ,�A ` B
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Logical rules of HLJS4
Logical rule
∆; Γ ` A ∆; Γ ` B ∧R∆; Γ ` A ∧ B
∆; Γ,Ai ` B∧L
∆; Γ,A1 ∧ A2 ` B
∆; Γ ` Ai ∨R∆; Γ ` A1 ∨ A2
∆; Γ,A ` C ∆; Γ,B ` C∨L
∆; Γ,A ∨ B ` C
∆; Γ,A ` B⊃ R
∆; Γ ` A ⊃ B
∆; Γ ` A ∆; Γ,B ` C⊃ L
∆; Γ,A ⊃ B ` C
∆; ∅ ` A�R
∆; ∅ ` �A∆,A; Γ ` B
�L∆; Γ,�A ` B
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Structural rules of HLJS4
Structural rule
∆; Γ ` BW
∆;Γ,A ` B∆; Γ ` B
�W∆,A; Γ ` B
∆; Γ,A,A ` BC
∆; Γ,A ` B
∆,A,A; Γ ` B�C
∆,A; Γ ` B
Cut rule
∆; Γ ` A ∆; Γ,A ` BCut
∆; Γ ` B∆; ∅ ` A ∆,A; Γ ` B
�Cut∆; Γ ` B
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On the cut-elimination procedure
While we can prove the cut-elimination theorem for HLJS4,the proof by the mix-elimination is problematic; because ...
Π∆; Γ ` A
Π′
∆′; Γ′,A,A ` BC
∆′; Γ′,A ` BCut
∆,∆′; Γ, Γ′ ` B
=⇒Π
∆; Γ ` A
∆′; Γ′,A,A ` B
Π′
∆′; Γ′,A,A ` BMix
∆,∆′; Γ, Γ′ ` B
In the elimination procedure,
it is “okay” if we consider the provability of the judgment; but
it is “not okay” if we consider the construction of the judgment
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On the cut-elimination procedure
While we can prove the cut-elimination theorem for HLJS4,the proof by the mix-elimination is problematic; because ...
Π∆; Γ ` A
Π′
∆′; Γ′,A,A ` BC
∆′; Γ′,A ` BCut
∆,∆′; Γ, Γ′ ` B
=⇒Π
∆; Γ ` A
∆′; Γ′,A,A ` B
Π′
∆′; Γ′,A,A ` BMix
∆,∆′; Γ, Γ′ ` B
In the elimination procedure,
it is “okay” if we consider the provability of the judgment; but
it is “not okay” if we consider the construction of the judgment
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On the cut-elimination procedure
While we can prove the cut-elimination theorem for HLJS4,the proof by the mix-elimination is problematic; because ...
Π∆; Γ ` A
Π′
∆′; Γ′,A,A ` BC
∆′; Γ′,A ` BCut
∆,∆′; Γ, Γ′ ` B
=⇒Π
∆; Γ ` A
∆′; Γ′,A,A ` B
Π′
∆′; Γ′,A,A ` BMix
∆,∆′; Γ, Γ′ ` B
In the elimination procedure,
it is “okay” if we consider the provability of the judgment; but
it is “not okay” if we consider the construction of the judgment
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G3-style sequent calculus
The G3-style [Kleene ’52][Dragalin ’88] is a style of formalization tomake a cut-free, or precisely, “structural-rule-free” system
The G3-style inference rules are defined in a somewhat tricky wayto derive the “height-preserving admissible” structural rules
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G3-style system for HLJS4, named G3-HLJS4
The G3-style inference rules are defined as follows:Ax
∆; Γ,A ` A�Ax
∆,A; Γ ` A
∆; Γ,A ` B⊃ R
∆; Γ ` A ⊃ B
∆; Γ,A ⊃ B ` A ∆; Γ,A ⊃ B,B ` C⊃ L
∆; Γ,A ⊃ B ` C
∆; Γ ` A ∆; Γ ` B ∧R∆; Γ ` A ∧ B
∆; Γ,A ∧ B,A,B ` C∧L
∆; Γ,A ∧ B ` C
