Towards a Logic-Based Framework for Analyzing Stream Reasoning - PowerPoint PPT Presentation

Towards a Logic-Based Framework for Analyzing Stream Reasoning Harald Beck Minh Dao-Tran Thomas Eiter Michael Fink 3rd International Workshop on Ordering and Reasoning October 20, 2014

Motivation Streams Logical Framework Conclusions What & Why “Towards a Logic-Based Framework for Analyzing Stream Reasoning” ◮ Stream Reasoning 1 / 11

Motivation Streams Logical Framework Conclusions What & Why “Towards a Logic-Based Framework for Analyzing Stream Reasoning” ◮ Stream Reasoning: Logical reasoning on streaming data 1 / 11

Motivation Streams Logical Framework Conclusions What & Why “Towards a Logic-Based Framework for Analyzing Stream Reasoning” ◮ Stream Reasoning: Logical reasoning on streaming data ◮ Streams = tuples (atoms) with timestamps ◮ Essential aspect: window functions 1 / 11

Motivation Streams Logical Framework Conclusions What & Why “Towards a Logic-Based Framework for Analyzing Stream Reasoning” ◮ Stream Reasoning: Logical reasoning on streaming data ◮ Streams = tuples (atoms) with timestamps ◮ Essential aspect: window functions ◮ Logic-Based 1 / 11

Motivation Streams Logical Framework Conclusions What & Why “Towards a Logic-Based Framework for Analyzing Stream Reasoning” ◮ Stream Reasoning: Logical reasoning on streaming data ◮ Streams = tuples (atoms) with timestamps ◮ Essential aspect: window functions ◮ Logic-Based: Lack of theory 1 / 11

Motivation Streams Logical Framework Conclusions What & Why “Towards a Logic-Based Framework for Analyzing Stream Reasoning” ◮ Stream Reasoning: Logical reasoning on streaming data ◮ Streams = tuples (atoms) with timestamps ◮ Essential aspect: window functions ◮ Logic-Based: Lack of theory ◮ Analysis 1 / 11

Motivation Streams Logical Framework Conclusions What & Why “Towards a Logic-Based Framework for Analyzing Stream Reasoning” ◮ Stream Reasoning: Logical reasoning on streaming data ◮ Streams = tuples (atoms) with timestamps ◮ Essential aspect: window functions ◮ Logic-Based: Lack of theory ◮ Analysis: Hard to predict, hard to compare 1 / 11

Motivation Streams Logical Framework Conclusions Example: Public Transportation Monitoring p 1 ℓ 1 p 3 p 2 ℓ 2 2 / 11

Motivation Streams Logical Framework Conclusions Example: Public Transportation Monitoring p 1 PLAN LINE ℓ 1 p 3 L X Y Z ID L ℓ 1 p 1 p 3 8 a 1 ℓ 1 p 2 ℓ 2 p 2 p 3 3 a 2 ℓ 2 ℓ 2 . . . . . . OLD a 1 . . . 2 / 11

Motivation Streams Logical Framework Conclusions Example: Public Transportation Monitoring p 1 PLAN LINE ℓ 1 p 3 L X Y Z ID L ℓ 1 p 1 p 3 8 a 1 ℓ 1 p 2 ℓ 2 p 2 p 3 3 a 2 ℓ 2 ℓ 2 . . . . . . OLD a 1 tram ( a 1 , p 1 ) tram ( a 2 , p 2 ) . . . t 36 40 2 / 11

Motivation Streams Logical Framework Conclusions Example: Public Transportation Monitoring p 1 PLAN LINE ℓ 1 p 3 L X Y Z ID L ℓ 1 p 1 p 3 8 a 1 ℓ 1 p 2 ℓ 2 p 2 p 3 3 a 2 ℓ 2 ℓ 2 . . . . . . OLD a 1 tram ( a 1 , p 1 ) tram ( a 2 , p 2 ) . . . t 36 40 43 44 ◮ Report trams’ expected arrival time. 2 / 11

