Download Definability in Boolean bunched logic

Definability in Boolean bunched logic
James Brotherston
Programming Principles, Logic and Verification Group
Dept. of Computer Science
University College London, UK
[email protected]
Logic Summer School, ANU, 11 December 2015
1/ 20
Introduction
• In mathematical logic, there is usually a trade-off between
expressivity and complexity of a logical language:
2/ 20
Introduction
• In mathematical logic, there is usually a trade-off between
expressivity and complexity of a logical language:
• weaker languages cannot capture interesting properties,
2/ 20
Introduction
• In mathematical logic, there is usually a trade-off between
expressivity and complexity of a logical language:
• weaker languages cannot capture interesting properties, but
• richer languages have higher complexity, may lack sensible
proof theories and may be unavoidably incomplete (cf.
Gödel).
2/ 20
Introduction
• In mathematical logic, there is usually a trade-off between
expressivity and complexity of a logical language:
• weaker languages cannot capture interesting properties, but
• richer languages have higher complexity, may lack sensible
proof theories and may be unavoidably incomplete (cf.
Gödel).
• Incompleteness manifests as a gap between two key
concepts:
2/ 20
Introduction
• In mathematical logic, there is usually a trade-off between
expressivity and complexity of a logical language:
• weaker languages cannot capture interesting properties, but
• richer languages have higher complexity, may lack sensible
proof theories and may be unavoidably incomplete (cf.
Gödel).
• Incompleteness manifests as a gap between two key
concepts:
• provability in some formal system for the logic
(which corresponds to validity in some class of models);
2/ 20
Introduction
• In mathematical logic, there is usually a trade-off between
expressivity and complexity of a logical language:
• weaker languages cannot capture interesting properties, but
• richer languages have higher complexity, may lack sensible
proof theories and may be unavoidably incomplete (cf.
Gödel).
• Incompleteness manifests as a gap between two key
concepts:
• provability in some formal system for the logic
(which corresponds to validity in some class of models); and
• validity in a (class of) intended model(s) of the logic.
2/ 20
Introduction (contd.)
Thus, given a logical language L, and an intended class C of
models for that language, there are at least two natural
questions:
3/ 20
Introduction (contd.)
Thus, given a logical language L, and an intended class C of
models for that language, there are at least two natural
questions:
1. Is the class C finitely axiomatisable, a.k.a. definable in L?
3/ 20
Introduction (contd.)
Thus, given a logical language L, and an intended class C of
models for that language, there are at least two natural
questions:
1. Is the class C finitely axiomatisable, a.k.a. definable in L?
2. Is there a complete proof system for L w.r.t. validity in C?
3/ 20
Introduction (contd.)
Thus, given a logical language L, and an intended class C of
models for that language, there are at least two natural
questions:
1. Is the class C finitely axiomatisable, a.k.a. definable in L?
2. Is there a complete proof system for L w.r.t. validity in C?
In the case of BBI, we are often interested in properties of the
heap models used in separation logic.
3/ 20
BBI, proof-theoretically
Recall:
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 ` B 1 A2 ` B 2
A∗B `C
A`B−
−∗ C
A1 ∗ A2 ` B1 ∗ B2
A`B−
−∗ C
A∗B `C
4/ 20
BBI, semantically (1)
Recall:
A BBI-model is given by hW, ◦, Ei, 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
def S
W1 ◦ W2 = w1 ∈W1 ,w2 ∈W2 w1 ◦ w2
• ◦ is commutative and associative;
• the set of units E ⊆ W satisfies w ◦ E = {w} for all w ∈ W .
A valuation for BBI-model M = hW, ◦, Ei is a function ρ from
propositional variables to P(W ).
5/ 20
BBI, semantically (2)
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 ⇔ ∀w0 , w00 ∈ W. if w00 ∈ w ◦ w0 and w0 |=ρ A
then w00 |=ρ B
A is valid in M iff w |=ρ A for all ρ and w ∈ W .
6/ 20
Definable properties
A property P of BBI-models is said to be definable if there
exists a formula A such that for all BBI-models M ,
A is valid in M ⇐⇒ M ∈ P.
7/ 20
Definable properties
A property P of BBI-models is said to be definable if there
exists a formula A such that for all BBI-models M ,
A is valid in M ⇐⇒ M ∈ P.
We’ll consider properties that feature in various models of
separation logic.
