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Stratification Criteria and Rewriting Techniques for Checking Chase Termination Stratification Criteria and Rewriting Techniques for Checking Chase Termination S. Greco, F . Spezzano and I. Trubitsyna DEIS, Universit` a della Calabria - Italy

  1. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria C-Stratification [DeutschNR08,MeierSL09] Builds a c-chase graph G c (Σ) = (Σ , E ) representing how constraints fire each other. ( r 1 , r 2 ) ∈ E if r 1 ≺ c r 2 (firing r 1 can cause r 2 to fire) ∗ , r 1 , h 1 K 2 , 1) K 1 → r 1 ≺ c r 2 if: 2) K 2 �| = h 2 ( r 2 ) , 3) K 1 | = h 2 ( r 2 ) . S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  2. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria C-Stratification [DeutschNR08,MeierSL09] (C-Stratification) A set of constraint Σ is c-stratified if every cycle in the c-chase graph is weakly acyclic. Example Σ = { r : T ( x , y ) ∧ T ( y , x ) → ∃ z T ( y , z ) ∧ T ( z , x ) } we have that r ≺ c r and { r } is not weakly acyclic Figure: Dependency graph Dep (Σ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  3. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria C-Stratification versus Stratification C-Stratification Stratification r 1 ≺ c r 2 if: r 1 ≺ r 2 if: ∗ , r 1 , h 1 K 2 , r 1 , h 1 K 2 , 1) K 1 → 1) K 1 → 2) K 2 �| = h 2 ( r 2 ) , 2) K 2 �| = h 2 ( r 2 ) , 3) K 1 | = h 2 ( r 2 ) . 3) K 1 | = h 2 ( r 2 ) . S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  4. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria C-Stratification versus Stratification C-Stratification Stratification r 1 ≺ c r 2 if: r 1 ≺ r 2 if: ∗ , r 1 , h 1 K 2 , r 1 , h 1 K 2 , 1) K 1 → 1) K 1 → 2) K 2 �| = h 2 ( r 2 ) , 2) K 2 �| = h 2 ( r 2 ) , 3) K 1 | = h 2 ( r 2 ) . 3) K 1 | = h 2 ( r 2 ) . Stratification guarantees the termination of at least one chase sequence. S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  5. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Stratification Example r 1 : R ( x ) → ∃ y T ( x , y ) Σ = r 2 : R ( x ) → T ( x , x ) r 3 : T ( x , y ) ∧ T ( x , x ) → R ( y ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  6. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Stratification Example r 1 : R ( x ) → ∃ y T ( x , y ) Σ = r 2 : R ( x ) → T ( x , x ) r 3 : T ( x , y ) ∧ T ( x , x ) → R ( y ) r 1 �≺ r 3 because S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  7. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Stratification Example r 1 : R ( x ) → ∃ y T ( x , y ) Σ = r 2 : R ( x ) → T ( x , x ) r 3 : T ( x , y ) ∧ T ( x , x ) → R ( y ) r 1 �≺ r 3 because if K 1 = { R ( a ) } then K 2 = { R ( a ) , T ( a , η 1 ) } | = r 3 1 (because T ( a , a ) �∈ K 2 ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  8. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Stratification Example r 1 : R ( x ) → ∃ y T ( x , y ) Σ = r 2 : R ( x ) → T ( x , x ) r 3 : T ( x , y ) ∧ T ( x , x ) → R ( y ) r 1 �≺ r 3 because if K 1 = { R ( a ) } then K 2 = { R ( a ) , T ( a , η 1 ) } | = r 3 1 (because T ( a , a ) �∈ K 2 ) if K 1 = { R ( a ) , T ( a , a ) } then K 1 | = Σ ( K 1 universal solution) 2 S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  9. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Stratification Example r 1 : R ( x ) → ∃ y T ( x , y ) Σ = r 2 : R ( x ) → T ( x , x ) r 3 : T ( x , y ) ∧ T ( x , x ) → R ( y ) chase sequence r 1 → r 2 → r 3 → . . . is not terminating S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  10. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Stratification Example r 1 : R ( x ) → ∃ y T ( x , y ) Σ = r 2 : R ( x ) → T ( x , x ) r 3 : T ( x , y ) ∧ T ( x , x ) → R ( y ) chase sequence r 1 → r 2 → r 3 → . . . is not terminating D = { R ( a ) } S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  11. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Stratification Example r 1 : R ( x ) → ∃ y T ( x , y ) Σ = r 2 : R ( x ) → T ( x , x ) r 3 : T ( x , y ) ∧ T ( x , x ) → R ( y ) chase sequence r 1 → r 2 → r 3 → . . . is not terminating D = { R ( a ) } r 1 , h 1 D 1 = { R ( a ) , T ( a , η 1 ) } D → S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  12. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Stratification Example r 1 : R ( x ) → ∃ y T ( x , y ) Σ = r 2 : R ( x ) → T ( x , x ) r 3 : T ( x , y ) ∧ T ( x , x ) → R ( y ) chase sequence r 1 → r 2 → r 3 → . . . is not terminating D = { R ( a ) } r 1 , h 1 D 1 = { R ( a ) , T ( a , η 1 ) } D → r 2 , h 2 D 2 = { R ( a ) , T ( a , η 1 ) , T ( a , a ) } D 1 → S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  13. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Stratification Example r 1 : R ( x ) → ∃ y T ( x , y ) Σ = r 2 : R ( x ) → T ( x , x ) r 3 : T ( x , y ) ∧ T ( x , x ) → R ( y ) chase sequence r 1 → r 2 → r 3 → . . . is not terminating D = { R ( a ) } r 1 , h 1 D 1 = { R ( a ) , T ( a , η 1 ) } D → r 2 , h 2 D 2 = { R ( a ) , T ( a , η 1 ) , T ( a , a ) } D 1 → r 3 , h 3 D 3 = { R ( a ) , T ( a , η 1 ) , T ( a , a ) , R ( η 1 ) } D 2 → S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  14. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Stratification Example r 1 : R ( x ) → ∃ y T ( x , y ) Σ = r 2 : R ( x ) → T ( x , x ) r 3 : T ( x , y ) ∧ T ( x , x ) → R ( y ) chase sequence r 1 → r 2 → r 3 → . . . is not terminating D = { R ( a ) } r 1 , h 1 D 1 = { R ( a ) , T ( a , η 1 ) } D → r 2 , h 2 D 2 = { R ( a ) , T ( a , η 1 ) , T ( a , a ) } D 1 → r 3 , h 3 D 3 = { R ( a ) , T ( a , η 1 ) , T ( a , a ) , R ( η 1 ) } D 2 → r 1 , h 4 D 4 = { R ( a ) , T ( a , η 1 ) , T ( a , a ) , R ( η 1 ) , T ( η 1 , η 2 ) } D 3 → . . . S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  15. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Stratification Example r 1 : R ( x ) → ∃ y T ( x , y ) Σ = r 2 : R ( x ) → T ( x , x ) r 3 : T ( x , y ) ∧ T ( x , x ) → R ( y ) chase sequence r 1 → r 2 → r 3 → . . . is not terminating D = { R ( a ) } r 1 , h 1 D 1 = { R ( a ) , T ( a , η 1 ) } D → r 2 , h 2 D 2 = { R ( a ) , T ( a , η 1 ) , T ( a , a ) } D 1 → r 3 , h 3 D 3 = { R ( a ) , T ( a , η 1 ) , T ( a , a ) , R ( η 1 ) } D 2 → r 1 , h 4 D 4 = { R ( a ) , T ( a , η 1 ) , T ( a , a ) , R ( η 1 ) , T ( η 1 , η 2 ) } D 3 → . . . chase sequence r 2 → r 3 → r 1 terminates S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  16. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria WA -stratification Builds a firing graph Γ(Σ) = (Σ , E ) representing how constraints fire each other. ( r 1 , r 2 ) ∈ E if r 1 < r 2 (firing r 1 can cause r 2 to fire) r 1 < r 2 if: r 1 ≺ r 2 if: r 1 , h 1 K 2 , r 1 , h 1 K 2 , 1) K 1 → 1) K 1 → 2) K 2 ∪ S �| 2) K 2 �| = h 2 ( r 2 ) , = h 2 ( r 2 ) , 3) K 1 ∪ S | = h 2 ( r 2 ) and 3) K 1 | = h 2 ( r 2 ) . 4) Null ( S ) ∩ ( Null ( K 2 ) − Null ( K 1 )) = ∅ . S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  17. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria WA -stratification Builds a firing graph Γ(Σ) = (Σ , E ) representing how constraints fire each other. ( r 1 , r 2 ) ∈ E if r 1 < r 2 (firing r 1 can cause r 2 to fire) r 1 < r 2 if: r 1 ≺ r 2 if: r 1 , h 1 K 2 , r 1 , h 1 K 2 , 1) K 1 → 1) K 1 → 2) K 2 ∪ S �| 2) K 2 �| = h 2 ( r 2 ) , = h 2 ( r 2 ) , 3) K 1 ∪ S | = h 2 ( r 2 ) and 3) K 1 | = h 2 ( r 2 ) . 4) Null ( S ) ∩ ( Null ( K 2 ) − Null ( K 1 )) = ∅ . Example r 1 : R ( x ) → ∃ y T ( x , y ) Σ = r 2 : R ( x ) → T ( x , x ) r 3 : T ( x , y ) ∧ T ( x , x ) → R ( y ) K 1 = { R ( a ) } and K 2 = { R ( a ) , T ( a , η 1 ) } S = { T ( a , a ) } r 3 : T ( a , η 1 ) ∧ T ( a , a ) → R ( η 1 ) r 3 is fired by r 1 , then we have r 1 < r 3 S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  18. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria WA -stratification (WA-Stratification) We say that Σ is WA-stratified ( W AStr ) iff the constraints in every nontrivial strongly connected component of the firing graph Γ(Σ) = (Σ , { ( r 1 , r 2 ) | r 1 < r 2 } ) are weakly acyclic. S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  19. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria WA -stratification (WA-Stratification) We say that Σ is WA-stratified ( W AStr ) iff the constraints in every nontrivial strongly connected component of the firing graph Γ(Σ) = (Σ , { ( r 1 , r 2 ) | r 1 < r 2 } ) are weakly acyclic. Example Σ = { r : T ( x , y ) ∧ T ( y , x ) → ∃ z T ( y , z ) ∧ T ( z , x ) } is WA -stratified, but not c-stratified. S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  20. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria WA -stratification (WA-Stratification) We say that Σ is WA-stratified ( W AStr ) iff the constraints in every nontrivial strongly connected component of the firing graph Γ(Σ) = (Σ , { ( r 1 , r 2 ) | r 1 < r 2 } ) are weakly acyclic. Example Σ = { r : T ( x , y ) ∧ T ( y , x ) → ∃ z T ( y , z ) ∧ T ( z , x ) } is WA -stratified, but not c-stratified. Proposition CS tr � WAS tr and SC ∦ WAS tr . S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  21. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria SC -stratification and SwA -stratification Definition Given a set of TDGs Σ , we say that Σ is SC-stratified ( SCStr ) if the constraints in every nontrivial strongly connected component of the firing graph Γ(Σ) are safe. SwA-stratified ( SwAStr ) if the constraints in every nontrivial strongly connected component of the firing graph Γ(Σ) are super-weak acyclic. S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  22. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria SC -stratification and SwA -stratification Definition Given a set of TDGs Σ , we say that Σ is SC-stratified ( SCStr ) if the constraints in every nontrivial strongly connected component of the firing graph Γ(Σ) are safe. SwA-stratified ( SwAStr ) if the constraints in every nontrivial strongly connected component of the firing graph Γ(Σ) are super-weak acyclic. Proposition Let Σ be a set of WA-stratified (resp. SC-stratified, SwA-stratified) TGDs and let D be a database. Then, the length of every chase sequence of Σ over D is polynomial in the size of D . S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  23. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria SwA -stratification Theorem 1 WAS tr � SCS tr � S w AS tr , for C ∈ { W A , SC , SwA } , C � C - S tr and 2 IR ∦ S w AS tr . 3 S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  24. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Local Stratification New chase termination condition: Local Stratification (LS) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  25. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Local Stratification New chase termination condition: Local Stratification (LS) Combine < -relation of WA - S tr , with SwA criterion, S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  26. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Local Stratification New chase termination condition: Local Stratification (LS) Combine < -relation of WA - S tr , with SwA criterion, ensures the termination of all chase sequences in polynomial data complexity, S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  27. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Local Stratification New chase termination condition: Local Stratification (LS) Combine < -relation of WA - S tr , with SwA criterion, ensures the termination of all chase sequences in polynomial data complexity, S w A - S tr � LS and IR � LS . S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  28. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Super-weak Acyclicity [Marnette09] Analyzes nulls propagation through constraints Example r 1 : N ( x ) → ∃ y ∃ z E ( x , y ) ∧ S ( z , y ) Σ = r 2 : E ( x 1 , y 1 ) ∧ S ( x 1 , y 1 ) → N ( y 1 ) r 3 : E ( x 2 , y 2 ) → E ( y 2 , x 2 ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  29. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Super-weak Acyclicity [Marnette09] Analyzes nulls propagation through constraints Example r 1 : N ( x ) → ∃ y ∃ z E ( x , y ) ∧ S ( z , y ) Σ = r 2 : E ( x 1 , y 1 ) ∧ S ( x 1 , y 1 ) → N ( y 1 ) r 3 : E ( x 2 , y 2 ) → E ( y 2 , x 2 ) p 1 ) → ∃ y ∃ z E ( x p 2 , y p 3 ) ∧ S ( z p 4 , y p 5 ) r 1 : N ( x p 6 p 7 p 8 p 9 p 10 Σ = r 2 : E ( x 1 , y 1 ) ∧ S ( x 1 , y 1 ) → N ( y 1 ) p 11 p 12 p 13 p 14 r 3 : E ( x 2 , y 2 ) → E ( y 2 , x 2 ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  30. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Super-weak Acyclicity [Marnette09] Analyzes nulls propagation through constraints Example p 1 ) → ∃ y ∃ z E ( x p 2 , y p 3 ) ∧ S ( z p 4 , y p 5 ) r 1 : N ( x p 6 p 8 p 9 p 10 p 7 Σ = r 2 : E ( x 1 , y 1 ) ∧ S ( x 1 , y 1 ) → N ( y 1 ) p 11 p 12 p 13 p 14 r 3 : E ( x 2 , y 2 ) → E ( y 2 , x 2 ) Move ( r 1 , y ) = { p 3 , p 5 } S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  31. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Super-weak Acyclicity [Marnette09] Analyzes nulls propagation through constraints Example p 1 ) → ∃ y ∃ z E ( x p 2 , y p 3 ) ∧ S ( z p 4 , y p 5 ) r 1 : N ( x p 6 p 7 ) ∧ S ( x p 8 p 9 ) → N ( y 1 p 10 ) Σ = r 2 : E ( x 1 , y 1 1 , y 1 p 11 p 12 p 13 p 14 r 3 : E ( x 2 , y 2 ) → E ( y 2 , x 2 ) Move ( r 1 , y ) = { p 3 , p 5 , p 10 } S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  32. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Super-weak Acyclicity [Marnette09] Analyzes nulls propagation through constraints Example p 1 ) → ∃ y ∃ z E ( x p 2 , y p 3 ) ∧ S ( z p 4 , y p 5 ) r 1 : N ( x p 6 p 8 p 9 p 10 p 7 Σ = r 2 : E ( x 1 , y 1 ) ∧ S ( x 1 , y 1 ) → N ( y 1 ) p 11 p 12 ) → E ( y 2 p 13 , x p 14 r 3 : E ( x 2 , y 2 2 ) Move ( r 1 , y ) = { p 3 , p 5 , p 10 , p 13 } S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  33. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Super-weak Acyclicity [Marnette09] Analyzes nulls propagation through constraints Example p 1 ) → ∃ y ∃ z E ( x p 2 , y p 3 ) ∧ S ( z p 4 , y p 5 ) r 1 : N ( x p 6 p 8 p 9 p 10 p 7 Σ = r 2 : E ( x 1 , y 1 ) ∧ S ( x 1 , y 1 ) → N ( y 1 ) p 11 p 12 p 13 p 14 r 3 : E ( x 2 , y 2 ) → E ( y 2 , x 2 ) Move ( r 1 , y ) = { p 3 , p 5 , p 10 , p 13 , p 2 } S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  34. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Super-weak Acyclicity [Marnette09] Analyzes nulls propagation through constraints Example p 1 ) → ∃ y ∃ z E ( x p 2 , y p 3 ) ∧ S ( z p 4 , y p 5 ) r 1 : N ( x p 6 p 8 p 9 p 10 p 7 Σ = r 2 : E ( x 1 , y 1 ) ∧ S ( x 1 , y 1 ) → N ( y 1 ) p 11 , y p 12 p 13 p 14 ) r 3 : E ( x 2 2 ) → E ( y 2 , x 2 Move ( r 1 , y )) = { p 3 , p 5 , p 10 , p 13 , p 2 , p 14 } S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  35. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Super-weak Acyclicity [Marnette09] Analyzes nulls propagation through constraints Example p 1 ) → ∃ y ∃ z E ( x p 2 , y p 3 ) ∧ S ( z p 4 , y p 5 ) r 1 : N ( x p 6 p 7 p 8 p 9 p 10 Σ = r 2 : E ( x 1 , y 1 ) ∧ S ( x 1 , y 1 ) → N ( y 1 ) p 11 p 12 p 13 p 14 r 3 : E ( x 2 , y 2 ) → E ( y 2 , x 2 ) Move ( r 1 , y ) = { p 3 , p 5 , p 10 , p 13 , p 2 , p 14 } S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  36. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Super-weak Acyclicity [Marnette09] Analyzes nulls propagation through constraints Example p 1 ) → ∃ y ∃ z E ( x p 2 , y p 3 ) ∧ S ( z p 4 , y p 5 ) r 1 : N ( x p 6 p 7 p 8 p 9 p 10 Σ = r 2 : E ( x 1 , y 1 ) ∧ S ( x 1 , y 1 ) → N ( y 1 ) p 11 p 12 p 13 p 14 r 3 : E ( x 2 , y 2 ) → E ( y 2 , x 2 ) Move ( r 1 , y ) = { p 3 , p 5 , p 10 , p 13 , p 2 , p 14 } Then, we say that r 1 � r 1 because the null introduced by r 1 ( p 3 and p 5 ) is copied again in the head of r 1 ( p 2 ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  37. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Example: Super-weak Acyclicity [Marnette09] Analyzes nulls propagation through constraints Example p 1 ) → ∃ y ∃ z E ( x p 2 , y p 3 ) ∧ S ( z p 4 , y p 5 ) r 1 : N ( x p 6 p 7 p 8 p 9 p 10 Σ = r 2 : E ( x 1 , y 1 ) ∧ S ( x 1 , y 1 ) → N ( y 1 ) p 11 p 12 p 13 p 14 r 3 : E ( x 2 , y 2 ) → E ( y 2 , x 2 ) Move ( r 1 , y ) = { p 3 , p 5 , p 10 , p 13 , p 2 , p 14 } Then, we say that r 1 � r 1 because the null introduced by r 1 ( p 3 and p 5 ) is copied again in the head of r 1 ( p 2 ) The relation � is cyclic and Σ is not in SwA S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  38. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Local Stratification IDEA: testing the firing relation < from WA -stratification in the MOVE construction Example p 1 ) → ∃ y ∃ z E ( x p 3 ) ∧ S ( z p 5 ) p 2 , y p 4 , y r 1 : N ( x p 6 p 7 p 8 p 9 p 10 Σ = r 2 : E ( x 1 , y 1 ) ∧ S ( x 1 , y 1 ) → N ( y 1 ) p 11 p 12 p 13 p 14 r 3 : E ( x 2 , y 2 ) → E ( y 2 , x 2 ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  39. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Local Stratification IDEA: testing the firing relation < from WA -stratification in the MOVE construction Example p 1 ) → ∃ y ∃ z E ( x p 3 ) ∧ S ( z p 5 ) p 2 , y p 4 , y r 1 : N ( x p 6 p 7 p 8 p 9 p 10 Σ = r 2 : E ( x 1 , y 1 ) ∧ S ( x 1 , y 1 ) → N ( y 1 ) p 11 p 12 p 13 p 14 r 3 : E ( x 2 , y 2 ) → E ( y 2 , x 2 ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  40. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Local Stratification IDEA: testing the firing relation < from WA -stratification in the MOVE construction Example p 1 ) → ∃ y ∃ z E ( x p 3 ) ∧ S ( z p 5 ) p 2 , y p 4 , y r 1 : N ( x p 6 p 7 p 8 p 9 p 10 Σ = r 2 : E ( x 1 , y 1 ) ∧ S ( x 1 , y 1 ) → N ( y 1 ) p 11 p 12 p 13 p 14 r 3 : E ( x 2 , y 2 ) → E ( y 2 , x 2 ) r 1 ≮ r 2 MOVE ( r 1 , y ) = { p 3 , p 5 } p 10 �∈ MOVE ( r 1 , y ) as r 1 ≮ r 2 S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  41. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Local Stratification IDEA: testing the firing relation < from WA -stratification in the MOVE construction Example p 1 ) → ∃ y ∃ z E ( x p 3 ) ∧ S ( z p 5 ) p 2 , y p 4 , y r 1 : N ( x p 6 p 7 p 8 p 9 p 10 Σ = r 2 : E ( x 1 , y 1 ) ∧ S ( x 1 , y 1 ) → N ( y 1 ) p 11 p 12 p 13 p 14 r 3 : E ( x 2 , y 2 ) → E ( y 2 , x 2 ) r 1 ≮ r 2 MOVE ( r 1 , y ) = { p 3 , p 5 } p 10 �∈ MOVE ( r 1 , y ) as r 1 ≮ r 2 S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  42. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Local Stratification S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  43. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Chase Termination Criteria Limitations Example Σ = ∀ x [ N ( x ) → ∃ y E ( x , y ) ] ∀ ( x , y ) [ S ( x ) ∧ E ( x , y ) → N ( y ) ] S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  44. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique Constraints Rewriting Technique S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  45. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique Constraints Rewriting Technique [GrecoS10] Idea: rewrite Σ into an ‘equivalent’ set Σ α and verify the structural properties for chase termination on Σ α introduction of adornments associated with predicates ≡ S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  46. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique Rewriting Algorithm Example N ( x ) → ∃ y E ( x , y ) Σ = S ( x ) ∧ E ( x , y ) → N ( y ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  47. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique Rewriting Algorithm Example N ( x ) → ∃ y E ( x , y ) Σ = S ( x ) ∧ E ( x , y ) → N ( y ) In (Σ) = N ( x ) → N b ( x ) S ( x ) → S b ( x ) E ( x , y ) → E bb ( x , y ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  48. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique Rewriting Algorithm Example N ( x ) → ∃ y E ( x , y ) Σ = S ( x ) ∧ E ( x , y ) → N ( y ) Base (Σ) = N b ( x ) → ∃ y E bf ( x , y ) In (Σ) = S b ( x ) ∧ E bb ( x , y ) → N b ( y ) N ( x ) → N b ( x ) S ( x ) → S b ( x ) E ( x , y ) → E bb ( x , y ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  49. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique Rewriting Algorithm Example N ( x ) → ∃ y E ( x , y ) Σ = S ( x ) ∧ E ( x , y ) → N ( y ) Base (Σ) = N b ( x ) → ∃ y E bf ( x , y ) In (Σ) = S b ( x ) ∧ E bb ( x , y ) → N b ( y ) N ( x ) → N b ( x ) S ( x ) → S b ( x ) Derived (Σ) = E ( x , y ) → E bb ( x , y ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  50. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique Rewriting Algorithm Example N ( x ) → ∃ y E ( x , y ) Σ = S ( x ) ∧ E ( x , y ) → N ( y ) Base (Σ) = N b ( x ) → ∃ y E bf ( x , y ) In (Σ) = S b ( x ) ∧ E bb ( x , y ) → N b ( y ) N ( x ) → N b ( x ) S ( x ) → S b ( x ) Derived (Σ) = E ( x , y ) → E bb ( x , y ) S b ( x ) ∧ E bf ( x , y ) → N f ( y ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  51. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique Rewriting Algorithm Example N ( x ) → ∃ y E ( x , y ) Σ = S ( x ) ∧ E ( x , y ) → N ( y ) Base (Σ) = N b ( x ) → ∃ y E bf ( x , y ) In (Σ) = S b ( x ) ∧ E bb ( x , y ) → N b ( y ) N ( x ) → N b ( x ) S ( x ) → S b ( x ) Derived (Σ) = E ( x , y ) → E bb ( x , y ) S b ( x ) ∧ E bf ( x , y ) → N f ( y ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  52. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique Rewriting Algorithm Example N ( x ) → ∃ y E ( x , y ) Σ = S ( x ) ∧ E ( x , y ) → N ( y ) Base (Σ) = N b ( x ) → ∃ y E bf ( x , y ) In (Σ) = S b ( x ) ∧ E bb ( x , y ) → N b ( y ) N ( x ) → N b ( x ) S ( x ) → S b ( x ) Derived (Σ) = E ( x , y ) → E bb ( x , y ) S b ( x ) ∧ E bf ( x , y ) → N f ( y ) N f ( x ) → ∃ y E ff ( x , y ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  53. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique Rewriting Algorithm Example N ( x ) → ∃ y E ( x , y ) Σ = S ( x ) ∧ E ( x , y ) → N ( y ) In (Σ) = Base (Σ) = N ( x ) → N b ( x ) N b ( x ) → ∃ y E bf ( x , y ) S ( x ) → S b ( x ) S b ( x ) ∧ E bb ( x , y ) → N b ( y ) E ( x , y ) → E bb ( x , y ) Derived (Σ) = Out (Σ) = N b ( x ) → ˆ S b ( x ) ∧ E bf ( x , y ) → N f ( y ) N ( x ) N f ( x ) → ˆ N f ( x ) → ∃ y E ff ( x , y ) N ( x ) S b ( x ) → ˆ S ( x ) E bb ( x , y ) → ˆ E ( x , y ) E bf ( x , y ) → ˆ E ( x , y ) E ff ( x , y ) → ˆ E ( x , y ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  54. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique Rewriting Algorithm Example D = { N ( a ) , S ( a ) } chase ( D , Adn (Σ)) = { chase ( D , Map (Σ)) = { N ( a ) S ( a ) N ( a ) S ( a ) E ( a , η 1 ) N ( η 1 ) E ( η 1 , η 2 ) N b ( a ) S b ( a ) E bf ( a , η 1 ) N f ( η 1 ) E ff ( η 1 , η 2 ) N ( a ) ˆ ˆ S ( a ) ˆ E ( a , η 1 ) ˆ ˆ N ( a ) ˆ ˆ S ( a ) ˆ N ( η 1 ) E ( η 1 , η 2 ) E ( a , η 1 ) ˆ ˆ } N ( η 1 ) E ( η 1 , η 2 ) } S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  55. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique Chase Termination Checking Test chase termination criteria over rewritten constraints Adn (Σ) Adn (Σ) = { Figure: Dep (Σ) . . . N b ( x ) → ∃ y E bf ( x , y ) S b ( x ) ∧ E bb ( x , y ) → N b ( y ) S b ( x ) ∧ E bf ( x , y ) → N f ( y ) N f ( x ) → ∃ y E ff ( x , y ) . . . } Figure: Dep ( Adn (Σ)) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  56. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique Chase Termination Checking Adn - T : class of sets of TGDs Σ s.t. Adn (Σ) is in T Theorem T � Adn −T for T ∈{WA , SC , CS tr , IR , S w A , WAS tr , . . . , LS} . Rewriting technique is orthogonal to termination conditions improves current chase termination criteria S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  57. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique Constraints Rewriting Technique Figure: Criteria Relationships S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  58. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique Improving Rewriting Algorithm Adn ++ S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  59. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Example r 1 : R ( x ) → S ( x , x ) r 2 : S ( x 1 , x 2 ) → ∃ z T ( x 2 , z ) ∧ Q ( x 2 ) Σ = r 3 : T ( x 1 , x 2 ) ∧ T ( x 1 , x 3 ) ∧ T ( x 3 , x 1 ) → R ( x 2 ) Adn (Σ) �∈ LS · · · R f ( x ) → S ff ( x , x ) Adn (Σ) = S ff ( x 1 , x 2 ) → ∃ z T ff ( x 2 , z ) ∧ Q f ( x 2 ) T ff ( x 1 , x 2 ) ∧ T ff ( x 1 , x 3 ) ∧ T ff ( x 3 , x 1 ) → R f ( x 2 ) chase ( D , Σ) terminates for all database D S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  60. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Adn ++ algorithm improves Adn S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  61. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Adn ++ algorithm improves Adn testing of the < -relation during the generation of adorned constraints, S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  62. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Adn ++ algorithm improves Adn testing of the < -relation during the generation of adorned constraints, the adornment f i is assigned in a more refined way (skolem function), S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  63. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Adn ++ algorithm improves Adn testing of the < -relation during the generation of adorned constraints, the adornment f i is assigned in a more refined way (skolem function), cyclicity detection ( Adn ++ returns a boolean value, cyc , saying if there is cyclicity among the adorned constraints). S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  64. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Example r 1 : N ( x ) → ∃ y E ( x , y ) Σ = r 2 : E ( x , y ) → N ( y ) r 1 < r 2 r 2 < r 1 S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  65. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Example r 1 : N ( x ) → ∃ y E ( x , y ) Σ = r 2 : E ( x , y ) → N ( y ) r 1 < r 2 r 2 < r 1 cyc = false S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  66. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Example Base (Σ) = r 1 : N ( x ) → ∃ y E ( x , y ) r i 1 : N b ( x ) → ∃ y E bf 1 ( x , y ) Σ = r 2 : E ( x , y ) → N ( y ) r i 2 : E bb ( x , y ) → N b ( y ) r 1 < r 2 r 2 < r 1 cyc = false S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  67. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Example Base (Σ) = r 1 : N ( x ) → ∃ y E ( x , y ) r i 1 : N b ( x ) → ∃ y E bf 1 ( x , y ) Σ = r 2 : E ( x , y ) → N ( y ) r i 2 : E bb ( x , y ) → N b ( y ) Derived (Σ) = r 1 < r 2 r 2 < r 1 cyc = false S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  68. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Example Base (Σ) = r 1 : N ( x ) → ∃ y E ( x , y ) r i 1 : N b ( x ) → ∃ y E bf 1 ( x , y ) Σ = r 2 : E ( x , y ) → N ( y ) r i 2 : E bb ( x , y ) → N b ( y ) Derived (Σ) = r 1 < r 2 r 2 < r 1 cyc = false S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  69. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Example Base (Σ) = r 1 : N ( x ) → ∃ y E ( x , y ) r i 1 : N b ( x ) → ∃ y E bf 1 ( x , y ) Σ = r 2 : E ( x , y ) → N ( y ) r i 2 : E bb ( x , y ) → N b ( y ) Derived (Σ) = r 1 < r 2 r 2 < r 1 cyc = false S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  70. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Example Base (Σ) = r 1 : N ( x ) → ∃ y E ( x , y ) r i 1 : N b ( x ) → ∃ y E bf 1 ( x , y ) Σ = r 2 : E ( x , y ) → N ( y ) r i 2 : E bb ( x , y ) → N b ( y ) Derived (Σ) = r 1 < r 2 r ii 2 : E bf 1 ( x , y ) → N f 1 ( y ) r 2 < r 1 cyc = false S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  71. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Example Base (Σ) = r 1 : N ( x ) → ∃ y E ( x , y ) r i 1 : N b ( x ) → ∃ y E bf 1 ( x , y ) Σ = r 2 : E ( x , y ) → N ( y ) r i 2 : E bb ( x , y ) → N b ( y ) Derived (Σ) = r 1 < r 2 r ii 2 : E bf 1 ( x , y ) → N f 1 ( y ) r 2 < r 1 cyc = false S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  72. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Example Base (Σ) = r 1 : N ( x ) → ∃ y E ( x , y ) r i 1 : N b ( x ) → ∃ y E bf 1 ( x , y ) Σ = r 2 : E ( x , y ) → N ( y ) r i 2 : E bb ( x , y ) → N b ( y ) Derived (Σ) = r 1 < r 2 r ii 2 : E bf 1 ( x , y ) → N f 1 ( y ) r 2 < r 1 cyc = false S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  73. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Example Base (Σ) = r 1 : N ( x ) → ∃ y E ( x , y ) r i 1 : N b ( x ) → ∃ y E bf 1 ( x , y ) Σ = r 2 : E ( x , y ) → N ( y ) r i 2 : E bb ( x , y ) → N b ( y ) Derived (Σ) = r 1 < r 2 r ii 2 : E bf 1 ( x , y ) → N f 1 ( y ) r 2 < r 1 r ii 1 : N f 1 ( x ) → ∃ y E f 1 f 2 ( x , y ) cyc = false S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  74. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Example Base (Σ) = r 1 : N ( x ) → ∃ y E ( x , y ) r i 1 : N b ( x ) → ∃ y E bf 1 ( x , y ) Σ = r 2 : E ( x , y ) → N ( y ) r i 2 : E bb ( x , y ) → N b ( y ) Derived (Σ) = r 1 < r 2 r ii 2 : E bf 1 ( x , y ) → N f 1 ( y ) r 2 < r 1 r ii 1 : N f 1 ( x ) → ∃ y E f 1 f 2 ( x , y ) cyc = false S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  75. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Example Base (Σ) = r 1 : N ( x ) → ∃ y E ( x , y ) r i 1 : N b ( x ) → ∃ y E bf 1 ( x , y ) Σ = r 2 : E ( x , y ) → N ( y ) r i 2 : E bb ( x , y ) → N b ( y ) Derived (Σ) = r 1 < r 2 r ii 2 : E bf 1 ( x , y ) → N f 1 ( y ) r 2 < r 1 r ii 1 : N f 1 ( x ) → ∃ y E f 1 f 2 ( x , y ) cyc = false S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  76. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Example Base (Σ) = r 1 : N ( x ) → ∃ y E ( x , y ) r i 1 : N b ( x ) → ∃ y E bf 1 ( x , y ) Σ = r 2 : E ( x , y ) → N ( y ) r i 2 : E bb ( x , y ) → N b ( y ) Derived (Σ) = r 1 < r 2 r ii 2 : E bf 1 ( x , y ) → N f 1 ( y ) r 2 < r 1 r ii 1 : N f 1 ( x ) → ∃ y E f 1 f 2 ( x , y ) cyc = false r iii 2 : E f 1 f 2 ( x , y ) → N f 2 ( y ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  77. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Example Base (Σ) = r i 1 : N b ( x ) → ∃ y E bf 1 ( x , y ) r 1 : N ( x ) → ∃ y E ( x , y ) Σ = r 2 : E ( x , y ) → N ( y ) r i 2 : E bb ( x , y ) → N b ( y ) Derived (Σ) = r 1 < r 2 r ii 2 : E bf 1 ( x , y ) → N f 1 ( y ) r 2 < r 1 r ii 1 : N f 1 ( x ) → ∃ y E f 1 f 2 ( x , y ) cyc = false r iii 2 : E f 1 f 2 ( x , y ) → N f 2 ( y ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

  78. