Sequent Calculi for Normal Update Logics Katsuhiko Sano 1 Minghui Ma - PowerPoint PPT Presentation

Sequent Calculi for Normal Update Logics Katsuhiko Sano 1 Minghui Ma 2 1 Graduate School of Letters, Hokkaido University, Japan 2 Institute of Logic and Coginition, Sun Yat-Sen University, China. ICLA 2019 @ IIT Delhi, 4th March 2019

Outline What are Normal Update Logics? 1 Background on Sequent Calculi for Modal Logics 2 Sequent Calculi for Normal Update Logics 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 G ψ ( ψ will be always the case ) for Tense Logic

Kripke Semantics of Tense Logic � ψ : “It was the case that ψ . ” (P ψ ) 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 .

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 , ( R X ) X ⊆ W , V ) where R X ⊆ W × W . M , w | = [ ϕ ] ψ ⇐ ⇒ For all v ( wR � ϕ � v implies M , v | = ψ ) , where � ϕ � := { x ∈ W | M , x | = ϕ } . wR X v : v is one of the most “ X -similar” states from w .

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 , ( R X ) X ⊆ W , V ) where R X ⊆ W × W . = � ϕ − � ψ M , w | ⇐ ⇒ For some v ( vR � ϕ � w and M , v | = ψ ) . where � ϕ � := { x ∈ W | M , x | = ϕ } . wR X v : v is one of the most “ X -similar” states from w .

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.

H CK 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 ϕ 1 ↔ ϕ 2 , we may infer [ ϕ 1 ] ψ ↔ [ ϕ 2 ] ψ .

H UCK for Normal Update Logic: Herzig 1998 To Hilbert system H CK for Conditional Logic, we add: (Conv1) ψ → [ ϕ ] � ϕ − � ψ . (Conv2) � ϕ − � [ ϕ ] ψ → ψ . (Mon �· − � ) From ψ 1 → ψ 2 we may infer � ϕ − � ψ 1 → � ϕ − � ψ 2 . (EQUA) From ϕ 1 ↔ ϕ 2 , we may infer � ϕ − 1 � ψ ↔ � ϕ − 2 � ψ . ⊢ H UCK � ϕ − � ψ → θ ⇐ ⇒ ⊢ H UCK ψ → [ ϕ ] θ Semantic Completeness of H UCK (Herzig 1998) ϕ is provable in H UCK iff ϕ is valid in all models. ( ∵ ) By Canonical Model Construction.

Background & Contribution of This Talk Main Question of This Talk Does H UCK enjoy the decidability or the finite model property (FMP)? This talk provides sequent calculus G UCK ; shows that it is equipollent with H UCK ; establishes the subformula property and FMP of G UCK to obtain the decidability.

What are Sequents? a sequent = a pair of finite sets of formulas ϕ 1 , . . . , ϕ m ⇒ ψ 1 , . . . , ψ n . “If all ϕ i s hold, then some ψ j holds. ” ( ϕ 1 ∧ · · · ∧ ϕ m ) → ( ψ 1 ∨ · · · ∨ ψ n )

Sequent Calculus G K 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, Σ ⇒ Π Γ ⇒ ∆ , ϕ → ψ ( ⇒→ ) ϕ → ψ, Γ , Σ ⇒ ∆ , Π ( →⇒ ) Cut Rule: Γ ⇒ ∆ , ϕ ϕ, Π ⇒ Σ ( Cut ) Γ , Π ⇒ ∆ , Σ Modal Rule: ψ 1 , . . . , ψ n ⇒ ϕ � ψ 1 , . . . , � ψ n ⇒ � ϕ ( � ) ( n � 0 )

Cut Elimination of G K If a sequent is provable in G K then it is provable in G K w/o (Cut). (Cor.) Subformula Property of G K If Γ ⇒ ∆ is provable in G K then it is provable in G K by a derivation which consists of subformulas of Γ , ∆ alone.

Sequent Calculus G Kt for Tense Logic To get G Kt , we replace the rule ( � ) in G K w/: � Θ , Γ ⇒ ϕ ϕ ⇒ Σ , � Θ Θ , � Γ ⇒ � ϕ ( � Kt ) � ϕ ⇒ � Σ , Θ ( � Kt ) (Nishimura 1980) ( Cut ) is indispensable in G Kt as: � ¬ p ⇒ � ¬ p p ⇒ p � � ¬ p ⇒ ¬ p ( � Kt ) ¬ p , p ⇒ ( ¬ ⇒ ) ( Cut ) p , � � ¬ p ⇒ Subformula Property of G Kt (Takano 1992) If Γ ⇒ ∆ is provable in G Kt then it is provable in G Kt by a derivation which consists of subformulas of Γ , ∆ alone.

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.