7/ 20
Definable properties
A property P of BBI-models is said to be definable if there
exists a formula A such that for all BBI-models M ,
A is valid in M ⇐⇒ M ∈ P.
We’ll consider properties that feature in various models of
separation logic.
To show a property is definable, just exhibit the defining
formula!
7/ 20
Definable properties
A property P of BBI-models is said to be definable if there
exists a formula A such that for all BBI-models M ,
A is valid in M ⇐⇒ M ∈ P.
We’ll consider properties that feature in various models of
separation logic.
To show a property is definable, just exhibit the defining
formula!
To show a property is not definable, we show it is not preserved
by some validity-preserving model construction.
7/ 20
Properties of (some) BBI-models
Partial functionality: w, w0 ∈ w1 ◦ w2 implies w = w0 ;
8/ 20
Properties of (some) BBI-models
Partial functionality: w, w0 ∈ w1 ◦ w2 implies w = w0 ;
Cancellativity: (w ◦ w1 ) ∩ (w ◦ w2 ) 6= ∅ implies w1 = w2 ;
8/ 20
Properties of (some) BBI-models
Partial functionality: w, w0 ∈ w1 ◦ w2 implies w = w0 ;
Cancellativity: (w ◦ w1 ) ∩ (w ◦ w2 ) 6= ∅ implies w1 = w2 ;
Single unit: w, w0 ∈ E implies w = w0 ;
8/ 20
Properties of (some) BBI-models
Partial functionality: w, w0 ∈ w1 ◦ w2 implies w = w0 ;
Cancellativity: (w ◦ w1 ) ∩ (w ◦ w2 ) 6= ∅ implies w1 = w2 ;
Single unit: w, w0 ∈ E implies w = w0 ;
Indivisible units: (w ◦ w0 ) ∩ E 6= ∅ implies w ∈ E;
8/ 20
Properties of (some) BBI-models
Partial functionality: w, w0 ∈ w1 ◦ w2 implies w = w0 ;
Cancellativity: (w ◦ w1 ) ∩ (w ◦ w2 ) 6= ∅ implies w1 = w2 ;
Single unit: w, w0 ∈ E implies w = w0 ;
Indivisible units: (w ◦ w0 ) ∩ E 6= ∅ implies w ∈ E;
Disjointness: w ◦ w 6= ∅ implies w ∈ E;
8/ 20
Properties of (some) BBI-models
Partial functionality: w, w0 ∈ w1 ◦ w2 implies w = w0 ;
Cancellativity: (w ◦ w1 ) ∩ (w ◦ w2 ) 6= ∅ implies w1 = w2 ;
Single unit: w, w0 ∈ E implies w = w0 ;
Indivisible units: (w ◦ w0 ) ∩ E 6= ∅ implies w ∈ E;
Disjointness: w ◦ w 6= ∅ implies w ∈ E;
Divisibility: for every w 6∈ E there are w1 , w2 ∈
/ E such that
w ∈ w1 ◦ w2 ;
8/ 20
Properties of (some) BBI-models
Partial functionality: w, w0 ∈ w1 ◦ w2 implies w = w0 ;
Cancellativity: (w ◦ w1 ) ∩ (w ◦ w2 ) 6= ∅ implies w1 = w2 ;
Single unit: w, w0 ∈ E implies w = w0 ;
Indivisible units: (w ◦ w0 ) ∩ E 6= ∅ implies w ∈ E;
Disjointness: w ◦ w 6= ∅ implies w ∈ E;
Divisibility: for every w 6∈ E there are w1 , w2 ∈
/ E such that
w ∈ w1 ◦ w2 ;
Cross-split property: whenever (a ◦ b) ∩ (c ◦ d) 6= ∅, there exist
ac, ad , bc, bd such that a ∈ ac ◦ ad , b ∈ bc ◦ bd ,
c ∈ ac ◦ bc and d ∈ ad ◦ bd .