∆; Γ ` Ai ∨R∆; Γ ` A1 ∨ A2
∆; Γ,A ∨ B,A ` C ∆; Γ,A ∨ B,B ` C∨L
∆; Γ,A ∨ B ` C
∆; ∅ ` A�R
∆; Γ ` �A∆,A; Γ,�A ` B
�L∆; Γ,�A ` B
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From the original rules to the G3-style
Idea: all we have to do is to get “height-preserving” structural rules
Original HLJS4� �Ax∅;A ` A
∆; Γ,Ai ` B∧L
∆; Γ,A1 ∧ A2 ` B
∆,A; Γ ` B�L
∆; Γ,�A ` B� �
=⇒
G3-style G3-HLJS4� �Ax
∆; Γ,A ` A
∆; Γ,A ∧ B,A,B ` C∧L
∆; Γ,A ∧ B ` C
∆,A; Γ,�A ` B�L
∆; Γ,�A ` B� �13 / 24
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Desired properties
Lemma (Height-preserving weakening/contraction)
The followings are height-preserving admissible rules in G3-HLJS4:
∆; Γ ` BW
∆; Γ,A ` B
∆; Γ ` B�W
∆,A; Γ ` B
∆; Γ,A,A ` BC
∆; Γ,A ` B
∆,A,A; Γ ` B�C
∆,A; Γ ` B
Theorem (Equivalence)
The provability of HLJS4 and G3-HLJS4 + Cut is equivalent
Theorem (Cut-elimination)
The cut rules Cut and �Cut are admissible in G3-HLJS4
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Desired properties
Lemma (Height-preserving weakening/contraction)
The followings are height-preserving admissible rules in G3-HLJS4:
∆; Γ ` BW
∆; Γ,A ` B
∆; Γ ` B�W
∆,A; Γ ` B
∆; Γ,A,A ` BC
∆; Γ,A ` B
∆,A,A; Γ ` B�C
∆,A; Γ ` B
Theorem (Equivalence)
The provability of HLJS4 and G3-HLJS4 + Cut is equivalent
Theorem (Cut-elimination)
The cut rules Cut and �Cut are admissible in G3-HLJS4
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Desired properties
Lemma (Height-preserving weakening/contraction)
The followings are height-preserving admissible rules in G3-HLJS4:
∆; Γ ` BW
∆; Γ,A ` B
∆; Γ ` B�W
∆,A; Γ ` B
∆; Γ,A,A ` BC
∆; Γ,A ` B
∆,A,A; Γ ` B�C
∆,A; Γ ` B
Theorem (Equivalence)
The provability of HLJS4 and G3-HLJS4 + Cut is equivalent
Theorem (Cut-elimination)
The cut rules Cut and �Cut are admissible in G3-HLJS4
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Term assignment for the modal sequent calculus
We propose a term assignment system for the G3-HLJS4, λ�seq,
to get the computational model
As [Ohori ’99] did for a G3-style prop. int. sequent calc.,we assign terms to G3-HLJS4 + Cut as follows:
Init/Right rules: assign λ-terms, as we do for N.D. system
Left/Cut rules: assign the so-called “let expression”
Good point: λ�seq does not use “meta-level” substitution!
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Term assignment for the modal sequent calculus
We propose a term assignment system for the G3-HLJS4, λ�seq,
to get the computational model
As [Ohori ’99] did for a G3-style prop. int. sequent calc.,we assign terms to G3-HLJS4 + Cut as follows:
Init/Right rules: assign λ-terms, as we do for N.D. system
Left/Cut rules: assign the so-called “let expression”
Good point: λ�seq does not use “meta-level” substitution!
