What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Stone Duality for Separation Logic Part 1: A Stone Duality Primer
Simon Docherty
University College London
Thursday 8th June 2017
1 / 33 Algebra, Logic and Topology
Stone Duality
Duality in Logic and Computer Science
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Outline
What Is Duality?
2 / 33 Stone Duality
Duality in Logic and Computer Science
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Outline
What Is Duality?
Algebra, Logic and Topology
2 / 33 Duality in Logic and Computer Science
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Outline
What Is Duality?
Algebra, Logic and Topology
Stone Duality
2 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Outline
What Is Duality?
Algebra, Logic and Topology
Stone Duality
Duality in Logic and Computer Science
2 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
What Is Duality?
3 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
A Really Informal Description of Duality
4 / 33 2. Every structure preserving map of type A can be systematically transformed into a structure preserving map of type B in the other direction. 3. These transformations are (essentially) inverse to each other.
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
An Slightly Less Informal Description of Duality
Duality relates two types of mathematical structure in a strong way. 1. Every structure of type A can be systematically transformed into a structure of type B (and vice versa).
5 / 33 3. These transformations are (essentially) inverse to each other.
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
An Slightly Less Informal Description of Duality
Duality relates two types of mathematical structure in a strong way. 1. Every structure of type A can be systematically transformed into a structure of type B (and vice versa). 2. Every structure preserving map of type A can be systematically transformed into a structure preserving map of type B in the other direction.
5 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
An Slightly Less Informal Description of Duality
Duality relates two types of mathematical structure in a strong way. 1. Every structure of type A can be systematically transformed into a structure of type B (and vice versa). 2. Every structure preserving map of type A can be systematically transformed into a structure preserving map of type B in the other direction. 3. These transformations are (essentially) inverse to each other.
5 / 33 I together with natural transformations : IdDop → FG and η : IdC → GF
I such that every component D : D → FG(D), ηC : C → GF(C) is an isomorphism.
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
A Formal Definition of Duality
A dual equivalence of categories is I a pair of functors F : C → Dop and G : Dop → C
6 / 33 I such that every component D : D → FG(D), ηC : C → GF(C) is an isomorphism.
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
A Formal Definition of Duality
A dual equivalence of categories is I a pair of functors F : C → Dop and G : Dop → C
I together with natural transformations : IdDop → FG and η : IdC → GF
6 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
A Formal Definition of Duality
A dual equivalence of categories is I a pair of functors F : C → Dop and G : Dop → C
I together with natural transformations : IdDop → FG and η : IdC → GF
I such that every component D : D → FG(D), ηC : C → GF(C) is an isomorphism.
6 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Algebra, Logic and Topology
7 / 33 Semantics A valuation v : Prop → {0, 1} assigns truth values to each propositional variable.
v extends to vˆ : Form → {0, 1}: vˆ(p) = v(p) vˆ(>) = 1 vˆ(⊥) = 0 vˆ(ϕ ∨ ψ) = max(vˆ(ϕ), vˆ(ψ)) vˆ(ϕ ∧ ψ) = min(vˆ(ϕ), vˆ(ψ)) vˆ(¬ϕ) = 1 − vˆ(ϕ) vˆ(ϕ → ψ) = max(1 − vˆ(ϕ), vˆ(ψ))
ϕ ψ if every v satisfying ϕ satisfies ψ.
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Classical Propositional Logic
Syntax Formulas generated by grammar p | > | ⊥ | ϕ∨ϕ | ϕ∧ϕ | ¬ϕ | ϕ → ϕ
Expressions ϕ ` ψ derived by rules η ` φ η ` ψ η ` φ ∧ ψ η ∧ φ ` ψ η ` φ → ψ η ` φ → ψ η ` φ η ` ψ
8 / 33 v extends to vˆ : Form → {0, 1}: vˆ(p) = v(p) vˆ(>) = 1 vˆ(⊥) = 0 vˆ(ϕ ∨ ψ) = max(vˆ(ϕ), vˆ(ψ)) vˆ(ϕ ∧ ψ) = min(vˆ(ϕ), vˆ(ψ)) vˆ(¬ϕ) = 1 − vˆ(ϕ) vˆ(ϕ → ψ) = max(1 − vˆ(ϕ), vˆ(ψ))
ϕ ψ if every v satisfying ϕ satisfies ψ.