a b
ac bc
ad bd
c
d
8/ 20
Two definable properties
Proposition
The following two properties are BBI-definable:
Indivisible units:
(w ◦ w0 ) ∩ E 6= ∅ implies w ∈ E
I ∧ (A ∗ B) ` A
Divisibility:
∀w 6∈ E.∃w1 , w2 ∈
/ E such that w ∈ w1 ◦ w2
¬I ` ¬I ∗ ¬I
9/ 20
Disjoint unions of BBI-models
Definition
If M1 = hW1 , ◦1 , E1 i and M2 = hW2 , ◦2 , E2 i are BBI-models and
W1 , W2 are disjoint then their disjoint union is given by
def
M1 ] M2 = hW1 ∪ W2 , ◦1 ∪ ◦2 , E1 ∪ E2 i
(where ◦1 ∪ ◦2 is lifted to W1 ∪ W2 in the obvious way)
10/ 20
Disjoint unions of BBI-models
Definition
If M1 = hW1 , ◦1 , E1 i and M2 = hW2 , ◦2 , E2 i are BBI-models and
W1 , W2 are disjoint then their disjoint union is given by
def
M1 ] M2 = hW1 ∪ W2 , ◦1 ∪ ◦2 , E1 ∪ E2 i
(where ◦1 ∪ ◦2 is lifted to W1 ∪ W2 in the obvious way)
Proposition
If A is valid in M1 and in M2 , and M1 ] M2 is defined, then it
is also valid in M1 ] M2 .
10/ 20
Disjoint unions of BBI-models
Definition
If M1 = hW1 , ◦1 , E1 i and M2 = hW2 , ◦2 , E2 i are BBI-models and
W1 , W2 are disjoint then their disjoint union is given by
def
M1 ] M2 = hW1 ∪ W2 , ◦1 ∪ ◦2 , E1 ∪ E2 i
(where ◦1 ∪ ◦2 is lifted to W1 ∪ W2 in the obvious way)
Proposition
If A is valid in M1 and in M2 , and M1 ] M2 is defined, then it
is also valid in M1 ] M2 .
Proof. Structural induction on A.
10/ 20
Undefinability of single-unit property
Lemma
Suppose that there exist BBI-models M1 and M2 such that
M1 , M2 ∈ P but M1 ] M2 6∈ P. Then P is not BBI-definable.
11/ 20
Undefinability of single-unit property
Lemma
Suppose that there exist BBI-models M1 and M2 such that
M1 , M2 ∈ P but M1 ] M2 6∈ P. Then P is not BBI-definable.
Proof. If P were definable via A say, then A would be true in
M1 and M2 but not in M1 ] M2 , contradicting previous
Proposition.
11/ 20
Undefinability of single-unit property
Lemma
Suppose that there exist BBI-models M1 and M2 such that
M1 , M2 ∈ P but M1 ] M2 6∈ P. Then P is not BBI-definable.
Proof. If P were definable via A say, then A would be true in
M1 and M2 but not in M1 ] M2 , contradicting previous
Proposition.
Theorem
The single unit property is not BBI-definable.
11/ 20
Undefinability of single-unit property
Lemma
Suppose that there exist BBI-models M1 and M2 such that
M1 , M2 ∈ P but M1 ] M2 6∈ P. Then P is not BBI-definable.
Proof. If P were definable via A say, then A would be true in
M1 and M2 but not in M1 ] M2 , 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.
11/ 20
Bounded morphisms on BBI-models
Definition
Let M = hW, ◦, Ei and M 0 = hW 0 , ◦0 , E 0 i be BBI-models.
12/ 20
Bounded morphisms on BBI-models
Definition
Let M = hW, ◦, Ei and M 0 = hW 0 , ◦0 , E 0 i be BBI-models. A
bounded morphism from M to M 0 is a function f : W → W 0 s.t.:
1. w ∈ E iff f (w) ∈ E 0 ;
12/ 20
Bounded morphisms on BBI-models
Definition
Let M = hW, ◦, Ei and M 0 = hW 0 , ◦0 , E 0 i be BBI-models. A
bounded morphism from M to M 0 is a function f : W → W 0 s.t.:
1. w ∈ E iff f (w) ∈ E 0 ;
2. w ∈ w1 ◦ w2 implies f (w) ∈ f (w1 ) ◦0 f (w2 );
12/ 20
Bounded morphisms on BBI-models
Definition
Let M = hW, ◦, Ei and M 0 = hW 0 , ◦0 , E 0 i be BBI-models. A
bounded morphism from M to M 0 is a function f : W → W 0 s.t.:
1. w ∈ E iff f (w) ∈ E 0 ;
2. w ∈ w1 ◦ w2 implies f (w) ∈ f (w1 ) ◦0 f (w2 );
3. f (w) ∈ w10 ◦0 w20 implies ∃w1 , w2 ∈ W. w ∈ w1 ◦ w2 and
f (w1 ) = w10 and f (w2 ) = w20 ;
12/ 20
Bounded morphisms on BBI-models
Definition
Let M = hW, ◦, Ei and M 0 = hW 0 , ◦0 , E 0 i be BBI-models. A
bounded morphism from M to M 0 is a function f : W → W 0 s.t.:
1. w ∈ E iff f (w) ∈ E 0 ;
2. w ∈ w1 ◦ w2 implies f (w) ∈ f (w1 ) ◦0 f (w2 );
3. f (w) ∈ w10 ◦0 w20 implies ∃w1 , w2 ∈ W. w ∈ w1 ◦ w2 and
f (w1 ) = w10 and f (w2 ) = w20 ;
4. w20 ∈ f (w) ◦0 w10 implies ∃w1 , w2 ∈ W. w2 ∈ w ◦ w1 and
f (w1 ) = w10 and f (w2 ) = w20 .
