An algebraic approach to canonical formulas Nick Bezhanishvili - - PowerPoint PPT Presentation

An algebraic approach to canonical formulas

Nick Bezhanishvili Imperial College London Joint work with Guram Bezhanishvili New Mexico State University

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Jankov formulas

Refutation patterns play an important role in developing axiomatic bases for intermediate and modal logics. First algebra-based formulas for finite subdirectly irreducible Heyting algebras were constructed by Jankov in 1963. These are now called Jankov formulas. The Jankov formula of A is refuted on a Heyting algebra B iff A is a subalgebra of a homomorphic image of B. The Jankov formula of A axiomatizes the greatest variety that does not contain A.

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De Jongh formulas

De Jongh (1968) introduced what we now call de Jongh formulas (also called frame formulas). They are defined for finite rooted intuitionistic Kripke frames. The de Jongh formula of a frame F is refuted on a frame G iff F is a p-morphic image of a generated subframe of G We will see that via Esakia duality Jankov formulas are dual to de Jongh formulas Jankov formulas ⇐ ⇒Esakia Duality de Jongh formulas

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Fine and Rautenberg formulas

Fine (1974) defined what we now call Fine formulas (also frame formulas or Jankov-Fine formulas) for rooted transitive

• frames. They have the same property as de Jongh formulas

expressed in terms of p-morphisms and generated subframes. Rautenberg (1980) gave a Jankov style purely algebraic description of these formulas in terms of modal algebra subalgebras and homomorphisms. Jankov-Rautenberg formulas ⇐ ⇒Modal duality Fine formulas

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Canonical formulas

The formulas discussed so far axiomatize large classes of logics. However, there exist intermediate and transitive modal logics that are not axiomatized by these formulas. Zakharyaschev (1992) introduced canonical formulas and showed that every intermediate and transitive modal logic is axiomatized by canonical formulas. Canonical formulas are defined for a finite rooted frame F and a set D of antichains F. Unlike de Jongh formulas and Fine formulas in canonical formulas Zakharyaschev uses partial p-morphisms and the so called closed domain condition (CDC).

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Canonical formulas

The aim of this talk is to fill in question marks in the following diagram (for both intermediate and modal logics). ?? formulas ⇐ ⇒?? duality canonical formulas This will give a purely algebraic account of canonical formulas. In the intuitionistic case the algebraic side of the duality will be the category of Heyting algebras and (∧, →)-preserving maps. In the modal case the algebraic side will be the category of modal algebras and relativized homomorphisms.

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Jankov-Rautenberg formulas

Let A be a finite subdirectly irreducible (s.i.) Heyting algebra. Recall that A is s.i. iff there exists the second largest element s of A; that is, there exists s ∈ A such that s is the largest element of A − {1}. For each a ∈ A Jankov introduced a new variable pa and defined the Jankov formula χ(A) as follows: χ(A) = [{pa∧b ↔ pa ∧ pb : a, b ∈ A}∧ {pa∨b ↔ pa ∨ pb : a, b ∈ A}∧ {pa→b ↔ pa → pb : a, b ∈ A}∧ {p¬a ↔ ¬pa : a ∈ A}] → ps

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Jankov-Rautenberg formulas

Let A be a finite subdirectly irreducible K4-algebra, H = +(A), and t be the second largest element of H. We recall that the Jankov-Rautenberg formula of A is χ(A) =+[

• {pa∨b ↔ pa ∨ pb : a, b ∈ A}∧
• {pa∧b ↔ pa ∧ pb : a, b ∈ A}∧
• {p¬a ↔ ¬pa : a ∈ A}∧
• {p♦a ↔ ♦pa : a ∈ A}] → pt

Jankov’s Theorem. For each (Heyting or modal) algebra B we have B | = χ(A) iff A ∈ HS(B).

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Fine-De Jongh formulas

Fine-de Jongh Theorem. For each finite rooted Kripke frame F there exists a formula J(F) such that for each frame G we have G | = J(F) iff F is a p-morphic image of a generated subframe of G. We will link Jankov-Rautenberg formulas and Fine-de Jongh formulas via Esakia duality.

