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

Dynamic Epistemic Logic Displayed

Giuseppe Greco & Alexander Kurz & Alessandra Palmigiano

April 19, 2013 —————— ALCOP

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1

Motivation Proof-theory meets coalgebra

2

From global- to local-rules calculi Axiomatic Calculi Natural Deduction Calculi Sequent Calculi Cut-elimination

3

From holistic to modular calculi Display Calculi Propositions- and Structures-Language Display Postulates and Display Property Structural Rules Operational Rules No-standard Rules

4

Conclusions 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]1 [¬A ∨ ¬B]2 [A ∧ B]3

E∧

A [¬A]4

I∧

A ∧ ¬A

3 I¬

¬(A ∧ B) [A ∧ B]5

E∧

B [¬B]6

I∧

B ∧ ¬B

5 I¬

¬(A ∧ B)

4,6 E∨

¬(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]1 [¬A ∨ ¬B]2 [A ∧ B]3

E∧

A [¬A]4

I∧

A ∧ ¬A

3 I¬

¬(A ∧ B) [A ∧ B]5

E∧

B [¬B]6

I∧

B ∧ ¬B

5 I¬

¬(A ∧ B)

4,6 E∨

¬(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]1 [¬A ∨ ¬B]2 [A ∧ B]3

E∧

A [¬A]4

I∧

A ∧ ¬A

3 I¬

¬(A ∧ B) [A ∧ B]5

E∧

B [¬B]6

I∧

B ∧ ¬B

5 I¬

¬(A ∧ B)

4,6 E∨

¬(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

• f 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

• f 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 Sequent Calculi

Sequent calculi ´ a la Gentzen are characterized by a single axiom (Identity), the use of assumptions and conclusions, by different inference rules for different connectives and for different structural

• perations.

Objects manipulated in such calculations are sequents: Γ ⊢ ∆ where Γ and ∆ are (possibly empty) sequences of formulas separated by a (poliadyc) comma. The meaning of logical symbols is explicitly defined (by Left/Right Introduction Rule). The structural rules allow a direct control of the ‘structure’.

A ⊢ A ⊥ ⊢ ⊥

W

A, ⊥ ⊢ ⊥ A, A → ⊥ ⊢ ⊥ A, ¬A ⊢ ⊥ A ∧ B, ¬A ⊢ ⊥ B ⊢ B ⊥ ⊢ ⊥

W

B, ⊥ ⊢ ⊥ B, B → ⊥ ⊢ ⊥ B, ¬B ⊢ ⊥ A ∧ B, ¬B ⊢ ⊥ A ∧ B, ¬A ∨ ¬B ⊢ ⊥

E

¬A ∨ ¬B, A ∧ B ⊢ ⊥ A ∧ B ⊢ ¬A ∨ ¬B → ⊥ A ∧ B ⊢ ¬(¬A ∨ ¬B)

8 / 43

From global- to local-rules calculi Sequent Calculi

Sequent calculi ´ a la Gentzen are characterized by a single axiom (Identity), the use of assumptions and conclusions, by different inference rules for different connectives and for different structural

• perations.

Objects manipulated in such calculations are sequents: Γ ⊢ ∆ where Γ and ∆ are (possibly empty) sequences of formulas separated by a (poliadyc) comma. The meaning of logical symbols is explicitly defined (by Left/Right Introduction Rule). The structural rules allow a direct control of the ‘structure’.

A ⊢ A ⊥ ⊢ ⊥

W

A, ⊥ ⊢ ⊥ A, A → ⊥ ⊢ ⊥ A, ¬A ⊢ ⊥ A ∧ B, ¬A ⊢ ⊥ B ⊢ B ⊥ ⊢ ⊥

W

B, ⊥ ⊢ ⊥ B, B → ⊥ ⊢ ⊥ B, ¬B ⊢ ⊥ A ∧ B, ¬B ⊢ ⊥ A ∧ B, ¬A ∨ ¬B ⊢ ⊥

E

¬A ∨ ¬B, A ∧ B ⊢ ⊥ A ∧ B ⊢ ¬A ∨ ¬B → ⊥ A ∧ B ⊢ ¬(¬A ∨ ¬B)

8 / 43

From global- to local-rules calculi Sequent Calculi

Sequent calculi ´ a la Gentzen are characterized by a single axiom (Identity), the use of assumptions and conclusions, by different inference rules for different connectives and for different structural

• perations.

Objects manipulated in such calculations are sequents: Γ ⊢ ∆ where Γ and ∆ are (possibly empty) sequences of formulas separated by a (poliadyc) comma. The meaning of logical symbols is explicitly defined (by Left/Right Introduction Rule). The structural rules allow a direct control of the ‘structure’.

A ⊢ A ⊥ ⊢ ⊥

W

A, ⊥ ⊢ ⊥ A, A → ⊥ ⊢ ⊥ A, ¬A ⊢ ⊥ A ∧ B, ¬A ⊢ ⊥ B ⊢ B ⊥ ⊢ ⊥

W

B, ⊥ ⊢ ⊥ B, B → ⊥ ⊢ ⊥ B, ¬B ⊢ ⊥ A ∧ B, ¬B ⊢ ⊥ A ∧ B, ¬A ∨ ¬B ⊢ ⊥

E

¬A ∨ ¬B, A ∧ B ⊢ ⊥ A ∧ B ⊢ ¬A ∨ ¬B → ⊥ A ∧ B ⊢ ¬(¬A ∨ ¬B)

8 / 43

From global- to local-rules calculi Sequent Calculi

Advantages: the derivations are local; the proofs in the system are automatizable (if the calculus enjoy cut-elimination); a distinction between connectives and structure is introduced (this is in the direction of proof-theoretic semantics). Disadvantages: the space of logics cannot be reconstructed in a modular way (if the calculus is non-standard, ı.e. as usual for modal logics); it is not simple to obtain sequent calculi for substructural or modal logics (with the sub-formula property).

9 / 43

From global- to local-rules calculi Sequent Calculi

Advantages: the derivations are local; the proofs in the system are automatizable (if the calculus enjoy cut-elimination); a distinction between connectives and structure is introduced (this is in the direction of proof-theoretic semantics). Disadvantages: the space of logics cannot be reconstructed in a modular way (if the calculus is non-standard, ı.e. as usual for modal logics); it is not simple to obtain sequent calculi for substructural or modal logics (with the sub-formula property).

9 / 43

From global- to local-rules calculi Cut-elimination

Common forms of the cut rule are the following:

Γ ⊢ C, ∆ Γ′, C ⊢ ∆′ Γ′, Γ ⊢ ∆′, ∆ Γ ⊢ C, ∆ Γ, C ⊢ ∆ Γ ⊢ ∆ Γ ⊢ C Γ′, C ⊢ ∆ Γ′, Γ ⊢ ∆ Γ ⊢ C, ∆ C ⊢ ∆′ Γ ⊢ ∆′, ∆

Theorem (Cut-elimination) If Γ ⊢ ∆ is derivable in calculus S with Cut, then it is in S without Cut. The cut-elimination is the most fundamental technique in proof theory and many important syntactic properties derive from it (e.g. decidability). A cut is an intermediate step in a deduction, by which a conclusion(s) ∆ can be proved from the assumption(s) Γ via the lemma C. ‘Eliminating the cut’ from such a proof generates a new (and lemma-free) proof of ∆, which exclusively employs syntactic material coming from Γ and ∆ (subformula property). Typically, syntactic proofs of cut-elimination are non-modular: if a new rule is added, cut-elimination must be proved from scratch.