12/ 20
Bounded morphisms on BBI-models
Definition
Let M = hW, ◦, Ei and M 0 = hW 0 , ◦0 , E 0 i be BBI-models. A
bounded morphism from M to M 0 is a function f : W → W 0 s.t.:
1. w ∈ E iff f (w) ∈ E 0 ;
2. w ∈ w1 ◦ w2 implies f (w) ∈ f (w1 ) ◦0 f (w2 );
3. f (w) ∈ w10 ◦0 w20 implies ∃w1 , w2 ∈ W. w ∈ w1 ◦ w2 and
f (w1 ) = w10 and f (w2 ) = w20 ;
4. w20 ∈ f (w) ◦0 w10 implies ∃w1 , w2 ∈ W. w2 ∈ w ◦ w1 and
f (w1 ) = w10 and f (w2 ) = w20 .
Proposition
Suppose there is a surjective bounded morphism from M to M 0 .
Then any formula valid in M is also valid in M 0 .
12/ 20
Undefinability via bounded morphisms
Lemma
Suppose there are BBI-models M and M 0 s.t. there is a
surjective bounded morphism from M to M 0 , and M ∈ P while
M0 ∈
/ P. Then P is not BBI-definable.
13/ 20
Undefinability via bounded morphisms
Lemma
Suppose there are BBI-models M and M 0 s.t. there is a
surjective bounded morphism from M to M 0 , and M ∈ P while
M0 ∈
/ P. Then P is not BBI-definable.
Proof. If P were definable via A say, then A would be true in
M but not in M 0 , contradicting previous Proposition.
13/ 20
Undefinability via bounded morphisms
Lemma
Suppose there are BBI-models M and M 0 s.t. there is a
surjective bounded morphism from M to M 0 , and M ∈ P while
M0 ∈
/ P. Then P is not BBI-definable.
Proof. If P were definable via A say, then A would be true in
M but not in M 0 , contradicting previous Proposition.
Theorem
None of the following properties is BBI-definable: (a) partial
functionality; (b) cancellativity; (c) disjointness.
13/ 20
Undefinability via bounded morphisms
Lemma
Suppose there are BBI-models M and M 0 s.t. there is a
surjective bounded morphism from M to M 0 , and M ∈ P while
M0 ∈
/ P. Then P is not BBI-definable.
Proof. If P were definable via A say, then A would be true in
M but not in M 0 , contradicting previous Proposition.
Theorem
None of the following properties is BBI-definable: (a) partial
functionality; (b) cancellativity; (c) disjointness.
Proof. In each case we build models M and M 0 such that there
is a bounded morphism from M to M 0 , but M has the property
while M 0 doesn’t.