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Esakia Duality

Duality theory for Heyting and S4-algebras was developed by Esakia (1934-2010) in 1970s. This provides the bridge between the algebraic and model-theoretic technique. The bridge is topological in nature. Recall that a modal space is a triple (X, τ, R), where:

1

(X, τ) is a compact, Hausdorff, zero-dimensional space.

2

R is a binary relation.

3

R(x) is closed for each x ∈ X. Here R(x) = {y ∈ X : xRy}.

4

If U is clopen (closed and open), then so is RU. Here RU = {x ∈ X : R(x) ∩ U = ∅}. If R is a partial order, then we call (X, τ, R) an Esakia space.

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Esakia Duality

Also recall that given two modal (Esakia) spaces X and Y, a map f : X → Y is a modal (Esakia) morphism if:

1

f is continuous.

2

f is a p-morphism; that is, xRz implies f(x)Rf(z) for each x, z ∈ X. For each x ∈ X and y ∈ Y, if f(x)Ry, then there exists z ∈ X such that xRz and f(z) = y.

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Let MS (Esa) denote the category of modal (Esakia) spaces and modal (Esakia) morphisms. Let also MA (Heyt) denote the category of modal (Heyting) algebras and modal (Heyting) algebra homomorphisms Theorem: (Esakia, 1974) Heyt is dually equivalent to Esa. MA is dually equivalent to MS. For (a modal or Heyting) algebra A we let A∗ be its dual space. For (a modal or Esakia) space X we let X∗ be its dual algebra.

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Duality

Homomorphic images ⇐ ⇒Duality Generated subframes Subalgebras ⇐ ⇒Duality p-morphic images Jankov formulas ⇐ ⇒Duality de Jongh formulas Jankov-Rautenberg form. ⇐ ⇒Duality Fine form.

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Generalized Esakia Duality

Now let A and B be two Heyting algebras and h : A → B be a

• map. In order for h to be a morphism of Heyt, h should preserve

all Heyting algebra operations. It is useful to work with such h : A → B which only preserve ∧, →, and 1. In fact, as long as h preserves →, it automatically preserves 1! Therefore, we are interested in (∧, →)-homomorphisms; that is, those h : A → B for which we have h(a ∧ b) = h(a) ∧ h(b) and h(a → b) = h(a) → h(b).

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Generalized Esakia Duality

Let h : A → B be a (∧, →)-homomorphism. If h(0) = 0, then we call h a (∧, →, 0)-homomorphism; and if h(a ∨ b) = h(a) ∨ h(b), then we call h a (∧, →, ∨)-homomorphism. Clearly h is a Heyting algebra homomorphism iff h is a (∧, →, ∨, 0)-homomorphism. This outlook provides us with three new categories:

1

Heyt(∧,→) is the category of Heyting algebras and (∧, →)-homomorphisms.

2

Heyt(∧,→,0) is the category of Heyting algebras and (∧, →, 0)-homomorphisms.

3

Heyt(∧,→,∨) is the category of Heyting algebras and (∧, →, ∨)-homomorphisms.

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Generalized Esakia Duality

Given Heyting algebras A and B, we need to give the dual description of a (∧, →)-homomorphism h : A → B. It turns out that (∧, →)-homomorphisms h : A → B can dually be characterized by special partial maps from the Esakia dual B∗

• f B to the Esakia dual A∗ of A.

Let h : A → B be a (∧, →)-homomorphism. Unlike the case of Heyting algebra homomorphisms, the inverse image h−1(x) of a prime filter x of B may not be a prime filter of A. And this is exactly why the dual of h is a partial function!

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Generalized Esakia Duality

Let X and Y be Esakia spaces and f : X → Y a partial map. Let dom(f) denote the domain of f. We call f a partial Esakia morphism provided that:

1

If x, z ∈ dom(f) and x ≤ z, then f(x) ≤ f(z).

2

If x ∈ dom(f), y ∈ Y, and f(x) ≤ y, then there exists z ∈ dom(f) such that x ≤ z and f(z) = f(y).

3

x ∈ dom(f) iff f[↑x] = ↑y for some y ∈ Y.

4

f[↑x] is a closed subset of Y.

5

If U is a clopen upset of Y, then X − ↓f −1(Y − U)) is a clopen upset of X.

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Generalized Esakia Duality

Let EsaP denote the category of Esakia spaces and partial Esakia

• morphisms. Note that the composition of partial Esakia

morphisms is not a composition of the partial maps. Theorem: Heyt(∧,→) is dually equivalent to EsaP.

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Subframe formulas

Let A be a finite subdirectly irreducible (s.i.) Heyting algebra with the second largest element s. α(A) = [{pa∧b ↔ pa ∧ pb : a, b ∈ A}∧ {pa→b ↔ pa → pb : a, b ∈ A}] → ps (∧, →)-Jankov formulas ⇐ ⇒Gen ES Duality subframe formulas Theorem.

1

A HA B | = α(A) iff there exists a homomorphic image C of B and a 1-1 (∧, →)-embedding of A into C.

2

An ES Y | = α(A) iff there exists a generated subframe Z of Y and a partial Esakia morphism from Y onto A∗.

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Cofinal morphisms

A subset Z of a poset X is called cofinal if for each x ∈ X there is y ∈ Y such that xRy. A partial map f : X → Y is called cofinal if dom(f) is cofinal.

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Cofinal subframe formulas

Let A be a finite subdirectly irreducible (s.i.) Heyting algebra with the second largest element s. α(A, ⊥) = [{pa∧b ↔ pa ∧ pb : a, b ∈ A}∧ {pa→b ↔ pa → pb : a, b ∈ A}∧ {p¬a ↔ ¬pa : a ∈ A}] → ps (∧, →, ⊥)-Jankov form. ⇐ ⇒Gen ES Duality cofinal subframe form. Theorem.

1

A HA B | = α(A, ⊥) iff there exists a homomorphic image C

• f B and a 1-1 (∧, →, 0)-embedding of A into C.

2

An ES Y | = α(A, ⊥) iff there exists a generated subframe Z

• f Y and a cofinal partial Esakia morphism from Y onto A∗.

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Closed Domain Condition

Let X and Y be Esakia spaces and let f : X → Y be a partial Esakia morphism. Let also D be a (possibly empty) set of antichains in Y. We say that f satisfies the closed domain condition (CDC) for D if: f(R(x)) = R(d) for some d ∈ D implies x ∈ dom(f).

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Canonical formulas

Let A be a finite subdirectly irreducible (s.i.) Heyting algebra with the second largest element s and D ⊆ A2 α(A, D, ⊥) = [{pa∧b ↔ pa ∧ pb : a, b ∈ A}∧ {pa→b ↔ pa → pb : a, b ∈ A}∧ {p¬a ↔ ¬pa : a ∈ A}∧ {pa∨b ↔ pa ∨ pb : (a, b) ∈ D}] → ps

• alg. canonical formulas ⇐

⇒Gen ES Duality Zakh. canonical formulas Theorem.

1

A HA B | = α(A, D, ⊥) iff there exists a homomorphic image C of B and a (∧, →, 0)-embedding h : A → C such that h(a ∨ b) = h(a) ∨ h(b) for each (a, b) ∈ D.

2

An ES Y | = α(A, D, ⊥) iff there exists a generated subframe Z of Y and a cofinal partial Esakia morphism from Y onto A∗ satisfying (CDC) for D = D∗.

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An Algebraic Account of Canonical Formulas

Let ϕ be a formula such that IPC ⊢ ϕ. Using the Diego theorem we can find finite s.i. Heyting algebras A1, . . . Ak and Di ⊆ A2

i such that

IPC + ϕ = IPC + α(A1, D1, ⊥) ∧ · · · ∧ α(Ak, Dk, ⊥) As a consequence we obtain a purely algebraic proof of Zakharyaschev’s theorem: Theorem: (Zakharyaschev) Each intermediate logic is axiomatizable by canonical formulas. Moreover, if an intermediate logic is finitely axiomatizable, then it is finitely axiomatizable by canonical formulas.

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References

• G. Bezhanishvili and S. Ghilardi, An algebraic approach to

subframe logics. Intuitionistic case. Ann. Pure Appl. Logic, 147(1-2):84100, 2007

• G. Bezhanishvili and N. Bezhanishvili, An algebraic

approach to canonical formulas: Intuitionistic case. Review

• f Symbolic Logic, vol 2, number 3, pp. 517-549, 2009.

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Modal Case

Let B be a Boolean algebra and s ∈ B. Then [0, s] = {x ∈ B : 0 x s} also forms a Boolean algebra which we denote by Bs. The Boolean operations on Bs are defined as follows:

1

x ∧s y = x ∧ y;

2

x ∨s y = x ∨ y;

3

0s = 0 and 1s = s;

4

¬sx = ¬x ∧ s. We call Bs the relativization of B to s.

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Relativized homomorphisms

Let A be a modal algebra and s ∈ A. We define ♦s : As → As by ♦sx = s ∧ ♦x for each x ∈ As. Then As is a modal algebra. Instead of modal algebra homomorphisms we will work with relativized modal algebra homomorphisms; that is, maps η : A → B such that η is a modal algebra homomorphism from A to the relativized modal algebra Bη(1).

• Theorem. η is a relativized modal algebra homomorphism iff η

preserves ∧, ∨, 0 and η(♦a) = ♦η(1)η(a) for each a ∈ A.

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Extended modal duality

Modal algebras and relativized homomorphisms form a category which we denote by MAR. Let X and Y be Stone spaces and f : X → Y be a partial map. We call f a partial continuous map if dom(f) is a clopen subset of X and f is a continuous map from dom(f) to Y. Let (X, R) and (Y, Q) be modal spaces and f : X → Y be a partial continuous map. We call f a partial continuous p-morphism if in addition f satisfies:

1

x, z ∈ dom(f) and xRz imply f(x)Qf(z),

2

x ∈ dom(f) and f(x)Qy imply there exists z ∈ dom(f) such that xRz and f(z) = y.

• Theorem. MAR is dually equivalent to MSP.

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Subframe formulas

Let A be a finite subdirectly irreducible K4-algebra. Then H = +(A) is a subdirectly irreducible Heyting algebra, hence H has the second largest element which we denote by t. Let A be a s.i. K4-algebra. For each a ∈ A we introduce a new variable pa and define the subframe formula of A as αs(A) =+[ (⊥ ↔ p0)∧

• {pa∨b ↔ pa ∨ pb : a, b ∈ A}∧
• {pa∧b ↔ pa ∧ pb : a, b ∈ A}∧
• {p♦a ↔ ♦p1pa : a ∈ A}] → (p1 → pt)

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Subframe formulas

Let A be a finite subdirectly irreducible K4-algebra.

1

For each K4-algebra B, we have B | = αs(A) iff there exist a homomorphic image C of B and a 1-1 relativized homomorphism from A into C.

2

For each transitive space X, we have X | = αs(A) iff there exist a generated subframe Y of X and a partial continuous p-morphism from Y onto A∗.

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Cofinal relativizations

We say that η is cofinal if ♦+η(1) = 1. A partial p-morphism f : X → Y is called cofinal if dom(f) is

• cofinal. That is, for each x ∈ dom(f) there exists y ∈ dom(f) with

xR+y. Let A and B be K4-algebras and let η : A → B be a relativized

• homomorphism. Then: η is cofinal iff η∗ = η−1 : B∗ → A∗ is

cofinal.

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Cofinal subframe formulas

The cofinal subframe formula of a finite subdirectly irreducible K4-algebra A is defined as αcs(A) =+[ (⊤ ↔ ♦+p1) ∧ (⊥ ↔ p0)∧

• {pa∨b ↔ pa ∨ pb : a, b ∈ A}∧
• {pa∧b ↔ pa ∧ pb : a, b ∈ A}∧
• {p♦a ↔ ♦p1pa : a ∈ A}] → (p1 → pt)

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Let A be a finite subdirectly irreducible K4-algebra.

1

For each K4-algebra B, we have B | = αcs(A) iff there exist a homomorphic image C of B and a 1-1 cofinal relativized homomorphism from A into C.

2

For each transitive space X, we have X | = αcs(A) iff there exist a generated subframe Y of X and a cofinal partial continuous p-morphism from Y onto A∗.