10 / 43

From global- to local-rules calculi Cut-elimination

Common forms of the cut rule are the following:

Γ ⊢ C, ∆ Γ′, C ⊢ ∆′ Γ′, Γ ⊢ ∆′, ∆ Γ ⊢ C, ∆ Γ, C ⊢ ∆ Γ ⊢ ∆ Γ ⊢ C Γ′, C ⊢ ∆ Γ′, Γ ⊢ ∆ Γ ⊢ C, ∆ C ⊢ ∆′ Γ ⊢ ∆′, ∆

Theorem (Cut-elimination) If Γ ⊢ ∆ is derivable in calculus S with Cut, then it is in S without Cut. The cut-elimination is the most fundamental technique in proof theory and many important syntactic properties derive from it (e.g. decidability). A cut is an intermediate step in a deduction, by which a conclusion(s) ∆ can be proved from the assumption(s) Γ via the lemma C. ‘Eliminating the cut’ from such a proof generates a new (and lemma-free) proof of ∆, which exclusively employs syntactic material coming from Γ and ∆ (subformula property). Typically, syntactic proofs of cut-elimination are non-modular: if a new rule is added, cut-elimination must be proved from scratch.

10 / 43

From global- to local-rules calculi Cut-elimination

Common forms of the cut rule are the following:

Γ ⊢ C, ∆ Γ′, C ⊢ ∆′ Γ′, Γ ⊢ ∆′, ∆ Γ ⊢ C, ∆ Γ, C ⊢ ∆ Γ ⊢ ∆ Γ ⊢ C Γ′, C ⊢ ∆ Γ′, Γ ⊢ ∆ Γ ⊢ C, ∆ C ⊢ ∆′ Γ ⊢ ∆′, ∆

Theorem (Cut-elimination) If Γ ⊢ ∆ is derivable in calculus S with Cut, then it is in S without Cut. The cut-elimination is the most fundamental technique in proof theory and many important syntactic properties derive from it (e.g. decidability). A cut is an intermediate step in a deduction, by which a conclusion(s) ∆ can be proved from the assumption(s) Γ via the lemma C. ‘Eliminating the cut’ from such a proof generates a new (and lemma-free) proof of ∆, which exclusively employs syntactic material coming from Γ and ∆ (subformula property). Typically, syntactic proofs of cut-elimination are non-modular: if a new rule is added, cut-elimination must be proved from scratch.

10 / 43

From global- to local-rules calculi Cut-elimination

Common forms of the cut rule are the following:

Γ ⊢ C, ∆ Γ′, C ⊢ ∆′ Γ′, Γ ⊢ ∆′, ∆ Γ ⊢ C, ∆ Γ, C ⊢ ∆ Γ ⊢ ∆ Γ ⊢ C Γ′, C ⊢ ∆ Γ′, Γ ⊢ ∆ Γ ⊢ C, ∆ C ⊢ ∆′ Γ ⊢ ∆′, ∆

Theorem (Cut-elimination) If Γ ⊢ ∆ is derivable in calculus S with Cut, then it is in S without Cut. The cut-elimination is the most fundamental technique in proof theory and many important syntactic properties derive from it (e.g. decidability). A cut is an intermediate step in a deduction, by which a conclusion(s) ∆ can be proved from the assumption(s) Γ via the lemma C. ‘Eliminating the cut’ from such a proof generates a new (and lemma-free) proof of ∆, which exclusively employs syntactic material coming from Γ and ∆ (subformula property). Typically, syntactic proofs of cut-elimination are non-modular: if a new rule is added, cut-elimination must be proved from scratch.

10 / 43

From holistic to modular calculi Display Calculi

Display calculi were introduced by Belnap [2] to provide a uniform account for cut-elimination; a ‘pure’ proof-theoretical analisys of logics; a tool useful to ‘merge’ different logics. Display calculi generalize sequent calculi allowing: different ‘structural connectives’ (not just the Gentzen’s comma), where the structures in X ⊢ Y are binary trees (not sequences); a set of structural rules named Display Postulates, that give the Display Property (essential in Belnap’s cut-elimination).

A ⊢ A A ; B ⊢ A A ∧ B ⊢ A ⊥ ⊢ ⊥ A → ⊥ ⊢ A ∧ B > ⊥ ¬A ⊢ A ∧ B > ⊥ B ⊢ B A ; B ⊢ B A ∧ B ⊢ B ⊥ ⊢ ⊥ B → ⊥ ⊢ A ∧ B > ⊥ ¬B ⊢ A ∧ B > ⊥ ¬A ∨ ¬B ⊢ (A ∧ B > ⊥) ; (A ∧ B > ⊥) ¬A ∨ ¬B ⊢ A ∧ B > ⊥

> ;

A ∧ B ; ¬A ∨ ¬B ⊢ ⊥ ¬A ∨ ¬B ; A ∧ B ⊢ ⊥

; >

A ∧ B ⊢ ¬A ∨ ¬B > ⊥ A ∧ B ⊢ ¬A ∨ ¬B → ⊥ A ∧ B ⊢ ¬(¬A ∨ ¬B)

11 / 43

From holistic to modular calculi Display Calculi

Display calculi were introduced by Belnap [2] to provide a uniform account for cut-elimination; a ‘pure’ proof-theoretical analisys of logics; a tool useful to ‘merge’ different logics. Display calculi generalize sequent calculi allowing: different ‘structural connectives’ (not just the Gentzen’s comma), where the structures in X ⊢ Y are binary trees (not sequences); a set of structural rules named Display Postulates, that give the Display Property (essential in Belnap’s cut-elimination).

A ⊢ A A ; B ⊢ A A ∧ B ⊢ A ⊥ ⊢ ⊥ A → ⊥ ⊢ A ∧ B > ⊥ ¬A ⊢ A ∧ B > ⊥ B ⊢ B A ; B ⊢ B A ∧ B ⊢ B ⊥ ⊢ ⊥ B → ⊥ ⊢ A ∧ B > ⊥ ¬B ⊢ A ∧ B > ⊥ ¬A ∨ ¬B ⊢ (A ∧ B > ⊥) ; (A ∧ B > ⊥) ¬A ∨ ¬B ⊢ A ∧ B > ⊥

> ;

A ∧ B ; ¬A ∨ ¬B ⊢ ⊥ ¬A ∨ ¬B ; A ∧ B ⊢ ⊥

; >

A ∧ B ⊢ ¬A ∨ ¬B > ⊥ A ∧ B ⊢ ¬A ∨ ¬B → ⊥ A ∧ B ⊢ ¬(¬A ∨ ¬B)

11 / 43

From holistic to modular calculi Display Calculi

Display calculi were introduced by Belnap [2] to provide a uniform account for cut-elimination; a ‘pure’ proof-theoretical analisys of logics; a tool useful to ‘merge’ different logics. Display calculi generalize sequent calculi allowing: different ‘structural connectives’ (not just the Gentzen’s comma), where the structures in X ⊢ Y are binary trees (not sequences); a set of structural rules named Display Postulates, that give the Display Property (essential in Belnap’s cut-elimination).

