Classifying Unification Problems in Preliminaries Algebraic - - PowerPoint PPT Presentation

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Classifying Unification Problems in Distributive Lattices and Kleene Algebras

Leonardo Manuel Cabrer University of Bern

(joint work with Simone Bova)

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Preliminaries

Algebraic Unifiers

[1]

• S. Ghilardi. Unification through Projectivity.

Journal Logic and computation 7(4), 1997.

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Preliminaries

Algebraic Unifiers

[1]

• S. Ghilardi. Unification through Projectivity.

Journal Logic and computation 7(4), 1997. Given an algebraic language L, a unification problem in the language L is a finite set of equations S = {(s1, t1), . . . , (sn, tn)} ⊆ Term2

L.

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Preliminaries

Algebraic Unifiers

[1]

• S. Ghilardi. Unification through Projectivity.

Journal Logic and computation 7(4), 1997. Given an algebraic language L, a unification problem in the language L is a finite set of equations S = {(s1, t1), . . . , (sn, tn)} ⊆ Term2

L.

Given a unification problem S and an equational theory E, an algebraic E-unifier for S is pair (h, P) where P is a projective algebra in the equational class determined by E and h: Fp(S) → P is a homomorphism.

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Preliminaries

Algebraic Unifiers

If (h1, P1), (h2, P2) are algebraic E-unifiers for S, we say that (h1, P1) is more general than (h2, P2) ((h2, P2) (h1, P1)) if there exists a homomorphism f : P1 → P2 such that Fp(S)

h1

• h2
• P1

f

• P2

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Preliminaries

Algebraic Unifiers

If (h1, P1), (h2, P2) are algebraic E-unifiers for S, we say that (h1, P1) is more general than (h2, P2) ((h2, P2) (h1, P1)) if there exists a homomorphism f : P1 → P2 such that Fp(S)

h1

• h2
• P1

f

• P2

We denote by UE(S) the pre-ordered set of algebraic E-unifiers for S.

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Preliminaries

Unification types

A unification problem S in an equational theory E is said to have type: UE(S) 1

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Preliminaries

Unification types

A unification problem S in an equational theory E is said to have type: UE(S) 1 ω

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Preliminaries

Unification types

A unification problem S in an equational theory E is said to have type: UE(S) 1 ω ∞

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Preliminaries

Unification types

A unification problem S in an equational theory E is said to have type: UE(S) 1 ω ∞

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Preliminaries

Unification types

A equational theory E is said to have type:

◮ 1 if every unification problem S has type 1,

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Preliminaries

Unification types

A equational theory E is said to have type:

◮ 1 if every unification problem S has type 1, ◮ ω if every unification problem S has type ω and at

least one S has not unification type 1,

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Preliminaries

Unification types

A equational theory E is said to have type:

◮ 1 if every unification problem S has type 1, ◮ ω if every unification problem S has type ω and at

least one S has not unification type 1,

◮ ∞ if every unification problem S has type 1, ω or ∞

and at least one S has unification type ∞,

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Preliminaries

Unification types

A equational theory E is said to have type:

◮ 1 if every unification problem S has type 1, ◮ ω if every unification problem S has type ω and at

least one S has not unification type 1,

◮ ∞ if every unification problem S has type 1, ω or ∞

and at least one S has unification type ∞,

◮ 0 if at least one S has unification type 0.

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Unifiers through duality

Duals of Unifiers

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Unifiers through duality

Duals of Unifiers

Fp(S)

u

P

• I

f

D(Fp(S))

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Unifiers through duality

Working strategy

(I) Description of finitely generated projective algebras. Injective objects.

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Unifiers through duality

Working strategy

(I) Description of finitely generated projective algebras. Injective objects. (II) Analysis of the unification type. Examples

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Unifiers through duality

Working strategy

(I) Description of finitely generated projective algebras. Injective objects. (II) Analysis of the unification type. Examples (III) Classification of a given unification problem. Analysis

• f the examples

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Bounded Distributive Lattices

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Bounded Distributive Lattices

(I) Description of finitely generated projectives

A finite bounded distributive lattice L is projective if and

• nly if J(L), ≤ is a lattice.

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Bounded Distributive Lattices

(II) Analysis of the unification type

The unification problem S = {x ∧ y ≈ z ∨ t} has nullary unification type.

