SLIDE 1 Sequent Calculi for Normal Update Logics
Katsuhiko Sano1 Minghui Ma2
1Graduate School of Letters, Hokkaido University, Japan 2Institute of Logic and Coginition, Sun Yat-Sen University, China.
ICLA 2019 @ IIT Delhi, 4th March 2019
SLIDE 2 Outline
1
What are Normal Update Logics?
2
Background on Sequent Calculi for Modal Logics
3
Sequent Calculi for Normal Update Logics
SLIDE 3
Kripke Semantics of Modal Logic
ψ: “It is necessary that ψ.” ϕ ::= p | ¬ϕ | ϕ → ϕ | ϕ. Let M = (W, R, V) where R ⊆ W × W. M, w | = ψ ⇐ ⇒ For all v (wRv implies M, v | = ψ). ψ is Gψ (ψ will be always the case) for Tense Logic
SLIDE 4
Kripke Semantics of Tense Logic
ψ: “It was the case that ψ. ” (Pψ) ϕ ::= p | ¬ϕ | ϕ → ϕ | ϕ | ϕ. Let M = (W, R, V) where R ⊆ W × W. M, w | = ψ ⇐ ⇒ For some v (vRw and M, v | = ψ). | = ϕ → ψ ⇐ ⇒ | = ϕ → ψ.
SLIDE 5
Kripke Sem. of Conditional Logic (Chellas 1975)
[ϕ]ψ: “If ϕ then (normally) ψ.” “The current belief base is updated by ϕ, ψ follows.” ϕ ::= p | ¬ϕ | ϕ → ϕ | [ϕ]ϕ. Let M = (W, (RX)X⊆W, V) where RX ⊆ W × W. M, w | = [ϕ]ψ ⇐ ⇒ For all v (wRϕv implies M, v | = ψ), where ϕ := { x ∈ W | M, x | = ϕ }. wRXv: v is one of the most “X-similar” states from w.
SLIDE 6
Kripke Sem. of Normal Update Logic
ϕ−ψ: “ψ has been updated by ϕ.” (cf. Herzig (1998)) ϕ ::= p | ¬ϕ | ϕ → ϕ | [ϕ]ϕ | ϕ−ϕ. Let M = (W, (RX)X⊆W, V) where RX ⊆ W × W. M, w | = ϕ−ψ ⇐ ⇒ For some v (vRϕw and M, v | = ψ). where ϕ := { x ∈ W | M, x | = ϕ }. wRXv: v is one of the most “X-similar” states from w.
SLIDE 7
Motivation for Normal Update Logic
ϕ−ψ: “ψ has been updated by ϕ.” (cf. Herzig (1998)) Let p be an input, q a current belief base, r a resulting belief base. p = “I come to New Delhi” q = “I suffer from a heavy jet-lag in Japan,” r = “My jet-lag becomes lighter.” p−q ⊢ r ⇐ ⇒ q ⊢ [p]r
Herzig (1998), p.193
we neither consider updates to be more basic than conditionals nor the contrary, and shall rather take the equivalence to be basic.
SLIDE 8
HCK for Conditional Logic: Chellas 1975
(Taut) All instances of propositional tautologies (MP) Modus Ponens (K) [ϕ](ψ → θ) → ([ϕ]ψ → [ϕ]θ) (Nec) From ψ we may infer [ϕ]ψ. (EQCA) From ϕ1 ↔ ϕ2, we may infer [ϕ1]ψ ↔ [ϕ2]ψ.
SLIDE 9 HUCK for Normal Update Logic: Herzig 1998
To Hilbert system HCK for Conditional Logic, we add: (Conv1) ψ → [ϕ]ϕ−ψ. (Conv2) ϕ−[ϕ]ψ → ψ. (Mon ·−) From ψ1 → ψ2 we may infer ϕ−ψ1 → ϕ−ψ2. (EQUA) From ϕ1 ↔ ϕ2, we may infer ϕ−
1 ψ ↔ ϕ− 2 ψ.
