Parametric completeness for separation theories (via hybrid logic) - PowerPoint PPT Presentation

Semantics of BBI Semantics of formula A wrt. BBI-model M = � W, ◦ , E � , valuation ρ , and w ∈ W given by relation M, w | = ρ A : 9/ 26

Semantics of BBI Semantics of formula A wrt. BBI-model M = � W, ◦ , E � , valuation ρ , and w ∈ W given by relation M, w | = ρ A : M, w | = ρ P ⇔ w ∈ ρ ( P ) 9/ 26

Semantics of BBI Semantics of formula A wrt. BBI-model M = � W, ◦ , E � , valuation ρ , and w ∈ W given by relation M, w | = ρ A : M, w | = ρ P ⇔ w ∈ ρ ( P ) M, w | = ρ A 1 ∧ A 2 ⇔ M, w | = ρ A 1 and M, w | = ρ A 2 9/ 26

Semantics of BBI Semantics of formula A wrt. BBI-model M = � W, ◦ , E � , valuation ρ , and w ∈ W given by relation M, w | = ρ A : M, w | = ρ P ⇔ w ∈ ρ ( P ) M, w | = ρ A 1 ∧ A 2 ⇔ M, w | = ρ A 1 and M, w | = ρ A 2 . . . M, w | = ρ I ⇔ w ∈ E 9/ 26

Semantics of BBI Semantics of formula A wrt. BBI-model M = � W, ◦ , E � , valuation ρ , and w ∈ W given by relation M, w | = ρ A : M, w | = ρ P ⇔ w ∈ ρ ( P ) M, w | = ρ A 1 ∧ A 2 ⇔ M, w | = ρ A 1 and M, w | = ρ A 2 . . . M, w | = ρ I ⇔ w ∈ E M, w | = ρ A 1 ∗ A 2 ⇔ w ∈ w 1 ◦ w 2 and M, w 1 | = ρ A 1 and M, w 2 | = ρ A 2 9/ 26

Semantics of BBI Semantics of formula A wrt. BBI-model M = � W, ◦ , E � , valuation ρ , and w ∈ W given by relation M, w | = ρ A : M, w | = ρ P ⇔ w ∈ ρ ( P ) M, w | = ρ A 1 ∧ A 2 ⇔ M, w | = ρ A 1 and M, w | = ρ A 2 . . . M, w | = ρ I ⇔ w ∈ E M, w | = ρ A 1 ∗ A 2 ⇔ w ∈ w 1 ◦ w 2 and M, w 1 | = ρ A 1 and M, w 2 | = ρ A 2 ∀ w ′ , w ′′ ∈ W. if w ′′ ∈ w ◦ w ′ and M, w ′ | M, w | = ρ A 1 − − ∗ A 2 ⇔ = ρ A 1 then M, w ′′ | = ρ A 2 9/ 26

Semantics of BBI Semantics of formula A wrt. BBI-model M = � W, ◦ , E � , valuation ρ , and w ∈ W given by relation M, w | = ρ A : M, w | = ρ P ⇔ w ∈ ρ ( P ) M, w | = ρ A 1 ∧ A 2 ⇔ M, w | = ρ A 1 and M, w | = ρ A 2 . . . M, w | = ρ I ⇔ w ∈ E M, w | = ρ A 1 ∗ A 2 ⇔ w ∈ w 1 ◦ w 2 and M, w 1 | = ρ A 1 and M, w 2 | = ρ A 2 ∀ w ′ , w ′′ ∈ W. if w ′′ ∈ w ◦ w ′ and M, w ′ | M, w | = ρ A 1 − − ∗ A 2 ⇔ = ρ A 1 then M, w ′′ | = ρ A 2 A is valid in M iff M, w | = ρ A for all ρ and w ∈ W . 9/ 26

Semantics of BBI Semantics of formula A wrt. BBI-model M = � W, ◦ , E � , valuation ρ , and w ∈ W given by relation M, w | = ρ A : M, w | = ρ P ⇔ w ∈ ρ ( P ) M, w | = ρ A 1 ∧ A 2 ⇔ M, w | = ρ A 1 and M, w | = ρ A 2 . . . M, w | = ρ I ⇔ w ∈ E M, w | = ρ A 1 ∗ A 2 ⇔ w ∈ w 1 ◦ w 2 and M, w 1 | = ρ A 1 and M, w 2 | = ρ A 2 ∀ w ′ , w ′′ ∈ W. if w ′′ ∈ w ◦ w ′ and M, w ′ | M, w | = ρ A 1 − − ∗ A 2 ⇔ = ρ A 1 then M, w ′′ | = ρ A 2 A is valid in M iff M, w | = ρ A for all ρ and w ∈ W . Theorem (Galmiche and Larchey-Wendling 2006) Provability in BBI coincides with validity in BBI -models. 9/ 26