13/ 20
Example:partial functionality
Define BBI-models M = hW, ◦, Ei and M 0 = hW 0 , ◦0 , E 0 i by
W = {e, v1 , v2 , x1 , x2 , y, z}
E = {e}
w ◦ e = e ◦ w = {w} for all w ∈ W
x1 ◦ v1 = v1 ◦ x1 = {y}
x1 ◦ v2 = v2 ◦ x1 = {y}
x2 ◦ v1 = v1 ◦ x2 = {z}
x2 ◦ v2 = v2 ◦ x2 = {z}
14/ 20
Example:partial functionality
Define BBI-models M = hW, ◦, Ei and M 0 = hW 0 , ◦0 , E 0 i by
W = {e, v1 , v2 , x1 , x2 , y, z}
E = {e}
w ◦ e = e ◦ w = {w} for all w ∈ W
x1 ◦ v1 = v1 ◦ x1 = {y}
x1 ◦ v2 = v2 ◦ x1 = {y}
x2 ◦ v1 = v1 ◦ x2 = {z}
x2 ◦ v2 = v2 ◦ x2 = {z}
W 0 = {e, v, x, y, z}
E 0 = {e}
w ◦0 e = e ◦0 w = {w} for all w ∈ W 0
x ◦0 v = v ◦0 x = {y, z}
14/ 20
Example:partial functionality
Define BBI-models M = hW, ◦, Ei and M 0 = hW 0 , ◦0 , E 0 i by
W = {e, v1 , v2 , x1 , x2 , y, z}
E = {e}
w ◦ e = e ◦ w = {w} for all w ∈ W
x1 ◦ v1 = v1 ◦ x1 = {y}
x1 ◦ v2 = v2 ◦ x1 = {y}
x2 ◦ v1 = v1 ◦ x2 = {z}
x2 ◦ v2 = v2 ◦ x2 = {z}
W 0 = {e, v, x, y, z}
E 0 = {e}
w ◦0 e = e ◦0 w = {w} for all w ∈ W 0
x ◦0 v = v ◦0 x = {y, z}
Easy to check M, M 0 are both BBI-models, and M is partial
functional but M 0 is not.
14/ 20
Example:partial functionality
Define BBI-models M = hW, ◦, Ei and M 0 = hW 0 , ◦0 , E 0 i by
W = {e, v1 , v2 , x1 , x2 , y, z}
E = {e}
w ◦ e = e ◦ w = {w} for all w ∈ W
x1 ◦ v1 = v1 ◦ x1 = {y}
x1 ◦ v2 = v2 ◦ x1 = {y}
x2 ◦ v1 = v1 ◦ x2 = {z}
x2 ◦ v2 = v2 ◦ x2 = {z}
W 0 = {e, v, x, y, z}
E 0 = {e}
w ◦0 e = e ◦0 w = {w} for all w ∈ W 0
x ◦0 v = v ◦0 x = {y, z}
Easy to check M, M 0 are both BBI-models, and M is partial
functional but M 0 is not. Our surjective morphism is:
f (v1 ) = f (v2 ) = v f (x1 ) = f (x2 ) = x
f (w) = w (w ∈ {e, y, z})
14/ 20
HyBBI: a hybrid extension of BBI
• So, BBI cannot define some natural properties.
15/ 20
HyBBI: a hybrid extension of BBI
• So, BBI cannot define some natural properties.
• Idea: conservatively increase the expressivity of BBI, using
machinery of hybrid logic.
15/ 20
HyBBI: a hybrid extension of BBI
• So, BBI cannot define some natural properties.
• Idea: conservatively increase the expressivity of BBI, using
machinery of hybrid logic.
• HyBBI extends the language of BBI by: any nominal ` is a
formula, and so is any formula of the form @` A.
15/ 20
HyBBI: a hybrid extension of BBI
• So, BBI cannot define some natural properties.
• Idea: conservatively increase the expressivity of BBI, using
machinery of hybrid logic.
• HyBBI extends the language of BBI by: any nominal ` is a
formula, and so is any formula of the form @` A.
• Valuations interpret nominals as individual worlds in a
BBI-model.
15/ 20
HyBBI: a hybrid extension of BBI
• So, BBI cannot define some natural properties.
• Idea: conservatively increase the expressivity of BBI, using
machinery of hybrid logic.
• HyBBI extends the language of BBI by: any nominal ` is a
formula, and so is any formula of the form @` A.
• Valuations interpret nominals as individual worlds in a
BBI-model.
• We extend the forcing relation by:
M, w |=ρ `
⇔
15/ 20
w = ρ(`)
HyBBI: a hybrid extension of BBI
• So, BBI cannot define some natural properties.
• Idea: conservatively increase the expressivity of BBI, using
machinery of hybrid logic.
• HyBBI extends the language of BBI by: any nominal ` is a
formula, and so is any formula of the form @` A.
• Valuations interpret nominals as individual worlds in a
BBI-model.
• We extend the forcing relation by:
M, w |=ρ `
M, w |=ρ @` A
⇔
⇔
15/ 20
w = ρ(`)
M, ρ(`) |=ρ A
HyBBI: a hybrid extension of BBI
• So, BBI cannot define some natural properties.
