A cut-free proof system for pseudo-transitive
modal logics
Sonia Marin
With Lutz Straßburger
Inria, LIX, École Polytechnique
Topology, Algebra, and Categories in Logic
June 22, 2015
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
1 / 10
Classical modal logic
• Formulas: A, B, ... ::= p | p̄ | A ∧ B | A ∨ B | A | ♦A
• Negation is defined via De Morgan duality and A → B ≡ Ā ∨ B
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
2 / 10
Classical modal logic
• Formulas: A, B, ... ::= p | p̄ | A ∧ B | A ∨ B | A | ♦A
• Negation is defined via De Morgan duality and A → B ≡ Ā ∨ B
• Axioms for K: classical propositional logic and
k : (A → B) → (A → B)
• Rules: modus ponens:
S.Marin, L.Straßburger (Inria)
A
A→B
−
−−−−−−−−−−
−
B
necessitation:
Pseudo-transitive modal logics
A
−−−
A
2 / 10
Classical modal logic
• Formulas: A, B, ... ::= p | p̄ | A ∧ B | A ∨ B | A | ♦A
• Negation is defined via De Morgan duality and A → B ≡ Ā ∨ B
• Axioms for K: classical propositional logic and
k : (A → B) → (A → B)
• Rules: modus ponens:
A
A→B
−
−−−−−−−−−−
−
B
necessitation:
A
−−−
A
• Theorem: The logic K is sound and complete wrt Kripke frames hW , Ri.
• W a non-empty set of worlds
• R ⊆ W × W the accessibility relation
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
2 / 10
A fine selection of modal axioms
t : A → A
reflexivity
∀w .wRw
4 : A → A
transitivity
∀xyw .xRy ∧ yRw → xRw
4∗ : A → A
pseudo-transitivity
∀xyzw .xRy ∧ yRz ∧ zRw → ∃u.xRu ∧ uRw
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
3 / 10
A fine selection of modal axioms
t : A → A
reflexivity
∀w .wRw
4 : A → A
transitivity
∀xyw .xRy ∧ yRw → xRw
4∗ : A → A
pseudo-transitivity
∀xyzw .xRy ∧ yRz ∧ zRw → ∃u.xRu ∧ uRw
4nm : n A → m A
(m,n)-transitivity
∀xw .xR m w → xR n w
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
3 / 10
Pure nested sequents
•
Sequent: Γ ::= A1 , . . . , Am
• Corresponding formula: fm(Γ) = A1 ∨ . . . ∨ Am
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
4 / 10
Pure nested sequents
• Nested Sequent: Γ ::= A1 , . . . , Am , [Γ1 ], . . . , [Γn ]
• Corresponding formula: fm(Γ) = A1 ∨ . . . ∨ Am ∨ fm(Γ1 ) ∨ . . . ∨ fm(Γn )
[Kashima, 1994], [Brünnler, 2009], [Poggiolesi, 2009]
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
4 / 10
Pure nested sequents
• Nested Sequent: Γ ::= A1 , . . . , Am , [Γ1 ], . . . , [Γn ]
• Corresponding formula: fm(Γ) = A1 ∨ . . . ∨ Am ∨ fm(Γ1 ) ∨ . . . ∨ fm(Γn )
• Sequent context: Γ{ }{ }{ } = A, [{ }], [B, { }, [{ }]]
[Kashima, 1994], [Brünnler, 2009], [Poggiolesi, 2009]
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
4 / 10
Pure nested sequents
• Nested Sequent: Γ ::= A1 , . . . , Am , [Γ1 ], . . . , [Γn ]
• Corresponding formula: fm(Γ) = A1 ∨ . . . ∨ Am ∨ fm(Γ1 ) ∨ . . . ∨ fm(Γn )
• Sequent context: Γ{C }{ }{ } = A, [C ], [B, { }, [{ }]]