A ⊢ A A ; B ⊢ A A ∧ B ⊢ A ⊥ ⊢ ⊥ A → ⊥ ⊢ A ∧ B > ⊥ ¬A ⊢ A ∧ B > ⊥ B ⊢ B A ; B ⊢ B A ∧ B ⊢ B ⊥ ⊢ ⊥ B → ⊥ ⊢ A ∧ B > ⊥ ¬B ⊢ A ∧ B > ⊥ ¬A ∨ ¬B ⊢ (A ∧ B > ⊥) ; (A ∧ B > ⊥) ¬A ∨ ¬B ⊢ A ∧ B > ⊥

> ;

A ∧ B ; ¬A ∨ ¬B ⊢ ⊥ ¬A ∨ ¬B ; A ∧ B ⊢ ⊥

; >

A ∧ B ⊢ ¬A ∨ ¬B > ⊥ A ∧ B ⊢ ¬A ∨ ¬B → ⊥ A ∧ B ⊢ ¬(¬A ∨ ¬B)

11 / 43

From holistic to modular calculi Display Calculi

Advantages: cut-elimination is a consequence of design principles, by the following: Theorem (Cut-elimination [2] [5]) If a logic is ‘properly displayable’, then it enjoys cut-elimination space of logics can be reconstructed in a modular way, because: Doˇ sen Principle [5] The rules for the logical operations are never changed: all changes are made in the structural rules a ‘real’ proof-theory is possible for substrucural and modal logics (e.g. separated, symmetrical and explicit introduction rules for the normal modal operators are available). Disadvantages: not amenable for proof-search (because of Display Postulates).

12 / 43

From holistic to modular calculi Display Calculi

Advantages: cut-elimination is a consequence of design principles, by the following: Theorem (Cut-elimination [2] [5]) If a logic is ‘properly displayable’, then it enjoys cut-elimination space of logics can be reconstructed in a modular way, because: Doˇ sen Principle [5] The rules for the logical operations are never changed: all changes are made in the structural rules a ‘real’ proof-theory is possible for substrucural and modal logics (e.g. separated, symmetrical and explicit introduction rules for the normal modal operators are available). Disadvantages: not amenable for proof-search (because of Display Postulates).

12 / 43

From holistic to modular calculi Propositions- and Structures-Language

We consider the display calculus (plus explicit negations) introduced in [3] for the Baltag-Moss-Solecki logic of Epistemic Actions and Knowledge EAK [1]. For each agent a ∈ Ag and action α ∈ Act, Propositions are built from a set of atomic propositional variables AtProp = {p, q, r, . . .} and two constants ⊥ and ⊤: A := p | ⊥ | ⊤ | A ∧ A | A ∨ A | A → A | A > A | ¬A | ∼A | ✸

aA | ✷aA | aA | aA | [α]A | αA |

• α
• A |

[ α ] A . Structures are built from formulas and one structural constant I: X :=

• I | A | X; X | X > X | ∗X |
• aX | ◦

aX | {α}X |

{ α } X .

13 / 43

From holistic to modular calculi Propositions- and Structures-Language

We consider the display calculus (plus explicit negations) introduced in [3] for the Baltag-Moss-Solecki logic of Epistemic Actions and Knowledge EAK [1]. For each agent a ∈ Ag and action α ∈ Act, Propositions are built from a set of atomic propositional variables AtProp = {p, q, r, . . .} and two constants ⊥ and ⊤: A := p | ⊥ | ⊤ | A ∧ A | A ∨ A | A → A | A > A | ¬A | ∼A | ✸

aA | ✷aA | aA | aA | [α]A | αA |

• α
• A |

[ α ] A . Structures are built from formulas and one structural constant I: X :=

• I | A | X; X | X > X | ∗X |
• aX | ◦

aX | {α}X |

{ α } X .

13 / 43

From holistic to modular calculi Propositions- and Structures-Language

We consider the display calculus (plus explicit negations) introduced in [3] for the Baltag-Moss-Solecki logic of Epistemic Actions and Knowledge EAK [1]. For each agent a ∈ Ag and action α ∈ Act, Propositions are built from a set of atomic propositional variables AtProp = {p, q, r, . . .} and two constants ⊥ and ⊤: A := p | ⊥ | ⊤ | A ∧ A | A ∨ A | A → A | A > A | ¬A | ∼A | ✸

aA | ✷aA | aA | aA | [α]A | αA |

• α
• A |

[ α ] A . Structures are built from formulas and one structural constant I: X :=

• I | A | X; X | X > X | ∗X |
• aX | ◦

aX | {α}X |

{ α } X .

13 / 43

From holistic to modular calculi Propositions- and Structures-Language

The structural connectives are contextual (as the Gentzen’s comma) and each of them is associated with a pair of logical connectives: Structural symb: I ; > ∗ Operational symb: ⊤ ⊥ ∧ ∨

>

→ ¬ ∼ Structural symb:

• a
• a

{α} { α } Operational symb: ✸

a

✷a

• a

a α [α]

• α
• [

α ] by the translations τ1 of precedent and τ2 of succedent into prop. :

τ1(A) := A τ2(A) := A τ1(I) := ⊤ τ2(I) := ⊥ τ1(X ; Y) := τ1(X) ∧ τ1(Y) τ2(X ; Y) := τ2(X) ∨ τ2(Y) τ1(X > Y) := τ2(X) > τ1(Y) τ2(X > Y) := τ1(X) → τ2(Y) τ1(∗X) := ∼τ2(X) τ2(∗X) := ¬τ1(X) τ1(◦

aX)

:= ✸

aτ1(X)

τ2(◦

aX)

:= ✷aτ2(X) τ1(•

aX)

:=

• aτ1(X)

τ2(•

aX)

:= aτ2(X) τ1({α}X) := ατ1(X) τ2({α}X) := ατ2(X) τ1( { α } X) :=

• α
• τ1(X)

τ2({α}X) := [ α ] τ2(X)

14 / 43

From holistic to modular calculi Propositions- and Structures-Language

The structural connectives are contextual (as the Gentzen’s comma) and each of them is associated with a pair of logical connectives: Structural symb: I ; > ∗ Operational symb: ⊤ ⊥ ∧ ∨

>

→ ¬ ∼ Structural symb:

• a
• a

{α} { α } Operational symb: ✸

a

✷a

• a

a α [α]

• α
• [

α ] by the translations τ1 of precedent and τ2 of succedent into prop. :

τ1(A) := A τ2(A) := A τ1(I) := ⊤ τ2(I) := ⊥ τ1(X ; Y) := τ1(X) ∧ τ1(Y) τ2(X ; Y) := τ2(X) ∨ τ2(Y) τ1(X > Y) := τ2(X) > τ1(Y) τ2(X > Y) := τ1(X) → τ2(Y) τ1(∗X) := ∼τ2(X) τ2(∗X) := ¬τ1(X) τ1(◦

aX)

:= ✸

aτ1(X)

τ2(◦

aX)

:= ✷aτ2(X) τ1(•

aX)

:=

• aτ1(X)

τ2(•

aX)

:= aτ2(X) τ1({α}X) := ατ1(X) τ2({α}X) := ατ2(X) τ1( { α } X) :=

• α
• τ1(X)

