General Methods in Proof Theory for Modal Logic { Lecture 2 Limits of the Sequent Framework Bj¨ornLellmann and Revantha Ramanayake TU Wien Tutorial co-located with TABLEAUX 2017, FroCoS 2017 and ITP 2017 Bras´ılia,Brasil, Sep 24 2017 Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination Outline Case Study: S5 Sequents for S5 Hypersequents for S5 Cut Elimination Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination Reminder: Modal Logics The formulae of modal logic are given by (V is a set of variables): F ∶∶= V S F ∧ F S F ∨ F S F → F S ¬F S ◻F with ÅA abbreviating the formula ¬ ◻ ¬A. A Kripke frame consists of a set W of worlds and an accessibility relation R ⊆ W × W . A Kripke model is a Kripke frame with a valuation V ∶ V → P(W ). Truth at a world w in a model M is defined via: M; w ⊩ p iff w ∈ V (p) M; w ⊩ ◻A iff ∀v ∈ W ∶ wRv ⇒ M; v ⊩ A M; w ⊩ ÅA iff ∃v ∈ W ∶ wRv & M; v ⊩ A Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination Modal Logic S5 Definition Modal logic S5 is the logic given by the class of Kripke frames with universal accessibility relation, i.e., frames (W ; R) with: ∀x; y ∈ W ∶ xRy : Thus S5-theorems are those modal formulae which are true in every world of every Kripke model with universal accessibility relation. R universalR universal ⍑ ; p ; ◻◻p p ; ◻p → p; ◻p → ◻◻p R universal ⍑ ⍑ ⍑ ; ◻Åp Åp ◻p ◻p Hilbert-style Definition: S5 is given by closing the axioms ◻(p → q) → (◻p → ◻q) p → ◻Åp ◻ p → p ◻ p → ◻◻ p and propositional axioms under uniform substitution and the rules AA → B modus ponens, MP A necessitation, nec B ◻A Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination Modal Logic S5 Example The formulae p → ◻Åp are theorems of S5: ⍑ p R universalR universal ⍑ ; p ; ◻◻p p ; ◻p → p; ◻p → ◻◻p ⍑ ⍑ ◻p ◻p Hilbert-style Definition: S5 is given by closing the axioms ◻(p → q) → (◻p → ◻q) p → ◻Åp ◻ p → p ◻ p → ◻◻ p and propositional axioms under uniform substitution and the rules AA → B modus ponens, MP A necessitation, nec B ◻A Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination Modal Logic S5 Example The formulae p → ◻Åp are theorems of S5: R universal ⍑ ⍑ p; ◻Åp Åp R universal ⍑ ; ◻◻p p ; ◻p → ◻◻p R universalR universal ⍑ ; p ◻p Hilbert-style Definition: S5 is given by closing the axioms ◻(p → q) → (◻p → ◻q) p → ◻Åp ◻ p → p ◻ p → ◻◻ p and propositional axioms under uniform substitution and the rules AA → B modus ponens, MP A necessitation, nec B ◻A Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination Modal Logic S5 Example The formulae p → ◻Åp; ◻p → p are theorems of S5: ⍑ ⍑ ⍑ p; ◻Åp Åp ◻p R universal ⍑ ; ◻◻p p ; ◻p → ◻◻p R universal ⍑ ◻p Hilbert-style Definition: S5 is given by closing the axioms ◻(p → q) → (◻p → ◻q) p → ◻Åp ◻ p → p ◻ p → ◻◻ p and propositional axioms under uniform substitution and the rules AA → B modus ponens, MP A necessitation, nec B ◻A Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination Modal Logic S5 Example The formulae p → ◻Åp; ◻p → p are theorems of S5: R universal ⍑ ⍑ ⍑ p; ◻Åp Åp ◻p; p R universalR universalR universal ⍑ ; ◻◻p p Hilbert-style Definition: S5 is given by closing the axioms ◻(p → q) → (◻p → ◻q) p → ◻Åp ◻ p → p ◻ p → ◻◻ p and propositional axioms under uniform substitution and the rules AA → B modus ponens, MP A necessitation, nec B ◻A Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination Modal Logic S5 Example The formulae p → ◻Åp; ◻p → p; ◻p → ◻◻p are theorems of S5: ⍑ ⍑ ⍑ ⍑ p; ◻Åp Åp ◻p; p ◻p R universalR universal Hilbert-style Definition: S5 is given by closing the axioms ◻(p → q) → (◻p → ◻q) p → ◻Åp ◻ p → p ◻ p → ◻◻ p and propositional axioms under uniform substitution and the rules AA → B modus ponens, MP A necessitation, nec B ◻A Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination Modal Logic S5 Example The formulae p → ◻Åp; ◻p → p; ◻p → ◻◻p are theorems of S5: R universal ⍑ ⍑ ⍑ ⍑ ⍑ p; ◻Åp Åp ◻p; p ◻p; ◻◻p p R universalR universalR universal ⍑ p Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination Modal Logic S5 Example The formulae p → ◻Åp; ◻p → p; ◻p → ◻◻p are theorems of S5: ⍑ ⍑ ⍑ ⍑ p; ◻Åp Åp ◻p; p ◻p; ◻◻p Hilbert-style Definition: S5 is given by closing the axioms ◻(p → q) → (◻p → ◻q) p → ◻Åp ◻ p → p ◻ p → ◻◻ p and propositional axioms under uniform substitution and the rules AA → B modus ponens, MP A necessitation, nec B ◻A Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination A Sequent Calculus for S5 Definition (Takano 1992) The sequent calculus sS5 contains the standard propositional rules and Γ; A ∆ Γ A; ∆ ⊢ T ◻ ⊢ ◻ 45 Γ; ◻A ⊢ ∆ ◻Γ ⊢ ◻A; ◻∆ Theorem sS5 is sound and complete (with cut) for S5. Proof. Derive axioms and rules of the Hilbert-system. E.g., for p → ◻Åp: init init ◻¬p ⊢ ◻¬p p ⊢ p ¬L ¬L ⊢ ¬ ◻ ¬p; ◻¬p ¬p; p ⊢ p; p 45 p; p T ⊢ ◻¬ ◻ ¬ ◻¬ ◻¬ ⊢ cut p ⊢ ◻¬ ◻ ¬p →R ⊢ p → ◻Åp Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination A Sequent Calculus for S5 Definition (Takano 1992) The sequent calculus sS5 contains the standard propositional rules and Γ; A ∆ Γ A; ∆ ⊢ T ◻ ⊢ ◻ 45 Γ; ◻A ⊢ ∆ ◻Γ ⊢ ◻A; ◻∆ Theorem sS5 is sound and complete (with cut) for S5. Proof. E.g. the modus ponens rule AA → B is simulated by: B A; B ⊢ BA ⊢ A; B A B A; A B B →L ⊢ → → ⊢ cut A A B ⊢ ⊢ cut ⊢ B Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination What about cut-free completeness? Our standard proof of cut elimination fails: . A; A A; A ⊢ ¬◻¬ ◻¬ 45 ¬ ⊢ T A; A A; A ⊢ ◻¬ ◻ ¬ ◻¬ ◻¬ ⊢ cut A ⊢ ◻¬ ◻ ¬A would need to reduce to: . A; A . ¬ ⊢ T A; A A; A ⊢ ¬ ◻ ¬ ◻¬ ◻¬ ⊢ cut A A ⊢ ¬ ◻ ¬ ?? But we can't apply rule 45 anymore since A is not boxed! Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination What about cut-free completeness? But could there be a different derivation? No! In fact we have: Theorem The sequent p ⊢ ◻Åp is not cut-free derivable in sS5. Proof. The only rules that can be applied in a cut-free derivation ending in p ⊢ ◻Åp are weakening and contraction, possibly followed by 45. Hence, such a derivation can only contain sequents of one of the forms m n p ⊢ ◻¬ ◻ ¬p m n k j ` ◻¬p ; ¬p ⊢ ◻¬ ◻ ¬p ; ¬ ◻ ¬p ; p i with m; n; k; `; j ≥ 0 and A = A;:::; A. Thus it cannot contain an ´¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¶ initial sequent. i-times Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination How to show that a logic does not have a cut-free sequent calculus? Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination Is there a cut-free sequent calculus for S5? Trivial answer: Of course! Take the rules A valid in S5 . { ⊢ A S } Non-trivial answer: That depends on the shape of the rules! General method for showing certain rule shapes cannot capture a semantically given modal logic even with cut: ▸ translate the rules into Hilbert-axioms of specific form ▸ connect Hilbert-style axiomatisability with frame definability ▸ show that the translations of the rules cannot define the frames for the logic. (The translation involves cut, so this shows a stronger statement.) Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination What Is a Rule? Let us call a sequent rule modal if it has the shape: Γ1; Σ1 ⊢ Π1; ∆1 ::: Γn; Σn ⊢ Πn; ∆n Γ; ◻Σ ⊢ ◻Π; ∆ where (writingΓ ◻ for the restriction of Γ to modal formulae) ▸ Σi ⊆ Σ; Πi ⊆ Π ◻ ▸ Γi is one of ∅; Γ; Γ ◻ ▸ ∆i is one of ∅; ∆; ∆ Example ◻ ◻ ◻ Σ A Γ; A ∆ Γ ; Σ A Γ A; ∆ ⊢ K ⊢ T ⊢ 4 ⊢ 45 Γ; ◻Σ ⊢ ◻A; ∆ Γ; ◻A ⊢ ∆ Γ; ◻Σ ⊢ ◻A; ∆ Γ ⊢ ◻A; ∆ are all modal rules (and equivalent to the rules considered earlier). Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination What Is a Rule? Let us call a sequent rule modal if it has the shape: Γ1; Σ1 ⊢ Π1; ∆1 ::: Γn; Σn ⊢ Πn; ∆n Γ; ◻Σ ⊢ ◻Π; ∆ where (writingΓ ◻ for the restriction of Γ to modal formulae) ▸ Σi ⊆ Σ; Πi ⊆ Π ◻ ▸ Γi is one of ∅; Γ; Γ ◻ ▸ ∆i is one of ∅; ∆; ∆ Example Γ◻; Σ; A A ◻ ⊢ GLR Γ; ◻Σ ⊢ ◻A; ∆ is not a modal rule (because the ◻A changes sides). Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination Mixed-cut-closed Rule Sets sS5 has modal rules in this sense, so we need something more. Definition A set of modal rules is mixed-cut-closed if principal-context cuts can be permuted up in the context. Example Γ◻; Σ A The set with modal rule ⊢ 4 is mixed-cut-closed: E.g.: Γ ◻ Σ ⊢ ◻A; ∆ Γ◻; Σ A A; Ω◻; Θ B ⊢ 4 ◻ ⊢ 4 Γ; Σ A; ∆ A; Ω; Θ B; Ξ ◻ ⊢ ◻ ◻ ◻ ⊢ ◻ cut Γ; ◻Σ; Ω; ◻Θ ⊢ ∆; ◻B; Ξ Γ◻; Σ A ⊢ 4 Γ◻; Σ A; ∆ A; Ω◻; Θ B ◻ ⊢ ◻ ◻ ⊢ cut ↝ Γ◻; Σ; Ω◻; Θ B ⊢ 4 Γ; ◻Σ; Ω; ◻Θ ⊢ ∆; ◻B; Ξ Modal Logic S5 Sequents for S5 Hypersequents for S5 Cut Elimination Mixed-cut-closed Rule Sets sS5 has modal rules in this sense, so we need something more. Definition A set of modal rules is mixed-cut-closed if principal-context cuts can be permuted up in the context.