Part III (Un)definable properties in BBI 10/ 26

Separation theories Applications of separation logic are typically based on BBI-models satisfying some collection of algebraic properties which we call a separation theory. 11/ 26

Separation theories Applications of separation logic are typically based on BBI-models satisfying some collection of algebraic properties which we call a separation theory. We consider the following: Partial functionality: w, w ′ ∈ w 1 ◦ w 2 implies w = w ′ ; 11/ 26

Separation theories Applications of separation logic are typically based on BBI-models satisfying some collection of algebraic properties which we call a separation theory. We consider the following: Partial functionality: w, w ′ ∈ w 1 ◦ w 2 implies w = w ′ ; Cancellativity: ( w ◦ w 1 ) ∩ ( w ◦ w 2 ) � = ∅ implies w 1 = w 2 ; 11/ 26

Separation theories Applications of separation logic are typically based on BBI-models satisfying some collection of algebraic properties which we call a separation theory. We consider the following: Partial functionality: w, w ′ ∈ w 1 ◦ w 2 implies w = w ′ ; Cancellativity: ( w ◦ w 1 ) ∩ ( w ◦ w 2 ) � = ∅ implies w 1 = w 2 ; Single unit: w, w ′ ∈ E implies w = w ′ ; 11/ 26

Separation theories Applications of separation logic are typically based on BBI-models satisfying some collection of algebraic properties which we call a separation theory. We consider the following: Partial functionality: w, w ′ ∈ w 1 ◦ w 2 implies w = w ′ ; Cancellativity: ( w ◦ w 1 ) ∩ ( w ◦ w 2 ) � = ∅ implies w 1 = w 2 ; Single unit: w, w ′ ∈ E implies w = w ′ ; Indivisible units: ( w ◦ w ′ ) ∩ E � = ∅ implies w ∈ E ; 11/ 26

Separation theories Applications of separation logic are typically based on BBI-models satisfying some collection of algebraic properties which we call a separation theory. We consider the following: Partial functionality: w, w ′ ∈ w 1 ◦ w 2 implies w = w ′ ; Cancellativity: ( w ◦ w 1 ) ∩ ( w ◦ w 2 ) � = ∅ implies w 1 = w 2 ; Single unit: w, w ′ ∈ E implies w = w ′ ; Indivisible units: ( w ◦ w ′ ) ∩ E � = ∅ implies w ∈ E ; Disjointness: w ◦ w � = ∅ implies w ∈ E ; 11/ 26

Separation theories Applications of separation logic are typically based on BBI-models satisfying some collection of algebraic properties which we call a separation theory. We consider the following: Partial functionality: w, w ′ ∈ w 1 ◦ w 2 implies w = w ′ ; Cancellativity: ( w ◦ w 1 ) ∩ ( w ◦ w 2 ) � = ∅ implies w 1 = w 2 ; Single unit: w, w ′ ∈ E implies w = w ′ ; Indivisible units: ( w ◦ w ′ ) ∩ E � = ∅ implies w ∈ E ; Disjointness: w ◦ w � = ∅ implies w ∈ E ; Divisibility: for every w �∈ E there are w 1 , w 2 / ∈ E such that w ∈ w 1 ◦ w 2 ; 11/ 26

Separation theories Applications of separation logic are typically based on BBI-models satisfying some collection of algebraic properties which we call a separation theory. We consider the following: Partial functionality: w, w ′ ∈ w 1 ◦ w 2 implies w = w ′ ; Cancellativity: ( w ◦ w 1 ) ∩ ( w ◦ w 2 ) � = ∅ implies w 1 = w 2 ; Single unit: w, w ′ ∈ E implies w = w ′ ; Indivisible units: ( w ◦ w ′ ) ∩ E � = ∅ implies w ∈ E ; Disjointness: w ◦ w � = ∅ implies w ∈ E ; Divisibility: for every w �∈ E there are w 1 , w 2 / ∈ E such that w ∈ w 1 ◦ w 2 ; Cross-split property: whenever ( a ◦ b ) ∩ ( c ◦ d ) � = ∅ , there exist ac , ad , bc , bd such that a ∈ ac ◦ ad , b ∈ bc ◦ bd , c ∈ ac ◦ bc and d ∈ ad ◦ bd . 11/ 26