τ2({α}X) := [ α ] τ2(X)

14 / 43

From holistic to modular calculi Display Postulates and Display Property

Display Postulates X ; Y ⊢ Z

; >

Y ⊢ X > Z Z ⊢ Y ; X

> ;

Y > Z ⊢ X

• aX ⊢ Y
• a
• a

X ⊢ •

aY

X ⊢ ◦

aY

• a
• a
• aX ⊢ Y

{α}X ⊢ Y

{α} { α }

X ⊢ { α } Y X ⊢ {α}Y

{ α } {α}

{ α } X ⊢ Y ∗X ⊢ Y

∗ ∗L ∗Y ⊢ X

Y ⊢ ∗X

∗ ∗R

X ⊢ ∗Y Z ⊢ Y ; X

; ∗ ;

∗Y ; Z ⊢ X X ; Y ⊢ Z

; ; ∗

Y ⊢ ∗X ; Z ∗∗X ⊢ Y

∗∗L

X ⊢ Y Y ⊢ ∗∗X

∗∗R

Y ⊢ X

15 / 43

From holistic to modular calculi Display Postulates and Display Property

Display Postulates X ; Y ⊢ Z

; >

Y ⊢ X > Z Z ⊢ Y ; X

> ;

Y > Z ⊢ X

• aX ⊢ Y
• a
• a

X ⊢ •

aY

X ⊢ ◦

aY

• a
• a
• aX ⊢ Y

{α}X ⊢ Y

{α} { α }

X ⊢ { α } Y X ⊢ {α}Y

{ α } {α}

{ α } X ⊢ Y ∗X ⊢ Y

∗ ∗L ∗Y ⊢ X

Y ⊢ ∗X

∗ ∗R

X ⊢ ∗Y Z ⊢ Y ; X

; ∗ ;

∗Y ; Z ⊢ X X ; Y ⊢ Z

; ; ∗

Y ⊢ ∗X ; Z ∗∗X ⊢ Y

∗∗L

X ⊢ Y Y ⊢ ∗∗X

∗∗R

Y ⊢ X

15 / 43

From holistic to modular calculi Display Postulates and Display Property

By definition, structural connectives form adjoint pairs as follows: ; ⊣ > > ⊣ ;

• a ⊣ •

a

• a ⊣ ◦

a

∗

• ∗

where

• means order-reversing adjoint (or Galois connection).

Related notion: ‘adjointness’ in category theory So, Display Postulates are ‘about the connection between left and right side of the turnstile’.

16 / 43

From holistic to modular calculi Display Postulates and Display Property

By definition, structural connectives form adjoint pairs as follows: ; ⊣ > > ⊣ ;

• a ⊣ •

a

• a ⊣ ◦

a

∗

• ∗

where

• means order-reversing adjoint (or Galois connection).

Related notion: ‘adjointness’ in category theory So, Display Postulates are ‘about the connection between left and right side of the turnstile’.

16 / 43

From holistic to modular calculi Display Postulates and Display Property

The Display Postulates allow to disassembly and reassembly structures and provide the following: Theorem (Display Property [2] [5]) Each substructure in a display-sequent is isolable or ‘displayable’ in precedent or, exclusively, succedent position. Note that ‘in precedent (succedent) position’ and ‘on the left (right) side

• f turnstile’ coincide in a Gentzen’s sequent calculus, but not in a

display calculus. E.g. In ‘Y ⊢ X > Z’, X is on the right of the turnstile but it is precedent structure, in fact it is displayable in the precedent position: Y ⊢ X > Z X ; Y ⊢ Z Y ; X ⊢ Z X ⊢ Y > Z

17 / 43

From holistic to modular calculi Display Postulates and Display Property

The Display Postulates allow to disassembly and reassembly structures and provide the following: Theorem (Display Property [2] [5]) Each substructure in a display-sequent is isolable or ‘displayable’ in precedent or, exclusively, succedent position. Note that ‘in precedent (succedent) position’ and ‘on the left (right) side

• f turnstile’ coincide in a Gentzen’s sequent calculus, but not in a

display calculus. E.g. In ‘Y ⊢ X > Z’, X is on the right of the turnstile but it is precedent structure, in fact it is displayable in the precedent position: Y ⊢ X > Z X ; Y ⊢ Z Y ; X ⊢ Z X ⊢ Y > Z

17 / 43

From holistic to modular calculi Structural Rules

Let be ⊙ ∈ {◦

a, • a}.

Structural Rules

• entry/exit rules -

Id p ⊢ p

X ⊢ A A ⊢ Y

Cut

X ⊢ Y X ⊢ Y

IL X ; I ⊢ Y

Y ⊢ X

IR

Y ⊢ I ; X X ⊢ Z

WL X ; Y ⊢ Z

Z ⊢ Y

WR

Z ⊢ Y ; X X ; X ⊢ Y

CL

X ⊢ Y Y ⊢ X ; X

CR

Y ⊢ X X ⊢ I

⊙ I

⊙X ⊢ I I ⊢ X

I ⊙

I ⊢ ⊙X I ⊢ X

∗

I

∗X ⊢ I X ⊢ I

I

∗

I ⊢ ∗X

18 / 43

From holistic to modular calculi Structural Rules

Let be ⊛ ∈ {∗, ◦

a, • a, {α},

{ α } }. ⊛X ; ⊛Y ⊢ Z

⊛ ;

⊛(X ; Y) ⊢ Z Z ⊢ ⊛Y ; ⊛X

; ⊛

Z ⊢ ⊛(Y ; X) ⊛X > ⊛Y ⊢ Z

⊛ > ⊛(X > Y) ⊢ Z

Z ⊢ ⊛Y > ⊛X

> ⊛

Z ⊢ ⊛(Y > X)

• manipulation rules -

Y ; X ⊢ Z

EL X ; Y ⊢ Z

Z ⊢ X ; Y

ER

Z ⊢ Y ; X X ; (Y ; Z) ⊢ W

AL (X ; Y) ; Z ⊢ W

W ⊢ (Z ; Y) ; X

AR

W ⊢ Z ; (Y ; X) X > (Y ; Z) ⊢ W

GrnL (X > Y) ; Z ⊢ W

W ⊢ X > (Y ; Z)

GrnR

W ⊢ (X > Y) ; Z

19 / 43

From holistic to modular calculi Structural Rules

Related notion: ‘naturality’ in category theory So, Structural Rules are ‘about the left side or, esclusively, the right side of the turnstile’. Note that the Excluded Middle is derivable by Grishin’s rules as follows:

A ⊢ A A ; I ⊢ A A ; I ⊢ ⊥ ; A I ⊢ A > (⊥ ; A)

Grn

I ⊢ (A > ⊥) ; A I ⊢ A ; (A > ⊥) A > I ⊢ A > ⊥ A > I ⊢ A → ⊥ A > I ⊢ ¬A I ⊢ A ; ¬A I ⊢ A ∨ ¬A

20 / 43

From holistic to modular calculi Structural Rules

Related notion: ‘naturality’ in category theory So, Structural Rules are ‘about the left side or, esclusively, the right side of the turnstile’. Note that the Excluded Middle is derivable by Grishin’s rules as follows:

A ⊢ A A ; I ⊢ A A ; I ⊢ ⊥ ; A I ⊢ A > (⊥ ; A)

Grn

I ⊢ (A > ⊥) ; A I ⊢ A ; (A > ⊥) A > I ⊢ A > ⊥ A > I ⊢ A → ⊥ A > I ⊢ ¬A I ⊢ A ; ¬A I ⊢ A ∨ ¬A

20 / 43

From holistic to modular calculi Operational Rules

Operational Rules

• translation rules -

⊥L ⊥ ⊢ I

X ⊢ I

⊥R

X ⊢ ⊥ I ⊢ X

⊤L ⊤ ⊢ X ⊤R

I ⊢ ⊤ A ; B ⊢ Z

∧L A ∧ B ⊢ Z

X ⊢ A Y ⊢ B

∧R

X ; Y ⊢ A ∧ B B ⊢ Y A ⊢ X

∨L

B ∨ A ⊢ Y ; X Z ⊢ B ; A

∨R

Z ⊢ B ∨ A X ⊢ A B ⊢ Y

→L

A → B ⊢ X > Y Z ⊢ A > B

→R

Z ⊢ A → B A > B ⊢ Z

> L A > B ⊢ Z

Y ⊢ B A ⊢ X

> R

X > Y ⊢ A > B

21 / 43

From holistic to modular calculi Operational Rules

Let be ⊙α ∈ {◦

a, • a, {α},

{ α } }, ✸ ·α ∈ {✸

a, a, α,

• α
• },

⊡α ∈ {✷a, a, [α], [ α ] }. ⊙αA ⊢ X

✸ ·αL

✸ ·αA ⊢ X X ⊢ A

✸ ·αR

⊙αX ⊢ ✸ ·αA A ⊢ X

⊡αL ⊡αA ⊢ ⊙αX

X ⊢ ⊙αA ⊡αR X ⊢ ⊡αA ∗A ⊢ X

∼L ∼A ⊢ X

A ⊢ X

∼R

∗X ⊢ ∼A X ⊢ A

¬L ¬A ⊢ ∗X

X ⊢ ∗A

¬R

X ⊢ ¬A Related notion: ‘functoriality’ in category theory So, (one half of the) Operational Rules are ‘about left and right side of the turnstile at the same time’.

22 / 43

From holistic to modular calculi Operational Rules

Let be ⊙α ∈ {◦

a, • a, {α},

{ α } }, ✸ ·α ∈ {✸

a, a, α,

• α
• },

⊡α ∈ {✷a, a, [α], [ α ] }. ⊙αA ⊢ X

✸ ·αL

✸ ·αA ⊢ X X ⊢ A

✸ ·αR

⊙αX ⊢ ✸ ·αA A ⊢ X

⊡αL ⊡αA ⊢ ⊙αX

X ⊢ ⊙αA ⊡αR X ⊢ ⊡αA ∗A ⊢ X

∼L ∼A ⊢ X

A ⊢ X

∼R

∗X ⊢ ∼A X ⊢ A

¬L ¬A ⊢ ∗X

X ⊢ ∗A

¬R

X ⊢ ¬A Related notion: ‘functoriality’ in category theory So, (one half of the) Operational Rules are ‘about left and right side of the turnstile at the same time’.

22 / 43

From holistic to modular calculi No-standard Rules

In a context whit Pre(α), we allow the following no-standard rules. Contextual Operational Rules

• translation rules -

Pre(α) ; {α}A ⊢ X

reverseL

Pre(α) ; [α]A ⊢ X X ⊢ Pre(α) > {α}A reverseR X ⊢ Pre(α) > αA

23 / 43

From holistic to modular calculi No-standard Rules

Contextual Structural Rules

• entry/exit rules -

X ⊢ Y

balance

{α}X ⊢ {α}Y

factsL {α} p ⊢ X −a factsR

X −a ⊢ {α} p Pre(α) ; {α}A ⊢ X

reduceL

{α}A ⊢ X X ⊢ Pre(α) > {α}A

reduceR

X ⊢ {α}A

• manipulation rules -

Pre(α) ; {α}◦

aX ⊢ Y

swap-inL Pre(α) ; ◦

a{β}αaβ X ⊢ Y

Y ⊢ Pre(α) > {α}◦

aX

swap-inR

Y ⊢ Pre(α) > ◦

a{β}αaβ X

• Pre(α) ; ◦

a{β} X ⊢ Y | αaβ

• s-outL

Pre(α) ; {α}◦

aX ⊢ ;

• Y | αaβ
• Y ⊢ Pre(α) > ◦

a{β} X | αaβ

• s-outR

;

• Y | αaβ
• ⊢ Pre(α) > {α}◦

aX

24 / 43

Conclusions Counterexample in Kripke semantics

u p, r v q Let α = r, ϕ = ✷p and ψ = q, so [ [✷p] ]M = ∅ ⊆ [ [ [ α ] q] ]M however, [ [α✷p] ]M = [ [α] ]M ∩ i[[ [✷p] ]Mα] = V(r) ∩ {u} = {u} ⊆ {v} = [ [q] ]M which shows that [ [ϕ] ]M ⊆ [ [ [ α ] ψ] ]M ⇒ [ [αϕ] ]M ⊆ [ [ψ] ]M. [ [✷p] ]M ⊆ [ [ [ α ] q] ]M holds independently of the way in which [ [ [ α ] q] ]M is defined, so ({α}, { α } ) is never sound for any interpretation of { α } X.

25 / 43

Conclusions Counterexample in Kripke semantics

u p, r v q Let α = r, ϕ = ✷p and ψ = q, so [ [✷p] ]M = ∅ ⊆ [ [ [ α ] q] ]M however, [ [α✷p] ]M = [ [α] ]M ∩ i[[ [✷p] ]Mα] = V(r) ∩ {u} = {u} ⊆ {v} = [ [q] ]M which shows that [ [ϕ] ]M ⊆ [ [ [ α ] ψ] ]M ⇒ [ [αϕ] ]M ⊆ [ [ψ] ]M. [ [✷p] ]M ⊆ [ [ [ α ] q] ]M holds independently of the way in which [ [ [ α ] q] ]M is defined, so ({α}, { α } ) is never sound for any interpretation of { α } X.

25 / 43

Conclusions Counterexample in Kripke semantics

u p, r v q Let α = r, ϕ = ✷p and ψ = q, so [ [✷p] ]M = ∅ ⊆ [ [ [ α ] q] ]M however, [ [α✷p] ]M = [ [α] ]M ∩ i[[ [✷p] ]Mα] = V(r) ∩ {u} = {u} ⊆ {v} = [ [q] ]M which shows that [ [ϕ] ]M ⊆ [ [ [ α ] ψ] ]M ⇒ [ [αϕ] ]M ⊆ [ [ψ] ]M. [ [✷p] ]M ⊆ [ [ [ α ] q] ]M holds independently of the way in which [ [ [ α ] q] ]M is defined, so ({α}, { α } ) is never sound for any interpretation of { α } X.

