[PPT] - Boolean bunched logic: its semantics and completeness James PowerPoint Presentation

SLIDE 1

Boolean bunched logic: its semantics and completeness

James Brotherston

Programming Principles, Logic and Verification Group

Dept. of Computer Science

University College London, UK J.Brotherston@ucl.ac.uk

Logic Summer School, ANU, 8 December 2015

1/ 18

SLIDE 2

Bunched logics

Bunched logics extend classical or intuitionistic logic with

various “linear” or multiplicative connectives.

2/ 18

SLIDE 3

Bunched logics

Bunched logics extend classical or intuitionistic logic with

various “linear” or multiplicative connectives.

Formulas can be understood as sets of “worlds” (often

“resources”) in an underlying model.

2/ 18

SLIDE 4

Bunched logics

Bunched logics extend classical or intuitionistic logic with

various “linear” or multiplicative connectives.

Formulas can be understood as sets of “worlds” (often

“resources”) in an underlying model.

The multiplicatives generally denote composition
perations on these worlds.

2/ 18

SLIDE 5

Bunched logics

Bunched logics extend classical or intuitionistic logic with

various “linear” or multiplicative connectives.

Formulas can be understood as sets of “worlds” (often

“resources”) in an underlying model.

The multiplicatives generally denote composition
perations on these worlds.
Bunched logics are closely related to relevant logics and can

also be seen as modal logics.

2/ 18

SLIDE 6

Boolean BI

In this course we focus on Boolean BI (from now on BBI)

3/ 18

SLIDE 7

Boolean BI

In this course we focus on Boolean BI (from now on BBI)
BBI extends classical propositional logic with the following

“multiplicative” connectives:

∗, a multiplicative conjunction;

3/ 18

SLIDE 8

Boolean BI

In this course we focus on Boolean BI (from now on BBI)
BBI extends classical propositional logic with the following

“multiplicative” connectives:

∗, a multiplicative conjunction;
—

∗ (“magic wand”), a multiplicative implication;

3/ 18

SLIDE 9

Boolean BI

In this course we focus on Boolean BI (from now on BBI)
BBI extends classical propositional logic with the following

“multiplicative” connectives:

∗, a multiplicative conjunction;
—

∗ (“magic wand”), a multiplicative implication;

I, a multiplicative unit.

3/ 18

SLIDE 10

Boolean BI

In this course we focus on Boolean BI (from now on BBI)
BBI extends classical propositional logic with the following

“multiplicative” connectives:

∗, a multiplicative conjunction;
—

∗ (“magic wand”), a multiplicative implication;

I, a multiplicative unit.
“Multiplicative” means ∗ does not satisfy weakening or

contraction: A ∗ B ⊢ A A ⊢ A ∗ A

3/ 18

SLIDE 11

Boolean BI

In this course we focus on Boolean BI (from now on BBI)
BBI extends classical propositional logic with the following

“multiplicative” connectives:

∗, a multiplicative conjunction;
—

∗ (“magic wand”), a multiplicative implication;

I, a multiplicative unit.
“Multiplicative” means ∗ does not satisfy weakening or

contraction: A ∗ B ⊢ A A ⊢ A ∗ A

The multiplicatives can be seen as modalities in modal

logic (more on that later).

3/ 18

SLIDE 12

Reading the multiplicatives

Intuitively, formulas in BBI can be read as properties of

resources.

4/ 18

SLIDE 13

Reading the multiplicatives

Intuitively, formulas in BBI can be read as properties of

resources.

A ∗ B can be read as “my current resource decomposes into

two parts that satisfy A and B respectively”.

4/ 18

SLIDE 14

Reading the multiplicatives

Intuitively, formulas in BBI can be read as properties of

resources.

A ∗ B can be read as “my current resource decomposes into

two parts that satisfy A and B respectively”.

I can be read as “my resource is empty / of unit type”.

4/ 18

SLIDE 15

Reading the multiplicatives

Intuitively, formulas in BBI can be read as properties of

resources.

A ∗ B can be read as “my current resource decomposes into

two parts that satisfy A and B respectively”.

I can be read as “my resource is empty / of unit type”.
A —

∗ B can be read as “if I add a resource satisfying A to my current resource, the whole thing satisfies B”.

