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Algorithmic correspondence and canonicity for non-distributive - - PowerPoint PPT Presentation

Mar 11, 2023 876 likes •1.5k views

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities Algorithmic correspondence and canonicity for non-distributive logics Willem Conradie 1 Alessandra Palmigiano 2 1 University of Johannesburg

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Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Algorithmic correspondence and canonicity for non-distributive logics

Willem Conradie1 Alessandra Palmigiano2

1University of Johannesburg 2ILLC, University of Amsterdam

TACL2011

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

(Classical) Modal Logic

Syntax

ϕ ::= ⊥ | p | ¬ϕ | ϕ1 ∧ ϕ2 | ϕ

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 3

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

(Classical) Modal Logic

Syntax

ϕ ::= ⊥ | p | ¬ϕ | ϕ1 ∧ ϕ2 | ϕ

Semantics Relational Algebraic Kripke frames BAO’s

F = (W, R) A = (A, ∧, ∨, −, 1, 0, )

Valuations: V : Var → ℘(W) Assignments: v : Var → A

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 4

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Correspondence: An example

On models:

(F, V) |= p → p

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 5

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Correspondence: An example

On models:

(F, V) |= p → p

iff

(F, V) |= ∀x(P(x) → ∃y(Rxy ∧ P(y)))

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 6

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Correspondence: An example

On models:

(F, V) |= p → p

iff

(F, V) |= ∀x(P(x) → ∃y(Rxy ∧ P(y)))

On frames:

F |= p → p

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 7

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Correspondence: An example

On models:

(F, V) |= p → p

iff

(F, V) |= ∀x(P(x) → ∃y(Rxy ∧ P(y)))

On frames:

F |= p → p

iff

F |= ∀P∀x(P(x) → ∃y(Rxy ∧ P(y)))

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 8

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Correspondence: An example

On models:

(F, V) |= p → p

iff

(F, V) |= ∀x(P(x) → ∃y(Rxy ∧ P(y)))

On frames:

F |= p → p

iff

F |= ∀P∀x(P(x) → ∃y(Rxy ∧ P(y)))

iff

F |= ∀xRxx

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 9

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Correspondence theory

Given a modal formula ϕ, does it always have a first order correspondent?

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 10

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Correspondence theory

Given a modal formula ϕ, does it always have a first order correspondent? NO.

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 11

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Correspondence theory

Given a modal formula ϕ, does it always have a first order correspondent? NO. Central question: Which modal fmls have first-order frame correspondents? [Sahhlqvist, van Benthem, 1970’s]

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 12

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Correspondence theory

Given a modal formula ϕ, does it always have a first order correspondent? NO. Central question: Which modal fmls have first-order frame correspondents? [Sahhlqvist, van Benthem, 1970’s] Syntactic classes: Sahlqvist formulas [Sahlqvist], inductive formulas [Goranko, Vakarelov], Complex formulas [Vakarelov]

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 13

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Correspondence theory

Given a modal formula ϕ, does it always have a first order correspondent? NO. Central question: Which modal fmls have first-order frame correspondents? [Sahhlqvist, van Benthem, 1970’s] Syntactic classes: Sahlqvist formulas [Sahlqvist], inductive formulas [Goranko, Vakarelov], Complex formulas [Vakarelov] Algorithms: SCAN [Gabbay, Olbach], DLS [Szalas], SQEMA [Conradie, Goranko, Vakarelov]

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 14

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Correspondence theory

Given a modal formula ϕ, does it always have a first order correspondent? NO. Central question: Which modal fmls have first-order frame correspondents? [Sahhlqvist, van Benthem, 1970’s] Syntactic classes: Sahlqvist formulas [Sahlqvist], inductive formulas [Goranko, Vakarelov], Complex formulas [Vakarelov] Algorithms: SCAN [Gabbay, Olbach], DLS [Szalas], SQEMA [Conradie, Goranko, Vakarelov] Strong relationship between correspondence and completeness / canonicity.

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

FAQ:

How can I prove that a formula does not correspond?

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 16

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

FAQ:

How can I prove that a formula does not correspond? With model-theoretic techniques (failure of L¨

  • wenheim-Skolem,

compactness, etc.)