Definable properties A property P of BBI-models is said to be L -definable if there exists an L -formula A such that for all BBI-models M , A is valid in M ⇐ ⇒ M ∈ P . 12/ 26

Definable properties A property P of BBI-models is said to be L -definable if there exists an L -formula A such that for all BBI-models M , A is valid in M ⇐ ⇒ M ∈ P . Proposition The following separation theory properties are BBI -definable: 12/ 26

Definable properties A property P of BBI-models is said to be L -definable if there exists an L -formula A such that for all BBI-models M , A is valid in M ⇐ ⇒ M ∈ P . Proposition The following separation theory properties are BBI -definable: I ∧ ( A ∗ B ) ⊢ A Indivisible units: 12/ 26

Definable properties A property P of BBI-models is said to be L -definable if there exists an L -formula A such that for all BBI-models M , A is valid in M ⇐ ⇒ M ∈ P . Proposition The following separation theory properties are BBI -definable: I ∧ ( A ∗ B ) ⊢ A Indivisible units: Divisibility: ¬ I ⊢ ¬ I ∗ ¬ I 12/ 26

Definable properties A property P of BBI-models is said to be L -definable if there exists an L -formula A such that for all BBI-models M , A is valid in M ⇐ ⇒ M ∈ P . Proposition The following separation theory properties are BBI -definable: I ∧ ( A ∗ B ) ⊢ A Indivisible units: Divisibility: ¬ I ⊢ ¬ I ∗ ¬ I Proof. Just directly verify the needed biimplication. 12/ 26

Undefinability via disjoint union To show a property is not BBI-definable, we show it is not preserved by some validity-preserving model construction. 13/ 26

Undefinability via disjoint union To show a property is not BBI-definable, we show it is not preserved by some validity-preserving model construction. Definition If M 1 = � W 1 , ◦ 1 , E 1 � and M 2 = � W 2 , ◦ 2 , E 2 � are BBI-models and W 1 , W 2 are disjoint then their disjoint union is given by 13/ 26

Undefinability via disjoint union To show a property is not BBI-definable, we show it is not preserved by some validity-preserving model construction. Definition If M 1 = � W 1 , ◦ 1 , E 1 � and M 2 = � W 2 , ◦ 2 , E 2 � are BBI-models and W 1 , W 2 are disjoint then their disjoint union is given by def M 1 ⊎ M 2 = � W 1 ∪ W 2 , ◦ 1 ∪ ◦ 2 , E 1 ∪ E 2 � (where ◦ 1 ∪ ◦ 2 is lifted to W 1 ∪ W 2 in the obvious way) 13/ 26

Undefinability via disjoint union To show a property is not BBI-definable, we show it is not preserved by some validity-preserving model construction. Definition If M 1 = � W 1 , ◦ 1 , E 1 � and M 2 = � W 2 , ◦ 2 , E 2 � are BBI-models and W 1 , W 2 are disjoint then their disjoint union is given by def M 1 ⊎ M 2 = � W 1 ∪ W 2 , ◦ 1 ∪ ◦ 2 , E 1 ∪ E 2 � (where ◦ 1 ∪ ◦ 2 is lifted to W 1 ∪ W 2 in the obvious way) Proposition If A is valid in M 1 and in M 2 , and M 1 ⊎ M 2 is defined, then it is also valid in M 1 ⊎ M 2 . 13/ 26

Undefinability via disjoint union To show a property is not BBI-definable, we show it is not preserved by some validity-preserving model construction. Definition If M 1 = � W 1 , ◦ 1 , E 1 � and M 2 = � W 2 , ◦ 2 , E 2 � are BBI-models and W 1 , W 2 are disjoint then their disjoint union is given by def M 1 ⊎ M 2 = � W 1 ∪ W 2 , ◦ 1 ∪ ◦ 2 , E 1 ∪ E 2 � (where ◦ 1 ∪ ◦ 2 is lifted to W 1 ∪ W 2 in the obvious way) Proposition If A is valid in M 1 and in M 2 , and M 1 ⊎ M 2 is defined, then it is also valid in M 1 ⊎ M 2 . Proof. Structural induction on A . 13/ 26