4/ 18

SLIDE 16

BBI, proof-theoretically

Provability in BBI is given by extending a Hilbert system for propositional classical logic by A ∗ B ⊢ B ∗ A A ∗ (B ∗ C) ⊢ (A ∗ B) ∗ C A ⊢ A ∗ I A ∗ I ⊢ A A1 ⊢ B1 A2 ⊢ B2 A1 ∗ A2 ⊢ B1 ∗ B2 A ∗ B ⊢ C A ⊢ B — ∗ C A ⊢ B — ∗ C A ∗ B ⊢ C These rules are exactly the usual ones for multiplicative intuitionistic linear logic (MILL).

5/ 18

SLIDE 17

BBI, semantically (1)

A BBI-model is given by W, ◦, E, where

W is a set (of “worlds”),

6/ 18

SLIDE 18

BBI, semantically (1)

A BBI-model is given by W, ◦, E, where

W is a set (of “worlds”),
◦ is a binary function W × W → P(W);

6/ 18

SLIDE 19

BBI, semantically (1)

A BBI-model is given by W, ◦, E, where

W is a set (of “worlds”),
◦ is a binary function W × W → P(W);we extend ◦ to

P(W) × P(W) → P(W) by W1 ◦ W2 =def

w1∈W1,w2∈W2 w1 ◦ w2

6/ 18

SLIDE 20

BBI, semantically (1)

A BBI-model is given by W, ◦, E, where

W is a set (of “worlds”),
◦ is a binary function W × W → P(W);we extend ◦ to

P(W) × P(W) → P(W) by W1 ◦ W2 =def

w1∈W1,w2∈W2 w1 ◦ w2
◦ is commutative and associative;

6/ 18

SLIDE 21

BBI, semantically (1)

A BBI-model is given by W, ◦, E, where

W is a set (of “worlds”),
◦ is a binary function W × W → P(W);we extend ◦ to

P(W) × P(W) → P(W) by W1 ◦ W2 =def

w1∈W1,w2∈W2 w1 ◦ w2
◦ is commutative and associative;
the set of units E ⊆ W satisfies w ◦ E = {w} for all w ∈ W.

6/ 18

SLIDE 22

BBI, semantically (1)

A BBI-model is given by W, ◦, E, where

W is a set (of “worlds”),
◦ is a binary function W × W → P(W);we extend ◦ to

P(W) × P(W) → P(W) by W1 ◦ W2 =def

w1∈W1,w2∈W2 w1 ◦ w2
◦ is commutative and associative;
the set of units E ⊆ W satisfies w ◦ E = {w} for all w ∈ W.

(Note that ◦ can equivalently be seen as a ternary relation,

⊆ W × W × W.)

6/ 18

SLIDE 23

BBI, semantically (2)

A valuation for BBI-model M = W, ◦, E is a function ρ from propositional variables to P(W).

7/ 18

SLIDE 24

BBI, semantically (2)

A valuation for BBI-model M = W, ◦, E is a function ρ from propositional variables to P(W). Given M, ρ, and w ∈ W, we define the forcing relation w | =ρ A by induction on formula A:

w | =ρ P ⇔ w ∈ ρ(P)

7/ 18

SLIDE 25

BBI, semantically (2)

A valuation for BBI-model M = W, ◦, E is a function ρ from propositional variables to P(W). Given M, ρ, and w ∈ W, we define the forcing relation w | =ρ A by induction on formula A:

w | =ρ P ⇔ w ∈ ρ(P) w | =ρ A → B ⇔ w | =ρ A implies w | =ρ B

7/ 18

SLIDE 26

BBI, semantically (2)

A valuation for BBI-model M = W, ◦, E is a function ρ from propositional variables to P(W). Given M, ρ, and w ∈ W, we define the forcing relation w | =ρ A by induction on formula A:

w | =ρ P ⇔ w ∈ ρ(P) w | =ρ A → B ⇔ w | =ρ A implies w | =ρ B . . . w | =ρ I ⇔ w ∈ E

7/ 18

SLIDE 27

BBI, semantically (2)

A valuation for BBI-model M = W, ◦, E is a function ρ from propositional variables to P(W). Given M, ρ, and w ∈ W, we define the forcing relation w | =ρ A by induction on formula A:

w | =ρ P ⇔ w ∈ ρ(P) w | =ρ A → B ⇔ w | =ρ A implies w | =ρ B . . . w | =ρ I ⇔ w ∈ E w | =ρ A ∗ B ⇔ w ∈ w1 ◦ w2 and w1 | =ρ A and w2 | =ρ B