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Classical Propositional Logic
Syntax Semantics Formulas generated by grammar A valuation v : Prop → {0, 1} p | > | ⊥ | ϕ∨ϕ | ϕ∧ϕ | ¬ϕ | ϕ → ϕ assigns truth values to each propositional variable. Expressions ϕ ` ψ derived by rules η ` φ η ` ψ η ` φ ∧ ψ η ∧ φ ` ψ η ` φ → ψ η ` φ → ψ η ` φ η ` ψ
8 / 33 ϕ ψ if every v satisfying ϕ satisfies ψ.
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Classical Propositional Logic
Syntax Semantics Formulas generated by grammar A valuation v : Prop → {0, 1} p | > | ⊥ | ϕ∨ϕ | ϕ∧ϕ | ¬ϕ | ϕ → ϕ assigns truth values to each propositional variable. Expressions ϕ ` ψ derived by rules v extends to vˆ : Form → {0, 1}: η ` φ η ` ψ vˆ(p) = v(p) vˆ(>) = 1 vˆ(⊥) = 0 η ` φ ∧ ψ vˆ(ϕ ∨ ψ) = max(vˆ(ϕ), vˆ(ψ)) vˆ(ϕ ∧ ψ) = min(vˆ(ϕ), vˆ(ψ)) η ∧ φ ` ψ vˆ(¬ϕ) = 1 − vˆ(ϕ) η ` φ → ψ vˆ(ϕ → ψ) = max(1 − vˆ(ϕ), vˆ(ψ)) η ` φ → ψ η ` φ η ` ψ
8 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Classical Propositional Logic
Syntax Semantics Formulas generated by grammar A valuation v : Prop → {0, 1} p | > | ⊥ | ϕ∨ϕ | ϕ∧ϕ | ¬ϕ | ϕ → ϕ assigns truth values to each propositional variable. Expressions ϕ ` ψ derived by rules v extends to vˆ : Form → {0, 1}: η ` φ η ` ψ vˆ(p) = v(p) vˆ(>) = 1 vˆ(⊥) = 0 η ` φ ∧ ψ vˆ(ϕ ∨ ψ) = max(vˆ(ϕ), vˆ(ψ)) vˆ(ϕ ∧ ψ) = min(vˆ(ϕ), vˆ(ψ)) η ∧ φ ` ψ vˆ(¬ϕ) = 1 − vˆ(ϕ) η ` φ → ψ vˆ(ϕ → ψ) = max(1 − vˆ(ϕ), vˆ(ψ)) η ` φ → ψ η ` φ ϕ ψ if every v satisfying ϕ η ` ψ satisfies ψ. 8 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Algebraizing Classical Propositional Logic
I Define ϕ ≡ ψ iff ϕ ` ψ and ψ ` ϕ provable. I [ϕ] = {ψ | ϕ ≡ ψ}
I AForm = {[ϕ] | ϕ ∈ Form}.
I AForm has the structure of a Boolean algebra when ∨, ∧, ¬ interpreted as join, meet and negation.
I If ϕ ` ψ not provable, then [ϕ → ψ] < > in AForm
9 / 33 An ultrafilter additionally satisfies 4. x ∨ y ∈ F implies x ∈ F or y ∈ F.
Theorem On a Boolean algebra A, a filter F is an ultrafilter iff F is maximal wrt ⊆ iff for all a ∈ A, a ∈ F or ¬a ∈ F.
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Ultrafilters
A filter on a Boolean algebra A is a subset F ⊆ A satisfying the following properties 1. ⊥ < F; 2. x, y ∈ F implies x ∧ y ∈ F; 3. x ∈ F and x ≤ y implies y ∈ F.