Motivation Streams Logical Framework Conclusions Example: Public Transportation Monitoring p 1 PLAN LINE ℓ 1 p 3 L X Y Z ID L ℓ 1 p 1 p 3 8 a 1 ℓ 1 p 2 ℓ 2 p 2 p 3 3 a 2 ℓ 2 ℓ 2 . . . . . . OLD waiting a 1 tram ( a 1 , p 1 ) tram ( a 2 , p 2 ) time . . . t 36 40 43 44 ◮ Report trams’ expected arrival time. ◮ Report good connections between two lines at a given stop. 2 / 11

Motivation Streams Logical Framework Conclusions Streams ◮ Data Stream D = ( T , υ ) T = [ 0 , 50 ] υ = { 36 �→ { tram ( a 1 , p 1 ) } , 40 �→ { tram ( a 2 , p 2 ) }} 3 / 11

Motivation Streams Logical Framework Conclusions Streams ◮ Data Stream D = ( T , υ ) T = [ 0 , 50 ] υ = { 36 �→ { tram ( a 1 , p 1 ) } , 40 �→ { tram ( a 2 , p 2 ) }} ◮ Interpretation Stream S ⋆ = ( T ⋆ , υ ⋆ ) ⊇ D T ⋆ = [ 0 , 50 ] � 36 �→ { tram ( a 1 , p 1 ) } , � 40 �→ { tram ( a 2 , p 2 ) } , υ ⋆ = 43 �→ { exp ( a 2 , p 3 ) } , 44 �→ { exp ( a 1 , p 3 ) } 3 / 11

Motivation Streams Logical Framework Conclusions Window Functions ) ) ) ) , p 1 , p 2 , p 3 , p 3 tram ( a 1 tram ( a 2 exp ( a 2 exp ( a 1 36 37 38 39 40 43 41 42 44 45 S ′ = w ι ( S , t ,� x ) 4 / 11

Motivation Streams Logical Framework Conclusions Window Functions ) ) ) ) , p 1 , p 2 , p 3 , p 3 tram ( a 1 tram ( a 2 exp ( a 2 exp ( a 1 36 37 38 39 40 43 41 42 44 45 • • • • ℓ u   40 �→ { tram ( a 2 , p 2 ) } , S ′ = w τ ( S ⋆ , 40 , ( 1 , 5 , 1 )) = ([ 39 , 45 ] ,   43 �→ { exp ( a 2 , p 3 ) } ,  ) 44 �→ { exp ( a 1 , p 3 ) }  4 / 11

Motivation Streams Logical Framework Conclusions Window Operators ⊞ � x ⇒ w ι ( ch ( S ⋆ , S ) , t ,� ι, ch ⇐ x ) 5 / 11

Motivation Streams Logical Framework Conclusions Window Operators ⊞ � x ⇒ w ι ( ch ( S ⋆ , S ) , t ,� ι, ch ⇐ x ) ◮ ch : stream choice ch 1 ( S ⋆ , S ) = S ⋆ ch 2 ( S ⋆ , S ) = S 5 / 11

Motivation Streams Logical Framework Conclusions Window Operators ⊞ � x ⇒ w ι ( ch ( S ⋆ , S ) , t ,� ι, ch ⇐ x ) ◮ ch : stream choice ch 1 ( S ⋆ , S ) = S ⋆ ch 2 ( S ⋆ , S ) = S τ = ⊞ 10 , 0 , 1 ◮ ⊞ 10 w τ ( ch 2 ( S ⋆ , S ) , t , ( 10 , 0 , 1 )) = w τ ( S , t , ( 10 , 0 , 1 )) τ, ch 2 5 / 11

Motivation Streams Logical Framework Conclusions Window Operators ⊞ � x ⇒ w ι ( ch ( S ⋆ , S ) , t ,� ι, ch ⇐ x ) ◮ ch : stream choice ch 1 ( S ⋆ , S ) = S ⋆ ch 2 ( S ⋆ , S ) = S = ⊞ 0 , 5 , 1 ◮ ⊞ + 5 w τ ( ch 2 ( S ⋆ , S ) , t , ( 0 , 5 , 1 )) = w τ ( S , t , ( 0 , 5 , 1 )) τ τ, ch 2 5 / 11

