Distributive lattice-structured ontologies

Transcription

1 Distributive lattice-structured ontologies Hans Bruun 1 Dion Coumans 2 Mai Gerhke 2 1 Technical University of Denmark, Denmark 2 Radboud University Nijmegen, The Netherlands CALCO, september 2009

2 Introduction Methods of knowledge representation: Traditional databases: information stored in tuples Name(Dion) - Nat(Dutch) - Prof(PhD-student) Ontologies: classification of concepts dogs animals

3 Introduction Methods of knowledge representation: Traditional databases: information stored in tuples Name(Dion) - Nat(Dutch) - Prof(PhD-student) Ontologies: classification of concepts dogs animals Aim: provide a common framework (Fischer Nilsson). Mathematical setting: distributive lattices with additional unary operations preserving, and.

4 Ontological Framework An Ontological Framework (OF), O = (C, A, Π), consists of: C set of generators (concept names) A set of unary operation names (attributes) Π set of terminological axioms Π T L A(C) T L A(C) (L A = {,,, {a} a A }) relations on the generators

5 Problem description Given O = (C,A,Π), find L = (L,,,,, {a} a A ) such that: (L,,,, ) is a distributive lattice L is generated by C for all a A, a( ) = a(x y) = a(x) a(y) a(x y) = a(x) a(y) a( ) = L satisfies the terminological axioms.

6 Problem description Given O = (C,A,Π), find L = (L,,,,, {a} a A ) such that: (L,,,, ) is a distributive lattice L is generated by C for all a A, a( ) = a(x y) = a(x) a(y) a(x y) = a(x) a(y) a( ) = L satisfies the terminological axioms. Quotients of F DLA (C) (= F DL (A (C)))

7 Solution of an ontological framework A solution of O = (C,A,Π) is a quotient h : F DLA (C) D s.t. h(r) = h(s) for all (r,s) Π.

8 Solution of an ontological framework A solution of O = (C,A,Π) is a quotient h : F DLA (C) D s.t. h(r) = h(s) for all (r,s) Π. For every ontological framework O = (C,A,Π) there exists a quotient h O : F DLA (C) F O (universal solution) st: 1 h O : F DLA (C) F O is a solution of O = (C,A,Π) 2 if h : F DLA (C) D is any solution of O = (C,A,Π), then there is a unique homomorphism h D : F O D so that F DLA (C) D ց ր commutes. F O Note: The solutions of O are exactly the quotients of F O.

9 Example C = {h,a,c,m,f}, A =, Π = {(h,a c),(h,m f)} h f c a m

10 Solutions of an ontological framework Solutions of an ontological framework O = (C, A, Π): h O : F DLA (C) F O universal solution of O huge! : F DLA (C) 1 most collapsed solution uninteresting Question: What is a useful solution?

11 Knowledge Base A Knowledge Base (KB), B = (C, A, Π, I), consists of an ontological framework (C, A, Π) a set I of inserted or inhabited terms I F DLA (C) terms of interest Aim: Find the smallest solution of (C,A,Π) in which all information relevant to the inserted terms is present.

12 Classification of a term Starting with O h : F DLA (C) D t F DLA (C) an ontological framework a solution of O a term We say t 1,...,t n F DLA (C) is a classification of t w.r.t. h iff t i t for all i h(t) = h(t 1 )... h(t n )

13 Classification of a term We say t 1,...,t n F DLA (C) is a classification of t w.r.t. h iff t i t for all i h(t) = h(t 1 )... h(t n ) Observe: A classification of t wrt the universal solution F O is also a classification of t wrt any other solution. For the trivial quotient, F DLA (C) 1, is by itself a classification of any term.

14 Solution of a knowledge base A solution of B = (C,A,Π,I) is a quotient h : F DLA (C) D s.t. h : F DLA (C) D is a solution of (C,A,Π) for all t I, every classification of t wrt h is a classification of t wrt h O

15 Solution of a knowledge base A solution of B = (C,A,Π,I) is a quotient h : F DLA (C) D s.t. h : F DLA (C) D is a solution of (C,A,Π) for all t I, every classification of t wrt h is a classification of t wrt h O Note: The universal solution (C,A,Π) is a solution of B. Theorem: Every KB has a least solution h B : F DLA (C) D B : F DLA (C) ց D ր D B

16 Example C = {h,a,c,m,f}, Π = {(h,a c),(h,m f)}, I = {h} h f c a m (a) Universal solution m a f c man boy girl woman (b) Terminal solution h

17 Duality for DLAs Distributive lattices with add al operations Ordered topological spaces with add al functions (D, {a} a A ) (P D,τ,, {f a } a A ) P D = prime filters of D τ = gen. by { ˆd,( ˆd) c : d D} ˆd = { P D : d }

18 Duality for DLAs Distributive lattices with add al operations Ordered topological spaces with add al functions (D, {a} a A ) (P D,τ,, {f a } a A ) P D = prime filters of D τ = gen. by { ˆd,( ˆd) c : d D} ˆd = { P D : d } DLA-quotient D E Topological closed subset closed under the maps f a P E P D