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 17

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

FAQ:

How can I prove that a formula does not correspond? With model-theoretic techniques (failure of L¨

  • wenheim-Skolem,

compactness, etc.) Is there a characterization of all the formulas that have a first

  • rder correspondent?

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 18

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

FAQ:

How can I prove that a formula does not correspond? With model-theoretic techniques (failure of L¨

  • wenheim-Skolem,

compactness, etc.) Is there a characterization of all the formulas that have a first

  • rder correspondent? No, and this class is an undecidable.

[Chagrova]

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Generalizing: Lattice based logics

Relational Algebraic RS Frames [Gehrke] Lattices with operators e.g. L = (L, ∧, ∨, ◦, ⋆, , , ⊳, ⊲, ⊥, ⊤)

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 20

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Generalizing: Lattice based logics

Relational Algebraic RS Frames [Gehrke] Lattices with operators e.g. L = (L, ∧, ∨, ◦, ⋆, , , ⊳, ⊲, ⊥, ⊤)

(L, ∧, ∨, ⊥, ⊤) is perfect if it is

1

complete,

2

completely join generated by its completely join irreducible elements J∞, and

3

completely meet generated by its completely meet irreducible elements the set M∞.

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Reflexivity, again

∀p[p ≤ p]

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Reflexivity, again

∀p[p ≤ p] ∀j∀m∀p

  • j ≤ p,

p ≤ m

  • ⇒ j ≤ m
  • Willem Conradie, Alessandra Palmigiano

Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 23

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Reflexivity, again

∀p[p ≤ p] ∀j∀m∀p

  • j ≤ p,

p ≤ m

  • ⇒ j ≤ m
  • ∀j∀m∀p
  • j ≤ m
  • ⇒ j ≤ m
  • Willem Conradie, Alessandra Palmigiano

Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 24

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Reflexivity, again

∀p[p ≤ p] ∀j∀m∀p

  • j ≤ p,

p ≤ m

  • ⇒ j ≤ m
  • ∀j∀m∀p
  • j ≤ m
  • ⇒ j ≤ m
  • j ≤ j

Concretely (on Kripke frames / BAO’s):

{x} ⊆ {y ∈ W | Rxy}, i.e., Rxx.

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Ackermann’s Lemma

L a perfect lattice with operators. α, β and γ terms such that

p VAR(α), β(p) positive in p, and γ(p) negative in p

TFAE:

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 26

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Ackermann’s Lemma

L a perfect lattice with operators. α, β and γ terms such that

p VAR(α), β(p) positive in p, and γ(p) negative in p

TFAE:

1

L, v |= β(α) ≤ γ(α)

2

there exists some v′ ∼p v such that

L, v′ |= α ≤ p and L, v′ |= β(p) ≤ γ(p).

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Adjunction

A and B complete lattices.

f : A → B and g : B → A.

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 28

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Adjunction

A and B complete lattices.

f : A → B and g : B → A. f ⊣ g if f(a) ≤ b iff a ≤ g(b) iff f( S) =

s∈S f(s) and g( S) = s∈S g(s)

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 29

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Adjunction

A and B complete lattices.

f : A → B and g : B → A. f ⊣ g if f(a) ≤ b iff a ≤ g(b) iff f( S) =

s∈S f(s) and g( S) = s∈S g(s)

Example

−1 ⊣ and ⊣ −1

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Confluence: p → p

∀p[p ≤ p]

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Confluence: p → p

∀p[p ≤ p] ∀p[

  • {j ∈ J∞ | j ≤ p} ≤
  • {m ∈ M∞ | p ≤ m}]

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 32

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Confluence: p → p

∀p[p ≤ p] ∀p[

  • {j ∈ J∞ | j ≤ p} ≤
  • {m ∈ M∞ | p ≤ m}]

∀p[

  • {j | j ∈ J∞ & j ≤ p} ≤
  • {m ∈ M∞ | p ≤ m}]

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 33

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Confluence: p → p

∀p[p ≤ p] ∀p[

  • {j ∈ J∞ | j ≤ p} ≤
  • {m ∈ M∞ | p ≤ m}]

