Quantifiers and Measures, Part II Tom a s Jakl & Luca Reggio - PowerPoint PPT Presentation

Quantifiers and Measures, Part II Tom´ aˇ s Jakl & Luca Reggio (j.w.w. Mai Gehrke) 26 March 2020

Taking over where Luca stopped... The image of R f : β ( Mod n − 1 ) → V ( Typ n ) is the Stone dual of B ∃ x n = �∃ x n ϕ | ϕ ∈ FO n � . 1

Taking over where Luca stopped... The image of R f : β ( Mod n − 1 ) → V ( Typ n ) is the Stone dual of B ∃ x n = �∃ x n ϕ | ϕ ∈ FO n � . The construction B � B ∃ x n works for any B ֒ → P ( Mod n ) dually given by f : β ( Mod n ) ։ X then we build R f : β ( Mod n − 1 ) → V ( X ) . And B ∃ x n can be identified with a subalgebra of P ( Mod n ). 1

The Boolean algebra of formulas Inductively, B ( n ) = QF ( x 1 , . . . , x n ) 0 B ( n ) B ( n ) ∃ x 1 + . . . + B ( n ) ∃ x n + B ( n ) i +1 = the image of → P ( Mod n ) i we build ∞ ∞ � � B ( n ) FO = i n =1 i =1 as a Boolean subalgebra of P ( Mod ω ). 2

Inductive constructions in domain theory In DTLF operators +, × , P P , P H , P S , → on the space side dually correspond to enrichments of logic. E.g. function space construction [ E → D ] adds a layer of implications to the logic. 3

Inductive constructions in domain theory In DTLF operators +, × , P P , P H , P S , → on the space side dually correspond to enrichments of logic. E.g. function space construction [ E → D ] adds a layer of implications to the logic. The solution of a domain equation D ∼ = σ ( D ) computed as a bilimit, dually adds logical connectives, step by step. 3

Vietoris as a space of measures Closed subsets of a Stone space X ← → finitely additive measures on X → 2 (where 2 = ( { 0 , 1 } , ∧ , ∨ , 0 , 1)) functions µ : Clp( X ) → 2 s.t. • µ ( ∅ ) = 0 • A ∩ B = ∅ = ⇒ µ ( A ∪ B ) = µ ( A ) ∨ µ ( B ) 4

Vietoris as a space of measures Closed subsets of a Stone space X ← → finitely additive measures on X → 2 (where 2 = ( { 0 , 1 } , ∧ , ∨ , 0 , 1)) functions µ : Clp( X ) → 2 s.t. • µ ( ∅ ) = 0 • A ∩ B = ∅ = ⇒ µ ( A ∪ B ) = µ ( A ) ∨ µ ( B ) 4

Vietoris as a space of measures Closed subsets of a Stone space X ← → finitely additive measures on X → 2 (where 2 = ( { 0 , 1 } , ∧ , ∨ , 0 , 1)) functions µ : Clp( X ) → 2 s.t. Via the correspondence • µ ( ∅ ) = 0  1 C ∩ A � = ∅  C �→ µ C • A ∩ B = ∅ = ⇒ µ ( A ∪ B ) = µ ( A ) ∨ µ ( B ) such that µ C ( A ) = 0 otherwise  This yields a homeomorphism V ( X ) ∼ = M ( X , 2 ). 4

Quantifiers ← → measures? • Existential quantifiers ← → space of measures X → 2 5

Quantifiers ← → measures? • Existential quantifiers ← → space of measures X → 2 • Semiring quantifiers ← → space of measures X → S (from Logic on Words, adaptable to arbitrary finite models) 5

Quantifiers ← → measures? • Existential quantifiers ← → space of measures X → 2 • Semiring quantifiers ← → space of measures X → S (from Logic on Words, adaptable to arbitrary finite models) e.g. for k ∈ S , ϕ ( x ) ∈ FO , A | = ∃ k x .ϕ ( x ) iff 1 + · · · + 1 = k in S � �� � for every a ∈ A s.t. A | = ϕ ( a ) 5

Quantifiers ← → measures? • Existential quantifiers ← → space of measures X → 2 • Semiring quantifiers ← → space of measures X → S (from Logic on Words, adaptable to arbitrary finite models) e.g. for k ∈ S , ϕ ( x ) ∈ FO , A | = ∃ k x .ϕ ( x ) iff 1 + · · · + 1 = k in S � �� � for every a ∈ A s.t. A | = ϕ ( a ) • “Probabilistic quantifiers” ← → probabilistic measures X → [0 , 1] (from structural limits) 5

Quantifiers ← → measures? • Existential quantifiers ← → space of measures X → 2 • Semiring quantifiers ← → space of measures X → S (from Logic on Words, adaptable to arbitrary finite models) e.g. for k ∈ S , ϕ ( x ) ∈ FO , A | = ∃ k x .ϕ ( x ) iff 1 + · · · + 1 = k in S � �� � for every a ∈ A s.t. A | = ϕ ( a ) • “Probabilistic quantifiers” ← → probabilistic measures X → [0 , 1] (from structural limits) • more... ? 5

Stone pairings [Neˇ setˇ ril, Ossona de Mendez, 2013] For a formula ϕ ( x 1 , . . . , x n ) and a finite structure A , |{ a ∈ A n | A | = ϕ ( a ) }| � ϕ, A � = (Stone pairing) | A | n 6