10 / 33 Theorem On a Boolean algebra A, a filter F is an ultrafilter iff F is maximal wrt ⊆ iff for all a ∈ A, a ∈ F or ¬a ∈ F.
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Ultrafilters
A filter on a Boolean algebra A is a subset F ⊆ A satisfying the following properties 1. ⊥ < F; 2. x, y ∈ F implies x ∧ y ∈ F; 3. x ∈ F and x ≤ y implies y ∈ F. An ultrafilter additionally satisfies 4. x ∨ y ∈ F implies x ∈ F or y ∈ F.
10 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Ultrafilters
A filter on a Boolean algebra A is a subset F ⊆ A satisfying the following properties 1. ⊥ < F; 2. x, y ∈ F implies x ∧ y ∈ F; 3. x ∈ F and x ≤ y implies y ∈ F. An ultrafilter additionally satisfies 4. x ∨ y ∈ F implies x ∈ F or y ∈ F.
Theorem On a Boolean algebra A, a filter F is an ultrafilter iff F is maximal wrt ⊆ iff for all a ∈ A, a ∈ F or ¬a ∈ F.
10 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Another Perspective on Ultrafilters
I For any Boolean algebra A, ultrafilters are in bijective correspondence with homomorphisms f : A → {0, 1}.
I Every homomorphism f : AForm → {0, 1} uniquely corresponds to a valuation v : Prop → {0, 1} with f = vˆ.
I Ultrafilters on the formula algebra AForm are in bijective correspondence with valuations.
11 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Stone’s Representation Theorem
Theorem (Stone 1936) 1 Every Boolean algebra is isomorphic to a Boolean algebra of sets.
Proof.
I Let A be a Boolean algebra and Uf(A) its set of ultrafilters. I Consider the power set algebra (P(Uf(A)), ∩, ∪, \, Uf(A), ∅). I The map h : A → P(Uf(A)), defined
h(a) = {F ∈ Uf(A) | a ∈ F},
is an embedding. 1M.H. Stone. The Theory of Representations of Boolean Algebras. Transactions of the American Mathematical Society 40: 37 – 111. 1936 12 / 33 I Suppose ϕ ` ψ not provable. Then ϕ → ψ < > in AForm. Hence h(ϕ → ψ) $ Val. I Thus there exists a valuation v such that vˆ(ϕ → ψ) = 0. I Hence vˆ(ϕ) = 1 and vˆ(ψ) = 0. So ϕ ψ does not hold.
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
From Representation, Completeness
Theorem (Completeness Theorem for Propositional Logic) ϕ ψ implies ϕ ` ψ. Proof.
I The representation theorem gives us an embedding h : AForm → P(Val) where AForm is the set of all valuations and h(ϕ) = {v | vˆ(ϕ) = 1}.
13 / 33 I Thus there exists a valuation v such that vˆ(ϕ → ψ) = 0. I Hence vˆ(ϕ) = 1 and vˆ(ψ) = 0. So ϕ ψ does not hold.
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
From Representation, Completeness
Theorem (Completeness Theorem for Propositional Logic) ϕ ψ implies ϕ ` ψ. Proof.
I The representation theorem gives us an embedding h : AForm → P(Val) where AForm is the set of all valuations and h(ϕ) = {v | vˆ(ϕ) = 1}.
I Suppose ϕ ` ψ not provable. Then ϕ → ψ < > in AForm. Hence h(ϕ → ψ) $ Val.
13 / 33 I Hence vˆ(ϕ) = 1 and vˆ(ψ) = 0. So ϕ ψ does not hold.
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
From Representation, Completeness
Theorem (Completeness Theorem for Propositional Logic) ϕ ψ implies ϕ ` ψ. Proof.
I The representation theorem gives us an embedding h : AForm → P(Val) where AForm is the set of all valuations and h(ϕ) = {v | vˆ(ϕ) = 1}.
