Just some space Axiomatizing a Reflexive Real-Valued Modal Logic - - PowerPoint PPT Presentation

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Just some space Axiomatizing a Reflexive Real-Valued Modal Logic - - PowerPoint PPT Presentation

Apr 16, 2023 514 likes •1.4k views

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Just some space Axiomatizing a Reflexive Real-Valued Modal Logic Laura Janina Schn uriger Axiomatizing a Reflexive Real-valued Modal Logic Just some space

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SLIDE 1

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work

Just some space Axiomatizing a Reflexive Real-Valued Modal Logic

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-valued Modal Logic Just some space TACL, Praha 29 June 2017

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 2

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work That’s the talk about A Real-Valued Modal Logic

That’s the talk about

Motivation

Many-valued logics and modal logics are well-established topics

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 3

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work That’s the talk about A Real-Valued Modal Logic

That’s the talk about

Motivation

Many-valued logics and modal logics are well-established topics But the connection is still not a very well-studied field of research

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 4

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work That’s the talk about A Real-Valued Modal Logic

A Real-Valued Modal Logic

universe

(=truth values): R

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 5

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work That’s the talk about A Real-Valued Modal Logic

A Real-Valued Modal Logic

universe

(=truth values): R

designated

(=true truth values): R+

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 6

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work That’s the talk about A Real-Valued Modal Logic

A Real-Valued Modal Logic

universe

(=truth values): R

designated

(=true truth values): R+

connectives

interpreted as arithmetic group operations +, −, 0

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 7

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work That’s the talk about A Real-Valued Modal Logic

A Real-Valued Modal Logic

universe

(=truth values): R

designated

(=true truth values): R+

connectives

interpreted as arithmetic group operations +, −, 0 plus modality

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 8

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work That’s the talk about A Real-Valued Modal Logic

Language

L

{&, ¬, 0, } = {→, }

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 9

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work That’s the talk about A Real-Valued Modal Logic

Language

L

{&, ¬, 0, } = {→, } = p0 → p0 p0 ∈ Var

¬ϕ

= ϕ → 0

ϕ&ψ

= ¬ϕ → ψ

♦ϕ

= ¬¬ϕ

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 10

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work That’s the talk about A Real-Valued Modal Logic

Language

L

{&, ¬, 0, } = {→, } = p0 → p0 p0 ∈ Var

¬ϕ

= ϕ → 0

ϕ&ψ

= ¬ϕ → ψ

♦ϕ

= ¬¬ϕ

0ϕ

= 0

(n + 1)ϕ

= ϕ&nϕ

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 11

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Frames and models

Kripke frames

frame

F = W , R

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 12

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Frames and models

Kripke frames

frame

F = W , R

W

non-empty set of elements called worlds

R

⊆ W × W a binary relation on W

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 13

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Frames and models

Kripke frames

frame

F = W , R

W

non-empty set of elements called worlds

R

⊆ W × W a binary relation on W

serial

a frame F is called serial if for all x ∈ W there is y ∈ W such that Rxy

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 14

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Frames and models

Kripke frames

frame

F = W , R

W

non-empty set of elements called worlds

R

⊆ W × W a binary relation on W

serial

a frame F is called serial if for all x ∈ W there is y ∈ W such that Rxy

reflexive

if for all x ∈ W : Rxx

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 15

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Frames and models

Models

Intuition

Talk about abelian groups at each world of the frames

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 16

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Frames and models

Models

model

M = W , R, V

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 17

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Frames and models

Models

model

M = W , R, V W , R is a frame

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 18

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Frames and models

Models

model

M = W , R, V W , R is a frame

valuation

V : Var ×W → [−r, r] for some r ∈ R+

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 19

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Frames and models

Models

model

M = W , R, V W , R is a frame

valuation

V : Var ×W → [−r, r] for some r ∈ R+ extends to V : Fm ×W → R via V (ϕ → ψ; x) = V (ψ; x) − V (ϕ; x) V (ϕ; x) = {V (ϕ; y) | Rxy}

