Declarative Merging of and Reasoning about Decision Diagrams Thomas - PowerPoint PPT Presentation

Declarative Merging of and Reasoning about Decision Diagrams Thomas Eiter Thomas Krennwallner Christoph Redl { eiter,tkren,redl } @kr.tuwien.ac.at September 12, 2011 Eiter T., Krennwallner T., Redl C. (TU Vienna) Dec. Merging of and Reasoning about Decision Diagrams September 12, 2011 1 / 26

Outline Motivation 1 Preliminaries: MELD 2 3 Merging of Decision Diagrams 4 Reasoning about Decision Diagrams Application: DNA Classification 5 Conclusion 6 Eiter T., Krennwallner T., Redl C. (TU Vienna) Dec. Merging of and Reasoning about Decision Diagrams September 12, 2011 2 / 26

Motivation Outline Motivation 1 Preliminaries: MELD 2 3 Merging of Decision Diagrams 4 Reasoning about Decision Diagrams Application: DNA Classification 5 Conclusion 6 Eiter T., Krennwallner T., Redl C. (TU Vienna) Dec. Merging of and Reasoning about Decision Diagrams September 12, 2011 3 / 26

Motivation Motivation Decision Diagrams Important means for decision making Intuitively understandable Not only for knowledge engineers G G A T C C T A G G C C G A A T G T C A G C G C v=(f 1 , ..., f 20 ) Examples feature1>12,50 feature1>12,50 feature1>12,50 feature1>12,50 Severity ratings feature8<0,06 feature8<0,06 feature8<0,06 feature8<0,06 feature8<0,06 feature8<0,06 feature8<0,06 feature8<0,06 (e.g. TNM system) feature3>7,50 feature3>7,50 feature3>7,50 feature3>7,50 feature20<1,42 feature20<1,42 feature20<1,42 feature20<1,42 feature20>1,42 feature20>1,42 feature20>1,42 feature20>1,42 feature3>7,50 feature3>7,50 feature3>7,50 feature3>7,50 Diagnosis of feature20>1,42 feature20>1,42 feature20>1,42 feature20<1,42 feature20<1,42 feature20<1,42 feature20<1,42 feature20<1,42 feature20<1,42 feature20>1,42 feature20>1,42 feature20>1,42 feature20>1,42 feature20<1,42 non- feature20<1,42 feature20>1,42 coding personality disorders non- coding coding coding DNA classification non- coding non- coding non- coding coding non- coding coding coding coding Eiter T., Krennwallner T., Redl C. (TU Vienna) Dec. Merging of and Reasoning about Decision Diagrams September 12, 2011 4 / 26

Motivation Multiple Diagrams Reasons Different opinions Randomized machine-learning algorithms Statistical impreciseness Question: How to combine them? Eiter T., Krennwallner T., Redl C. (TU Vienna) Dec. Merging of and Reasoning about Decision Diagrams September 12, 2011 5 / 26

Motivation Multiple Diagram Integration The DDM System Integration process declaratively described Ingredients: input decision diagrams 1 merging algorithms 2 (predefined or user-defined) Focus: process formalization experimenting with different (combinations of) merging algorithms declarative reasoning for controlling the merging process We do not focus: concrete merging strategies accuracy improvement Eiter T., Krennwallner T., Redl C. (TU Vienna) Dec. Merging of and Reasoning about Decision Diagrams September 12, 2011 6 / 26

Preliminaries: MELD Outline Motivation 1 Preliminaries: MELD 2 3 Merging of Decision Diagrams 4 Reasoning about Decision Diagrams Application: DNA Classification 5 Conclusion 6 Eiter T., Krennwallner T., Redl C. (TU Vienna) Dec. Merging of and Reasoning about Decision Diagrams September 12, 2011 7 / 26

Preliminaries: MELD MELD Task Collection of knowledge bases: KB = KB 1 , . . . , KB n Associated collections of belief sets: BS ( KB 1 ) , . . . , BS ( KB n ) ∈ B Σ Goal: Integrate them into a single set of belief sets Method: Merging Operators ◦ n , m : � 2 B Σ � n → 2 B Σ × A 1 × . . . × A m � �� � � �� � operator arguments collections of belief sets Example Operator definition: ◦ 2 , 0 ∪ ( B 1 , B 2 ) = { B 1 ∪ B 2 | B 1 ∈ B 1 , B 2 ∈ B 2 , ∄ A : { A , ¬ A } ⊆ ( B 1 ∪ B 2 ) } , Application: B 1 = {{ a , b , c } , {¬ a , c }} , B 2 = {{¬ a , d } , { c , d }} ◦ 2 , 0 ∪ ( B 1 , B 2 ) = {{ a , b , c , d } , {¬ a , c , d }} Eiter T., Krennwallner T., Redl C. (TU Vienna) Dec. Merging of and Reasoning about Decision Diagrams September 12, 2011 8 / 26

Preliminaries: MELD MELD Merging Plan Hierarchical arrangement of merging operators Example ◦ \ ◦ ∪ ◦ ∪ BS ( KB 4 ) BS ( KB 5 ) ◦ ¬ BS ( KB 2 ) BS ( KB 3 ) BS ( KB 1 ) Eiter T., Krennwallner T., Redl C. (TU Vienna) Dec. Merging of and Reasoning about Decision Diagrams September 12, 2011 9 / 26

Preliminaries: MELD MELD Merging Tasks User provides belief bases with associated collections of belief sets merging plan optional: user-defined merging operators MELD: automated evaluation Advantages Reuse of operators Quick restructuring of merging plan Eiter T., Krennwallner T., Redl C. (TU Vienna) Dec. Merging of and Reasoning about Decision Diagrams September 12, 2011 10 / 26