25 / 43

Conclusions Counterexample in Kripke semantics

u p, r v q Let α = r, ϕ = ✷p and ψ = q, so [ [✷p] ]M = ∅ ⊆ [ [ [ α ] q] ]M however, [ [α✷p] ]M = [ [α] ]M ∩ i[[ [✷p] ]Mα] = V(r) ∩ {u} = {u} ⊆ {v} = [ [q] ]M which shows that [ [ϕ] ]M ⊆ [ [ [ α ] ψ] ]M ⇒ [ [αϕ] ]M ⊆ [ [ψ] ]M. [ [✷p] ]M ⊆ [ [ [ α ] q] ]M holds independently of the way in which [ [ [ α ] q] ]M is defined, so ({α}, { α } ) is never sound for any interpretation of { α } X.

25 / 43

Conclusions Counterexample in Kripke semantics

u p, r v q Let α = r, ϕ = ✷p and ψ = q, so [ [✷p] ]M = ∅ ⊆ [ [ [ α ] q] ]M however, [ [α✷p] ]M = [ [α] ]M ∩ i[[ [✷p] ]Mα] = V(r) ∩ {u} = {u} ⊆ {v} = [ [q] ]M which shows that [ [ϕ] ]M ⊆ [ [ [ α ] ψ] ]M ⇒ [ [αϕ] ]M ⊆ [ [ψ] ]M. [ [✷p] ]M ⊆ [ [ [ α ] q] ]M holds independently of the way in which [ [ [ α ] q] ]M is defined, so ({α}, { α } ) is never sound for any interpretation of { α } X.

25 / 43

Conclusions Interpretation in final coalgebra

Whereas the semantics of classical modal logic can be described equally w.r.t. models/frames/coalgebras and w.r.t. the final coalgebra, this is no longer the case in the presence of dynamic modalities. In particular, the dynamic modalities do not come in adjoint pairs if they are given a coalgebra instead of a final coalgebra semantics. In other words, Belnap’s display postulates do not hold for dynamic modalities if considered w.r.t. to the standard semantics.

26 / 43

Conclusions Interpretation in final coalgebra

Whereas the semantics of classical modal logic can be described equally w.r.t. models/frames/coalgebras and w.r.t. the final coalgebra, this is no longer the case in the presence of dynamic modalities. In particular, the dynamic modalities do not come in adjoint pairs if they are given a coalgebra instead of a final coalgebra semantics. In other words, Belnap’s display postulates do not hold for dynamic modalities if considered w.r.t. to the standard semantics.

26 / 43

Conclusions Interpretation in final coalgebra

Whereas the semantics of classical modal logic can be described equally w.r.t. models/frames/coalgebras and w.r.t. the final coalgebra, this is no longer the case in the presence of dynamic modalities. In particular, the dynamic modalities do not come in adjoint pairs if they are given a coalgebra instead of a final coalgebra semantics. In other words, Belnap’s display postulates do not hold for dynamic modalities if considered w.r.t. to the standard semantics.

26 / 43

Conclusions Interpretation in final coalgebra

The interpretation of dynamic modalities is given in terms of the actions parametrizing them. Actions are semantically represented as transformations of Kripke models, i.e., as relations between states of different Kripke models. From the viewpoint of the final coalgebra, we can then interpret action symbols α as binary relations αZ on the final coalgebra Z. Let us first recall a proposition showing how relations give rise to modal operators.

27 / 43

Conclusions Interpretation in final coalgebra

The interpretation of dynamic modalities is given in terms of the actions parametrizing them. Actions are semantically represented as transformations of Kripke models, i.e., as relations between states of different Kripke models. From the viewpoint of the final coalgebra, we can then interpret action symbols α as binary relations αZ on the final coalgebra Z. Let us first recall a proposition showing how relations give rise to modal operators.

27 / 43

Conclusions Interpretation in final coalgebra

Every relation R ⊆ X × Y gives rise to the following modal operators R, [R] : PY → PX and R◦, [R◦] : PX → PY defined, for every V ⊆ X and every U ⊆ Y, RU = {x ∈ X | ∃y . xRy & y ∈ U} (1) [R]U = {x ∈ X | ∀y . xRy ⇒ y ∈ U} (2) R◦V = {y ∈ Y | ∃x . xRy & x ∈ V} (3) [R◦]V = {y ∈ Y | ∀x . xRy ⇒ x ∈ V} (4) which come in adjoint pairs: RU ⊆ V iff U ⊆ [R◦]V (5) R◦V ⊆ U iff V ⊆ [R]U (6) R preserves the top-element iff R is total (since R⊤ = dom(R)); R preserves binary intersections iff R is single-valued. [R] preserves the empty set iff R is total; [R] preserves directed unions iff it is image-finite; [R] preserves non-empty unions iff it is single-valued.

28 / 43

Conclusions Interpretation in final coalgebra

Every relation R ⊆ X × Y gives rise to the following modal operators R, [R] : PY → PX and R◦, [R◦] : PX → PY defined, for every V ⊆ X and every U ⊆ Y, RU = {x ∈ X | ∃y . xRy & y ∈ U} (1) [R]U = {x ∈ X | ∀y . xRy ⇒ y ∈ U} (2) R◦V = {y ∈ Y | ∃x . xRy & x ∈ V} (3) [R◦]V = {y ∈ Y | ∀x . xRy ⇒ x ∈ V} (4) which come in adjoint pairs: RU ⊆ V iff U ⊆ [R◦]V (5) R◦V ⊆ U iff V ⊆ [R]U (6) R preserves the top-element iff R is total (since R⊤ = dom(R)); R preserves binary intersections iff R is single-valued. [R] preserves the empty set iff R is total; [R] preserves directed unions iff it is image-finite; [R] preserves non-empty unions iff it is single-valued.

28 / 43

Conclusions Interpretation in final coalgebra

The operators αZ, [αZ], αZ◦, [αZ◦], respectively interpreting the dynamic modalities α, [α],

• α
• ,

[ α ] in the final coalgebra Z, are the

• nes given by the proposition in the special case where X = Y is the

carrier Z of Z.

29 / 43

References

[1] A. Baltag, L.S. Moss, S. Solecki, The logic of public announcements, common knowledge and private suspicions, TARK, 43-56, 1998 [2] N. Belnap, Display logic, Journal of Philosophical Logic, 11: 375-417, 1982 [3] G. Greco, A. Kurz, A. Palmigiano, Dynamic Epistemic Logic Displayed, Submitted, 2013. [4] R. Gor´ e, L. Postniece, A. Tiu, Cut-elimination and Proof Search for Bi-Intuitionistic Tense Logic, Proc. Adv. in Modal Logic, 156-177, 2010 [5] H. Wansing, Displaying modal logic, Kluwer Academic Publishers, 1998

30 / 43

IEAK axiomatized

Interaction axioms Constants Preservation of facts α⊥ = ⊥, α⊤ = α αp = α ∧ p [α]⊤ = ⊤, [α]⊥ = ¬α [α]p = α → p Disjunction Conjunction α(φ ∨ ψ) = αφ ∨ αψ α(φ ∧ ψ) = αφ ∧ αψ [α](φ ∨ ψ) = α → (αφ ∨ αψ) [α](φ ∧ ψ) = [α]φ ∧ [α]ψ Implication α(φ → ψ) = α ∧ (αφ → αψ) [α](φ → ψ) = αφ → αψ Diamond α✸A ↔ Pre(α) ∧ {✸αjA | kαj} [α]✸A ↔ Pre(α) → {✸αjA | kαj} Box [α]✷A ↔ Pre(α) → {✷[αj]A | kαj} α✷A ↔ Pre(α) ∧ {✷[αj]A | kαj}

31 / 43

IEAK axiomatized D.IEAK is complete

p ⊢ p {α} p ⊢ p α ⊢ α {α} p ; α ⊢ p ∧ α α ; {α} p ⊢ p ∧ α α p ⊢ p ∧ α p ⊢ p p ⊢ {α} p α ; p ⊢ {α} p p ⊢ α > {α} p p ⊢ α > α p α ; p ⊢ α p p ∧ α ⊢ α p p ⊢ p {α} p ⊢ p α ; {α} p ⊢ p α ; [α] p ⊢ p [α] p ⊢ α > p [α] p ⊢ α → p α ⊢ α p ⊢ p p ⊢ {α} p α → p ⊢ α > {α} p α → p ⊢ [α] p

⊥ ⊢ ⊥ {α}⊥ ⊢ ⊥ α⊥ ⊢ ⊥ ⊥ ⊢ I ⊥ ⊢ α⊥ ; I ⊥ ⊢ α⊥ α ⊢ α α ; {α}⊤ ⊢ α α⊤ ⊢ α I ⊢ ⊤ I ⊢ {α}⊤ α ; I ⊢ {α}⊤ I ⊢ α > {α}⊤ I ⊢ α > α⊤ α ; I ⊢ α⊤ α ⊢ α⊤

32 / 43

IEAK axiomatized D.IEAK is complete

I ⊢ ⊤ I ; [α]⊤ ⊢ ⊤ [α]⊤ ⊢ ⊤ ⊤ ⊢ ⊤ ⊤ ⊢ {α}⊤ ⊤ ⊢ [α]⊤ ⊥ ⊢ ⊥ {α}⊥ ⊢ ⊥ α ; {α}⊥ ⊢ ⊥ α ; [α]⊥ ⊢ ⊥ [α]⊥ ⊢ α > ⊥ [α]⊥ ⊢ α → ⊥ [α]⊥ ⊢ ¬α α ⊢ α ⊥ ⊢ ⊥ ⊥ ⊢ {α}⊥ α → ⊥ ⊢ α > {α}⊥ ¬α ⊢ α > {α}⊥ ¬α ⊢ [α]⊥

33 / 43

IEAK axiomatized D.IEAK is complete

A ⊢ A A ; B ⊢ A A ∧ B ⊢ A [α](A ∧ B) ⊢ {α}A [α](A ∧ B) ⊢ [α]A B ⊢ B A ; B ⊢ B A ∧ B ⊢ B [α](A ∧ B) ⊢ {α}B [α](A ∧ B) ⊢ [α]B [α](A ∧ B) ; [α](A ∧ B) ⊢ [α]A ∧ [α]B [α](A ∧ B) ⊢ [α]A ∧ [α]B

34 / 43

IEAK axiomatized D.IEAK is complete

A ⊢ A [α]A ⊢ {α}A [α]A ; [α]B ⊢ {α}A [α]A ∧ [α]B ⊢ {α}A { α } [α]A ∧ [α]B ⊢ A B ⊢ B [α]B ⊢ {α}B [α]A ; [α]B ⊢ {α}B [α]A ∧ [α]B ⊢ {α}B { α } [α]A ∧ [α]B ⊢ B { α } [α]A ∧ [α]B ; { α } [α]A ∧ [α]B ⊢ A ∧ B { α } ([α]A ∧ [α]B ; [α]A ∧ [α]B) ⊢ A ∧ B [α]A ∧ [α]B ; [α]A ∧ [α]B ⊢ {α}A ∧ B [α]A ∧ [α]B ⊢ {α}A ∧ B [α]A ∧ [α]B ; [α]A ∧ [α]B ⊢ {α}A ∧ B [α]A ∧ [α]B ⊢ {α}A ∧ B [α]A ∧ [α]B ⊢ [α](A ∧ B)

35 / 43

IEAK axiomatized D.IEAK is complete

A ⊢ A A ; B ⊢ A A ∧ B ⊢ A {α}A ∧ B ⊢ αA α(A ∧ B) ⊢ αA B ⊢ B A ; B ⊢ B A ∧ B ⊢ B {α}A ∧ B ⊢ αB α(A ∧ B) ⊢ αB α(A ∧ B) ; α(A ∧ B) ⊢ αA ∧ αB α(A ∧ B) ⊢ αA ∧ αB

A ⊢ A {α}A ⊢ {α}A { α } {α}A ⊢ A B ⊢ B {α}B ⊢ {α}B { α } {α}B ⊢ B { α } {α}A ; { α } {α}B ⊢ A ∧ B { α } ({α}A ; {α}B) ⊢ A ∧ B {α}A ; {α}B ⊢ {α}A ∧ B {α}A ; {α}B ⊢ Pre(α) > {α}A ∧ B {α}A ; {α}B ⊢ Pre(α) > αA ∧ B Pre(α) ; ({α}A ; {α}B) ⊢ αA ∧ B (Pre(α) ; {α}A) ; {α}B ⊢ αA ∧ B Pre(α) ; {α}A ⊢ αA ∧ B < {α}B {α}A ⊢ αA ∧ B < {α}B αA ⊢ αA ∧ B < {α}B αA ; {α}B ⊢ αA ∧ B {α}B ⊢ αA > αA ∧ B αB ⊢ αA > αA ∧ B αA ; αB ⊢ αA ∧ B αA ∧ αB ⊢ αA ∧ B 36 / 43

IEAK axiomatized D.IEAK is complete

A ⊢ A {α}A ⊢ αA B ⊢ B {α}B ⊢ αB {α}A ∨ B ⊢ αA ; αB α(A ∨ B) ⊢ αA ; αB α(A ∨ B) ⊢ αA ∨ αB A ⊢ A A ⊢ A ; B A ⊢ A ∨ B {α}A ⊢ α(A ∨ B) αA ⊢ α(A ∨ B) B ⊢ B B ⊢ A ; B B ⊢ A ∨ B {α}B ⊢ α(A ∨ B) αB ⊢ α(A ∨ B) αA ∨ αB ⊢ α(A ∨ B) ; α(A ∨ B) αA ∨ αB ⊢ α(A ∨ B) A ⊢ A {α}A ⊢ αA B ⊢ B {α}B ⊢ αB {α}A ∨ B ⊢ αA ; αB {α}A ∨ B ⊢ αA ∨ αB α ; {α}A ∨ B ⊢ αA ∨ αB α ; [α](A ∨ B) ⊢ αA ∨ αB [α](A ∨ B) ⊢ α > αA ∨ αB [α](A ∨ B) ⊢ α → αA ∨ αB α ⊢ α A ⊢ A {α}A ⊢ {α}A αA ⊢ {α}A B ⊢ B {α}B ⊢ {α}B αB ⊢ {α}B αA ∨ αB ⊢ {α}A ; {α}B αA ∨ αB ⊢ {α}(A ; B) αA ∨ αB ⊢ {α}A ∨ B α → αA ∨ αB ⊢ α > {α}A ∨ B α → αA ∨ αB ⊢ [α](A ∨ B)

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IEAK axiomatized D.IEAK is complete