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 20

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Frames and models

Models

V (0; x) = 0 V (¬ϕ; x) = −V (ϕ; x) V (ϕ&ψ; x) = V (ϕ; x) + V (ψ; x) V (♦ϕ; x) = {V (ϕ; y) | Rxy}

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 21

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Frames and models

Models

K(A)-model

if R is serial

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 22

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Frames and models

Models

K(A)-model

if R is serial

KT(A)-model

if R is reflexive

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 23

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Frames and models

Validity

valid

ϕ ∈ Fm is K(A)/KT(A)-valid if V (ϕ; x) ≥ 0 for all x ∈ W in all K(A)/KT(A)-models M = W , R, V

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 24

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Frames and models

Validity

valid

ϕ ∈ Fm is K(A)/KT(A)-valid if V (ϕ; x) ≥ 0 for all x ∈ W in all K(A)/KT(A)-models M = W , R, V

notation

K(A) ϕ respectively KT(A) ϕ

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 25

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Frames and models

Remarks

K(A)

seriality is not necessary if ∅ = ∅ := 0

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 26

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Frames and models

Remarks

K(A)

seriality is not necessary if ∅ = ∅ := 0 K(A) ϕ iff ϕ is valid in all models without restricting R to be serial

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 27

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Frames and models

Remarks

K(A)

seriality is not necessary if ∅ = ∅ := 0 K(A) ϕ iff ϕ is valid in all models without restricting R to be serial

KT(A)

reflexive ⇒ serial

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Axiom systems Main Theorem 1

Aim

so far

semantical definition of K(A) and KT(A)

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 29

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Axiom systems Main Theorem 1

Aim

so far

semantical definition of K(A) and KT(A)

aim

syntactical description

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 30

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Axiom systems Main Theorem 1

Aim

so far

semantical definition of K(A) and KT(A)

aim

syntactical description

find

axiomatizations K and KT such that ϕ is derivable in K ⇔ K(A) ϕ ϕ is derivable in KT ⇔ KT(A) ϕ

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 31

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Axiom systems Main Theorem 1

Axiom system K

Axiomatization for multiplicative fragment of Abelian logic (logic of abelian groups) A plus

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 32

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Axiom systems Main Theorem 1

Axiom system K

Axiomatization for multiplicative fragment of Abelian logic (logic of abelian groups) A plus

axioms (K)

(ϕ → ψ) → (ϕ → ψ)

(Dn)

(nϕ) → nϕ (n ≥ 2)

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 33

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Axiom systems Main Theorem 1

Axiom system K

Axiomatization for multiplicative fragment of Abelian logic (logic of abelian groups) A plus

axioms (K)

(ϕ → ψ) → (ϕ → ψ)

(Dn)

(nϕ) → nϕ (n ≥ 2)

rules (nec)

ϕ/ϕ

(conn)

nϕ/ϕ (n ≥ 2)

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 34

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Axiom systems Main Theorem 1

Axiom system KT

K plus

axiom (T)

ϕ → ϕ

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 35

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Axiom systems Main Theorem 1

Main Theorem 1

Theorem

For any ϕ ∈ Fm :

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 36

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Axiom systems Main Theorem 1

Main Theorem 1

Theorem

For any ϕ ∈ Fm : ⊢K ϕ ⇔ K(A) ϕ

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 37

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Axiom systems Main Theorem 1

Main Theorem 1

Theorem

For any ϕ ∈ Fm : ⊢K ϕ ⇔ K(A) ϕ

Proof ⇒

easy:

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 38

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Axiom systems Main Theorem 1

Main Theorem 1

Theorem

For any ϕ ∈ Fm : ⊢K ϕ ⇔ K(A) ϕ

Proof ⇒

easy: check that axioms are K(A)-valid rules preserve validity

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 39

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Axiom systems Main Theorem 1

Main Theorem 1

Theorem

For any ϕ ∈ Fm : ⊢K ϕ ⇔ K(A) ϕ

Proof ⇒

easy: check that axioms are K(A)-valid rules preserve validity

⇐

not easy:

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 40

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Axiom systems Main Theorem 1