⊢HUCK ϕ−ψ → θ ⇐ ⇒ ⊢HUCK ψ → [ϕ]θ
Semantic Completeness of HUCK (Herzig 1998)
ϕ is provable in HUCK iff ϕ is valid in all models. (∵) By Canonical Model Construction.
SLIDE 10
Background & Contribution of This Talk
Main Question of This Talk
Does HUCK enjoy the decidability or the finite model property (FMP)? This talk provides sequent calculus GUCK; shows that it is equipollent with HUCK; establishes the subformula property and FMP of GUCK to obtain the decidability.
SLIDE 11
What are Sequents?
a sequent = a pair of finite sets of formulas ϕ1, . . . , ϕm ⇒ ψ1, . . . , ψn. “If all ϕis hold, then some ψj holds. ” (ϕ1 ∧ · · · ∧ ϕm) → (ψ1 ∨ · · · ∨ ψn)
SLIDE 12
Sequent Calculus GK for Modal Logic K
Axioms: ϕ ⇒ ϕ Weakening rules Logical Rules: ϕ, Γ ⇒ ∆ Γ ⇒ ∆, ¬ϕ (⇒ ¬) Γ ⇒ ∆, ϕ ¬ϕ, Γ ⇒ ∆ (¬ ⇒) ϕ, Γ ⇒ ∆, ψ Γ ⇒ ∆, ϕ → ψ (⇒→) Γ ⇒ ∆, ϕ ψ, Σ ⇒ Π ϕ → ψ, Γ, Σ ⇒ ∆, Π (→⇒) Cut Rule: Γ ⇒ ∆, ϕ ϕ, Π ⇒ Σ Γ, Π ⇒ ∆, Σ (Cut) Modal Rule: ψ1, . . . , ψn ⇒ ϕ ψ1, . . . , ψn ⇒ ϕ () (n 0)
SLIDE 13
Cut Elimination of GK
If a sequent is provable in GK then it is provable in GK w/o (Cut).
(Cor.) Subformula Property of GK
If Γ ⇒ ∆ is provable in GK then it is provable in GK by a derivation which consists of subformulas of Γ, ∆ alone.
SLIDE 14
Sequent Calculus GKt for Tense Logic
To get GKt, we replace the rule () in GK w/: Θ, Γ ⇒ ϕ Θ, Γ ⇒ ϕ (Kt) ϕ ⇒ Σ, Θ ϕ ⇒ Σ, Θ (Kt) (Nishimura 1980) (Cut) is indispensable in GKt as: ¬p ⇒ ¬p ¬p ⇒ ¬p (Kt) p ⇒ p ¬p, p ⇒ (¬ ⇒) p, ¬p ⇒ (Cut)
Subformula Property of GKt (Takano 1992)
If Γ ⇒ ∆ is provable in GKt then it is provable in GKt by a derivation which consists of subformulas of Γ, ∆ alone.
SLIDE 15
Takano’s Methods for Subformual Property
Syntactic Method: Very much like cut-elimination.
Takano (1992) Subformula property as a substitute for cut-elimination in modal propositional logics, Math Jpn, 37(6),1129-1145.
Semantic Method: Show that system w/ analytic cut is semantically complete.
Takano (2018) A semantical analysis of cut-free calculi for modal logics, Rep. Math. Logic, 54, 43-65.
SLIDE 16
- Seq. Calc. GCK for Conditional Logic
To get GCK, add the following to the Boolean part of GK: ϕ0 ⇔ · · · ⇔ ϕn ψ1, . . . , ψn ⇒ ψ0 [ϕ1]ψ1, . . . , [ϕn]ψn ⇒ [ϕ0]ψ0 ([·]) (Pattinson et al. 2011)
Cut Elimination of GCK (Pattinson et al. 2011)
If a sequent is provable in GCK then it is provable in GCK w/o (Cut).