Undefinability of single-unit property Lemma Let P be a property of BBI -models, and suppose that there exist BBI -models M 1 and M 2 such that M 1 , M 2 ∈ P but M 1 ⊎ M 2 �∈ P . Then P is not BBI -definable. 14/ 26

Undefinability of single-unit property Lemma Let P be a property of BBI -models, and suppose that there exist BBI -models M 1 and M 2 such that M 1 , M 2 ∈ P but M 1 ⊎ M 2 �∈ P . Then P is not BBI -definable. Proof. If P were definable via A say, then A would be true in M 1 and M 2 but not in M 1 ⊎ M 2 , contradicting previous Proposition. 14/ 26

Undefinability of single-unit property Lemma Let P be a property of BBI -models, and suppose that there exist BBI -models M 1 and M 2 such that M 1 , M 2 ∈ P but M 1 ⊎ M 2 �∈ P . Then P is not BBI -definable. Proof. If P were definable via A say, then A would be true in M 1 and M 2 but not in M 1 ⊎ M 2 , contradicting previous Proposition. Theorem The single unit property is not BBI -definable. 14/ 26

Undefinability of single-unit property Lemma Let P be a property of BBI -models, and suppose that there exist BBI -models M 1 and M 2 such that M 1 , M 2 ∈ P but M 1 ⊎ M 2 �∈ P . Then P is not BBI -definable. Proof. If P were definable via A say, then A would be true in M 1 and M 2 but not in M 1 ⊎ M 2 , contradicting previous Proposition. Theorem The single unit property is not BBI -definable. Proof. The disjoint union of any two single-unit BBI-models (e.g. two copies of N under addition) is not a single-unit model, so we are done by the above Lemma. 14/ 26

Undefinability via bounded morphisms We adapt the notion of bounded morphism from modal logic to BBI-models, and can show it is also validity-preserving. 15/ 26

Undefinability via bounded morphisms We adapt the notion of bounded morphism from modal logic to BBI-models, and can show it is also validity-preserving. Theorem None of the following separation theory properties (or any combination thereof) is BBI -definable: 15/ 26

Undefinability via bounded morphisms We adapt the notion of bounded morphism from modal logic to BBI-models, and can show it is also validity-preserving. Theorem None of the following separation theory properties (or any combination thereof) is BBI -definable: • functionality; 15/ 26

Undefinability via bounded morphisms We adapt the notion of bounded morphism from modal logic to BBI-models, and can show it is also validity-preserving. Theorem None of the following separation theory properties (or any combination thereof) is BBI -definable: • functionality; • cancellativity; 15/ 26

Undefinability via bounded morphisms We adapt the notion of bounded morphism from modal logic to BBI-models, and can show it is also validity-preserving. Theorem None of the following separation theory properties (or any combination thereof) is BBI -definable: • functionality; • cancellativity; • disjointness. 15/ 26

Undefinability via bounded morphisms We adapt the notion of bounded morphism from modal logic to BBI-models, and can show it is also validity-preserving. Theorem None of the following separation theory properties (or any combination thereof) is BBI -definable: • functionality; • cancellativity; • disjointness. Proof. E.g., for functionality, we build models M and M ′ such that there is a bounded morphism from M to M ′ , but M is functional while M ′ is not. See paper for details. 15/ 26

Part IV Hybrid extensions of BBI 16/ 26

HyBBI : a hybrid extension of BBI • We saw that BBI is not expressive enough to accurately capture many separation theories. 17/ 26

HyBBI : a hybrid extension of BBI • We saw that BBI is not expressive enough to accurately capture many separation theories. • Idea: conservatively increase the expressivity of BBI, using machinery of hybrid logic. 17/ 26

HyBBI : a hybrid extension of BBI • We saw that BBI is not expressive enough to accurately capture many separation theories. • Idea: conservatively increase the expressivity of BBI, using machinery of hybrid logic. • HyBBI extends the language of BBI by: any nominal ℓ is a formula, and so is any formula of the form @ ℓ A . 17/ 26

HyBBI : a hybrid extension of BBI • We saw that BBI is not expressive enough to accurately capture many separation theories. • Idea: conservatively increase the expressivity of BBI, using machinery of hybrid logic. • HyBBI extends the language of BBI by: any nominal ℓ is a formula, and so is any formula of the form @ ℓ A . • Valuations interpret nominals as individual worlds in a BBI-model. 17/ 26