Main Theorem 1

Theorem

For any ϕ ∈ Fm : ⊢K ϕ ⇔ K(A) ϕ

Proof ⇒

easy: check that axioms are K(A)-valid rules preserve validity

⇐

not easy: via systems LK(A) and GK(A)

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 41

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Axiom systems Main Theorem 1

Main Theorem 1

Theorem

For any ϕ ∈ Fm : ⊢K ϕ ⇔ K(A) ϕ

Proof ⇒

easy: check that axioms are K(A)-valid rules preserve validity

⇐

not easy: via systems LK(A) and GK(A) K(A) ϕ ⇔ ⊢LK(A) ϕ labelled calculus ⇓ ⊢K(A) ϕ ⇔ ⊢GK(A)⇒ ϕ sequent calculus

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 42

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Axiom systems Main Theorem 1

Main Theorem 1

Theorem

For any ϕ ∈ Fm : ⊢K ϕ ⇔ K(A) ϕ

Proof ⇒

easy: check that axioms are K(A)-valid rules preserve validity

⇐

not easy: via systems LK(A) and GK(A) K(A) ϕ ⇔ ⊢LK(A) ϕ labelled calculus ⇓ ⊢K(A) ϕ ⇔ ⊢GK(A)⇒ ϕ sequent calculus K(A) ϕ ⇔ ⊢K ϕ ⇔

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 43

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Axiom systems Main Theorem 1

Main Theorem 1

Theorem

For any ϕ ∈ Fm : ⊢K ϕ ⇔ K(A) ϕ

Proof ⇒

easy: check that axioms are K(A)-valid rules preserve validity

⇐

not easy: via systems LK(A) and GK(A) K(A) ϕ ⇔ ⊢LK(A) ϕ labelled calculus ⇓ ⊢K(A) ϕ ⇔ ⊢GK(A)⇒ ϕ sequent calculus K(A) ϕ ⇔ ⊢K ϕ ⇔ ⇓

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 44

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work A Semantical Calculus A Syntactical Calculus

The labelled tableau system LK(A)

Notation

multisets Γ = [ϕ1, . . . , ϕn], ∆ = [ψ1, . . . , ψm] and for k1, . . . , kn, l1, . . . , lm ∈ N, let (Γ)k = [(ϕ1)k1, . . . , (ϕn)kn], (∆)l = [(ψ1)l1, . . . , (ψm)lm]

tableau nodes

are of the forms (Γ)k ⊲ (∆)l for ⊲∈ {>, ≥} rij for i, j ∈ N

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 45

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work A Semantical Calculus A Syntactical Calculus

The labelled tableau system LK(A)

Notation

multisets Γ = [ϕ1, . . . , ϕn], ∆ = [ψ1, . . . , ψm] and for k1, . . . , kn, l1, . . . , lm ∈ N, let (Γ)k = [(ϕ1)k1, . . . , (ϕn)kn], (∆)l = [(ψ1)l1, . . . , (ψm)lm]

tableau nodes

are of the forms (Γ)k ⊲ (∆)l for ⊲∈ {>, ≥} rij for i, j ∈ N

tableau

for a formula ϕ is a finite sequence of nodes with root [] > [(ϕ)1] r12

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 46

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work A Semantical Calculus A Syntactical Calculus

The rules

(Γ)k, (ψ)i ⊲ (ϕ)i, (∆)l (Γ)k, (ϕ → ψ)i ⊲ (∆)l (→⊲) (Γ)k, (ϕ)i ⊲ (ψ)i, (∆)l (Γ)k ⊲ (ϕ → ψ)i, (∆)l (⊲→) (ϕ)j ≥ (ϕ)i (Γ)k, (ϕ)i ⊲ (∆)l rij ( ⊲) rij (ϕ)i ≥ (ϕ)j (Γ)k ⊲ (ϕ)i, (∆)l (⊲ ) (j ∈ N new)

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 47

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work A Semantical Calculus A Syntactical Calculus