∀p[

  • {j | j ∈ J∞ & j ≤ p} ≤
  • {m ∈ M∞ | p ≤ m}]

∀p∀j∀m[(j ≤ p & p ≤ m) ⇒ j ≤ m]

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 34

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Confluence: p → p

∀p[p ≤ p] ∀p[

  • {j ∈ J∞ | j ≤ p} ≤
  • {m ∈ M∞ | p ≤ m}]

∀p[

  • {j | j ∈ J∞ & j ≤ p} ≤
  • {m ∈ M∞ | p ≤ m}]

∀p∀j∀m[(j ≤ p & p ≤ m) ⇒ j ≤ m] ∀p∀j∀m[(j ≤ p & p ≤ m) ⇒ j ≤ m]

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 35

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Confluence: p → p

∀p[p ≤ p] ∀p[

  • {j ∈ J∞ | j ≤ p} ≤
  • {m ∈ M∞ | p ≤ m}]

∀p[

  • {j | j ∈ J∞ & j ≤ p} ≤
  • {m ∈ M∞ | p ≤ m}]

∀p∀j∀m[(j ≤ p & p ≤ m) ⇒ j ≤ m] ∀p∀j∀m[(j ≤ p & p ≤ m) ⇒ j ≤ m] ∀j∀m[(j ≤ m) ⇒ j ≤ m]

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 36

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Confluence: p → p

∀p[p ≤ p] ∀p[

  • {j ∈ J∞ | j ≤ p} ≤
  • {m ∈ M∞ | p ≤ m}]

∀p[

  • {j | j ∈ J∞ & j ≤ p} ≤
  • {m ∈ M∞ | p ≤ m}]

∀p∀j∀m[(j ≤ p & p ≤ m) ⇒ j ≤ m] ∀p∀j∀m[(j ≤ p & p ≤ m) ⇒ j ≤ m] ∀j∀m[(j ≤ m) ⇒ j ≤ m] ∀j[j ≤ j]

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 37

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Confluence: p → p — Translation

On Kripke frames

∀j[j ≤ j]

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 38

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Confluence: p → p — Translation

On Kripke frames

∀j[j ≤ j] ∀y[{y} ≤ −1{y}]

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 39

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Confluence: p → p — Translation

On Kripke frames

∀j[j ≤ j] ∀y[{y} ≤ −1{y}] ∀y[Rxy → ∀z(Rxz → ∃u(Rzu ∧ Ryu))]

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 40

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Confluence: p → p — Translation

On Kripke frames

∀j[j ≤ j] ∀y[{y} ≤ −1{y}] ∀y[Rxy → ∀z(Rxz → ∃u(Rzu ∧ Ryu))]

On DML-frames / intuitionistic-ML frames

∀y[{y}↑ ≤ {y}↑]

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 41

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

ALBA algorithm

Algorithm / calculus: based on Ackermann, approximation, and residuation rules.

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 42

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

ALBA algorithm

Algorithm / calculus: based on Ackermann, approximation, and residuation rules. eliminates propositional variables from inequalities for sake of special variables ranging over J∞ and M∞.

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 43

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

ALBA algorithm

Algorithm / calculus: based on Ackermann, approximation, and residuation rules. eliminates propositional variables from inequalities for sake of special variables ranging over J∞ and M∞. Theorem All ALBA-reducible inequalities are elementary on the relational semantics.

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 44

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

ALBA algorithm

Algorithm / calculus: based on Ackermann, approximation, and residuation rules. eliminates propositional variables from inequalities for sake of special variables ranging over J∞ and M∞. Theorem All ALBA-reducible inequalities are elementary on the relational semantics. Theorem All ALBA-reducible inequalities are canonical.

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 45

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Canononical Extension

Aσ A J∞ M∞ 1

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 46

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Canonicity

Admissible assignment v for L+ on Aσ: v : PROP → A, v : J → J∞(Aσ), and v : M → M∞(Aσ).

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 47

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Canonicity

Admissible assignment v for L+ on Aσ: v : PROP → A, v : J → J∞(Aσ), and v : M → M∞(Aσ). Notation: Validity under admissible assignments: Aσ |=A ϕ ≤ ψ.