HyBBI : a hybrid extension of BBI • We saw that BBI is not expressive enough to accurately capture many separation theories. • Idea: conservatively increase the expressivity of BBI, using machinery of hybrid logic. • HyBBI extends the language of BBI by: any nominal ℓ is a formula, and so is any formula of the form @ ℓ A . • Valuations interpret nominals as individual worlds in a BBI-model. • We extend the forcing relation by: M, w | = ρ ℓ ⇔ w = ρ ( ℓ ) 17/ 26

HyBBI : a hybrid extension of BBI • We saw that BBI is not expressive enough to accurately capture many separation theories. • Idea: conservatively increase the expressivity of BBI, using machinery of hybrid logic. • HyBBI extends the language of BBI by: any nominal ℓ is a formula, and so is any formula of the form @ ℓ A . • Valuations interpret nominals as individual worlds in a BBI-model. • We extend the forcing relation by: M, w | = ρ ℓ ⇔ w = ρ ( ℓ ) M, w | = ρ @ ℓ A ⇔ M, ρ ( ℓ ) | = ρ A 17/ 26

HyBBI : a hybrid extension of BBI • We saw that BBI is not expressive enough to accurately capture many separation theories. • Idea: conservatively increase the expressivity of BBI, using machinery of hybrid logic. • HyBBI extends the language of BBI by: any nominal ℓ is a formula, and so is any formula of the form @ ℓ A . • Valuations interpret nominals as individual worlds in a BBI-model. • We extend the forcing relation by: M, w | = ρ ℓ ⇔ w = ρ ( ℓ ) M, w | = ρ @ ℓ A ⇔ M, ρ ( ℓ ) | = ρ A Easy to see that HyBBI is a conservative extension of BBI. 17/ 26

Definable properties in HyBBI A formula is pure if it contains no propositional variables. Pure formulas have particularly nice properties wrt. completeness. 18/ 26

Definable properties in HyBBI A formula is pure if it contains no propositional variables. Pure formulas have particularly nice properties wrt. completeness. Theorem The following separation theory properties are HyBBI -definable, using pure formulas: 18/ 26

Definable properties in HyBBI A formula is pure if it contains no propositional variables. Pure formulas have particularly nice properties wrt. completeness. Theorem The following separation theory properties are HyBBI -definable, using pure formulas: @ ℓ ( j ∗ k ) ∧ @ ℓ ′ ( j ∗ k ) ⊢ @ ℓ ℓ ′ Functionality: 18/ 26

Definable properties in HyBBI A formula is pure if it contains no propositional variables. Pure formulas have particularly nice properties wrt. completeness. Theorem The following separation theory properties are HyBBI -definable, using pure formulas: @ ℓ ( j ∗ k ) ∧ @ ℓ ′ ( j ∗ k ) ⊢ @ ℓ ℓ ′ Functionality: Cancellativity: ℓ ∗ j ∧ ℓ ∗ k ⊢ @ j k 18/ 26

Definable properties in HyBBI A formula is pure if it contains no propositional variables. Pure formulas have particularly nice properties wrt. completeness. Theorem The following separation theory properties are HyBBI -definable, using pure formulas: @ ℓ ( j ∗ k ) ∧ @ ℓ ′ ( j ∗ k ) ⊢ @ ℓ ℓ ′ Functionality: Cancellativity: ℓ ∗ j ∧ ℓ ∗ k ⊢ @ j k Single unit: @ ℓ 1 I ∧ @ ℓ 2 I ⊢ @ ℓ 1 ℓ 2 18/ 26

Definable properties in HyBBI A formula is pure if it contains no propositional variables. Pure formulas have particularly nice properties wrt. completeness. Theorem The following separation theory properties are HyBBI -definable, using pure formulas: @ ℓ ( j ∗ k ) ∧ @ ℓ ′ ( j ∗ k ) ⊢ @ ℓ ℓ ′ Functionality: Cancellativity: ℓ ∗ j ∧ ℓ ∗ k ⊢ @ j k Single unit: @ ℓ 1 I ∧ @ ℓ 2 I ⊢ @ ℓ 1 ℓ 2 Disjointness: ℓ ∗ ℓ ⊢ I ∧ ℓ 18/ 26