The rules

(Γ)k, (ψ)i ⊲ (ϕ)i, (∆)l (Γ)k, (ϕ → ψ)i ⊲ (∆)l (→⊲) (Γ)k, (ϕ)i ⊲ (ψ)i, (∆)l (Γ)k ⊲ (ϕ → ψ)i, (∆)l (⊲→) (ϕ)j ≥ (ϕ)i (Γ)k, (ϕ)i ⊲ (∆)l rij ( ⊲) rij (ϕ)i ≥ (ϕ)j (Γ)k ⊲ (ϕ)i, (∆)l (⊲ ) rkj rik (ex) (j ∈ N new) (j ∈ N new)

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 48

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work A Semantical Calculus A Syntactical Calculus

Tableaux

inequations

The system of inequations associated to a tableau consists of all inequations over labelled variables and boxed formulas for each branch, where “,” is “+”.

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 49

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work A Semantical Calculus A Syntactical Calculus

Tableaux

inequations

The system of inequations associated to a tableau consists of all inequations over labelled variables and boxed formulas for each branch, where “,” is “+”.

closed

a tableau is closed if its associated systems of inequations are all inconsistent over R

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 50

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work A Semantical Calculus A Syntactical Calculus

Tableaux

inequations

The system of inequations associated to a tableau consists of all inequations over labelled variables and boxed formulas for each branch, where “,” is “+”.

closed

a tableau is closed if its associated systems of inequations are all inconsistent over R

derivable

in LK(A), ⊢LK(A) ϕ, if there is a closed tableau for ϕ

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 51

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work A Semantical Calculus A Syntactical Calculus

Theorem

Theorem

For any ϕ ∈ Fm K(A) ϕ ⇔ ⊢LK(A) ϕ

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 52

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work A Semantical Calculus A Syntactical Calculus

Theorem

Theorem

For any ϕ ∈ Fm K(A) ϕ ⇔ ⊢LK(A) ϕ

Theorem

K(A)-validity is in EXPTIME

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 53

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work A Semantical Calculus A Syntactical Calculus

The sequent calculus GK(R)

Notation

multiset union Γ, ∆ nΓ := Γ, . . . , Γ

n times

Γ := [ϕ | ϕ ∈ Γ]

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 54

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work A Semantical Calculus A Syntactical Calculus

The sequent calculus GK(R)

Notation

multiset union Γ, ∆ nΓ := Γ, . . . , Γ

n times

Γ := [ϕ | ϕ ∈ Γ]

sequent

Γ ⇒ ∆

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 55

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work A Semantical Calculus A Syntactical Calculus

The rules

∆ ⇒ ∆ (ID) Γ, ϕ ⇒ ∆ Π ⇒ ϕ, Σ Γ, Π ⇒ Σ, ∆ (CUT) Γ ⇒ ∆ Π ⇒ Σ Γ, Π ⇒ Σ, ∆ (MIX) nΓ ⇒ n∆ Γ ⇒ ∆ (scn)

(n ≥ 2)

Γ, ψ ⇒ ϕ, ∆ Γ, ϕ → ψ ⇒ ∆ (→⇒) Γ, ϕ ⇒ ψ, ∆ Γ ⇒ ϕ → ψ, ∆ (⇒→) Γ ⇒ n[ϕ] Γ ⇒ n[ϕ] (n)

(n ≥ 0)

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 56

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work A Semantical Calculus A Syntactical Calculus

Theorems

translation

sequent formula: I(Γ ⇒ ∆)

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 57

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work A Semantical Calculus A Syntactical Calculus

Theorems

translation

sequent formula: I(Γ ⇒ ∆)

Theorem

For any sequent Γ ⇒ ∆ ⊢K I(Γ ⇒ ∆) ⇔ ⊢GK(A) Γ ⇒ ∆

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 58

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work A Semantical Calculus A Syntactical Calculus

Theorems

translation

sequent formula: I(Γ ⇒ ∆)