7/ 18

SLIDE 28

BBI, semantically (2)

A valuation for BBI-model M = W, ◦, E is a function ρ from propositional variables to P(W). Given M, ρ, and w ∈ W, we define the forcing relation w | =ρ A by induction on formula A:

w | =ρ P ⇔ w ∈ ρ(P) w | =ρ A → B ⇔ w | =ρ A implies w | =ρ B . . . w | =ρ I ⇔ w ∈ E w | =ρ A ∗ B ⇔ w ∈ w1 ◦ w2 and w1 | =ρ A and w2 | =ρ B w | =ρ A — ∗ B ⇔ ∀w′, w′′ ∈ W. if w′′ ∈ w ◦ w′ and w′ | =ρ A then w′′ | =ρ B

7/ 18

SLIDE 29

BBI, semantically (2)

A valuation for BBI-model M = W, ◦, E is a function ρ from propositional variables to P(W). Given M, ρ, and w ∈ W, we define the forcing relation w | =ρ A by induction on formula A:

w | =ρ P ⇔ w ∈ ρ(P) w | =ρ A → B ⇔ w | =ρ A implies w | =ρ B . . . w | =ρ I ⇔ w ∈ E w | =ρ A ∗ B ⇔ w ∈ w1 ◦ w2 and w1 | =ρ A and w2 | =ρ B w | =ρ A — ∗ B ⇔ ∀w′, w′′ ∈ W. if w′′ ∈ w ◦ w′ and w′ | =ρ A then w′′ | =ρ B A is valid in M iff w | =ρ A for all ρ and w ∈ W.

7/ 18

SLIDE 30

Soundness and completeness

Theorem (Galmiche and Larchey-Wendling, 2006) A formula is BBI-provable iff it is valid in all BBI-models.

8/ 18

SLIDE 31

Soundness and completeness

Theorem (Galmiche and Larchey-Wendling, 2006) A formula is BBI-provable iff it is valid in all BBI-models.

Soundness (⇒) is straightforward: just show that each

proof rule preserves validity. (Easy exercise!)

8/ 18

SLIDE 32

Soundness and completeness

Theorem (Galmiche and Larchey-Wendling, 2006) A formula is BBI-provable iff it is valid in all BBI-models.

Soundness (⇒) is straightforward: just show that each

proof rule preserves validity. (Easy exercise!)

Completeness (⇐) is much harder.

8/ 18

SLIDE 33

Soundness and completeness

Theorem (Galmiche and Larchey-Wendling, 2006) A formula is BBI-provable iff it is valid in all BBI-models.

Soundness (⇒) is straightforward: just show that each

proof rule preserves validity. (Easy exercise!)

Completeness (⇐) is much harder.
Several different approaches are possible; I am going to try

to show you the simplest one, based on the Sahlqvist completeness theorem for modal logic.

8/ 18

SLIDE 34

Outline of the approach

We translate BBI into a normal modal logic over

“diamond” modalities I, ∗, ⊸, satisfying a set of well-behaved Sahlqvist axioms. A ⊸ B will come out as ¬(A — ∗ ¬B).

9/ 18

SLIDE 35

Outline of the approach

We translate BBI into a normal modal logic over

“diamond” modalities I, ∗, ⊸, satisfying a set of well-behaved Sahlqvist axioms. A ⊸ B will come out as ¬(A — ∗ ¬B).

Then the Sahlqvist completeness theorem says that this

modal logic is complete for its models in modal logic.

9/ 18

SLIDE 36

Outline of the approach

We translate BBI into a normal modal logic over

“diamond” modalities I, ∗, ⊸, satisfying a set of well-behaved Sahlqvist axioms. A ⊸ B will come out as ¬(A — ∗ ¬B).

Then the Sahlqvist completeness theorem says that this

modal logic is complete for its models in modal logic.

By suitable translations t and u between BBI and this

modal logic, we get A valid in BBI

9/ 18

SLIDE 37

Outline of the approach

We translate BBI into a normal modal logic over

“diamond” modalities I, ∗, ⊸, satisfying a set of well-behaved Sahlqvist axioms. A ⊸ B will come out as ¬(A — ∗ ¬B).