Merging of Decision Diagrams Outline Motivation 1 Preliminaries: MELD 2 3 Merging of Decision Diagrams 4 Reasoning about Decision Diagrams Application: DNA Classification 5 Conclusion 6 Eiter T., Krennwallner T., Redl C. (TU Vienna) Dec. Merging of and Reasoning about Decision Diagrams September 12, 2011 11 / 26

Merging of Decision Diagrams Decision Diagrams Definition (Decision Diagram) A decision diagram over D and C is a labelled rooted directed acyclic graph D = � V , E , ℓ C , ℓ E � V . . . nonempty set of nodes with unique root node r D ∈ V E ⊆ V × V . . . set of directed edges ℓ C : V → C . . . partial function assigning a class to all leafs ℓ E : E → Q . . . assign queries Q ( z ) : D → { true , false } to edges Query language: O 1 ◦ O 2 with operands O 1 , O 2 and ◦ ∈ { <, ≤ , = , � = , ≥ , > } or “else” Example r D D = { 1 , 2 , 3 , 4 , 5 } z < 3 C = { c 1 , c 2 } else v 1 v 2 z < 4 z < 2 else else v 3 c 1 c 2 v 4 Eiter T., Krennwallner T., Redl C. (TU Vienna) Dec. Merging of and Reasoning about Decision Diagrams September 12, 2011 12 / 26

Merging of Decision Diagrams Decision Diagrams Definition (Decision Diagram) A decision diagram over D and C is a labelled rooted directed acyclic graph D = � V , E , ℓ C , ℓ E � V . . . nonempty set of nodes with unique root node r D ∈ V E ⊆ V × V . . . set of directed edges ℓ C : V → C . . . partial function assigning a class to all leafs ℓ E : E → Q . . . assign queries Q ( z ) : D → { true , false } to edges Query language: O 1 ◦ O 2 with operands O 1 , O 2 and ◦ ∈ { <, ≤ , = , � = , ≥ , > } or “else” Example r D D = { 1 , 2 , 3 , 4 , 5 } z < 3 C = { c 1 , c 2 } else Classify: 4 v 1 v 2 z < 4 z < 2 else else v 3 c 1 c 2 v 4 Eiter T., Krennwallner T., Redl C. (TU Vienna) Dec. Merging of and Reasoning about Decision Diagrams September 12, 2011 12 / 26

Merging of Decision Diagrams Decision Diagrams Definition (Decision Diagram) A decision diagram over D and C is a labelled rooted directed acyclic graph D = � V , E , ℓ C , ℓ E � V . . . nonempty set of nodes with unique root node r D ∈ V E ⊆ V × V . . . set of directed edges ℓ C : V → C . . . partial function assigning a class to all leafs ℓ E : E → Q . . . assign queries Q ( z ) : D → { true , false } to edges Query language: O 1 ◦ O 2 with operands O 1 , O 2 and ◦ ∈ { <, ≤ , = , � = , ≥ , > } or “else” Example r D D = { 1 , 2 , 3 , 4 , 5 } z < 3 C = { c 1 , c 2 } else Classify: 4 v 1 v 2 z < 4 z < 2 else else v 3 c 1 c 2 v 4 Eiter T., Krennwallner T., Redl C. (TU Vienna) Dec. Merging of and Reasoning about Decision Diagrams September 12, 2011 12 / 26

Merging of Decision Diagrams Decision Diagrams Definition (Decision Diagram) A decision diagram over D and C is a labelled rooted directed acyclic graph D = � V , E , ℓ C , ℓ E � V . . . nonempty set of nodes with unique root node r D ∈ V E ⊆ V × V . . . set of directed edges ℓ C : V → C . . . partial function assigning a class to all leafs ℓ E : E → Q . . . assign queries Q ( z ) : D → { true , false } to edges Query language: O 1 ◦ O 2 with operands O 1 , O 2 and ◦ ∈ { <, ≤ , = , � = , ≥ , > } or “else” Example r D D = { 1 , 2 , 3 , 4 , 5 } z < 3 C = { c 1 , c 2 } else Classify: 4 ⇒ c 2 v 1 v 2 z < 4 z < 2 else else v 3 c 1 c 2 c 2 v 4 Eiter T., Krennwallner T., Redl C. (TU Vienna) Dec. Merging of and Reasoning about Decision Diagrams September 12, 2011 12 / 26

Merging of Decision Diagrams Decision Diagrams Definition (Decision Diagram) A decision diagram over D and C is a labelled rooted directed acyclic graph D = � V , E , ℓ C , ℓ E � V . . . nonempty set of nodes with unique root node r D ∈ V E ⊆ V × V . . . set of directed edges ℓ C : V → C . . . partial function assigning a class to all leafs ℓ E : E → Q . . . assign queries Q ( z ) : D → { true , false } to edges Query language: O 1 ◦ O 2 with operands O 1 , O 2 and ◦ ∈ { <, ≤ , = , � = , ≥ , > } or “else” Example r D D = { 1 , 2 , 3 , 4 , 5 } z < 3 C = { c 1 , c 2 } else Classify: 4 ⇒ c 2 v 1 v 2 z < 4 z < 2 else else v 3 c 1 c 2 v 4 c 2 Note: D may consist of composed objects, e.g. Q ( z ) = z . TSH > 4 . 5 mU / l Eiter T., Krennwallner T., Redl C. (TU Vienna) Dec. Merging of and Reasoning about Decision Diagrams September 12, 2011 12 / 26