Seq. Calc. G CK for Conditional Logic To get G CK , add the following to the Boolean part of G K : ϕ 0 ⇔ · · · ⇔ ϕ n ψ 1 , . . . , ψ n ⇒ ψ 0 ([ · ]) [ ϕ 1 ] ψ 1 , . . . , [ ϕ n ] ψ n ⇒ [ ϕ 0 ] ψ 0 (Pattinson et al. 2011) Cut Elimination of G CK (Pattinson et al. 2011) If a sequent is provable in G CK then it is provable in G CK w/o (Cut).

Seq. Calc. G UCK for Normal Update Logic To the Boolean part of G K , we add the following two: � ϕ − 1 � θ 1 , . . . , � ϕ − ϕ 0 ⇔ · · · ⇔ ϕ n n � θ n , ψ 1 , . . . , ψ n ⇒ ψ 0 ([ · ] UCK ) θ 1 , . . . , θ n , [ ϕ 1 ] ψ 1 , . . . , [ ϕ n ] ψ n ⇒ [ ϕ 0 ] ψ 0 ϕ 0 ⇔ · · · ⇔ ϕ n ψ 0 ⇒ ψ 1 , . . . , ψ n , [ ϕ 1 ] θ 1 , . . . , [ ϕ n ] θ n ( �· − � UCK ) � ϕ − 0 � ψ 0 ⇒ � ϕ − 1 � ψ 1 , . . . , � ϕ − n � ψ n , θ 1 , . . . , θ n

Equipollence Result ϕ is provable in H UCK iff ⇒ ϕ is provable in G UCK . ( Cut ) is indepensable in G UCK : [ q ] ¬ p ⇒ [ q ] ¬ p p ⇒ p � q − � [ q ] ¬ p ⇒ ¬ p ( �· − � UCK ) ¬ p , p ⇒ ( ¬ ⇒ ) ( Cut ) p , � q − � [ q ] ¬ p ⇒

Γ ⇒ ∆ is Ξ -provable in G UCK if there is a derivation D of the sequent such that D consists of formulas from Ξ . Main Result of This Talk TFAE: Γ ⇒ ∆ is Sub (Γ , ∆) -provable in G UCK . 1 Γ ⇒ ∆ is provable in G UCK . 2 � Γ → � ∆ is valid in all finite models. 3 ∵ (1) ⇒ (2) & (2) ⇒ (3) are easy. We focus on (3) ⇒ (1) below. Corollary G UCK enjoys the subformula property and FMP hence decidability. Therefore, H UCK is also decidable.

Proof Outline of (3) ⇒ (1) We prove the contrapositive implication. Suppose: Γ ⇒ ∆ is not Sub (Γ , ∆) -provable in G UCK . 1 Put Ξ := Sub (Γ , ∆) (finite!). 2 Extend Γ ⇒ ∆ to a Ξ -complete Γ + ⇒ ∆ + , 3 where “ Ξ -complete” means: Γ + ∪ ∆ + = Ξ . Γ + ⇒ ∆ + is still not Ξ -provable in G UCK . Define M Ξ = ( W Ξ , ( R Ξ X ) X ⊆ W Ξ , V Ξ ) as: 4 W Ξ = all Ξ -complete sequents (finite!). R Ξ X is defined via ([ · ] UCK ) and ( �· − � UCK ) . Π ⇒ Σ ∈ V Ξ ( p ) iff p ∈ Π . � Γ → � ∆ is falsified in the finite M Ξ . 5

Main Result of This Talk TFAE: Γ ⇒ ∆ is Sub (Γ , ∆) -provable in G UCK . 1 Γ ⇒ ∆ is provable in G UCK . 2 � Γ → � ∆ is valid in all finite models. 3 Corollary G UCK enjoys the subformula property and FMP hence decidability. Therefore, H UCK is also decidable.

Further Direction The paper contains the results on H UCK 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 G UCK ? Craig Interpolation Theorem for G UCK ?

Sequent Calculus G S5 for Modal Logic S5 To get G S5 , we replace the rule ( � ) in G K w/: � Γ ⇒ � ∆ , ψ ϕ, Γ ⇒ ∆ � Γ ⇒ � ∆ , � ψ ( ⇒ � S5 ) � ϕ, Γ ⇒ ∆ ( � ⇒ ) ( Cut ) is indispensable in G S5 as: ¬ p , p ⇒ ⇒ � ¬ p , ¬ � ¬ p � ¬ p , p ⇒ ( � S5 ⇒ ) ⇒ � ¬ p , � ¬ � ¬ p ( ⇒ � S5 ) p ⇒ ¬ � ¬ p ( ⇒ ¬ ) ¬ � ¬ p ⇒ � ¬ � ¬ p ( ¬ ⇒ ) ( Cut ) p ⇒ � ¬ � ¬ p Subformula Property of G S5 (Takano 1992) If Γ ⇒ ∆ is provable in G S5 then it is provable in G S5 by a derivation which consists of subformulas of Γ , ∆ alone.