α ⊢ α A ⊢ A {α}A ⊢ {α}A αA ⊢ {α}A B ⊢ B {α}B ⊢ αB {α}A → B ⊢ αA > αB {α}A → B ⊢ αA → αB α ; {α}A → B ⊢ α ∧ (αA → αB) α(A → B) ⊢ α ∧ (αA → αB) A ⊢ A {α}A ⊢ αA B ⊢ B {α}B ⊢ {α}B αB ⊢ {α}B αA → αB ⊢ {α}A > {α}B αA → αB ⊢ {α}(A > B) αA → αB ⊢ {α}A → B α ; αA → αB ⊢ {α}A → B αA → αB ⊢ α > {α}A → B αA → αB ⊢ α > α(A → B) α ; αA → αB ⊢ α(A → B) α ∧ (αA → αB) ⊢ α(A → B)

38 / 43

IEAK axiomatized D.IEAK is complete

A ⊢ A {α}A ⊢ {α}A B ⊢ B {α}B ⊢ αB {α}A → B ⊢ {α}A > αB α ; {α}A → B ⊢ {α}A > αB α ; [α](A → B) ⊢ {α}A > αB {α}A ; (α ; [α](A → B)) ⊢ αB ({α}A ; α) ; [α](A → B) ⊢ αB [α](A → B) ; ({α}A ; α) ⊢ αB {α}A ; α ⊢ [α](A → B) > αB α ; {α}A ⊢ [α](A → B) > αB αA ⊢ [α](A → B) > αB [α](A → B) ; αA ⊢ αB αA ; [α](A → B) ⊢ αB [α](A → B) ⊢ αA > αB [α](A → B) ⊢ αA → αB A ⊢ A {α}A ⊢ αA B ⊢ B {α}B ⊢ {α}B αB ⊢ {α}B αA → αB ⊢ {α}A > {α}B αA → αB ⊢ {α}(A > B) αA → αB ⊢ {α}A → B αA → αB ⊢ [α](A → B)

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IEAK axiomatized D.IEAK is complete

α ⊢ α A ⊢ A {α}A ⊢ αA

• {α}A ⊢ ✸αA

α ; ◦{α}A ⊢ α ∧ ✸αA α ; {α}◦A ⊢ α ∧ ✸αA {α}◦A ⊢ α > α ∧ ✸αA {α}✸A ⊢ α > α ∧ ✸αA α ; {α}✸A ⊢ α ∧ ✸αA α✸A ⊢ α ∧ ✸αA A ⊢ A

• A ⊢ ✸A

{α}◦A ⊢ α✸A α ; {α}◦A ⊢ α✸A α ; ◦{α}A ⊢ α✸A

• {α}A ⊢ α > α✸A

{α}A ⊢ •(α > α✸A) αA ⊢ •(α > α✸A)

• αA ⊢ α > α✸A

✸αA ⊢ α > α✸A α ; ✸αA ⊢ α✸A α ∧ ✸αA ⊢ α✸A

40 / 43

IEAK axiomatized D.IEAK is complete A ⊢ A {α}A ⊢ αA

• {α}A ⊢ ✸αA

α ; ◦{α}A ⊢ ✸αA α ; {α}◦A ⊢ ✸αA {α}◦A ⊢ α > ✸αA {α}✸A ⊢ α > ✸αA α ; {α}✸A ⊢ ✸αA α ; [α]✸A ⊢ ✸αA [α]✸A ⊢ α > ✸αA [α]✸A ⊢ α → ✸αA α ⊢ α A ⊢ A

• A ⊢ ✸A

{α}◦A ⊢ {α}✸A α ; {α}◦A ⊢ {α}✸A α ; ◦{α}A ⊢ {α}✸A

• {α}A ⊢ α > {α}✸A

{α}A ⊢ •(α > {α}✸A) αA ⊢ •(α > {α}✸A)

• αA ⊢ α > {α}✸A

✸αA ⊢ α > {α}✸A α → ✸αA ⊢ α > (α > {α}✸A) α ; α → ✸αA ⊢ α > {α}✸A α ; (α ; α → ✸αA) ⊢ {α}✸A (α ; α) ; α → ✸αA ⊢ {α}✸A α → ✸αA ; (α ; α) ⊢ {α}✸A α ; α ⊢ α → ✸αA > {α}✸A α ⊢ α → ✸αA > {α}✸A α → ✸αA ; α ⊢ {α}✸A α ; α → ✸αA ⊢ {α}✸A α → ✸αA ⊢ α > {α}✸A α → ✸αA ⊢ [α]✸A 41 / 43

IEAK axiomatized D.IEAK is complete

A ⊢ A ✷A ⊢ ◦A [α]✷A ⊢ {α}◦A [α]✷A ⊢ α > {α}◦A [α]✷A ⊢ α > ◦{α}A α ; [α]✷A ⊢ ◦{α}A

• (α ; [α]✷A) ⊢ {α}A
• (α ; [α]✷A) ⊢ [α]A

α ; [α]✷A ⊢ ◦[α]A α ; [α]✷A ⊢ ✷[α]A [α]✷A ⊢ α > ✷[α]A [α]✷A ⊢ α → ✷[α]A α ⊢ α A ⊢ A [α]A ⊢ {α}A ✷[α]A ⊢ ◦{α}A α → ✷[α]A ⊢ α > ◦{α}A α → ✷[α]A ⊢ α > {α}◦A α ; α → ✷[α]A ⊢ {α}◦A α ; α → ✷[α]A ⊢ {α}✷A α → ✷[α]A ⊢ α > {α}✷A α → ✷[α]A ⊢ [α]✷A

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IEAK axiomatized D.IEAK is complete

α ⊢ α A ⊢ A ✷A ⊢ ◦A {α}✷A ⊢ {α}◦A {α}✷A ⊢ α > {α}◦A {α}✷A ⊢ α > ◦{α}A α ; {α}✷A ⊢ ◦{α}A

• (α ; {α}✷A) ⊢ {α}A
• (α ; {α}✷A) ⊢ [α]A

α ; {α}✷A ⊢ ◦[α]A α ; {α}✷A ⊢ ✷[α]A α ; (α ; {α}✷A) ⊢ α ∧ ✷[α]A (α ; α) ; {α}✷A ⊢ α ∧ ✷[α]A {α}✷A ; (α ; α) ⊢ α ∧ ✷[α]A α ; α ⊢ {α}✷A > α ∧ ✷[α]A α ⊢ {α}✷A > α ∧ ✷[α]A {α}✷A ; α ⊢ α ∧ ✷[α]A α ; {α}✷A ⊢ α ∧ ✷[α]A α✷A ⊢ α ∧ ✷[α]A A ⊢ A [α]A ⊢ {α}A ✷[α]A ⊢ ◦{α}A α ; ✷[α]A ⊢ ◦{α}A ✷[α]A ⊢ α > ◦{α}A ✷[α]A ⊢ α > {α}◦A α ; ✷[α]A ⊢ {α}◦A { α } (α ; ✷[α]A) ⊢ ◦A α ; ✷[α]A ⊢ {α}✷A ✷[α]A ⊢ α > {α}✷A ✷[α]A ⊢ α > α✷A α ; ✷[α]A ⊢ α✷A α ∧ ✷[α]A ⊢ α✷A

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