Theorem

For any sequent Γ ⇒ ∆ ⊢K I(Γ ⇒ ∆) ⇔ ⊢GK(A) Γ ⇒ ∆

Theorem

GK(A) admits cut elimination

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

slide-59
SLIDE 59

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work

To show

Theorem 1

For any ϕ ∈ Fm : ⊢K ϕ ⇔ K(A) ϕ

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 60

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work

To show

Theorem 1

For any ϕ ∈ Fm : ⊢K ϕ ⇔ K(A) ϕ

known

K(A) ϕ ⇔ ⊢LK(A) ϕ ⊢K ϕ ⇔ ⊢GK(A)⇒ ϕ

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 61

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work

To show

Theorem 1

For any ϕ ∈ Fm : ⊢K ϕ ⇔ K(A) ϕ

known

K(A) ϕ ⇔ ⊢LK(A) ϕ ⊢K ϕ ⇔ ⊢GK(A)⇒ ϕ

left to show

For any ϕ ∈ Fm : ⊢LK(A) ϕ ⇒ ⊢GK(A)⇒ ϕ

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 62

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work

To show

Theorem 1

For any ϕ ∈ Fm : ⊢K ϕ ⇔ K(A) ϕ

known

K(A) ϕ ⇔ ⊢LK(A) ϕ ⊢K ϕ ⇔ ⊢GK(A)⇒ ϕ

left to show

For any ϕ ∈ Fm : ⊢LK(A) ϕ ⇒ ⊢GK(A)⇒ ϕ

  • r

⊢LK(A) I(Γ ⇒ ∆) ⇒ ⊢GK(A) Γ ⇒ ∆

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

slide-63
SLIDE 63

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work

Proof intuitive idea

induction

  • n sum complexities of Γ ⇒ ∆

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

slide-64
SLIDE 64

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work

Proof intuitive idea

induction

  • n sum complexities of Γ ⇒ ∆

base case Γ = ∆ immediate ψ → χ ∈ Γ or ψ → χ ∈ ∆ easy

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

slide-65
SLIDE 65

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work

Proof intuitive idea

induction

  • n sum complexities of Γ ⇒ ∆

base case Γ = ∆ immediate ψ → χ ∈ Γ or ψ → χ ∈ ∆ easy interesting case: Γ′ ⇒ ∆′

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

slide-66
SLIDE 66

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work

Proof intuitive idea

⊢LK(A) I(Γ′ ⇒ ∆′)

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 67

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work

Proof intuitive idea

⊢LK(A) I(Γ′ ⇒ ∆′)

⇒

there is a tableau with inconsistent set of inequations

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

slide-68
SLIDE 68

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work

Proof intuitive idea

⊢LK(A) I(Γ′ ⇒ ∆′)

⇒

there is a tableau with inconsistent set of inequations

⇒

linear programming problem over R

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

slide-69
SLIDE 69

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work

Proof intuitive idea

⊢LK(A) I(Γ′ ⇒ ∆′)

⇒

there is a tableau with inconsistent set of inequations

⇒

linear programming problem over R

⇒

gives formulas I(Σi ⇒ Πi) derivable in LK(A) for some i = 1, . . . , m

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

slide-70
SLIDE 70

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work

Proof intuitive idea

⊢LK(A) I(Γ′ ⇒ ∆′)

⇒

there is a tableau with inconsistent set of inequations

⇒

linear programming problem over R

⇒

gives formulas I(Σi ⇒ Πi) derivable in LK(A) for some i = 1, . . . , m

⇒

apply induction hypothesis

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

slide-71
SLIDE 71

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work

Proof intuitive idea

⊢LK(A) I(Γ′ ⇒ ∆′)

⇒

there is a tableau with inconsistent set of inequations

⇒

linear programming problem over R

⇒

gives formulas I(Σi ⇒ Πi) derivable in LK(A) for some i = 1, . . . , m

⇒

apply induction hypothesis ⊢GK(A) Σi ⇒ Πi

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

slide-72
SLIDE 72

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work

Proof intuitive idea

⊢LK(A) I(Γ′ ⇒ ∆′)