Stone pairings [Neˇ setˇ ril, Ossona de Mendez, 2013] For a formula ϕ ( x 1 , . . . , x n ) and a finite structure A , |{ a ∈ A n | A | = ϕ ( a ) }| � ϕ, A � = (Stone pairing) | A | n Mapping A �→ �− , A � defines an embedding Fin ֒ → M ( Typ , [0 , 1]) Recall that Typ is dual to FO , i.e. clopens are of the form � ϕ � for ϕ ∈ FO . 6

Stone pairings [Neˇ setˇ ril, Ossona de Mendez, 2013] For a formula ϕ ( x 1 , . . . , x n ) and a finite structure A , |{ a ∈ A n | A | = ϕ ( a ) }| � ϕ, A � = (Stone pairing) | A | n Mapping A �→ �− , A � defines an embedding Fin ֒ → M ( Typ , [0 , 1]) which lifts uniquely to β ( Fin ) → M ( Typ , [0 , 1]) Recall that Typ is dual to FO , i.e. clopens are of the form � ϕ � for ϕ ∈ FO . Motivation: The limit of ( A i ) i is computed as lim i →∞ �− , A � in M ( Typ , [0 , 1]). 6

The dual space of the image? What is the dual of X ? → M ( Typ , [0 , 1]) β ( Fin ) ։ X ֒ ⇒ Clp( X ) ∼ M ( Typ , [0 , 1]) has no non-trivial clopen! = = 2 7

The dual space of the image? What is the dual of X ? → M ( Typ , [0 , 1]) β ( Fin ) ։ X ֒ ⇒ Clp( X ) ∼ M ( Typ , [0 , 1]) has no non-trivial clopen! = = 2 Two possible solutions: 1. Describe X in terms of geometric logic, logic of proximity lattices or de Vries algebras, ... 2. Replace [0 , 1] to retain classical logic. 7

The dual space of the image? What is the dual of X ? → M ( Typ , [0 , 1]) β ( Fin ) ։ X ֒ ⇒ Clp( X ) ∼ M ( Typ , [0 , 1]) has no non-trivial clopen! = = 2 Two possible solutions: 1. Describe X in terms of geometric logic, logic of proximity lattices or de Vries algebras, ... 2. Replace [0 , 1] to retain classical logic. � Our choice today! 7

The Stone space Γ (motivation) Problem: We need to replace [0,1] by a similar space Γ s.t. 1. we can define measures X → Γ 2. the space M ( X , Γ ) is compact 0-dimensional 3. Stone pairing �− , −� : Fin → M ( Typ , Γ ) definable and is “comparable” with the original Stone pairing 8

The Stone space Γ (motivation) Problem: We need to replace [0,1] by a similar space Γ s.t. 1. we can define measures X → Γ 2. the space M ( X , Γ ) is compact 0-dimensional 3. Stone pairing �− , −� : Fin → M ( Typ , Γ ) definable and is “comparable” with the original Stone pairing Observe: For ϕ ( v 1 , . . . , v k ), the Stone pairing � ϕ, A � takes values in a finite chain � � n < · · · < n − 1 0 < 1 n < 2 I n = < 1 n where n = | A | k . 8

The Stone space Γ (motivation) Problem: We need to replace [0,1] by a similar space Γ s.t. 1. we can define measures X → Γ 2. the space M ( X , Γ ) is compact 0-dimensional 3. Stone pairing �− , −� : Fin → M ( Typ , Γ ) definable and is “comparable” with the original Stone pairing Observe: For ϕ ( v 1 , . . . , v k ), the Stone pairing � ϕ, A � takes values in a finite chain � � n < · · · < n − 1 0 < 1 n < 2 I n = < 1 n where n = | A | k . = ⇒ Define Γ as an inverse limit of those (discrete) posets! 8

The Stone space Γ (description) Define nm ( a nm ) = ⌊ a / m ⌋ Γ = lim { f n f n nm : I nm ։ I n } n , m ∈ N where . n Elements of Γ are vectors � ( x n ) n ∈ I n n such that f n nm ( x nm ) = x n , for every n , m ∈ N . 9

The Stone space Γ (description) Define nm ( a nm ) = ⌊ a / m ⌋ Γ = lim { f n f n nm : I nm ։ I n } n , m ∈ N where . n Elements of Γ are vectors � ( x n ) n ∈ I n n such that f n nm ( x nm ) = x n , for every n , m ∈ N . Intuitively: coordinates represent approximations of numbers in [0,1] from bottom. The larger the n the better the approximation.  one representation of irrational numbers: r −  This gives two representations of rational numbers: q − , q ◦  q − q ◦ 0 ◦ r − 1 − 1 ◦ Γ = 9

Properties of Γ • The subspace topology Γ ⊆ � n I n is compact 0-dimensional • Retraction-section maps Γ [0 , 1] • Semicontinuous partial operations − and ∼ on Γ − , ∼ : Γ × Γ ⇀ Γ 10

Properties of Γ • The subspace topology Γ ⊆ � n I n is compact 0-dimensional • Retraction-section maps Γ [0 , 1] • Semicontinuous partial operations − and ∼ on Γ − , ∼ : Γ × Γ ⇀ Γ allow to define measures X → Γ monotone functions µ : Clp( X ) → Γ s.t. • µ ( ∅ ) = 0 ◦ , µ ( X ) = 1 ◦ • µ ( A ) ∼ µ ( A ∩ B ) ≤ µ ( A ∪ B ) − µ ( B ) • µ ( A ) − µ ( A ∩ b ) ≥ µ ( A ∪ B ) ∼ µ ( B ) 10