Two Views on Building Conceptual Hierarchies Rainer Osswald Intelligente Informations- and Kommunikationssysteme Fachbereich Informatik FernUniversit¨ at in Hagen http://pi7.fernuni-hagen.de/osswald
Building Conceptual Lattices – Version I Formal Concept Analysis (FCA) [Ganter & Wille 1999] ◮ Formal context K = � U , Σ , � K � U set of objects, Σ set of attributes, � K ⊆ U × Σ satisfaction relation V ⊲ = { p ∈ Σ | x � K p for all x ∈ V } intent of V ⊆ U X ⊳ = { x ∈ U | x � K p for all p ∈ X } extent of X ⊆ Σ ◮ Formal concept: � V , X � with V ⊲ = X , X ⊳ = V ◮ Concept lattice: ordered set of formal concepts �{ J , M , F , A } , ∅ � � V , X � ≤ � W , Y � iff V ⊆ W , X ⊇ Y ◮ Example human biped featherless �{ J , M , A } , { b }� �{ J , M , F } , { f }� John x x x x x x Mary Fido x Alex x �{ J , M } , { h , f , b }� Complete implicational theory of context: human → biped ∧ featherless , biped ∧ featherless → human ◮ Basic Theorem of FCA Every concept lattice is a complete lattice and every complete lattice is representable as the concept lattice of a formal context. – 1 – Rainer Osswald
Building Conceptual Lattices – Version IIa O NTOLOG [Oldager 2000, Fischer-Nilsson 2001] ◮ Logico-algebraic language for specifying taxonomies/ontologies p ∈ Σ (set of basic concepts), null , univ , univ φ + ψ (conceptual sum), φ × ψ (conceptual product), φ ≤ ψ (subsumption), φ = ψ (equality) featherless + biped ◮ Distributive lattice of concepts via equational specification Example human = biped × featherless biped featherless ◮ Attributive descriptions (not taken into account here) human r : φ (Peirce Product) ◮ Relation to FCA? null “In some sense this approach can be compared to Formal Concept Analysis, since both are mathematical methods for generating concept lattices. However, our approach is fundamentally different, because O NTOLOG is a language for specifying taxonomies while the Formal Concept Analysis is a mathematical theory for analysis of conceptual data (the relations between objects and their attributes).” [Oldager 2000] – 2 – Rainer Osswald
Building Conceptual Lattices – Version IIb Oles (2000): An Application of Lattice Theory to Knowledge Representation ⊤ ◮ Knowledge base (KB): set of terminological axioms newconcept I ( I primitive identifier), b ∨ f subconcept C 1 C 2 ( C i built by ∧ , ∨ from primitives, ⊤ , ⊥ ) ◮ Example newconcept human ; f b newconcept biped ; newconcept featherless ; subconcept human ( biped ∧ featherless ) ; h subconcept ( biped ∧ featherless ) human ; ◮ Lattice of semantic concepts: ⊥ distributive lattice presented by generators and relations given by KB “Birkhoff implementation” by ordered set of ∨ -irreducibles ◮ Relation to FCA? “Right from the start, let us say what this paper is not about. It is not about the formal concept analysis of Wille, which deals with the derivation, from tables of attributes for individuals, of hierarchies of what are also called concepts. Wille’s hierarchies are lattices, but they are not necessarily distributive. There are no clear connections between Wille’s work and the contents of this paper.” [Oles 2000] – 3 – Rainer Osswald
Overview ◮ A Simple Propositional Framework ◮ Theories and Information Domains ◮ Formal Concept Analysis Reanalyzed ◮ The Lindenbaum Algebra of a Theory ◮ The Predicational Viewpoint ◮ Perspectives – 4 – Rainer Osswald
A Simple Propositional Framework Working hypothesis: concepts are subsets of a set Σ of attributes ◮ Regard p ∈ Σ as atomic proposition (propositional variable) ◮ Boolean formulas B [ Σ ] over Σ : inductively built by ∧ , ∨ , ¬ , ⊤ , ⊥ φ → ψ for ¬ φ ∨ ψ , φ ↔ ψ for φ → ψ ∧ ψ → φ Positive/affirmative formula: no ¬ Implication: � P → � Q , P , Q ⊆ Σ finite ( � ∅ = ⊤ , � ∅ = ⊥ ) Conditional (biconditional) form: φ → ψ ( φ ↔ ψ ), φ , ψ positive Conditional normal form: � P → � Q ◮ Theory over Σ : set of Boolean formulas over Σ ◮ Interpretation: function m : Σ → { 0 , 1 } , canonical extension to B [ Σ ] equivalent: subset X ⊆ Σ , via characteristic function χ X X satisfies φ iff χ X ( φ ) = 1, notation: X � φ ( X � p iff p ∈ X , X � ¬ φ iff X � φ , . . . ) ◮ X ⊆ Σ model of theory Γ iff X � φ for all φ ∈ Γ – 5 – Rainer Osswald