Then the Sahlqvist completeness theorem says that this

modal logic is complete for its models in modal logic.

By suitable translations t and u between BBI and this

modal logic, we get A valid in BBI ⇒ t(A) valid in modal logic

9/ 18

SLIDE 38

Outline of the approach

We translate BBI into a normal modal logic over

“diamond” modalities I, ∗, ⊸, satisfying a set of well-behaved Sahlqvist axioms. A ⊸ B will come out as ¬(A — ∗ ¬B).

Then the Sahlqvist completeness theorem says that this

modal logic is complete for its models in modal logic.

By suitable translations t and u between BBI and this

modal logic, we get A valid in BBI ⇒ t(A) valid in modal logic ⇒ t(A) provable in modal logic (Sahlqvist)

9/ 18

SLIDE 39

Outline of the approach

We translate BBI into a normal modal logic over

“diamond” modalities I, ∗, ⊸, satisfying a set of well-behaved Sahlqvist axioms. A ⊸ B will come out as ¬(A — ∗ ¬B).

Then the Sahlqvist completeness theorem says that this

modal logic is complete for its models in modal logic.

By suitable translations t and u between BBI and this

modal logic, we get A valid in BBI ⇒ t(A) valid in modal logic ⇒ t(A) provable in modal logic (Sahlqvist) ⇒ u(t(A)) provable in BBI

9/ 18

SLIDE 40

Outline of the approach

We translate BBI into a normal modal logic over

“diamond” modalities I, ∗, ⊸, satisfying a set of well-behaved Sahlqvist axioms. A ⊸ B will come out as ¬(A — ∗ ¬B).

Then the Sahlqvist completeness theorem says that this

modal logic is complete for its models in modal logic.

By suitable translations t and u between BBI and this

modal logic, we get A valid in BBI ⇒ t(A) valid in modal logic ⇒ t(A) provable in modal logic (Sahlqvist) ⇒ u(t(A)) provable in BBI ⇒ A provable in BBI

9/ 18

SLIDE 41

BBI as a modal logic

A MLBBI formula is built from propositional variables using the classical connectives, constant I and binary modalities ∗ and ⊸.

10/ 18

SLIDE 42

BBI as a modal logic

A MLBBI formula is built from propositional variables using the classical connectives, constant I and binary modalities ∗ and ⊸. Provability in the normal modal logic for MLBBI is given by extending classical propositional logic with the following axioms and rules, where ⊗ ∈ {∗, ⊸}:

10/ 18

SLIDE 43

BBI as a modal logic

A MLBBI formula is built from propositional variables using the classical connectives, constant I and binary modalities ∗ and ⊸. Provability in the normal modal logic for MLBBI is given by extending classical propositional logic with the following axioms and rules, where ⊗ ∈ {∗, ⊸}: ⊥ ⊗ A ⊢ ⊥ and A ⊗ ⊥ ⊢ ⊥ A1 ⊢ A2 B1 ⊢ B2 A1 ⊗ B1 ⊢ A2 ⊗ B2 (A ∨ B) ⊗ C ⊢ (A ⊗ C) ∨ (B ⊗ C) A ⊗ (B ∨ C) ⊢ (A ⊗ B) ∨ (A ⊗ C)

10/ 18

SLIDE 44

BBI as a modal logic (2)

A MLBBI frame is given by W, ◦, ⊸, E, where E ⊆ W and

, ⊸: W × W → P(W) (like in BBI).

11/ 18

SLIDE 45

BBI as a modal logic (2)

A MLBBI frame is given by W, ◦, ⊸, E, where E ⊆ W and

, ⊸: W × W → P(W) (like in BBI).

Now we give the forcing relation w | =ρ A: w | =ρ P ⇔ w ∈ ρ(P)

11/ 18

SLIDE 46

BBI as a modal logic (2)

A MLBBI frame is given by W, ◦, ⊸, E, where E ⊆ W and

, ⊸: W × W → P(W) (like in BBI).

Now we give the forcing relation w | =ρ A: w | =ρ P ⇔ w ∈ ρ(P) w | =ρ A → B ⇔ w | =ρ A implies w | =ρ B

11/ 18

SLIDE 47

BBI as a modal logic (2)

A MLBBI frame is given by W, ◦, ⊸, E, where E ⊆ W and

, ⊸: W × W → P(W) (like in BBI).