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Term assignment for init/right rules
Assign the modal λ-term [Pfenning+ ’01] to the init/right rules:
Ax∆; Γ, x : A ` x : A
�Ax∆, u : A; Γ ` u : A
∆; Γ ` M : A ∆; Γ ` N : B ∧R∆; Γ ` 〈M,N〉 : A ∧ B
∆; Γ, x : A ` M : B⊃ R
∆; Γ ` λx : A.M : A ⊃ B
∆; ∅ ` M : A�R
∆; Γ ` boxM : �A
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Term assignment for left rule of conjunction
Assign “let-expression” to the left conjunction rule:
∆; Γ, x : A ∧ B, y : A, z : B ` M : C∧L
∆; Γ, x : A ∧ B ` let 〈y , z〉 = x inM : C
The reduction intuitively proceeeds, e.g., as:
(let 〈y , z〉 = 〈N , L〉 inM) M[y := N , z := L]
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Term assignment for left rule of conjunction
Assign “let-expression” to the left conjunction rule:
∆; Γ, x : A ∧ B, y : A, z : B ` M : C∧L
∆; Γ, x : A ∧ B ` let 〈y , z〉 = x inM : C
The reduction intuitively proceeeds, e.g., as:
(let 〈y , z〉 = 〈N , L〉 inM) M[y := N , z := L]
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Term assignment for left rule of conjunction
Assign “let-expression” to the left conjunction rule:
∆; Γ, x : A ∧ B, y : A, z : B ` M : C∧L
∆; Γ, x : A ∧ B ` let 〈y , z〉 = x inM : C
The reduction intuitively proceeeds, e.g., as:
(let 〈y , z〉 = 〈N , L〉 inM) M[y := N , z := L]
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Term assignment for left rule of conjunction
Assign “let-expression” to the left conjunction rule:
∆; Γ, x : A ∧ B, y : A, z : B ` M : C∧L
∆; Γ, x : A ∧ B ` let 〈y , z〉 = x inM : C
The reduction intuitively proceeeds, e.g., as:
(let 〈y , z〉 = 〈N , L〉 inM) M[y := N , z := L]
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Term assignment for the other left rules
The rules for the other left rules are defined similarly:
∆; Γ, x : A ⊃ B ` M : A ∆; Γ, x : A ⊃ B, y : B ` N : C⊃ L
∆; Γ, x : A ⊃ B ` let y = x M inN : C
∆, u : A; Γ, x : �A ` M : B�L
∆; Γ, x : �A ` let box u = x inM : B
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Term assignment for cut rules
The term assignment for cut rules are defined as a “composition”of two constructions, again by using let-expressions:
∆; Γ ` M : A ∆; Γ, x : A ` N : BCut
∆; Γ ` let x = M inN : B
∆; ∅ ` M : A ∆, u : A; Γ ` N : B�Cut
∆; Γ ` let u = M inN : B
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Cut-elimination in terms of λ�seq
Let us consider the cut-elimination for conjunction:
` M : A ` N : B ∧R` 〈M,N〉 : A ∧ B
x : A ∧ B, y : A, z : B ` L : C∧L
x : A ∧ B ` let 〈y , z〉 = x in L : CCut` let x = 〈M,N〉 in let 〈y , z〉 = x in L : C
To eliminate cuts, all we have to do is to compute:
(let x = 〈M,N〉 in let 〈y , z〉 = x in L) L[y := M, z := N, x := 〈M,N〉]
but we do not want to use “meta-level” substitution
Fortunatelly, the (local) cut-elimination step defined in the G3-styleis exactly what we want!
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Cut-elimination in terms of λ�seq
Let us consider the cut-elimination for conjunction:
` M : A ` N : B ∧R` 〈M,N〉 : A ∧ B
x : A ∧ B, y : A, z : B ` L : C∧L
x : A ∧ B ` let 〈y , z〉 = x in L : CCut` let x = 〈M,N〉 in let 〈y , z〉 = x in L : C
To eliminate cuts, all we have to do is to compute:
(let x = 〈M,N〉 in let 〈y , z〉 = x in L) L[y := M, z := N, x := 〈M,N〉]
but we do not want to use “meta-level” substitution
Fortunatelly, the (local) cut-elimination step defined in the G3-styleis exactly what we want!
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Cut-elimination in terms of λ�seq
Let us consider the cut-elimination for conjunction:
` M : A ` N : B ∧R` 〈M,N〉 : A ∧ B
x : A ∧ B, y : A, z : B ` L : C∧L
x : A ∧ B ` let 〈y , z〉 = x in L : CCut` let x = 〈M,N〉 in let 〈y , z〉 = x in L : C
To eliminate cuts, all we have to do is to compute:
(let x = 〈M,N〉 in let 〈y , z〉 = x in L) L[y := M, z := N, x := 〈M,N〉]
but we do not want to use “meta-level” substitution
Fortunatelly, the (local) cut-elimination step defined in the G3-styleis exactly what we want!