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Bounded Distributive Lattices

(II) Analysis of the unification type

The unification problem S = {x ∧ y ≈ z ∨ t} has nullary unification type. x y z t 1

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Bounded Distributive Lattices

(II) Analysis of the unification type

The unification problem S = {x ∧ y ≈ z ∨ t} has nullary unification type. x y z t 1 a b c d

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Bounded Distributive Lattices

(III): Causes of nullarity

Lemma

Let S be a unification problem in the language of bounded

• lattices. If there exist x, a, b, c, d, y ∈ J(Fp(S)) satisfying:

(i) x ≤ a, b ≤ c, d ≤ y, and (ii) it does not exist e ∈ J(Fp(S)) such that a, b ≤ e ≤ c, d, then the unification type of S in the equational theory of distributive lattices is nullary.

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Bounded Distributive Lattices

Classification

Theorem

Let S be a unification problem in the language of bounded lattices. Then the unification type of S is: Unitary if and only if J(Fp(S)) is a lattice,

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Bounded Distributive Lattices

Classification

Theorem

Let S be a unification problem in the language of bounded lattices. Then the unification type of S is: Unitary if and only if J(Fp(S)) is a lattice, Finitary if and only if for every x, y ∈ J(Fp(S)) the interval [x, y] is a lattice,

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Bounded Distributive Lattices

Classification

Theorem

Let S be a unification problem in the language of bounded lattices. Then the unification type of S is: Unitary if and only if J(Fp(S)) is a lattice, Finitary if and only if for every x, y ∈ J(Fp(S)) the interval [x, y] is a lattice, Nullary otherwise.

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

Definition

A Kleene algebra A = (A, ∧, ∨, ¬, 0, 1) is a bounded distributive lattice equipped with a unary operation, ¬x, satisfying: x = ¬¬x, x ∧ y = ¬(¬x ∨ ¬y), x ∧ ¬x ≤ y ∨ ¬y.

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

Definition

A Kleene algebra A = (A, ∧, ∨, ¬, 0, 1) is a bounded distributive lattice equipped with a unary operation, ¬x, satisfying: x = ¬¬x, x ∧ y = ¬(¬x ∨ ¬y), x ∧ ¬x ≤ y ∨ ¬y. 1 a K

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

Natural duality

[2] Davey, B.A. & Werner, H. Piggyback-Dualitäten,

• Bull. Austral. Math. Soc.32, 1-32 (1985).

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

Natural duality

[2] Davey, B.A. & Werner, H. Piggyback-Dualitäten,

• Bull. Austral. Math. Soc.32, 1-32 (1985).

Definition

A structure X = X, ≤, ∼, Y, τ is called a Kleene space if it satisfies the following conditions: (i) X, ≤, τ is a Priestley space,

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

Natural duality

[2] Davey, B.A. & Werner, H. Piggyback-Dualitäten,

• Bull. Austral. Math. Soc.32, 1-32 (1985).

Definition

A structure X = X, ≤, ∼, Y, τ is called a Kleene space if it satisfies the following conditions: (i) X, ≤, τ is a Priestley space, (ii) ∼ is a closed binary relation, i.e., ∼ is a closed subset of X 2,

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

Natural duality

[2] Davey, B.A. & Werner, H. Piggyback-Dualitäten,

• Bull. Austral. Math. Soc.32, 1-32 (1985).

Definition

A structure X = X, ≤, ∼, Y, τ is called a Kleene space if it satisfies the following conditions: (i) X, ≤, τ is a Priestley space, (ii) ∼ is a closed binary relation, i.e., ∼ is a closed subset of X 2, (iii) Y is a closed subset of X,

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

Natural duality

[2] Davey, B.A. & Werner, H. Piggyback-Dualitäten,

• Bull. Austral. Math. Soc.32, 1-32 (1985).

Definition

A structure X = X, ≤, ∼, Y, τ is called a Kleene space if it satisfies the following conditions: (i) X, ≤, τ is a Priestley space, (ii) ∼ is a closed binary relation, i.e., ∼ is a closed subset of X 2, (iii) Y is a closed subset of X, and (iv) for every x, y, z ∈ X:

(a) x ∼ x, (b) if x ∼ y and x ∈ Y, then x ≤ y, (c) if x ∼ y and y ≤ z, then z ∼ x.

