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 ! f0; 1g assigns truth values to each propositional variable. v extends to vˆ : Form ! f0; 1g: vˆ(p) = v(p) vˆ(>) = 1 vˆ(?) = 0 vˆ(' _ ) = max(vˆ('); vˆ( )) vˆ(' ^ ) = min(vˆ('); vˆ( )) vˆ(:') = 1 − vˆ(') vˆ(' ! ) = max(1 − vˆ('); vˆ( )) ' if every v satisfying ' satisfies . What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science Classical Propositional Logic Syntax Formulas generated by grammar p j > j ? j '_' j '^' j :' j ' ! ' Expressions ' ` derived by rules η ` φ η ` η ` φ ^ η ^ φ ` η ` φ ! η ` φ ! η ` φ η ` 8 / 33 v extends to vˆ : Form ! f0; 1g: vˆ(p) = v(p) vˆ(>) = 1 vˆ(?) = 0 vˆ(' _ ) = max(vˆ('); vˆ( )) vˆ(' ^ ) = min(vˆ('); vˆ( )) vˆ(:') = 1 − vˆ(') vˆ(' ! ) = max(1 − vˆ('); vˆ( )) ' if every v satisfying ' satisfies . 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 ! f0; 1g p j > j ? j '_' j '^' j :' j ' ! ' assigns truth values to each propositional variable. Expressions ' ` derived by rules η ` φ η ` η ` φ ^ η ^ φ ` η ` φ ! η ` φ ! η ` φ η ` 8 / 33 ' if every v satisfying ' satisfies . 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 ! f0; 1g p j > j ? j '_' j '^' j :' j ' ! ' assigns truth values to each propositional variable. Expressions ' ` derived by rules v extends to vˆ : Form ! f0; 1g: η ` φ η ` vˆ(p) = v(p) vˆ(>) = 1 vˆ(?) = 0 η ` φ ^ vˆ(' _ ) = max(vˆ('); vˆ( )) vˆ(' ^ ) = min(vˆ('); vˆ( )) η ^ φ ` vˆ(:') = 1 − vˆ(') η ` φ ! vˆ(' ! ) = max(1 − vˆ('); vˆ( )) η ` φ ! η ` φ η ` 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 ! f0; 1g p j > j ? j '_' j '^' j :' j ' ! ' assigns truth values to each propositional variable. Expressions ' ` derived by rules v extends to vˆ : Form ! f0; 1g: η ` φ η ` vˆ(p) = v(p) vˆ(>) = 1 vˆ(?) = 0 η ` φ ^ vˆ(' _ ) = max(vˆ('); vˆ( )) vˆ(' ^ ) = min(vˆ('); vˆ( )) η ^ φ ` vˆ(:') = 1 − vˆ(') η ` φ ! vˆ(' ! ) = max(1 − vˆ('); vˆ( )) η ` φ ! η ` φ ' if every v satisfying ' η ` satisfies . 8 / 33 What Is Duality? Algebra, Logic and Topology Stone Duality Duality in Logic and Computer Science Algebraizing Classical Propositional Logic I Define ' ≡ iff ' ` and ` ' provable. I ['] = f j ' ≡ g I AForm = f['] j ' 2 Formg. I AForm has the structure of a Boolean algebra when _; ^; : interpreted as join, meet and negation. I If ' ` not provable, then [' ! ] < > in AForm 9 / 33 An ultrafilter additionally satisfies 4. x _ y 2 F implies x 2 F or y 2 F. Theorem On a Boolean algebra A, a filter F is an ultrafilter iff F is maximal wrt ⊆ iff for all a 2 A, a 2 F or :a 2 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 2 F implies x ^ y 2 F; 3. x 2 F and x ≤ y implies y 2 F. 10 / 33 Theorem On a Boolean algebra A, a filter F is an ultrafilter iff F is maximal wrt ⊆ iff for all a 2 A, a 2 F or :a 2 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 2 F implies x ^ y 2 F; 3. x 2 F and x ≤ y implies y 2 F. An ultrafilter additionally satisfies 4. x _ y 2 F implies x 2 F or y 2 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 2 F implies x ^ y 2 F; 3. x 2 F and x ≤ y implies y 2 F. An ultrafilter additionally satisfies 4. x _ y 2 F implies x 2 F or y 2 F. Theorem On a Boolean algebra A, a filter F is an ultrafilter iff F is maximal wrt ⊆ iff for all a 2 A, a 2 F or :a 2 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 ! f0; 1g. I Every homomorphism f : AForm ! f0; 1g uniquely corresponds to a valuation v : Prop ! f0; 1g 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)); \; [; n; Uf(A); ;).