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Local cut-elimination as program-simplification
(A part of) translation rules are obtained as follows:
Optimization (let x = M in x) M
(let x = M in y) y
Flattening
(letw = (let 〈y , z〉 = x inM) inN) (let 〈y , z〉 = x in letw = M inN)
(let y = (let box u = x inM) inN) (let box u = x in let y = M inN)
Decomposition
(let x = 〈M,N〉 in let 〈y , z〉 = x in L) (let y = M in let z = N in let x = 〈y , z〉 in L)(let x = boxM in let box u = x inN) (let u = M in let x = box u inN)
These translation corresponds to “A-normal form compilation” inthe theory of programming languages
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Local cut-elimination as program-simplification
(A part of) translation rules are obtained as follows:
Optimization (let x = M in x) M
(let x = M in y) y
Flattening
(letw = (let 〈y , z〉 = x inM) inN) (let 〈y , z〉 = x in letw = M inN)
(let y = (let box u = x inM) inN) (let box u = x in let y = M inN)
Decomposition
(let x = 〈M,N〉 in let 〈y , z〉 = x in L) (let y = M in let z = N in let x = 〈y , z〉 in L)(let x = boxM in let box u = x inN) (let u = M in let x = box u inN)
These translation corresponds to “A-normal form compilation” inthe theory of programming languages
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Properties of λ�seq and the cut-elimination theorem
Theorem (Subject reduction)
If ∆; Γ ` M : A and M M ′, then ∆; Γ ` M ′ : A
Theorem (Strong normalization)
Every typable term is strongly normalizing
Corollary (Cut-elimination theorem)
λ�seq enjoys the cut-elimination theorem, which also yields thatevery typable term can be reduced to the unique normal form
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Embedding from modal calculus
The following tells us that λ�seq can be used as a basis of model forthe existing theory:
Theorem (Embedding from modal typed λ-calculus)
The modal λ-calc. λ� [Pfenning+ ’01] can be embeded into λ�seq:
If ∆; Γ ` M : A in λ�, then ∆; Γ ` JMK : A in λ�seq
If M M ′ in λ�, then JMK JM ′K in λ�seq
where J−K means the translation mapping from λ� to λ�seq
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Conclusion and future work
Conclusion
A cut-free higher-arity sequent calc. for intuitionistic S4:HLJS4 and G3-HLJS4(A cut-free higher-arity sequent calc. for classical S4:HLKS4 and G3-HLKS4)The corresponding term calculus for G3-HLJS4
Future work
The corresponding term calculus for the classical version,following the work of λµ-calculus for modal logic [Kimura+ ’11](Ongoing work with Akira Yoshimizu):Geometry of Interaction semantics for modal logic in terms ofMELL, following the work of GoI semantics for PCF [Mackie ’95]
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Appendix
Appendix
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Cut-elimination (1)
let x = y inM M[x := y ]
let x = u inM M[x := u]
let u = v inM M[u := v ]
let x = M in x M
let x = M in y y
let u = M in x x
let u = M in u M
let u = M in v v
let x = M in u u
let z = (let y = x M inN) in L let y = x M in let z = N in L
letw = (let 〈y , z〉 = x inM) inN let 〈y , z〉 = x in letw = M inN
letw = (case x of [y ]M or [z]N) in L case x of [y ](letw = M in L) or [z](letw = N in L)
let y = (let box u = x inM) inN let box u = x in let y = M inN
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Cut-elimination (2)
let x = L in let z = y M inN let z = y (let x = L inM) in let x = L inN
let x = N in let 〈y , z〉 = w inM let 〈y , z〉 = w in let x = N inM
let x = L in casew of [y ]M or [z]N casew of [y ](let x = L inM) or [z](let x = L inN)
let x = N in let box u = y inM let box u = y in let x = N inM
let y = λx : A.M in let z = y N in L let y = λx : A.M in let x = N in let z = M in L
let x = 〈M,N〉 in let 〈y , z〉 = x in L let y = M in let z = N in let x = 〈y , z〉 in L
let x = ιA∨Bl (M) in case x of [y ]N or [z]L let y = M in let x = ιA∨B
l (y) inN
let x = ιA∨Br (M) in case x of [y ]N or [z]L let z = M in let x = ιA∨B
r (z) in L
let x = boxM in let box u = x inN let u = M in let x = box u inN
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