Computing with Formal Concepts Bernhard Ganter Institut f ur - PowerPoint PPT Presentation

Outline Concept lattices Attribute logic Many valued contexts Conclusion Computing with Formal Concepts Bernhard Ganter Institut f¨ ur Algebra Dresden University of Technology D-01062 Dresden bernhard.ganter@tu-dresden.de EPCL Basic Training Camp November 8, 2011 Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion Outline 1 Concept lattices Data from a hospital Formal definitions More examples Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion Outline 1 Concept lattices Data from a hospital Formal definitions More examples 2 Attribute logic Checking completeness Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion Outline 1 Concept lattices Data from a hospital Formal definitions More examples 2 Attribute logic Checking completeness 3 Many valued contexts Scaling Turtoise logic Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion Outline 1 Concept lattices Data from a hospital Formal definitions More examples 2 Attribute logic Checking completeness 3 Many valued contexts Scaling Turtoise logic 4 Conclusion Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion Data from a hospital Interview data from a treatment of Anorexia nervosa easily offended over-sensitive self-confident apprehensive withdrawn superficial ambitious attentive sensitive difficult dutiful cordial chatty calm × × × × × × × × × × Myself My Ideal × × × × × × × × Father × × × × × × × × × × × × Mother × × × × × × × × × × × Sister × × × × × × × × × × × × × × × × × Brother-in-law Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion Data from a hospital A biplot of the interview data superficial Brother-in-law Mother complicated uncomplicated Father difficult valiant apprehensive thick-skinned Sister over-sensitive Myself My Ideal sympathetic attentive Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion Data from a hospital The concept lattice of the interview data ambitious cordial over-sensitive dutiful calm attentive sensitive super ficial apprehensive difficult self-confident withdrawn chatty easily offended Brother-in-law Mother My Sister Ideal Myself Father Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion Formal definitions Unfolding data in a concept lattice The basic procedure of Formal Concept Analysis: Data is represented in a very basic data type, called formal context . Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion Formal definitions Unfolding data in a concept lattice The basic procedure of Formal Concept Analysis: Data is represented in a very basic data type, called formal context . Each formal context is transformed into a mathematical structure called concept lattice . Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion Formal definitions Unfolding data in a concept lattice The basic procedure of Formal Concept Analysis: Data is represented in a very basic data type, called formal context . Each formal context is transformed into a mathematical structure called concept lattice . The information contained in the formal context is preserved. Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion Formal definitions Unfolding data in a concept lattice The basic procedure of Formal Concept Analysis: Data is represented in a very basic data type, called formal context . Each formal context is transformed into a mathematical structure called concept lattice . The information contained in the formal context is preserved. The concept lattice is the basis for further data analysis. It may be represented graphically to support communication, or it may be investigated with with algebraic methods to unravel its structure. Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion More examples The divisor lattice of 200 200 40 100 8 20 50 4 10 25 2 5 1 Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion More examples Recommended serving temperatures of some red wines 16 ◦ C 17 ◦ C 15 ◦ C 18 ◦ C Trollinger Negroamaro 19 ◦ C Beaujolais Brunello Barbera Burgundy Barolo Bordeaux Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion More examples Recommended serving temperatures for white wines 7 ◦ C 8 ◦ C 9 ◦ C 10 ◦ C 6 ◦ C 11 ◦ C Moscato 12 ◦ C Cava 13 ◦ C Prosecco Pinot Vernaccia Riesling 14 ◦ C Sauvignon grigio blanc Chardonnay Gew¨ urz Champagne traminer white white Rhone Burgundy Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion More examples An example about airlines . . . Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion More examples . . . and its concept lattice Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion More examples Formal contexts A formal context ( G , M , I ) consists of sets G , M and a binary relation I ⊆ G × M . For A ⊆ G and B ⊆ M , define A ′ := { m ∈ M | g I m for all g ∈ A } B ′ := { g ∈ G | g I m for all m ∈ B } . The mappings X → X ′′ are closure operators. Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion More examples Formal concepts ( A , B ) is a formal concept of ( G , M , I ) iff A ′ = B , A = B ′ . A ⊆ G , B ⊆ M , A is the extent and B is the intent of ( A , B ). Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion More examples Concept lattice Formal concepts can be ordered by ( A 1 , B 1 ) ≤ ( A 2 , B 2 ) : ⇐ ⇒ A 1 ⊆ A 2 . The set B ( G , M , I ) of all formal concepts of ( G , M , I ), with this order, is a complete lattice, called the concept lattice of ( G , M , I ). Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion More examples A formal context suckles its offspring needs water to live can move around needs chlorophyll two seed leaves lives in water one seed leaf lives on land has limbs Leech × × × Bream × × × × × × × × × Frog Dog × × × × × Spike-weed × × × × Reed × × × × × Bean × × × × Maize × × × × Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion More examples and its concept lattice needs water to live can move around needs chlorophyll lives lives in on land water has limbs one seed leaf leech spike-weed bream maize suckles its two seed offspring leaves dog frog bean reed Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion More examples The basic theorem Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion More examples Applications? Formal Concept Analysis has recently been applied in Description Logics, for checking completeness of knowledge bases, Linguistics, for the investigation of thesauri and ontologies, Software Engineering, for modelling type hierarchies with role types, Biomathematics, for analysing gene expression data, Machine Learning, for discovering website duplicates, Data Mining, for pattern matching problems, Rough Set Theory, for studying granular data, et cetera . . . Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion Checking completeness Particles under a microscope Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion Checking completeness Pairs of squares Bernhard Ganter, Dresden University Concept lattices

Outline Concept lattices Attribute logic Many valued contexts Conclusion Checking completeness But did we consider all possible cases? How can we decide if our selection of examples is complete? Bernhard Ganter, Dresden University Concept lattices