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 48

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Canonicity

Admissible assignment v for L+ on Aσ: v : PROP → A, v : J → J∞(Aσ), and v : M → M∞(Aσ). Notation: Validity under admissible assignments: Aσ |=A ϕ ≤ ψ. Outline of the canonicity proof:

A |= ϕ ≤ ψ Aσ |= ϕ ≤ ψ

  • Aσ |=A ϕ ≤ ψ
  • Aσ |=A ALBA(ϕ ≤ ψ)

⇔ Aσ |= ALBA(ϕ ≤ ψ)

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 49

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Justifying the Ackermann rule

We need an Ackermann lemma which says: TFAE:

1

Aσ, v |= β(α) ≤ γ(α)

2

there exists some v′ ∼p v such that

Aσ, v′ |= α ≤ p and L, v′ |= β(p) ≤ γ(p).

for admissible assignments v and v′, where

α, β, γ ∈ L+,

with positivity/negativity as before.

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 50

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Justifying the Ackermann rule

We need an Ackermann lemma which says: TFAE:

1

Aσ, v |= β(α) ≤ γ(α)

2

there exists some v′ ∼p v such that

Aσ, v′ |= α ≤ p and L, v′ |= β(p) ≤ γ(p).

for admissible assignments v and v′, where

α, β, γ ∈ L+,

with positivity/negativity as before. PROBLEM: v(α) A.

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 51

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Justifying the Ackermann rule (2)

This would we possible if we could prove the following equivalent:

β(v(α)) ≤ γ(v(α))

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

slide-52
SLIDE 52

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Justifying the Ackermann rule (2)

This would we possible if we could prove the following equivalent:

β(v(α)) ≤ γ(v(α)) β(

  • {a ∈ A | v(α) ≤ a})

≤ γ(

  • {a ∈ A | v(α) ≤ a})

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

slide-53
SLIDE 53

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Justifying the Ackermann rule (2)

This would we possible if we could prove the following equivalent:

β(v(α)) ≤ γ(v(α)) β(

  • {a ∈ A | v(α) ≤ a})

≤ γ(

  • {a ∈ A | v(α) ≤ a})
  • {β(a) | v(α) ≤ a ∈ A}

≤

  • {γ(a) | v(α) ≤ a ∈ A})

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 54

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Justifying the Ackermann rule (2)

This would we possible if we could prove the following equivalent:

β(v(α)) ≤ γ(v(α)) β(

  • {a ∈ A | v(α) ≤ a})

≤ γ(

  • {a ∈ A | v(α) ≤ a})
  • {β(a) | v(α) ≤ a ∈ A}

≤

  • {γ(a) | v(α) ≤ a ∈ A})

β(a0) ≤ γ(a0)

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 55

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Justifying the Ackermann rule (2)

This would we possible if we could prove the following equivalent:

β(v(α)) ≤ γ(v(α)) β(

  • {a ∈ A | v(α) ≤ a})

≤ γ(

  • {a ∈ A | v(α) ≤ a})
  • {β(a) | v(α) ≤ a ∈ A}

≤

  • {γ(a) | v(α) ≤ a ∈ A})

β(a0) ≤ γ(a0)

Use classification of L+-terms as syntactically open / closed.

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

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SLIDE 56

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Sahlqvist inequalities

ϕ ≤ ψ

with ǫp = 1 and ǫq = ∂.

+ϕ −ψ +q −p −q +p +p +q

join preserving right adjoint

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

slide-57
SLIDE 57

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Inductive Inequalities

ϕ ≤ ψ

with ǫp = 1, and ǫq = ∂, and q <Ω p.

+ϕ −ψ +q −p −q +p +q +q +p

join preserving right adjoint right residual

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

slide-58
SLIDE 58

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities

Completeness for inductive inequalities

Theorem ALBA successfully reduces all inductive (and hence Sahlqvist) inequalities. Corollary All inductive (and hence Sahlqvist) inequalities are elementary and canonical.

Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics

slide-59
SLIDE 59

Correspondence Theory ALBA and the non-distributive lattice setting Sahlqvist and Inductive Inequalities Willem Conradie, Alessandra Palmigiano Algorithmic correspondence and canonicity for non-distributive logics