[Kashima, 1994], [Brünnler, 2009], [Poggiolesi, 2009]
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
4 / 10
Pure nested sequents
• Nested Sequent: Γ ::= A1 , . . . , Am , [Γ1 ], . . . , [Γn ]
• Corresponding formula: fm(Γ) = A1 ∨ . . . ∨ Am ∨ fm(Γ1 ) ∨ . . . ∨ fm(Γn )
• Sequent context: Γ{C }{[D]}{ } = A, [C ], [B, [D], [{ }]]
[Kashima, 1994], [Brünnler, 2009], [Poggiolesi, 2009]
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
4 / 10
Pure nested sequents
• Nested Sequent: Γ ::= A1 , . . . , Am , [Γ1 ], . . . , [Γn ]
• Corresponding formula: fm(Γ) = A1 ∨ . . . ∨ Am ∨ fm(Γ1 ) ∨ . . . ∨ fm(Γn )
• Sequent context: Γ{C }{[D]}{A, [C ]} = A, [C ], [B, [D], [A, [C ]]]
[Kashima, 1994], [Brünnler, 2009], [Poggiolesi, 2009]
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
4 / 10
Pure nested sequents
• Nested Sequent: Γ ::= A1 , . . . , Am , [Γ1 ], . . . , [Γn ]
• Corresponding formula: fm(Γ) = A1 ∨ . . . ∨ Am ∨ fm(Γ1 ) ∨ . . . ∨ fm(Γn )
• Sequent context: Γ{ }{ }{ } = A, [{ }], [B, { }, [{ }]]
• System KN:
id −−−−−−−−
Γ{a, ā}
Γ{A, B}
∨ −−−−−−−−−−−
Γ{A ∨ B}
Γ{A} Γ{B}
∧ −−−−−−−−−−−−−−−
Γ{A ∧ B}
Γ{[A]}
Γ{♦A, [A, ∆]}
−−−−−−−− ♦ −−−−−−−−−−−−−−−−
Γ{A}
Γ{♦A, [∆]}
[Kashima, 1994], [Brünnler, 2009], [Poggiolesi, 2009]
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
4 / 10
Pure nested sequents
• Nested Sequent: Γ ::= A1 , . . . , Am , [Γ1 ], . . . , [Γn ]
• Corresponding formula: fm(Γ) = A1 ∨ . . . ∨ Am ∨ fm(Γ1 ) ∨ . . . ∨ fm(Γn )
• Sequent context: Γ{ }{ }{ } = A, [{ }], [B, { }, [{ }]]
• System KN:
id −−−−−−−−
Γ{a, ā}
Γ{A, B}
∨ −−−−−−−−−−−
Γ{A ∨ B}
Γ{A} Γ{B}
∧ −−−−−−−−−−−−−−−
Γ{A ∧ B}
Γ{[A]}
Γ{♦A, [A, ∆]}
−−−−−−−− ♦ −−−−−−−−−−−−−−−−
Γ{A}
Γ{♦A, [∆]}
• Theorem: System KN is sound and complete for the logic K.
[Kashima, 1994], [Brünnler, 2009], [Poggiolesi, 2009]
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
4 / 10
Indexed nested sequents
• Indexed Sequent: Γ ::= A1 , . . . , Am , [
w1
w
Γ1 ], . . . , [ n Γn ]
• No corresponding formula in the general case
2
1
2
2
1
2
• Indexed context: Γ { } { } { } = A, [ { }], [ B, { }, [ { }] ]
• System iKN:
id −−−−−−−−
Γ{a, ā}
Γ{A, B}
∨ −−−−−−−−−−−
Γ{A ∨ B}
Γ{A}
Γ{B}
∧ −−−−−−−−−−−−−−−
Γ{A ∧ B}
v
Γ{[ A]}
−−−−−−−−−
Γ{A}
u
Γ{♦A, [ A, ∆]}
−
♦ −−−−−−−−−−−
u−−−−−
Γ{♦A, [ ∆]}
[Fitting, 2015]
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
5 / 10
Indexed nested sequents
• Indexed Sequent: Γ ::= A1 , . . . , Am , [
w1
w
Γ1 ], . . . , [ n Γn ]
• No corresponding formula in the general case
2
1
2
2
1
2
• Indexed context: Γ {C } { } { } = A, [ C ], [ B, { }, [ { }] ]
• System iKN:
id −−−−−−−−
Γ{a, ā}
Γ{A, B}
∨ −−−−−−−−−−−
Γ{A ∨ B}
Γ{A}
Γ{B}
∧ −−−−−−−−−−−−−−−
Γ{A ∧ B}
v
Γ{[ A]}
−−−−−−−−−
Γ{A}
u
Γ{♦A, [ A, ∆]}
−
♦ −−−−−−−−−−−
u−−−−−
Γ{♦A, [ ∆]}
[Fitting, 2015]
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
5 / 10
