Dynamic Epistemic Logic Displayed Giuseppe Greco & Alexander - PowerPoint PPT Presentation

Dynamic Epistemic Logic Displayed Giuseppe Greco & Alexander Kurz & Alessandra Palmigiano April 19, 2013 —————— ALCOP 1 / 43

Motivation 1 Proof-theory meets coalgebra From global- to local-rules calculi 2 Axiomatic Calculi Natural Deduction Calculi Sequent Calculi Cut-elimination From holistic to modular calculi 3 Display Calculi Propositions- and Structures-Language Display Postulates and Display Property Structural Rules Operational Rules No-standard Rules Conclusions 4 Counterexample in Kripke semantics Interpretation in final coalgebra 2 / 43

Motivation Proof-theory meets coalgebra We introduce a display calculus for the Baltag-Moss-Solecki logic of Epistemic Actions and Knowledge (EAK). This calculus is cut-free and complete w.r.t. the standard Hilbert-style presentation of EAK, and moreover, it features a richer language than EAK, in which all logical operations have adjoints . Some of the additional dynamic logical operators do not have an interpretation in the standard Kripke semantics of EAK, but do have a natural interpretation in the final coalgebra . This proof-theoretic motivation revives the interest in the global semantics for dynamic epistemic logics. 3 / 43

Motivation Proof-theory meets coalgebra We introduce a display calculus for the Baltag-Moss-Solecki logic of Epistemic Actions and Knowledge (EAK). This calculus is cut-free and complete w.r.t. the standard Hilbert-style presentation of EAK, and moreover, it features a richer language than EAK, in which all logical operations have adjoints . Some of the additional dynamic logical operators do not have an interpretation in the standard Kripke semantics of EAK, but do have a natural interpretation in the final coalgebra . This proof-theoretic motivation revives the interest in the global semantics for dynamic epistemic logics. 3 / 43

Motivation Proof-theory meets coalgebra We introduce a display calculus for the Baltag-Moss-Solecki logic of Epistemic Actions and Knowledge (EAK). This calculus is cut-free and complete w.r.t. the standard Hilbert-style presentation of EAK, and moreover, it features a richer language than EAK, in which all logical operations have adjoints . Some of the additional dynamic logical operators do not have an interpretation in the standard Kripke semantics of EAK, but do have a natural interpretation in the final coalgebra . This proof-theoretic motivation revives the interest in the global semantics for dynamic epistemic logics. 3 / 43

From global- to local-rules calculi Axiomatic Calculi Axiomatic calculi ´ a la Hilbert were the first to appear and, typically, are characterized by ‘more’ axioms and ‘few’ inference rules, at the limit only one (Modus Ponens). The objects manipulated in such calculi are formulas . The meaning of logical symbols is implicitly defined by the axioms that, also, set their mutual relations. Again, the axioms allow only an indirect control of the ‘structure’. 1 ( A → (( A → A ) → A )) → (( A → ( A → A )) → ( A → A )) 2 A → (( A → A ) → A ) 3 ( A → ( A → A )) → ( A → A ) 4 A → ( A → A ) 5 A → A 1 2 MP 3 4 MP 5 where the leaves are all instantiations of axioms. 4 / 43

From global- to local-rules calculi Axiomatic Calculi Axiomatic calculi ´ a la Hilbert were the first to appear and, typically, are characterized by ‘more’ axioms and ‘few’ inference rules, at the limit only one (Modus Ponens). The objects manipulated in such calculi are formulas . The meaning of logical symbols is implicitly defined by the axioms that, also, set their mutual relations. Again, the axioms allow only an indirect control of the ‘structure’. 1 ( A → (( A → A ) → A )) → (( A → ( A → A )) → ( A → A )) 2 A → (( A → A ) → A ) 3 ( A → ( A → A )) → ( A → A ) 4 A → ( A → A ) 5 A → A 1 2 MP 3 4 MP 5 where the leaves are all instantiations of axioms. 4 / 43

From global- to local-rules calculi Axiomatic Calculi Axiomatic calculi ´ a la Hilbert were the first to appear and, typically, are characterized by ‘more’ axioms and ‘few’ inference rules, at the limit only one (Modus Ponens). The objects manipulated in such calculi are formulas . The meaning of logical symbols is implicitly defined by the axioms that, also, set their mutual relations. Again, the axioms allow only an indirect control of the ‘structure’. 1 ( A → (( A → A ) → A )) → (( A → ( A → A )) → ( A → A )) 2 A → (( A → A ) → A ) 3 ( A → ( A → A )) → ( A → A ) 4 A → ( A → A ) 5 A → A 1 2 MP 3 4 MP 5 where the leaves are all instantiations of axioms. 4 / 43

From global- to local-rules calculi Axiomatic Calculi Advantages: proofs on the system are simplified for systems with few and simple inference rules; the space of logics can be reconstructed in a modular way: adding incremently axioms to a previous axiomatization give other logics. Disadvantages: the proofs in the system are long and often unnatural; the meaning of connectives is global: e.g. the axiom ( A → B ) → (( C → B ) → ( A ∨ C → B )) involves more connectives; the derivations are global: e.g. only Modus Ponens is used to prove all theorems. 5 / 43