I Suppose ϕ ` ψ not provable. Then ϕ → ψ < > in AForm. Hence h(ϕ → ψ) $ Val. I Thus there exists a valuation v such that vˆ(ϕ → ψ) = 0.
13 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
From Representation, Completeness
Theorem (Completeness Theorem for Propositional Logic) ϕ ψ implies ϕ ` ψ. Proof.
I The representation theorem gives us an embedding h : AForm → P(Val) where AForm is the set of all valuations and h(ϕ) = {v | vˆ(ϕ) = 1}.
I Suppose ϕ ` ψ not provable. Then ϕ → ψ < > in AForm. Hence h(ϕ → ψ) $ Val. I Thus there exists a valuation v such that vˆ(ϕ → ψ) = 0. I Hence vˆ(ϕ) = 1 and vˆ(ψ) = 0. So ϕ ψ does not hold.
13 / 33 "A cardinal principle of modern mathematical research may be stated as a maxim: ’One must always topologize’" - Marshall H. Stone.
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
What about semantics?
We’ve generalized the syntax of propositional logic to obtain a category of algebras; can we similarly generalize the semantics?
14 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
What about semantics?
We’ve generalized the syntax of propositional logic to obtain a category of algebras; can we similarly generalize the semantics?
"A cardinal principle of modern mathematical research may be stated as a maxim: ’One must always topologize’" - Marshall H.
Stone. 14 / 33 For a A ⊆ X, A is... I Open if A ∈ O I Closed if X \ A ∈ O I Clopen if both open and closed.
Define hϕ = {v ∈ Val | vˆ(ϕ) = 1}. The set B = {hϕ | ϕ ∈ L} generates a topology on the set of valuations. We call this the Valuation Space.
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Topologizing Classical Propositional Logic
A topological space is a pair (X, O) s.t. X is a set and O ⊆ P(X) such that I ∅, X ∈ O, I O closed under finite intersections, I O closed under arbitrary unions.
15 / 33 Define hϕ = {v ∈ Val | vˆ(ϕ) = 1}. The set B = {hϕ | ϕ ∈ L} generates a topology on the set of valuations. We call this the Valuation Space.
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Topologizing Classical Propositional Logic
A topological space is a pair (X, O) s.t. X is a set and O ⊆ P(X) such that I ∅, X ∈ O, I O closed under finite intersections, I O closed under arbitrary unions. For a A ⊆ X, A is... I Open if A ∈ O I Closed if X \ A ∈ O I Clopen if both open and closed.
15 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Topologizing Classical Propositional Logic
A topological space is a pair (X, O) s.t. X is a set and O ⊆ P(X) such that I ∅, X ∈ O, I O closed under finite intersections, I O closed under arbitrary unions. For a A ⊆ X, A is... I Open if A ∈ O I Closed if X \ A ∈ O I Clopen if both open and closed.
Define hϕ = {v ∈ Val | vˆ(ϕ) = 1}. The set B = {hϕ | ϕ ∈ L} generates a topology on the set of valuations. We call this the Valuation Space.
15 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Properties of the Valuation Space
A topological space is...
I compact iff for every set of closed sets {Ci | i ∈ I}, if every finite T subset has non-empty intersection then i{Ci | i ∈ I} , ∅. I Hausdorff iff every pair of distinct points is separated by disjoint open sets. I zero-dimensional iff it is generated by clopen sets. I a Stone space if it is compact, Hausdorff and zero-dimensional. Theorem The valuation space is a Stone space.
16 / 33 Proof.
I Let Σ be finitely satisfiable. Define C = {hϕ | ϕ ∈ Σ}: each member is a clopen, thus closed, set.
I Given ϕ1, . . . , ϕn ∈ Σ, by finite satisfiability there exists v ∈ Tn satisfying ϕ1, . . . , ϕn so v i hϕi . 0 T I Hence by topological compactness there exists v ∈ C, thus v0 satisfies Σ.
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Compactness begets Compactness
Theorem (Compactness Theorem) Given a (possibly infinite) set of propositional formulas Σ, Σ is satisfiable iff every finite subset of Σ is satisfiable.
