Classifying Unification Problems in Preliminaries Algebraic - PowerPoint PPT Presentation

Classifying Unification Problems in Distributive Lattices and Kleene Algebras L.M. Cabrer Classifying Unification Problems in Preliminaries Algebraic Unifiers Distributive Lattices and Kleene Algebras Unification types Unifiers through duality Duals of Unifiers Working strategy Leonardo Manuel Cabrer Bounded Distributive Lattices (I) Description of finitely generated projectives (II) Analysis of the University of Bern unification type (III) : Causes of nullarity Classification (joint work with Simone Bova) Kleene Algebras Natural duality (I) Duals of Projectives (II) : Nullarity (III) : Necessary Conditions for Nullarity Classification

Classifying Preliminaries Unification Problems in Algebraic Unifiers Distributive Lattices and Kleene Algebras L.M. Cabrer [1] S. Ghilardi. Unification through Projectivity. Journal Logic and computation 7(4), 1997. 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 Preliminaries Unification Problems in Algebraic Unifiers Distributive Lattices and Kleene Algebras L.M. Cabrer [1] S. Ghilardi. Unification through Projectivity. Journal Logic and computation 7(4), 1997. Preliminaries Algebraic Unifiers Unification types Unifiers through duality Given an algebraic language L , a unification problem in Duals of Unifiers Working strategy the language L is a finite set of equations Bounded S = { ( s 1 , t 1 ) , . . . , ( s n , t n ) } ⊆ Term 2 L . 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 Preliminaries Unification Problems in Algebraic Unifiers Distributive Lattices and Kleene Algebras L.M. Cabrer [1] S. Ghilardi. Unification through Projectivity. Journal Logic and computation 7(4), 1997. Preliminaries Algebraic Unifiers Unification types Unifiers through duality Given an algebraic language L , a unification problem in Duals of Unifiers Working strategy the language L is a finite set of equations Bounded S = { ( s 1 , t 1 ) , . . . , ( s n , t n ) } ⊆ Term 2 L . Distributive Lattices (I) Description of finitely generated projectives (II) Analysis of the unification type (III) : Causes of nullarity Given a unification problem S and an equational theory Classification E , an algebraic E -unifier for S is pair ( h , P ) where P is a Kleene Algebras Natural duality projective algebra in the equational class determined by (I) Duals of Projectives (II) : Nullarity E and h : Fp ( S ) → P is a homomorphism. (III) : Necessary Conditions for Nullarity Classification

� � � Classifying Preliminaries Unification Problems in Algebraic Unifiers Distributive Lattices and Kleene Algebras L.M. Cabrer If ( h 1 , P 1 ) , ( h 2 , P 2 ) are algebraic E -unifiers for S , we say Preliminaries that ( h 1 , P 1 ) is more general than ( h 2 , P 2 ) Algebraic Unifiers ( ( h 2 , P 2 ) � ( h 1 , P 1 ) ) if there exists a homomorphism Unification types Unifiers through f : P 1 → P 2 such that duality Duals of Unifiers Working strategy h 1 Fp ( S ) P 1 Bounded Distributive � Lattices � � � f (I) Description of finitely � h 2 � generated projectives � � (II) Analysis of the P 2 unification type (III) : Causes of nullarity Classification Kleene Algebras Natural duality (I) Duals of Projectives (II) : Nullarity (III) : Necessary Conditions for Nullarity Classification

� � � Classifying Preliminaries Unification Problems in Algebraic Unifiers Distributive Lattices and Kleene Algebras L.M. Cabrer If ( h 1 , P 1 ) , ( h 2 , P 2 ) are algebraic E -unifiers for S , we say Preliminaries that ( h 1 , P 1 ) is more general than ( h 2 , P 2 ) Algebraic Unifiers ( ( h 2 , P 2 ) � ( h 1 , P 1 ) ) if there exists a homomorphism Unification types Unifiers through f : P 1 → P 2 such that duality Duals of Unifiers Working strategy h 1 Fp ( S ) P 1 Bounded Distributive � Lattices � � � f (I) Description of finitely � h 2 � generated projectives � � (II) Analysis of the P 2 unification type (III) : Causes of nullarity Classification Kleene Algebras Natural duality We denote by U E ( S ) the pre-ordered set of algebraic (I) Duals of Projectives (II) : Nullarity E -unifiers for S . (III) : Necessary Conditions for Nullarity Classification

Classifying Preliminaries Unification Problems in Unification types Distributive Lattices and A unification problem S in an equational theory E is said Kleene Algebras to have type: L.M. Cabrer Preliminaries U E ( S ) Algebraic Unifiers Unification types Unifiers through duality 1 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 Preliminaries Unification Problems in Unification types Distributive Lattices and A unification problem S in an equational theory E is said Kleene Algebras to have type: L.M. Cabrer Preliminaries U E ( S ) Algebraic Unifiers Unification types Unifiers through duality 1 Duals of Unifiers Working strategy Bounded Distributive Lattices (I) Description of finitely ω 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 Preliminaries Unification Problems in Unification types Distributive Lattices and A unification problem S in an equational theory E is said Kleene Algebras to have type: L.M. Cabrer Preliminaries U E ( S ) Algebraic Unifiers Unification types Unifiers through duality 1 Duals of Unifiers Working strategy Bounded Distributive Lattices (I) Description of finitely ω 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 Preliminaries Unification Problems in Unification types Distributive Lattices and A unification problem S in an equational theory E is said Kleene Algebras to have type: L.M. Cabrer Preliminaries U E ( S ) Algebraic Unifiers Unification types Unifiers through duality 1 Duals of Unifiers Working strategy Bounded Distributive Lattices (I) Description of finitely ω generated projectives (II) Analysis of the unification type (III) : Causes of nullarity Classification ∞ Kleene Algebras Natural duality (I) Duals of Projectives (II) : Nullarity 0 (III) : Necessary Conditions for Nullarity Classification

Classifying Preliminaries Unification Problems in Unification types Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries Algebraic Unifiers A equational theory E is said to have type: Unification types Unifiers through ◮ 1 if every unification problem S has type 1, 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 Preliminaries Unification Problems in Unification types Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries Algebraic Unifiers A equational theory E is said to have type: Unification types Unifiers through ◮ 1 if every unification problem S has type 1, duality Duals of Unifiers ◮ ω if every unification problem S has type ω and at Working strategy Bounded least one S has not unification type 1, 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 Preliminaries Unification Problems in Unification types Distributive Lattices and Kleene Algebras L.M. Cabrer Preliminaries Algebraic Unifiers A equational theory E is said to have type: Unification types Unifiers through ◮ 1 if every unification problem S has type 1, duality Duals of Unifiers ◮ ω if every unification problem S has type ω and at Working strategy Bounded least one S has not unification type 1, Distributive Lattices ◮ ∞ if every unification problem S has type 1, ω or ∞ (I) Description of finitely generated projectives and at least one S has unification type ∞ , (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