Motivation Streams Logical Framework Conclusions Formulas α ::= 6 / 11

Motivation Streams Logical Framework Conclusions Formulas α ::= a | ¬ α | α ∧ α | α ∨ α | α → α 6 / 11

Motivation Streams Logical Framework Conclusions Formulas α ::= a | ¬ α | α ∧ α | α ∨ α | α → α | ♦ α | � α | @ t α ◮ various ways for time references 6 / 11

Motivation Streams Logical Framework Conclusions Formulas α ::= a | ¬ α | α ∧ α | α ∨ α | α → α | ♦ α | � α | @ t α | ⊞ � x ι, ch α ◮ various ways for time references ◮ nesting of window operators ⊞ 60 τ � ⊞ 5 τ ♦ tramAt ( p 1 ) 6 / 11

Motivation Streams Logical Framework Conclusions Formulas α ::= a | ¬ α | α ∧ α | α ∨ α | α → α | ♦ α | � α | @ t α | ⊞ � x ι, ch α ◮ various ways for time references ◮ nesting of window operators ⊞ 60 τ � ⊞ 5 τ ♦ tramAt ( p 1 ) ◮ but need rules: tramAt ( P ) ← tram ( X , P ) . 6 / 11

Motivation Streams Logical Framework Conclusions Entailment ◮ Structure M = � T ⋆ , υ ⋆ , W , B � 7 / 11

Motivation Streams Logical Framework Conclusions Entailment ◮ Structure M = � T ⋆ , υ ⋆ , W , B � ◮ Substream S = ( T , υ ) of S ⋆ : currently considered window ◮ Time point t ∈ T 7 / 11

Motivation Streams Logical Framework Conclusions Entailment ◮ Structure M = � T ⋆ , υ ⋆ , W , B � ◮ Substream S = ( T , υ ) of S ⋆ : currently considered window ◮ Time point t ∈ T iff a ∈ υ ( t ) , M , S , t � a 7 / 11

Motivation Streams Logical Framework Conclusions Entailment ◮ Structure M = � T ⋆ , υ ⋆ , W , B � ◮ Substream S = ( T , υ ) of S ⋆ : currently considered window ◮ Time point t ∈ T iff a ∈ υ ( t ) , M , S , t � a M , S , t � ¬ α iff M , S , t � α, M , S , t � α ∧ β iff M , S , t � α and M , S , t � β, M , S , t � α ∨ β iff M , S , t � α or M , S , t � β, M , S , t � α → β iff M , S , t � α or M , S , t � β, 7 / 11

Motivation Streams Logical Framework Conclusions Entailment ◮ Structure M = � T ⋆ , υ ⋆ , W , B � ◮ Substream S = ( T , υ ) of S ⋆ : currently considered window ◮ Time point t ∈ T iff a ∈ υ ( t ) , M , S , t � a M , S , t � ¬ α iff M , S , t � α, M , S , t � α ∧ β iff M , S , t � α and M , S , t � β, M , S , t � α ∨ β iff M , S , t � α or M , S , t � β, M , S , t � α → β iff M , S , t � α or M , S , t � β, M , S , t ′ � α for some t ′ ∈ T , M , S , t � ♦ α iff 7 / 11

Motivation Streams Logical Framework Conclusions Entailment ◮ Structure M = � T ⋆ , υ ⋆ , W , B � ◮ Substream S = ( T , υ ) of S ⋆ : currently considered window ◮ Time point t ∈ T iff a ∈ υ ( t ) , M , S , t � a M , S , t � ¬ α iff M , S , t � α, M , S , t � α ∧ β iff M , S , t � α and M , S , t � β, M , S , t � α ∨ β iff M , S , t � α or M , S , t � β, M , S , t � α → β iff M , S , t � α or M , S , t � β, M , S , t ′ � α for some t ′ ∈ T , M , S , t � ♦ α iff M , S , t ′ � α for all t ′ ∈ T , M , S , t � � α iff 7 / 11