17 / 33 I Given ϕ1, . . . , ϕn ∈ Σ, by finite satisfiability there exists v ∈ Tn satisfying ϕ1, . . . , ϕn so v i hϕi . 0 T I Hence by topological compactness there exists v ∈ C, thus v0 satisfies Σ.
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Compactness begets Compactness
Theorem (Compactness Theorem) Given a (possibly infinite) set of propositional formulas Σ, Σ is satisfiable iff every finite subset of Σ is satisfiable. Proof.
I Let Σ be finitely satisfiable. Define C = {hϕ | ϕ ∈ Σ}: each member is a clopen, thus closed, set.
17 / 33 0 T I Hence by topological compactness there exists v ∈ C, thus v0 satisfies Σ.
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Compactness begets Compactness
Theorem (Compactness Theorem) Given a (possibly infinite) set of propositional formulas Σ, Σ is satisfiable iff every finite subset of Σ is satisfiable. Proof.
I Let Σ be finitely satisfiable. Define C = {hϕ | ϕ ∈ Σ}: each member is a clopen, thus closed, set.
I Given ϕ1, . . . , ϕn ∈ Σ, by finite satisfiability there exists v ∈ Tn satisfying ϕ1, . . . , ϕn so v i hϕi .
17 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Compactness begets Compactness
Theorem (Compactness Theorem) Given a (possibly infinite) set of propositional formulas Σ, Σ is satisfiable iff every finite subset of Σ is satisfiable. Proof.
I Let Σ be finitely satisfiable. Define C = {hϕ | ϕ ∈ Σ}: each member is a clopen, thus closed, set.
I Given ϕ1, . . . , ϕn ∈ Σ, by finite satisfiability there exists v ∈ Tn satisfying ϕ1, . . . , ϕn so v i hϕi . 0 T I Hence by topological compactness there exists v ∈ C, thus v0 satisfies Σ.
17 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Connecting The Perspectives
I Boolean algebras are an abstraction of the syntax of classical propositional logic. I Similarly, Stone spaces are an abstraction of the semantics of classical propositional logic. I Syntax and semantics are connected by soundness and completeness theorems. I Stone duality is the generalization of soundness and completeness to the level of Boolean algebras and Stone spaces.
18 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Stone Duality
19 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Recap
I Formulas quotiented by logical equivalence give Boolean algebra AForm. I Representation Theorem: h : A → P(Uf(A)) gives an embedding where h(a) = {F | a ∈ F}.
I hϕ = {v | vˆ(ϕ) = 1} generates Stone topology on set of valuations.
I h(ϕ) = hϕ when A = AForm.
20 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
The Stone Space of a Boolean Algebra
Theorem Given a Boolean algebra A, the set Uf(A) topologised by {h(a) | a ∈ A} is a Stone space. We denote the Stone dual of a Boolean algebra by S(A).
21 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
The Boolean Algebra of a Stone Space
Theorem Given a Stone space X, the set of clopen sets of X carries the structure of a Boolean algebra. We denote the Boolean dual of a Stone space X by A(X).
22 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Transforming Morphisms
Theorem 1. If f : A → A0 is a Boolean homomorphism, then
f −1 : S(A0) → S(A)
is a continuous map of Stone spaces. 2. If g : X → X0 is a continuous map of Stone spaces, then
g−1 : A(X0) → A(X)
is a Boolean homomorphism.
Note the change in direction of arrows!
23 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Retreiving the Original Structures
Theorem 1. For all Boolean algebras A, the map
a 7→ {F ∈ Uf(A) | a ∈ F}
is an isomorphism between A and AS(A). 2. For all Stone spaces X, the map
x 7→ {C | C clopen and x ∈ C}
is an isomoprhism between X and SA(X).
24 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Stone Duality
Theorem The categories of Boolean algebras and Stone spaces are dually equivalent.