SLIDE 17
- Seq. Calc. GUCK for Normal Update Logic
To the Boolean part of GK, we add the following two: ϕ0 ⇔ · · · ⇔ ϕn ϕ−
1 θ1, . . . , ϕ− n θn, ψ1, . . . , ψn ⇒ ψ0
θ1, . . . , θn, [ϕ1]ψ1, . . . , [ϕn]ψn ⇒ [ϕ0]ψ0 ([·]UCK) ϕ0 ⇔ · · · ⇔ ϕn ψ0 ⇒ ψ1, . . . , ψn, [ϕ1]θ1, . . . , [ϕn]θn ϕ−
0 ψ0 ⇒ ϕ− 1 ψ1, . . . , ϕ− n ψn, θ1, . . . , θn
(·−UCK)
SLIDE 18
Equipollence Result
ϕ is provable in HUCK iff ⇒ ϕ is provable in GUCK. (Cut) is indepensable in GUCK: [q]¬p ⇒ [q]¬p q−[q]¬p ⇒ ¬p (·−UCK) p ⇒ p ¬p, p ⇒ (¬ ⇒) p, q−[q]¬p ⇒ (Cut)
SLIDE 19 Γ ⇒ ∆ is Ξ-provable in GUCK if there is a derivation D of the sequent such that D consists of formulas from Ξ.
Main Result of This Talk
TFAE:
1
Γ ⇒ ∆ is Sub(Γ, ∆)-provable in GUCK.
2
Γ ⇒ ∆ is provable in GUCK.
3
Γ → ∆ is valid in all finite models. ∵ (1) ⇒ (2) & (2) ⇒ (3) are easy. We focus on (3) ⇒ (1) below.
Corollary
GUCK enjoys the subformula property and FMP hence
- decidability. Therefore, HUCK is also decidable.
SLIDE 20 Proof Outline of (3) ⇒ (1)
We prove the contrapositive implication.
1
Suppose: Γ ⇒ ∆ is not Sub(Γ, ∆)-provable in GUCK.
2
Put Ξ := Sub(Γ, ∆) (finite!).
3
Extend Γ ⇒ ∆ to a Ξ-complete Γ+ ⇒ ∆+, where “Ξ-complete” means:
Γ+ ∪ ∆+ = Ξ. Γ+ ⇒ ∆+ is still not Ξ-provable in GUCK.
4
Define MΞ = (W Ξ, (RΞ
X)X⊆W Ξ, V Ξ) as:
W Ξ = all Ξ-complete sequents (finite!). RΞ
X is defined via ([·]UCK) and (·−UCK).
Π ⇒ Σ ∈ V Ξ(p) iff p ∈ Π.
5
Γ → ∆ is falsified in the finite MΞ.
SLIDE 21 Main Result of This Talk
TFAE:
1
Γ ⇒ ∆ is Sub(Γ, ∆)-provable in GUCK.
2
Γ ⇒ ∆ is provable in GUCK.
3
Γ → ∆ is valid in all finite models.
Corollary
GUCK enjoys the subformula property and FMP hence
- decidability. Therefore, HUCK is also decidable.
SLIDE 22
Further Direction
The paper contains the results on HUCK extended w/: (CID) [ϕ]ϕ and/or (CMP) [ϕ]ψ → (ϕ → ψ). Further extension, say w/ (CLEM) [ϕ]ψ ∨ [ϕ]¬ψ. Syntactic proof of the subformula property of GUCK? Craig Interpolation Theorem for GUCK?
SLIDE 23
Sequent Calculus GS5 for Modal Logic S5
To get GS5, we replace the rule () in GK w/: Γ ⇒ ∆, ψ Γ ⇒ ∆, ψ (⇒ S5) ϕ, Γ ⇒ ∆ ϕ, Γ ⇒ ∆ ( ⇒) (Cut) is indispensable in GS5 as: ¬p, p ⇒ ¬p, p ⇒ (S5 ⇒) p ⇒ ¬¬p (⇒ ¬) ⇒ ¬p, ¬¬p ⇒ ¬p, ¬¬p (⇒ S5) ¬¬p ⇒ ¬¬p (¬ ⇒) p ⇒ ¬¬p (Cut)
Subformula Property of GS5 (Takano 1992)
If Γ ⇒ ∆ is provable in GS5 then it is provable in GS5 by a derivation which consists of subformulas of Γ, ∆ alone.