Indexed nested sequents
• Indexed Sequent: Γ ::= A1 , . . . , Am , [
w1
w
Γ1 ], . . . , [ n Γn ]
• No corresponding formula in the general case
2
1
3
2
2
1
3
2
• Indexed context: Γ {C } {[ D ]} { } = A, [ C ], [ B, [ D ], [ { }] ]
• System iKN:
id −−−−−−−−
Γ{a, ā}
Γ{A, B}
∨ −−−−−−−−−−−
Γ{A ∨ B}
Γ{A}
Γ{B}
∧ −−−−−−−−−−−−−−−
Γ{A ∧ B}
v
Γ{[ A]}
−−−−−−−−−
Γ{A}
u
Γ{♦A, [ A, ∆]}
−
♦ −−−−−−−−−−−
u−−−−−
Γ{♦A, [ ∆]}
[Fitting, 2015]
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
5 / 10
Indexed nested sequents
• Indexed Sequent: Γ ::= A1 , . . . , Am , [
w1
w
Γ1 ], . . . , [ n Γn ]
• No corresponding formula in the general case
2
1
3
2
4
2
1
3
2
4
• Indexed context: Γ {C } {[ D ]} {A, [ C ]} = A, [ C ], [ B, [ D ], [ A, [ C ] ] ]
• System iKN:
id −−−−−−−−
Γ{a, ā}
Γ{A, B}
∨ −−−−−−−−−−−
Γ{A ∨ B}
Γ{A}
Γ{B}
∧ −−−−−−−−−−−−−−−
Γ{A ∧ B}
v
Γ{[ A]}
−−−−−−−−−
Γ{A}
u
Γ{♦A, [ A, ∆]}
−
♦ −−−−−−−−−−−
u−−−−−
Γ{♦A, [ ∆]}
[Fitting, 2015]
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
5 / 10
Indexed nested sequents
• Indexed Sequent: Γ ::= A1 , . . . , Am , [
w1
w
Γ1 ], . . . , [ n Γn ]
• No corresponding formula in the general case
2
1
2
2
1
2
• Indexed context: Γ { } { } { } = A, [ { }], [ B, { }, [ { }] ]
• System iKN:
id −−−−−−−−
Γ{a, ā}
Γ{A, B}
∨ −−−−−−−−−−−
Γ{A ∨ B}
w
Γ{A}
Γ{B}
∧ −−−−−−−−−−−−−−−
Γ{A ∧ B}
w
Γ {∅} {A}
−
tp −−−
w−−−−−−
w−−−
Γ {A} {∅}
w
u
v
Γ{[ A]}
−−−−−−−−−
Γ{A}
u
w
Γ{♦A, [ A, ∆]}
−
♦ −−−−−−−−−−−
u−−−−−
Γ{♦A, [ ∆]}
u
Γ {[ ∆]} {[ ]}
−−−−−
−
bc −−−−w−−−−
u−−−−−−
w
Γ {[ ∆]} {∅}
[Fitting, 2015]
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
5 / 10
Indexed nested sequents
• Indexed Sequent: Γ ::= A1 , . . . , Am , [
w1
w
Γ1 ], . . . , [ n Γn ]
• No corresponding formula in the general case
2
1
2
2
1
2
• Indexed context: Γ { } { } { } = A, [ { }], [ B, { }, [ { }] ]
• System iKN:
id −−−−−−−−
Γ{a, ā}
Γ{A, B}
∨ −−−−−−−−−−−
Γ{A ∨ B}
w
Γ{A}
Γ{B}
∧ −−−−−−−−−−−−−−−
Γ{A ∧ B}
w
Γ {∅} {A}
−
tp −−−
w−−−−−−
w−−−
Γ {A} {∅}
w
u
v
Γ{[ A]}
−−−−−−−−−
Γ{A}
u
w
Γ{♦A, [ A, ∆]}
−
♦ −−−−−−−−−−−
u−−−−−
Γ{♦A, [ ∆]}
u
Γ {[ ∆]} {[ ]}
−−−−−
−
bc −−−−w−−−−
u−−−−−−
w
Γ {[ ∆]} {∅}
• Theorem: System iKN is sound and complete for the logic K.
[Fitting, 2015]
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
5 / 10
Why are we doing this?
t : A → A
Γ{[∆]}
ṫ −−−−−−−−
Γ{∆}
4 : A → A
Γ{[∆]}
4̇ −−−−−−−−−
Γ{[[∆]]}
Theorem: System KN + ṫ is sound and complete for the logic K + t.
Problem: System KN + 4̇ is not complete for the logic K + 4.
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
6 / 10
Why are we doing this?
t : A → A
Γ{[∆]}
ṫ −−−−−−−−
Γ{∆}
t♦
4 : A → A
Γ{[∆]}
4̇ −−−−−−−−−
Γ{[[∆]]}
4♦
Γ{♦A, A}
−
−−−−−−−−
−
Γ{♦A}
Γ{♦A, [♦A, ∆]}
−
−−−−−−−−−−−−−−−
−
Γ{♦A, [∆]}
Theorem: System KN + t♦ is sound and complete for the logic K + t.
Theorem: System KN + 4♦ is sound and complete for the logic K + 4.