25 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Duality in Logic and Computer Science
26 / 33 Algebra Topological Space Distributive lattice Priestley space Heyting algebra Esakia space Boolean algebra Stone space BAO Descriptive frame Relevant algebra Urquhart space MV algebra Tychonoff space Piron lattice Quantum dynamic frame BANoNa Nomimal Stone space Hybrid algebra 2-Sort Desc. Frame Modal µ-algebra Modal µ-frame Aumann algebra Markov process
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
A Cornucopia of Logical Dualities
Stone duality has been generalized to a variety of logical settings.
Logic Positive Intuitionistic Classical Modal Relevant Many-valued Quantum action Nominal Classical Hybrid Modal µ-calculus Markovan
27 / 33 Topological Space Priestley space Esakia space Stone space Descriptive frame Urquhart space Tychonoff space Quantum dynamic frame Nomimal Stone space 2-Sort Desc. Frame Modal µ-frame Markov process
What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
A Cornucopia of Logical Dualities
Stone duality has been generalized to a variety of logical settings.
Logic Algebra Positive Distributive lattice Intuitionistic Heyting algebra Classical Boolean algebra Modal BAO Relevant Relevant algebra Many-valued MV algebra Quantum action Piron lattice Nominal Classical BANoNa Hybrid Hybrid algebra Modal µ-calculus Modal µ-algebra Markovan Aumann algebra
27 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
A Cornucopia of Logical Dualities
Stone duality has been generalized to a variety of logical settings.
Logic Algebra Topological Space Positive Distributive lattice Priestley space Intuitionistic Heyting algebra Esakia space Classical Boolean algebra Stone space Modal BAO Descriptive frame Relevant Relevant algebra Urquhart space Many-valued MV algebra Tychonoff space Quantum action Piron lattice Quantum dynamic frame Nominal Classical BANoNa Nomimal Stone space Hybrid Hybrid algebra 2-Sort Desc. Frame Modal µ-calculus Modal µ-algebra Modal µ-frame Markovan Aumann algebra Markov process
27 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Applications of Logical Dualities
Dualities can be used to prove serious results about logics. Theorem 2 Every Sahlqvist formula of modal logic is canonical and corresponds to a first-order frame condition.
Theorem 3 Interpolation fails in relevant logic.
Proof Sketch. In both cases: reduce problem to an algebraic or topological property and transfer along the dual equivalence.
2G. Sambin and V. Vaccaro. A New Proof of Sahlqvist’s Theorem on Modal Definability and Completeness. Journal of Symbolic Logic 54, 1989 3A. Urquhart. Failure of Interpolation in Relevant Logics. Journal of Philosophical Logic 22(5), 1993 28 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Automata Theory
29 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Coalgebra
I Coalgebra give a common mathematical framework for investigating state-based dynamics I Ingredients: state space X and transition structure TX. I A coalgebra is a map α : X → TX. I Examples: automata, transition systems, Kripke frames, petri nets, etc.
30 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Coalgebra
I Stone-type duality gives a machine to generate logics of coalgebraic structures4
4M. M. Bonsangue and A. Kurz. Duality for Logics of Transition Systems. FOSSACS 2005 31 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
Program Semantics
I Plotkin/Smyth5: Duality between state transformation semantics and predicate-transformer semantics. I Kozen6: Duality between two kinds of probablistic program semantics. I Abramsky7: Duality between domain theoretic semantics and operational semantics. Framework for outputting program logics.
5G.D. Plotkin. Dijkstra’s predicate transformers and Smyth’s powerdomains. Abstract Software Specifications, LNCS 86, 1980 6D. Kozen, A Probabilistic PDL, STOC ’83, ACM, 1983 7S. Abramsky. Domain Theory in Logical Form. Annals of Pure and Applied Logic 51(1-2). 1991 32 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science
The Moral of The Story
I Dualities are everywhere in computer science. I Dualities provide a general framework for syntax, semantics, soundness and completeness in logic. I Dualities provide new perspectives to prove things with. I Dualities provide a mathematical foundation for established theory. I Dualities facilitate the transfer of theory from one field to another.
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