⇒

there is a tableau with inconsistent set of inequations

⇒

linear programming problem over R

⇒

gives formulas I(Σi ⇒ Πi) derivable in LK(A) for some i = 1, . . . , m

⇒

apply induction hypothesis ⊢GK(A) Σi ⇒ Πi

⇒

apply (sck), (MIX) and (n) to get ⊢GK(R) Γ′ ⇒ ∆′

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 73

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Main Theorem 2 Calculi

Main Theorem 2

Recall

KT is K plus ϕ → ϕ

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 74

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Main Theorem 2 Calculi

Main Theorem 2

Recall

KT is K plus ϕ → ϕ

Theorem 2

For any ϕ ∈ Fm : ⊢KT ϕ ⇔ KT(A) ϕ

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 75

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Main Theorem 2 Calculi

Main Theorem 2

Recall

KT is K plus ϕ → ϕ

Theorem 2

For any ϕ ∈ Fm : ⊢KT ϕ ⇔ KT(A) ϕ

Proof

same idea as for K(A)

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 76

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Main Theorem 2 Calculi

The two calculi

LKT(A)

replace (ex) by rjj rij (ref )

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 77

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Main Theorem 2 Calculi

The two calculi

LKT(A)

replace (ex) by rjj rij (ref )

GKT(A)

add Γ, ϕ ⇒ ∆ Γ, ϕ ⇒ ∆ ( ⇒)

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 78

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Main Theorem 2 Calculi

Theorems

Theorem

For any ϕ ∈ Fm KT(A) ϕ ⇔ ⊢LKT(A) ϕ

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 79

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Main Theorem 2 Calculi

Theorems

Theorem

For any ϕ ∈ Fm KT(A) ϕ ⇔ ⊢LKT(A) ϕ

Theorem

For any ϕ ∈ Fm KT I(Γ ⇒ ∆) ⇔ ⊢GKT(A) Γ ⇒ ∆

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

slide-80
SLIDE 80

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work

What’s different to before?

before

’s were removed simultaneously on the left and the right hand side of a sequent

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

slide-81
SLIDE 81

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work

What’s different to before?

before

’s were removed simultaneously on the left and the right hand side of a sequent

now

can be removed just on left hand side by ( ⇒)

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

slide-82
SLIDE 82

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work

What’s different to before?

before

’s were removed simultaneously on the left and the right hand side of a sequent

now

can be removed just on left hand side by ( ⇒)

⇒

it is more complicated to apply induction hypothesis

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 83

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Appendix

Further Work

Find axiomatizations for transitive (and symmetric) case

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 84

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Appendix

Further Work

Find axiomatizations for transitive (and symmetric) case add lattice operators ∧ and ∨

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 85

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Appendix

Further Work

Find axiomatizations for transitive (and symmetric) case add lattice operators ∧ and ∨ Find complexitiy class of decidability for K(A) (is it EXPTIME-complete?)

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 86

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Appendix

References

  • D. Diaconescu, G. Metcalfe and L. Schn¨

uriger. Axiomatizing a Real-Valued Modal Logic. In Proceedings of AiML 2016, Kings College Publications (2016), pp. 236-251.

  • D. Diaconescu, G. Metcalfe and L. Schn¨

uriger. A Real-Valued Modal Logic. submitted in LMCE.

  • G. Hansoul and B. Teheux. Extending

Lukasiewicz logics with a modality: Algebraic approach to relational

  • semantics. Studia logica 101 (2013), pp. 505-545.
  • G. Metcalfe, N. Olivetti and D. Gabbay. Sequent and

hypersequent calculi for abelian and Lukasiewicz logics. ACM Transactions on Computational Logic 6 (2005),

  • pp. 587-613.

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic

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SLIDE 87

About Kripke Semantics Aim Calculi Proof Reflexive Case Proof Further Work Appendix

Axiom system K(A)

axioms (B)

(ϕ → ψ) → ((ψ → χ) → (ϕ → χ))

(C)

(ϕ → (ψ → χ)) → (ψ → (ϕ → χ))

(I)

ϕ → ϕ

(A)

((ϕ → ψ) → ψ) → ϕ

(K)

(ϕ → ψ) → (ϕ → ψ)

(Dn)

(nϕ) → nϕ (n ≥ 2)

rules (mp)

{ϕ, ϕ → ψ}/ψ

(nec)

ϕ/ϕ

(conn)

nϕ/ϕ (n ≥ 2)

Laura Janina Schn¨ uriger Axiomatizing a Reflexive Real-Valued Modal Logic