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

Natural duality

K a 1 a 1 ˜ K {≤, ∼, Y}

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

(I) Duals of Projectives

Theorem

Let A be a finite Kleene Algebra. Then the following statements are equivalent: (i) A is projective, (ii) XK(A) = {XA, ≤A, ∼A, YA, τA} satisfies the following conditions:

(a) XA, ≤A is a meet semi-lattice, (b) YA = Max(XA, ≤A), (c) XA is 2-conditionally complete, (d) x ∼A y if and only if there exists z ∈ XA such that x, y ≤ z.

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

(II): Nullarity

The unification problem S = {x ∧ ¬x ≈ y ∨ z} has nullary unification type.

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

(II): Nullarity

The unification problem S = {x ∧ ¬x ≈ y ∨ z} has nullary unification type. x ¬x y z

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

(II): Nullarity

The unification problem S = {x ∧ ¬x ≈ y ∨ z} has nullary unification type. x ¬x y z

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

(III): Necessary Conditions for Nullarity

Lemma

Let S be a unification problem in the language of kleene

• algebras. If there exist x, a, b, c, d, y, z ∈ XK(Fp(S))

satisfying: (i) x ≤ a, b ≤ c, d, c ≤ y and d ≤ z, (ii) y, z ∈ Y, and (iii) it does not exist e ∈ XK(Fp(S)) such that a, b ≤ e ≤ c, d, then the unification type of S is nullary.

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

(III): Necessary Conditions for Nullarity

Lemma

Let S be a unification problem in the language of kleene algebras with. If there exist w, a, b, c, d, e, f, x, y, z ∈ XK(Fp(S)) satisfying: (i) w ≤ a, b, c; a ≤ d, e; b ≤ d, f; c ≤ e, f; d ≤ x; e ≤ y; and f ≤ z, (ii) x, y, z ∈ Y, and (iii) it does not exists g ∈ XK(Fp(S)) such that a, b, c ≤ g, then the unification type of S is nullary.

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

(III): Necessary Conditions for Nullarity

Lemma

Let S be a unification problem in the language of kleene algebras with. If there exist w, a, b, c, d, e, f, x, y, z ∈ XK(Fp(S)) satisfying: (i) w ≤ a, b, c; a ≤ d, e; b ≤ d, f; c ≤ e, f; d ≤ x; e ≤ y; and f ≤ z, (ii) x, y, z ∈ Y, and (iii) it does not exists g ∈ XK(Fp(S)) such that a, b, c ≤ g, then the unification type of S is nullary.

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

Classification

Theorem

Let S be a unification problem. Then the unification type

• f S over the equational theory of Kleene algebras is:

unitary if and only if the set K = {x ∈ XK(Fp(S)) | ∃y ∈ Y, x ≤ y} is a 2-conditionally complete meet semilattice,

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

Classification

Theorem

Let S be a unification problem. Then the unification type

• f S over the equational theory of Kleene algebras is:

unitary if and only if the set K = {x ∈ XK(Fp(S)) | ∃y ∈ Y, x ≤ y} is a 2-conditionally complete meet semilattice, finitary if and only if for x ∈ K the set {y ∈ K | x ≤ y} is a 2-conditionally complete meet semilattice,

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Kleene Algebras

Classification

Theorem

Let S be a unification problem. Then the unification type

• f S over the equational theory of Kleene algebras is:

unitary if and only if the set K = {x ∈ XK(Fp(S)) | ∃y ∈ Y, x ≤ y} is a 2-conditionally complete meet semilattice, finitary if and only if for x ∈ K the set {y ∈ K | x ≤ y} is a 2-conditionally complete meet semilattice, nullary otherwise.

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries

Algebraic Unifiers Unification types

Unifiers through duality

Duals of Unifiers Working strategy

Bounded Distributive Lattices

(I) Description of finitely generated projectives (II) Analysis of the unification type (III): Causes of nullarity Classification

Kleene Algebras

Natural duality (I) Duals of Projectives (II): Nullarity (III): Necessary Conditions for Nullarity Classification

Classifying Unification Problems in Distributive Lattices and Kleene Algebras Thank you for your attention!

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