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Closed Domain Condition

It is well known that the relativization As of A to s is also a K4-algebra. Let X and Y be transitive spaces and let f : X → Y be a partial continuous p-morphism. Let also D be a (possibly empty) set of antichains in Y. We say that f satisfies the closed domain condition (CDC) for D if: f(R(x)) = R+(d) for some d ∈ D implies x ∈ dom(f). R+ is the reflexive closure of R.

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Closed Domain Condition

Let (X, R), (Y, Q) be transitive spaces, f : X → Y be a partial continuous p-morphism, and U be a clopen subset of Y. We let DU = {minfR(x) : fR(x) ∩ U = ∅}. Then the following two conditions are equivalent:

1

♦Rf −1(U) ⊆ f −1♦Q(U).

2

f satisfies (CDC) for DU. Let A and B be K4-algebras, η : A → B be a relativized homomorphism, and D ⊆ A. Then the following two conditions are equivalent:

1

η(♦a) = ♦η(a) for each a ∈ D.

2

η∗ : B∗ → A∗ satisfies (CDC) for D = {Dϕ(a) : a ∈ D}.

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Canonical formulas

Let D be a subset of A. We define the canonical formula α(A, D) associated with A and D as follows: α(A, D) =+[ (⊤ ↔ ♦+p1) ∧ (⊥ ↔ p0)∧

• {pa∨b ↔ pa ∨ pb : a, b ∈ A}∧
• {pa∧b ↔ pa ∧ pb : a, b ∈ A}∧
• {p♦a ↔ ♦p1pa : a ∈ A}∧
• {p♦a ↔ ♦pa : a ∈ D}] → (p1 → pt)

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Canonical formulas

• Theorem. Let A be a finite subdirectly irreducible K4-algebra,

D ⊆ A.

1

Then for each K4-algebra B we have B | = α(A, D) iff there exist a homomorphic image C of B and a 1-1 modal algebra homomorphism η from A into a cofinal relativization Cs of C such that η(♦a) = ♦η(a) for each a ∈ D.

2

Let D = {Dϕ(a) : a ∈ D} be the set of antichains in A∗ associated with D. Then for each transitive space X, we have X | = α(A, D) iff there exist a closed upset Y of X and an onto cofinal partial continuous p-morphism f : Y → A∗ such that f satisfies (CDC) for D.

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Canonical formulas

Let ϕ be a formula such that K4 ⊢ ϕ. We can find finite s.i. K4-algebras A1, . . . Ak and Di ⊆ Ai such that K4 + ϕ = K4 + α(A1, D1, ⊥) ∧ · · · ∧ α(Ak, Dk, ⊥) The proof is the same as in (G. Bezhanishvili, Ghilardi, Jibladze, 2011). It uses essentially Diego’s theorem. As a consequence we obtain a purely algebraic proof of Zakharyaschev’s theorem: Theorem: (Zakharyaschev) Each transitive modal logic is axiomatizable by canonical formulas. Moreover, if a transitive modal logic is finitely axiomatizable, then it is finitely axiomatizable by canonical formulas.

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Jankov-Rautenberg formulas

α(A, A) is equivalent to the Jankov-Rautenberg formula χ(A). Let χ′(A) =+[ (⊤ ↔ p1) ∧ (⊥ ↔ p0)∧

• {pa∨b ↔ pa ∨ pb : a, b ∈ A}∧
• {pa∧b ↔ pa ∧ pb : a, b ∈ A}∧
• {p♦a ↔ ♦pa : a ∈ A}] → pt

Let A be a finite subdirectly irreducible K4-algebra and let B be a K4-algebra. The following three conditions are equivalent

1

B | = χ(A),

2

B | = χ′(A),

3

B | = α(A, A).

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References

• G. Bezhanishvili, S. Ghilardi, M. Jibladze, An algebraic

approach to subframe logics. Modal case. Notre Dame Journal of Formal Logic, 52(2):187202, 2011.

• G. Bezhanishvili and N. Bezhanishvili, An algebraic

approach to canonical formulas. Modal case. Submitted, 2011.

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Conclusions and future work

We developed an algebraic approach to canonical formulas for Heyting algebras and K4-algebras. Can we generalize this approach for logics below K4? wK4! We also have a different version of canonical formulas when we take ∧, ∨ as primitives and treat → as an additional

• perator. The analogous technique works for modal logic.

Do similar results hold for other non-classical logics?

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