Theories and Information Domains (I) ◮ Information domain C ( Γ ) of theory Γ : models of Γ ordered by set inclusion C ( Γ ) = { X ⊆ Σ | X � φ for all φ ∈ Γ } ◮ Examples { h , f , b } Σ = { h , b , f } { b } { f } Γ = { h ↔ b ∧ f } ∅ { a , b , c } { a , b , d } Σ = { a , b , c , d } Γ = {⊤ → a ∨ b , c ∧ d → ⊥ , a ∧ b ↔ c ∨ d } Σ { a } { b } X 2 Y Σ = { a 1 , a 2 ,... } Γ = � n ≥ 1 { a n → a 1 ∨ a n + 1 , a n + 1 → a n } X 1 X 0 X n = { a m | m ≤ n } , Y = { a n | n > 1 } – 6 – Rainer Osswald
Theories and Information Domains (II) Classification of information domains Class of formulas of Γ Closure properties of C ( Γ ) C ( Γ ) as ordered set � P → � Q local membership � P → q , � P → ⊥ nonempty intersection, Scott domain (Horn) directed union � P → q intersection, complete algebraic lattice (implication) directed union p → q intersection, union complete coprime-algebraic (simple implication) lattice � P → ⊥ subsets, bounded-complete atomic (contradiction) finitely bounded union dcpo with completely coprime atoms – 7 – Rainer Osswald
Theories and Information Domains (III) ◮ Canonical C -theory Th C ( U ) associated with subset system U ⊆ ℘ ( Σ ) Th C ( U ) = { φ ∈ C | X � φ for all X ∈ U } C ∈ { Boolean formulas , implications , ... } ◮ � Th C , C � is Galois connection between ℘ ( ℘ ( Σ )) and ℘ ( C ) induced by � Consequence: C ◦ Th C is closure operator on ℘ ( Σ ) ◮ Γ is complete C -theory of U iff Γ ⊆ Th C ( U ) and Γ ⊢ Th C ( U ) ◮ Example { a , b , c , d , e } U C ( Th impl ( U )) { a , b , c , d } { a , b , e } { a , b , c , d } { a , b , e } { a , c , d } { a , c , d } { a , b } { a , c } { a , c } { a } { b } { a } { b } Complete implicational theory of U : d → c , c → a , e → a ∧ b , b ∧ c → d ∅ – 8 – Rainer Osswald
Formal Concept Analysis Reanalyzed ◮ Formal context K = � U , Σ , � K � V ⊲ = { p ∈ Σ | x � K p for all x ∈ V } = � { x ⊲ | x ∈ V } (with x ⊲ for { x } ⊲ ) ◮ Complete implicational theory of K : complete implicational theory of object intents U K = { x ⊲ | x ∈ U } ◮ Theorem For finite attribute sets, there is an order-reversing one-to-one corre- spondence between the concept lattice of a formal context and the informa- tion domain of any complete implicational theory of that context. ◮ Example { a , b , c , d } { a , b , e } K U K a b c d e x 4 , x 6 x 1 { a , c , d } x x x x 1 x 7 x 2 x x 3 x x 4 x x x x x 5 x 5 x x { a , c } x 2 x 3 x 6 x x x x x x x { a } { b } x 7 – 9 – Rainer Osswald
The Lindenbaum Algebra of a Theory ◮ Lindenbaum algebra L ( Γ ) of theory Γ L ( Γ ) = A [ Σ ] / ≃ Γ (with φ ≃ Γ ψ iff Γ ⊢ φ ↔ ψ ) L ( Γ ) is distributive lattice with zero and unit ◮ Theorem (Variant of Birkhoff’s Representation Theorem) Suppose Σ is finite. There is an order-reversing one-to-one correspondence between C ( Γ ) (i) and the ordered set of ∨ -irreducible elements of L ( Γ ) . (ii) There is a one-to-one correspondence between L ( Γ ) and the antichains in C ( Γ ) , where an antichain S corresponds to the equivalence class of � { � X | X ∈ S } . [ a ∨ b ] L ( Γ ) ◮ Example [ b ∨ c ] C ( Γ ) [ a ] { a , b , c , d } { a , b , e } [ b ∨ d ] { a , c , d } [ c ∨ e ] [ c ] [ d ∨ e ] [ b ] { a , c } [ a ∧ b ] { a } { b } [ d ] [ e ] [ b ∧ d ] Γ = { d → c , c → a , e → a ∧ b , b ∧ c → d c ∧ e → ⊥ , a ∧ b → c ∨ e , ⊤ → a ∨ b } [ ⊥ ] – 10 – Rainer Osswald
The Predicational Viewpoint ◮ Natural formulation within monadic predicate logic attributes are monadic predicates statements are universally quantified (e.g. ∀ x ( human x → biped x ) ) ◮ Formalization Σ set of (atomic) monadic predicates Boolean predicates by ∧ , ∨ , ¬ via abstraction (i.e. ( φ ∧ ψ ) x is φ x ∧ ψ x , etc) theory = set of universal statements ∀ φ (for ∀ x ( φ x ) ) ◮ Interpretation of Σ : universe U , function M : Σ → ℘ ( U ) Model of theory Γ : interpretation of Σ such that Γ is true ◮ Information domain of theory Γ : C ( Γ ) = { X ⊆ Σ | X � φ for all ∀ φ ∈ Γ } Canonical model of Γ with universe C ( Γ ) : p ∈ Σ �→ { X ∈ C ( Γ ) | p ∈ X } ◮ Theorem The canonical model of Γ is universal in the sense that it embeds every other model of Γ satisfying identity of indiscernibles. – 11 – Rainer Osswald
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