Definable properties in HyBBI A formula is pure if it contains no propositional variables. Pure formulas have particularly nice properties wrt. completeness. Theorem The following separation theory properties are HyBBI -definable, using pure formulas: @ ℓ ( j ∗ k ) ∧ @ ℓ ′ ( j ∗ k ) ⊢ @ ℓ ℓ ′ Functionality: Cancellativity: ℓ ∗ j ∧ ℓ ∗ k ⊢ @ j k Single unit: @ ℓ 1 I ∧ @ ℓ 2 I ⊢ @ ℓ 1 ℓ 2 Disjointness: ℓ ∗ ℓ ⊢ I ∧ ℓ Proof. Easy verifications! 18/ 26

A word about cross-split We have brushed over the cross-split property: ( a ◦ b ) ∩ ( c ◦ d ) � = ∅ , implies ∃ ac , ad , bc , bd with a ∈ ac ◦ ad, b ∈ bc ◦ bd, c ∈ ac ◦ bc, d ∈ ad ◦ bd. 19/ 26

A word about cross-split We have brushed over the cross-split property: ( a ◦ b ) ∩ ( c ◦ d ) � = ∅ , implies ∃ ac , ad , bc , bd with a ∈ ac ◦ ad, b ∈ bc ◦ bd, c ∈ ac ◦ bc, d ∈ ad ◦ bd. bc c ac a b d ad bd 19/ 26

A word about cross-split We have brushed over the cross-split property: ( a ◦ b ) ∩ ( c ◦ d ) � = ∅ , implies ∃ ac , ad , bc , bd with a ∈ ac ◦ ad, b ∈ bc ◦ bd, c ∈ ac ◦ bc, d ∈ ad ◦ bd. bc c ac a b d ad bd We conjecture this is not definable in BBI or in HyBBI. 19/ 26

A word about cross-split We have brushed over the cross-split property: ( a ◦ b ) ∩ ( c ◦ d ) � = ∅ , implies ∃ ac , ad , bc , bd with a ∈ ac ◦ ad, b ∈ bc ◦ bd, c ∈ ac ◦ bc, d ∈ ad ◦ bd. bc c ac a b d ad bd We conjecture this is not definable in BBI or in HyBBI. If we add the ↓ binder to HyBBI, defined by M, w | = ρ ↓ ℓ. A ⇔ M, w | = ρ [ ℓ := w ] A 19/ 26

A word about cross-split We have brushed over the cross-split property: ( a ◦ b ) ∩ ( c ◦ d ) � = ∅ , implies ∃ ac , ad , bc , bd with a ∈ ac ◦ ad, b ∈ bc ◦ bd, c ∈ ac ◦ bc, d ∈ ad ◦ bd. bc c ac a b d ad bd We conjecture this is not definable in BBI or in HyBBI. If we add the ↓ binder to HyBBI, defined by M, w | = ρ ↓ ℓ. A ⇔ M, w | = ρ [ ℓ := w ] A then cross-split is definable as the pure formula ( a ∗ b ) ∧ ( c ∗ d ) ⊢ @ a ( ⊤ ∗ ↓ ac . @ a ( ⊤ ∗ ↓ ad . @ a ( ac ∗ ad ) ∧ @ b ( ⊤ ∗ ↓ bc . @ b ( ⊤ ∗ ↓ bd . @ b ( bc ∗ bd ) ∧ @ c ( ac ∗ bc ) ∧ @ d ( ad ∗ bd ))))) 19/ 26

Part V Parametric completeness for HyBBI( ↓ ) 20/ 26

Axiomatic proof systems for HyBBI( ↓ ) Our axiom system K HyBBI( ↓ ) is chosen to make the completeness proof as clean as possible. 21/ 26

Axiomatic proof systems for HyBBI( ↓ ) Our axiom system K HyBBI( ↓ ) is chosen to make the completeness proof as clean as possible. Some example axioms and rules: ( K @ ) @ ℓ ( A → B ) ⊢ @ ℓ A → @ ℓ B 21/ 26

Axiomatic proof systems for HyBBI( ↓ ) Our axiom system K HyBBI( ↓ ) is chosen to make the completeness proof as clean as possible. Some example axioms and rules: ( K @ ) @ ℓ ( A → B ) ⊢ @ ℓ A → @ ℓ B (@-intro) ℓ ∧ A ⊢ @ ℓ A 21/ 26

Axiomatic proof systems for HyBBI( ↓ ) Our axiom system K HyBBI( ↓ ) is chosen to make the completeness proof as clean as possible. Some example axioms and rules: ( K @ ) @ ℓ ( A → B ) ⊢ @ ℓ A → @ ℓ B (@-intro) ℓ ∧ A ⊢ @ ℓ A (Bridge ∗ ) @ ℓ ( k ∗ k ′ ) ∧ @ k A ∧ @ k ′ B ⊢ @ ℓ ( A ∗ B ) 21/ 26