• Idea: conservatively increase the expressivity of BBI, using
machinery of hybrid logic.
• HyBBI extends the language of BBI by: any nominal ` is a
formula, and so is any formula of the form @` A.
• Valuations interpret nominals as individual worlds in a
BBI-model.
• We extend the forcing relation by:
M, w |=ρ `
M, w |=ρ @` A
⇔
⇔
w = ρ(`)
M, ρ(`) |=ρ A
Easy to see that HyBBI is a conservative extension of BBI.
15/ 20
Definable properties in HyBBI
Theorem
The following properties are HyBBI-definable:
Functionality:
@` (j ∗ k) ∧ @`0 (j ∗ k) ` @` `0
16/ 20
Definable properties in HyBBI
Theorem
The following properties are HyBBI-definable:
Functionality: @` (j ∗ k) ∧ @`0 (j ∗ k) ` @` `0
Cancellativity:
` ∗ j ∧ ` ∗ k ` @j k
16/ 20
Definable properties in HyBBI
Theorem
The following properties are HyBBI-definable:
Functionality: @` (j ∗ k) ∧ @`0 (j ∗ k) ` @` `0
Cancellativity:
` ∗ j ∧ ` ∗ k ` @j k
Single unit:
@`1 I ∧ @`2 I ` @`1 `2
16/ 20
Definable properties in HyBBI
Theorem
The following properties are HyBBI-definable:
Functionality: @` (j ∗ k) ∧ @`0 (j ∗ k) ` @` `0
Cancellativity:
` ∗ j ∧ ` ∗ k ` @j k
Single unit:
@`1 I ∧ @`2 I ` @`1 `2
Disjointness:
`∗``I∧`
16/ 20
Definable properties in HyBBI
Theorem
The following properties are HyBBI-definable:
Functionality: @` (j ∗ k) ∧ @`0 (j ∗ k) ` @` `0
Cancellativity:
` ∗ j ∧ ` ∗ k ` @j k
Single unit:
@`1 I ∧ @`2 I ` @`1 `2
Disjointness:
`∗``I∧`
Proof.
Easy verifications!
16/ 20
A word about cross-split
We have brushed over the cross-split property:
(a ◦ b) ∩ (c ◦ d) 6= ∅, implies ∃ac, ad , bc, bd with
a ∈ ac ◦ ad , b ∈ bc ◦ bd , c ∈ ac ◦ bc, d ∈ ad ◦ bd .
a b
c
d
ac bc
ad bd
17/ 20
A word about cross-split
We have brushed over the cross-split property:
(a ◦ b) ∩ (c ◦ d) 6= ∅, implies ∃ac, ad , bc, bd with
a ∈ ac ◦ ad , b ∈ bc ◦ bd , c ∈ ac ◦ bc, d ∈ ad ◦ bd .
a b
c
d
ac bc
ad bd
We conjecture this is not definable in BBI or in HyBBI.
17/ 20
A word about cross-split
We have brushed over the cross-split property:
(a ◦ b) ∩ (c ◦ d) 6= ∅, implies ∃ac, ad , bc, bd with
a ∈ ac ◦ ad , b ∈ bc ◦ bd , c ∈ ac ◦ bc, d ∈ ad ◦ bd .
a b
c
d
ac bc
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 ↓`. A
⇔
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M, w |=ρ[`:=w] A
A word about cross-split
We have brushed over the cross-split property:
(a ◦ b) ∩ (c ◦ d) 6= ∅, implies ∃ac, ad , bc, bd with
a ∈ ac ◦ ad , b ∈ bc ◦ bd , c ∈ ac ◦ bc, d ∈ ad ◦ bd .
a b
c
d
ac bc
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 ↓`. A
⇔
M, w |=ρ[`:=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 )))))
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Statement of completeness
We can write down a (quite complex) Hilbert-style proof system
for HyBBI by adding rules for the hybrid operators. Soundness
is easy, as usual.
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Statement of completeness
We can write down a (quite complex) Hilbert-style proof system
for HyBBI by adding rules for the hybrid operators. Soundness
is easy, as usual.
Following an approach based on a Lindenbaum construction
using maximal consistent sets we obtain the following
completeness result:
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Statement of completeness
We can write down a (quite complex) Hilbert-style proof system
for HyBBI by adding rules for the hybrid operators. Soundness
is easy, as usual.