19 Solution of a knowledge base h : F DLA (C) D is a solution of B = (C,A,Π,I) iff h : F DLA (C) D is a solution of (C,A,Π) F DLA (C) F O D Quotient of F O P D P O P Topologically closed subspace of P O closed under f a s for all t I, every clas. of t wrt h is a clas. of t wrt h O for all t I,s F DLA (C), for all t I, h(t) h(s) h O (t) h O (s) max(ĥo(t)) P D

20 Terminal solution of a knowledge base For every knowledge base B = (C, A, Π, I) there exists a quotient h B : F DLA (C) D B (terminal solution) s.t. h B : F DLA (C) D B is a solution of B if h : F DLA (C) D is a solution of B, then there exists a unique homomorphism D D B making D B is described dually by F DLA (C) D B ց ր commute D P B = ( {fw (max(ĥo(t))) w A, t I} )

21 Dual frame of F DLA (C) c1 c2 a(c1) a(c2) c1a(c1) c1c2 c1c2a(c1) P(C) P(A 1 (C)) a 2 (c1) a 2 (c2) c1a 2 (c1) c1c2a 2 (c1) ca(c)a 2 (c)... a 100 (c)a 101 (c)... ca(c)a 2 (c)a 100 (c)a 101 (c)... F DLA (C) = F DL (A (C)) prime filters of F DLA (C) subsets of (conjunctions over) A (C)

22 Dual frame of F DLA (C) c1 c2 a(c1) a(c2) c1a(c1) c1c2 c1c2a(c1) P(C) P(A 1 (C)) a 2 (c1) a 2 (c2) c1a 2 (c1) c1c2a 2 (c1) ca(c)a 2 (c)... a 100 (c)a 101 (c)... ca(c)a 2 (c)a 100 (c)a 101 (c)... P O = {p (r,s) Π [p r p s ]}

23 Dual frame of F DLA (C) c1 c2 a(c1) a(c2) c1a(c1) c1c2 c1c2a(c1) P(C) P(A 1 (C)) a 2 (c1) a 2 (c2) c1a 2 (c1) c1c2a 2 (c1) ca(c)a 2 (c)... a 100 (c)a 101 (c)... ca(c)a 2 (c)a 100 (c)a 101 (c)... P O = {p (r,s) Π w A [p w(r) p w(s)]}

24 Dual frame of F DLA (C) c1 c2 a(c1) a(c2) c1a(c1) c1c2 c1c2a(c1) P(C) P(A 1 (C)) a 2 (c1) a 2 (c2) c1a 2 (c1) c1c2a 2 (c1) ca(c)a 2 (c)... a 100 (c)a 101 (c)... ca(c)a 2 (c)a 100 (c)a 101 (c)... P O = {p (r,s) Π w A [p w(r) p w(s)]} ĥ O (t) = t P O

25 Computing max(h O (t)) Attribute-free setting Let (r,s) T 2 L A, r s t finite conjunction over A (C) h (r,s) (t) = t P (r,s) r t s h (r,s)(t) P (r,s) = {p p r p s} { max( h (r,s) )(t) = t max({t p p s}) c1 c2 if t r if t r c1c2... cn cn {p p r p s} To find max(ĥo(t)) repeatedly apply this basic step.

26 Computing max(h O (t)) Problem: in the setting with attributes you might not reach max(ĥo(t)) in finitely many steps. Example C = {c}, A = {a}, Π = {(c,c a(c))}, I = {c} c c a(c) (c, c a(c)) (a(c), a(c) a 2 (c)) c a(c) a 2 (c)... (a 2 (c), a 2 (c) a 3 (c))

27 Computing max(h O (t)) Problem: in the setting with attributes you might not reach max(ĥo(t)) in finitely many steps. Example C = {c}, A = {a}, Π = {(c,c a(c))}, I = {c} c c f a c a(c) (c, c a(c)) (a(c), a(c) a 2 (c)) c a(c) a 2 (c)... (a 2 (c), a 2 (c) a 3 (c))

28 Open problems Using the notion of a classification of a term we showed that every knowledge base has a smallest meaningful solution. Open problems: finding a sharp termination condition extension to non-deterministic modalities understanding the relation to description logic

29 Complexity First problem: a sharp termination condition. But, under the assumption that P B is finite (e.g. in attribute-free setting): Worst case: size of P B is exponential. In practice the algorithm works well (Oles, Ontoquery project).

30 Relation to Description Logic We address a different problem: Description Logic: given two terms s,t, decide whether h O (s) h O (t). Our work: give a decomposition of each relevant term in terms of join irreducibles and thus give a global solution. This in particular generates the relevant join-irreducibles. Example C = {h,a,c,m,f}, Π = {(h,a c),(h,m f)}, I = {h}. h m a f c man boy girl woman