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
6 / 10
Why are we doing this?
t : A → A
Γ{[∆]}
ṫ −−−−−−−−
Γ{∆}
t♦
4 : A → A
Γ{[∆]}
4̇ −−−−−−−−−
Γ{[[∆]]}
4♦
4∗ : A → A
4̇∗
Γ{[[∆]]}
−
−−−−−−−−−
−
Γ{[[[∆]]]}
Γ{♦A, A}
−
−−−−−−−−
−
Γ{♦A}
Γ{♦A, [♦A, ∆]}
−
−−−−−−−−−−−−−−−
−
Γ{♦A, [∆]}
no ♦-rule!
Theorem: System KN + t♦ is sound and complete for the logic K + t.
Theorem: System KN + 4♦ is sound and complete for the logic K + 4.
Problem: complete system for K + 4∗ ?
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
6 / 10
Why are we doing this?
w w
Γ {[ ]}
1 −
−
4̇0 −−−
w−−−−
t : A → A
Γ {∅}
4̇12
4 : A → A
4nm : n A → m A
4̇nm
w
x w
Γ{[ ], [ [ ∆1 ], ∆2 ]}
−
−−−−−−−
−
x−−
w−−−−−−−−−−−−−−
Γ{[ [ ∆1 ], ∆2 ]}
u
u w
x
x w
] ] ], [ m · · · [ 2 [ ∆1 ], ∆2 ], · · · ∆m ]}
−
−−−−−−−−−−−−−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−
−−−−−−−−−−−−−−−−−−−−−−−−
−
x
x w
Γ{[ m · · · [ 2 [ ∆1 ], ∆2 ], · · · ∆m ]}
Γ{[ n · · · [ 2 [
Theorem: System iKN + 4̇nm is sound and complete for the logic K + 4nm .
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
7 / 10
Proof via syntactic cut-elimination
Theorem: If a sequent is derivable in iNK + 4̇nm together with the cut-rule
Γ{A} Γ{Ā}
cut −−−−−−−−−−−−−−
Γ{∅}
then it is also derivable in iNK + 4̇nm without cut.
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
8 / 10
Ongoing work
• decidability problems
• extension to a larger selection of modal axioms
• extension to intuitionistic modal logics
• ...
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
9 / 10
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
10 / 10
Kripke semantics
• Kripke model M = hW , R, V i:
• a non-empty set W of worlds
• an accessibility relation R ⊆ W × W ,
• a valuation function V : W → 2A
•
w
w
w
w
w
w
p
p̄
A∧B
A∨B
A
♦A
S.Marin, L.Straßburger (Inria)
iff w ∈ V (p)
iff w 1 p
iff w A and w B
iff w A or w B
iff for all w 0 if w 0 Ru, we have u A
iff there is a u ∈ W such that wRu and u A
Pseudo-transitive modal logics
11 / 10
♦q, [q], [[q], q, p], [[q, p], q, p̄, ♦p], [[q, p̄, ♦p], ♦p̄, ♦♦p]
cont, 4̇ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
♦q, [[q], q, p], [[q, p], q, p̄, ♦p], [[q, p̄, ♦p], ♦p̄, ♦♦p]
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
♦q, [q, q, p], [[q, p], q, p̄, ♦p], [[q, p̄, ♦p], ♦p̄, ♦♦p]
cont, ♦ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
♦q, [q, p], [[q, p], q, p̄, ♦p], [[q, p̄, ♦p], ♦p̄, ♦♦p]
cont, 4̇ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
♦q, [[q, p], q, p̄, ♦p], [[q, p̄, ♦p], ♦p̄, ♦♦p]
cont, ♦ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
♦q, [[q], q, p̄, ♦p], [[q, p̄, ♦p], ♦p̄, ♦♦p]
−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
♦q, [q, q, p̄, ♦p], [[q, p̄, ♦p], ♦p̄, ♦♦p]
cont, ♦ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
♦q, [q, p̄, ♦p], [[q, p̄, ♦p], ♦p̄, ♦♦p]
cont, 4̇ −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−
♦q, [[q, p̄, ♦p], ♦p̄, ♦♦p]
cont, ♦, ♦ −−−−−−−−−−−−−−−−−−−−−−−−−−−−
♦q, [[q], ♦p̄, ♦♦p]
−−−−−−−−−−−−−−−−−−−−−
♦q, [q, ♦p̄, ♦♦p]
cont, ♦ −−−−−−−−−−−−−−−−−−−−−
♦q, [♦p̄, ♦♦p]
∨, , ∨ −−−−−−−−−−−−−−−−−−−−−−
♦q ∨ (♦p̄ ∨ ♦♦p)
S.Marin, L.Straßburger (Inria)
Pseudo-transitive modal logics
12 / 10