Now we give the forcing relation w | =ρ A: w | =ρ P ⇔ w ∈ ρ(P) w | =ρ A → B ⇔ w | =ρ A implies w | =ρ B . . . w | =ρ I ⇔ w ∈ E

11/ 18

SLIDE 48

BBI as a modal logic (2)

A MLBBI frame is given by W, ◦, ⊸, E, where E ⊆ W and

, ⊸: W × W → P(W) (like in BBI).

Now we give the forcing relation w | =ρ A: w | =ρ P ⇔ w ∈ ρ(P) w | =ρ A → B ⇔ w | =ρ A implies w | =ρ B . . . w | =ρ I ⇔ w ∈ E w | =ρ A ∗ B ⇔ w ∈ w1 ◦ w2 and w1 | =ρ A and w2 | =ρ B

11/ 18

SLIDE 49

BBI as a modal logic (2)

A MLBBI frame is given by W, ◦, ⊸, E, where E ⊆ W and

, ⊸: W × W → P(W) (like in BBI).

Now we give the forcing relation w | =ρ A: w | =ρ P ⇔ w ∈ ρ(P) w | =ρ A → B ⇔ w | =ρ A implies w | =ρ B . . . w | =ρ I ⇔ w ∈ E w | =ρ A ∗ B ⇔ w ∈ w1 ◦ w2 and w1 | =ρ A and w2 | =ρ B w | =ρ A ⊸ B ⇔ w ∈ w1 ⊸ w2 and w1 | =ρ A and w2 | =ρ B

11/ 18

SLIDE 50

BBI as a modal logic (2)

A MLBBI frame is given by W, ◦, ⊸, E, where E ⊆ W and

, ⊸: W × W → P(W) (like in BBI).

Now we give the forcing relation w | =ρ A: w | =ρ P ⇔ w ∈ ρ(P) w | =ρ A → B ⇔ w | =ρ A implies w | =ρ B . . . w | =ρ I ⇔ w ∈ E w | =ρ A ∗ B ⇔ w ∈ w1 ◦ w2 and w1 | =ρ A and w2 | =ρ B w | =ρ A ⊸ B ⇔ w ∈ w1 ⊸ w2 and w1 | =ρ A and w2 | =ρ B A is valid in M iff w | =ρ A for all w ∈ W and valuations ρ. Same as BBI, except for — ∗ versus ⊸!

11/ 18

SLIDE 51

Sahlqvist axioms for MLBBI

Define a set ABBI of MLBBI-formulas as follows:

12/ 18

SLIDE 52

Sahlqvist axioms for MLBBI

Define a set ABBI of MLBBI-formulas as follows: (1) A ∧ (B ∗ C) ⊢ (B ∧ (C ⊸ A)) ∗ ⊤ (2) A ∧ (B ⊸ C) ⊢ ⊤ ⊸ (C ∧ (A ∗ B)) (3) A ∗ B ⊢ B ∗ A (4) A ∗ (B ∗ C) ⊢ (A ∗ B) ∗ C (5) A ∗ I ⊢ A (6) A ⊢ A ∗ I

12/ 18

SLIDE 53

Sahlqvist axioms for MLBBI

Define a set ABBI of MLBBI-formulas as follows: (1) A ∧ (B ∗ C) ⊢ (B ∧ (C ⊸ A)) ∗ ⊤ (2) A ∧ (B ⊸ C) ⊢ ⊤ ⊸ (C ∧ (A ∗ B)) (3) A ∗ B ⊢ B ∗ A (4) A ∗ (B ∗ C) ⊢ (A ∗ B) ∗ C (5) A ∗ I ⊢ A (6) A ⊢ A ∗ I These are all of a form called Sahlqvist formulas, and so we have by the Sahlqvist completeness theorem:

12/ 18

SLIDE 54

Sahlqvist axioms for MLBBI

Define a set ABBI of MLBBI-formulas as follows: (1) A ∧ (B ∗ C) ⊢ (B ∧ (C ⊸ A)) ∗ ⊤ (2) A ∧ (B ⊸ C) ⊢ ⊤ ⊸ (C ∧ (A ∗ B)) (3) A ∗ B ⊢ B ∗ A (4) A ∗ (B ∗ C) ⊢ (A ∗ B) ∗ C (5) A ∗ I ⊢ A (6) A ⊢ A ∗ I These are all of a form called Sahlqvist formulas, and so we have by the Sahlqvist completeness theorem: Theorem (Sahlqvist) If B is valid in the MLBBI frames satisfying ABBI, then it is provable in MLBBI + ABBI.