Axiomatic proof systems for HyBBI( ↓ ) Our axiom system K HyBBI( ↓ ) is chosen to make the completeness proof as clean as possible. Some example axioms and rules: ( K @ ) @ ℓ ( A → B ) ⊢ @ ℓ A → @ ℓ B (@-intro) ℓ ∧ A ⊢ @ ℓ A (Bridge ∗ ) @ ℓ ( k ∗ k ′ ) ∧ @ k A ∧ @ k ′ B ⊢ @ ℓ ( A ∗ B ) (Bind ↓ . ) ⊢ @ j ( ↓ ℓ. B ↔ B [ j/ℓ ]) 21/ 26

Axiomatic proof systems for HyBBI( ↓ ) Our axiom system K HyBBI( ↓ ) is chosen to make the completeness proof as clean as possible. Some example axioms and rules: ( K @ ) @ ℓ ( A → B ) ⊢ @ ℓ A → @ ℓ B (@-intro) ℓ ∧ A ⊢ @ ℓ A (Bridge ∗ ) @ ℓ ( k ∗ k ′ ) ∧ @ k A ∧ @ k ′ B ⊢ @ ℓ ( A ∗ B ) (Bind ↓ . ) ⊢ @ j ( ↓ ℓ. B ↔ B [ j/ℓ ]) @ ℓ ( k ∗ k ′ ) ∧ @ k A ∧ @ k ′ B ⊢ C k, k ′ not in A , B , C or { ℓ } (Paste ∗ ) @ ℓ ( A ∗ B ) ⊢ C 21/ 26

Axiomatic proof systems for HyBBI( ↓ ) Our axiom system K HyBBI( ↓ ) is chosen to make the completeness proof as clean as possible. Some example axioms and rules: ( K @ ) @ ℓ ( A → B ) ⊢ @ ℓ A → @ ℓ B (@-intro) ℓ ∧ A ⊢ @ ℓ A (Bridge ∗ ) @ ℓ ( k ∗ k ′ ) ∧ @ k A ∧ @ k ′ B ⊢ @ ℓ ( A ∗ B ) (Bind ↓ . ) ⊢ @ j ( ↓ ℓ. B ↔ B [ j/ℓ ]) @ ℓ ( k ∗ k ′ ) ∧ @ k A ∧ @ k ′ B ⊢ C k, k ′ not in A , B , C or { ℓ } (Paste ∗ ) @ ℓ ( A ∗ B ) ⊢ C Proposition (Soundness) Any K HyBBI( ↓ ) -provable sequent is valid in all BBI -models. 21/ 26

Completeness Standard modal logic approach to completeness via maximal consistent sets (MCSs): 22/ 26

Completeness Standard modal logic approach to completeness via maximal consistent sets (MCSs): 1. Show that any consistent set of formulas can be extended to an MCS (known as the Lindenbaum construction); 22/ 26

Completeness Standard modal logic approach to completeness via maximal consistent sets (MCSs): 1. Show that any consistent set of formulas can be extended to an MCS (known as the Lindenbaum construction); 2. Define a canonical model whose worlds are MCSs, and a valuation s.t. proposition P is true at Γ iff P ∈ Γ. 22/ 26

Completeness Standard modal logic approach to completeness via maximal consistent sets (MCSs): 1. Show that any consistent set of formulas can be extended to an MCS (known as the Lindenbaum construction); 2. Define a canonical model whose worlds are MCSs, and a valuation s.t. proposition P is true at Γ iff P ∈ Γ. 3. Truth Lemma: A is true at Γ iff A ∈ Γ for any formula A . 22/ 26

Completeness Standard modal logic approach to completeness via maximal consistent sets (MCSs): 1. Show that any consistent set of formulas can be extended to an MCS (known as the Lindenbaum construction); 2. Define a canonical model whose worlds are MCSs, and a valuation s.t. proposition P is true at Γ iff P ∈ Γ. 3. Truth Lemma: A is true at Γ iff A ∈ Γ for any formula A . 4. Now, if A is unprovable, {¬ A } is consistent so there is an MCS Γ ⊃ {¬ A } . Then A is false at Γ in the canonical model, hence invalid. 22/ 26