Following an approach based on a Lindenbaum construction
using maximal consistent sets we obtain the following
completeness result:
Theorem (Completeness)
Let Ax be a set of axioms not containing any propositional
variables (nominals are OK).
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Statement of completeness
We can write down a (quite complex) Hilbert-style proof system
for HyBBI by adding rules for the hybrid operators. Soundness
is easy, as usual.
Following an approach based on a Lindenbaum construction
using maximal consistent sets we obtain the following
completeness result:
Theorem (Completeness)
Let Ax be a set of axioms not containing any propositional
variables (nominals are OK).
Suppose that A is valid in the class of BBI-models satisfying Ax.
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Statement of completeness
We can write down a (quite complex) Hilbert-style proof system
for HyBBI by adding rules for the hybrid operators. Soundness
is easy, as usual.
Following an approach based on a Lindenbaum construction
using maximal consistent sets we obtain the following
completeness result:
Theorem (Completeness)
Let Ax be a set of axioms not containing any propositional
variables (nominals are OK).
Suppose that A is valid in the class of BBI-models satisfying Ax.
Then A is provable in the Hilbert system for HyBBI, extended
with Ax.
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Conclusions and future work
• BBI is insufficiently expressive to capture important classes
of models.
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Conclusions and future work
• BBI is insufficiently expressive to capture important classes
of models.
• We can gain this expressivity by deploying naming
machinery from hybrid logic.
19/ 20
Conclusions and future work
• BBI is insufficiently expressive to capture important classes
of models.
• We can gain this expressivity by deploying naming
machinery from hybrid logic.
• In HyBBI, we have parametric completeness for any set of
axioms expressed as pure formulas.
19/ 20
Conclusions and future work
• BBI is insufficiently expressive to capture important classes
of models.
• We can gain this expressivity by deploying naming
machinery from hybrid logic.
• In HyBBI, we have parametric completeness for any set of
axioms expressed as pure formulas.
• In particular, this yields complete proof systems for
previously undefinable classes of BBI-models.
19/ 20
Conclusions and future work
• BBI is insufficiently expressive to capture important classes
of models.
• We can gain this expressivity by deploying naming
machinery from hybrid logic.
• In HyBBI, we have parametric completeness for any set of
axioms expressed as pure formulas.
• In particular, this yields complete proof systems for
previously undefinable classes of BBI-models.
• Future work on our hybrid logics could include
19/ 20
Conclusions and future work
• BBI is insufficiently expressive to capture important classes
of models.
• We can gain this expressivity by deploying naming
machinery from hybrid logic.
• In HyBBI, we have parametric completeness for any set of
axioms expressed as pure formulas.
• In particular, this yields complete proof systems for
previously undefinable classes of BBI-models.
• Future work on our hybrid logics could include
• identification of decidable fragments;
19/ 20
Conclusions and future work
• BBI is insufficiently expressive to capture important classes
of models.
• We can gain this expressivity by deploying naming
machinery from hybrid logic.
• In HyBBI, we have parametric completeness for any set of
axioms expressed as pure formulas.
• In particular, this yields complete proof systems for
previously undefinable classes of BBI-models.
• Future work on our hybrid logics could include
• identification of decidable fragments;
• search for nice structural proof theories;
19/ 20
Conclusions and future work
• BBI is insufficiently expressive to capture important classes
of models.
• We can gain this expressivity by deploying naming
machinery from hybrid logic.
• In HyBBI, we have parametric completeness for any set of
axioms expressed as pure formulas.
• In particular, this yields complete proof systems for
previously undefinable classes of BBI-models.
• Future work on our hybrid logics could include
• identification of decidable fragments;
• search for nice structural proof theories;
• investigate possible applications to program analysis.
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Further reading
J. Brotherston and J. Villard.
Parametric completeness for separation theories.
In Proc. POPL-41. ACM, 2014.
P. Blackburn, M. de Rijke and Y. Venema.
Modal Logic.
Cambridge University Press, 2001.
Z. Hóu, R. Clouston, R. Goré and A. Tiu.
Proof search for propositional abstract separation logics via labelled
sequents.
In Proc. POPL-41. ACM, 2014.
D. Larchey-Wendling and D. Galmiche.
Exploring the relation between intuitionistic BI and Boolean BI: an
unexpected embedding.
In Math. Struct. in Comp. Sci., vol. 19. Cambridge University Press,
2009.
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