12/ 18

SLIDE 55

Modal frames are BBI-models

Lemma (1) Let M = W, ◦, ⊸, E be a modal frame satisfying axioms (1) and (2) of ABBI. Then we have, for any w, w1, w2 ∈ W: w ∈ w1 ⊸ w2 ⇔ w2 ∈ w ◦ w1. Proof.

13/ 18

SLIDE 56

Modal frames are BBI-models

Lemma (1) Let M = W, ◦, ⊸, E be a modal frame satisfying axioms (1) and (2) of ABBI. Then we have, for any w, w1, w2 ∈ W: w ∈ w1 ⊸ w2 ⇔ w2 ∈ w ◦ w1. Proof. Hint: (⇐) uses axiom (1), (⇒) uses axiom 2.

13/ 18

SLIDE 57

Modal frames are BBI-models

Lemma (1) Let M = W, ◦, ⊸, E be a modal frame satisfying axioms (1) and (2) of ABBI. Then we have, for any w, w1, w2 ∈ W: w ∈ w1 ⊸ w2 ⇔ w2 ∈ w ◦ w1. Proof. Hint: (⇐) uses axiom (1), (⇒) uses axiom 2. So, when axioms (1) and (2) are satisfied, Lemma 1 gives us: w | =ρ A ⊸ B ⇔ w ∈ w1 ⊸ w2 and w1 | =ρ A and w2 | =ρ B

13/ 18

SLIDE 58

Modal frames are BBI-models

Lemma (1) Let M = W, ◦, ⊸, E be a modal frame satisfying axioms (1) and (2) of ABBI. Then we have, for any w, w1, w2 ∈ W: w ∈ w1 ⊸ w2 ⇔ w2 ∈ w ◦ w1. Proof. Hint: (⇐) uses axiom (1), (⇒) uses axiom 2. So, when axioms (1) and (2) are satisfied, Lemma 1 gives us: w | =ρ A ⊸ B ⇔ w ∈ w1 ⊸ w2 and w1 | =ρ A and w2 | =ρ B ⇔ w2 ∈ w ◦ w1 and w1 | =ρ A and w2 | =ρ B

13/ 18

SLIDE 59

Modal frames are BBI-models

Lemma (1) Let M = W, ◦, ⊸, E be a modal frame satisfying axioms (1) and (2) of ABBI. Then we have, for any w, w1, w2 ∈ W: w ∈ w1 ⊸ w2 ⇔ w2 ∈ w ◦ w1. Proof. Hint: (⇐) uses axiom (1), (⇒) uses axiom 2. So, when axioms (1) and (2) are satisfied, Lemma 1 gives us: w | =ρ A ⊸ B ⇔ w ∈ w1 ⊸ w2 and w1 | =ρ A and w2 | =ρ B ⇔ w2 ∈ w ◦ w1 and w1 | =ρ A and w2 | =ρ B ⇔ w | =ρ ¬(A — ∗ ¬B)

13/ 18

SLIDE 60

Translating between BBI and MLBBI

Given a BBI-formula A, write t(A) for the MLBBI formula
btained by replacing every formula of the form B —

∗ C by ¬(B ⊸ ¬C).

14/ 18

SLIDE 61

Translating between BBI and MLBBI

Given a BBI-formula A, write t(A) for the MLBBI formula
btained by replacing every formula of the form B —

∗ C by ¬(B ⊸ ¬C).

Conversely, given MLBBI formula A, write u(A) for the

BBI-formula obtained by replacing every formula of the form B ⊸ C by ¬(B — ∗ ¬C).

14/ 18

SLIDE 62

Translating between BBI and MLBBI

Given a BBI-formula A, write t(A) for the MLBBI formula
btained by replacing every formula of the form B —

∗ C by ¬(B ⊸ ¬C).

Conversely, given MLBBI formula A, write u(A) for the

BBI-formula obtained by replacing every formula of the form B ⊸ C by ¬(B — ∗ ¬C). Lemma (2) If u(t(A)) is provable in BBI then so is A.