From global- to local-rules calculi Axiomatic Calculi Advantages: proofs on the system are simplified for systems with few and simple inference rules; the space of logics can be reconstructed in a modular way: adding incremently axioms to a previous axiomatization give other logics. Disadvantages: the proofs in the system are long and often unnatural; the meaning of connectives is global: e.g. the axiom ( A → B ) → (( C → B ) → ( A ∨ C → B )) involves more connectives; the derivations are global: e.g. only Modus Ponens is used to prove all theorems. 5 / 43

From global- to local-rules calculi Natural Deduction Calculi Natural deduction calculi ´ a la the Gentzen are characterized by the use of assumptions (introduced by an explicit rule) and different inference rules for different connectives. The objects manipulated in such calculi are formulas . The meaning of the logical symbols is explicitly defined (by Intr/Elim Rule): an operational content corresponds to each connective. Introduction Rules for implication and negation discharge assumptions: appropriate restrictions allow some control of the ‘structure’. [ A ∧ B ] 3 [ A ∧ B ] 5 E ∧ E ∧ [ ¬ A ] 4 [ ¬ B ] 6 A B I ∧ I ∧ A ∧ ¬ A B ∧ ¬ B 3 I ¬ 5 I ¬ [ ¬ A ∨ ¬ B ] 2 ¬ ( A ∧ B ) ¬ ( A ∧ B ) 4,6 E ∨ [ A ∧ B ] 1 ¬ ( A ∧ B ) I ∧ ( A ∧ B ) ∧ ¬ ( A ∧ B ) 2 I ¬ ¬ ( ¬ A ∨ ¬ B ) 1,3,5 I → A ∧ B → ¬ ( ¬ A ∨ ¬ B ) 6 / 43

From global- to local-rules calculi Natural Deduction Calculi Natural deduction calculi ´ a la the Gentzen are characterized by the use of assumptions (introduced by an explicit rule) and different inference rules for different connectives. The objects manipulated in such calculi are formulas . The meaning of the logical symbols is explicitly defined (by Intr/Elim Rule): an operational content corresponds to each connective. Introduction Rules for implication and negation discharge assumptions: appropriate restrictions allow some control of the ‘structure’. [ A ∧ B ] 3 [ A ∧ B ] 5 E ∧ E ∧ [ ¬ A ] 4 [ ¬ B ] 6 A B I ∧ I ∧ A ∧ ¬ A B ∧ ¬ B 3 I ¬ 5 I ¬ [ ¬ A ∨ ¬ B ] 2 ¬ ( A ∧ B ) ¬ ( A ∧ B ) 4,6 E ∨ [ A ∧ B ] 1 ¬ ( A ∧ B ) I ∧ ( A ∧ B ) ∧ ¬ ( A ∧ B ) 2 I ¬ ¬ ( ¬ A ∨ ¬ B ) 1,3,5 I → A ∧ B → ¬ ( ¬ A ∨ ¬ B ) 6 / 43

From global- to local-rules calculi Natural Deduction Calculi Natural deduction calculi ´ a la the Gentzen are characterized by the use of assumptions (introduced by an explicit rule) and different inference rules for different connectives. The objects manipulated in such calculi are formulas . The meaning of the logical symbols is explicitly defined (by Intr/Elim Rule): an operational content corresponds to each connective. Introduction Rules for implication and negation discharge assumptions: appropriate restrictions allow some control of the ‘structure’. [ A ∧ B ] 3 [ A ∧ B ] 5 E ∧ E ∧ [ ¬ A ] 4 [ ¬ B ] 6 A B I ∧ I ∧ A ∧ ¬ A B ∧ ¬ B 3 I ¬ 5 I ¬ [ ¬ A ∨ ¬ B ] 2 ¬ ( A ∧ B ) ¬ ( A ∧ B ) 4,6 E ∨ [ A ∧ B ] 1 ¬ ( A ∧ B ) I ∧ ( A ∧ B ) ∧ ¬ ( A ∧ B ) 2 I ¬ ¬ ( ¬ A ∨ ¬ B ) 1,3,5 I → A ∧ B → ¬ ( ¬ A ∨ ¬ B ) 6 / 43

From global- to local-rules calculi Natural Deduction Calculi Advantages: the proofs in the system are natural; the connectives are introduced one by one (this is in the direction of proof-theoretic semantics); Disadvantages: the derivations are global: assumptions tipically are discharged after many steps in a derivation; it is not simple to reconstruct the space of the logics; it is difficult to obtain natural deduction calculi for non-classical or modal logics. 7 / 43

From global- to local-rules calculi Natural Deduction Calculi Advantages: the proofs in the system are natural; the connectives are introduced one by one (this is in the direction of proof-theoretic semantics); Disadvantages: the derivations are global: assumptions tipically are discharged after many steps in a derivation; it is not simple to reconstruct the space of the logics; it is difficult to obtain natural deduction calculi for non-classical or modal logics. 7 / 43