14/ 18

SLIDE 63

Translating between BBI and MLBBI

Given a BBI-formula A, write t(A) for the MLBBI formula
btained by replacing every formula of the form B —

∗ C by ¬(B ⊸ ¬C).

Conversely, given MLBBI formula A, write u(A) for the

BBI-formula obtained by replacing every formula of the form B ⊸ C by ¬(B — ∗ ¬C). Lemma (2) If u(t(A)) is provable in BBI then so is A. Proof. Structural induction on A.

14/ 18

SLIDE 64

Validity translation lemma

Lemma (3) Let M = W, ◦, ⊸, E be a MLBBI frame satisfying axioms (3)–(6) of ABBI. Then W, ◦, E is a BBI-model.

15/ 18

SLIDE 65

Validity translation lemma

Lemma (3) Let M = W, ◦, ⊸, E be a MLBBI frame satisfying axioms (3)–(6) of ABBI. Then W, ◦, E is a BBI-model. Proof. Easy exercise!

15/ 18

SLIDE 66

Validity translation lemma

Lemma (3) Let M = W, ◦, ⊸, E be a MLBBI frame satisfying axioms (3)–(6) of ABBI. Then W, ◦, E is a BBI-model. Proof. Easy exercise! Lemma (4) If A is valid in BBI, then t(A) is valid in every MLBBI frame satisfying ABBI.

15/ 18

SLIDE 67

Validity translation lemma

Lemma (3) Let M = W, ◦, ⊸, E be a MLBBI frame satisfying axioms (3)–(6) of ABBI. Then W, ◦, E is a BBI-model. Proof. Easy exercise! Lemma (4) If A is valid in BBI, then t(A) is valid in every MLBBI frame satisfying ABBI. Proof. Uses Lemmas 1 and 3.

15/ 18

SLIDE 68

Proof translation lemma

Lemma (5) If B is provable in MLBBI + ABBI, then u(B) is provable in BBI.

16/ 18

SLIDE 69

Proof translation lemma

Lemma (5) If B is provable in MLBBI + ABBI, then u(B) is provable in BBI. Proof. By induction on the proof of B in MLBBI + ABBI. We have to show that every proof rule in MLBBI + ABBI is derivable in BBI under the translation u(−).

16/ 18

SLIDE 70

Proof of completeness

Theorem If A is BBI-valid then it is BBI-provable. Proof. Let A be BBI-valid.

17/ 18

SLIDE 71

Proof of completeness

Theorem If A is BBI-valid then it is BBI-provable. Proof. Let A be BBI-valid. By Lemma 4, t(A) is valid in the class of MLBBI frames satisfying ABBI.

17/ 18

SLIDE 72

Proof of completeness

Theorem If A is BBI-valid then it is BBI-provable. Proof. Let A be BBI-valid. By Lemma 4, t(A) is valid in the class of MLBBI frames satisfying ABBI. By the Sahlqvist Theorem, t(A) is provable in MLBBI + ABBI.

17/ 18

SLIDE 73

Proof of completeness

Theorem If A is BBI-valid then it is BBI-provable. Proof. Let A be BBI-valid. By Lemma 4, t(A) is valid in the class of MLBBI frames satisfying ABBI. By the Sahlqvist Theorem, t(A) is provable in MLBBI + ABBI. By Lemma 5, u(t(A)) is provable in BBI.

17/ 18

SLIDE 74

Proof of completeness

Theorem If A is BBI-valid then it is BBI-provable. Proof. Let A be BBI-valid. By Lemma 4, t(A) is valid in the class of MLBBI frames satisfying ABBI. By the Sahlqvist Theorem, t(A) is provable in MLBBI + ABBI. By Lemma 5, u(t(A)) is provable in BBI. Finally, by Lemma 2, A is provable in BBI.

17/ 18

SLIDE 75

Proof of completeness

Theorem If A is BBI-valid then it is BBI-provable. Proof. Let A be BBI-valid. By Lemma 4, t(A) is valid in the class of MLBBI frames satisfying ABBI. By the Sahlqvist Theorem, t(A) is provable in MLBBI + ABBI. By Lemma 5, u(t(A)) is provable in BBI. Finally, by Lemma 2, A is provable in BBI. Exercise: fill in the proofs of Lemmas 1–5!

17/ 18

SLIDE 76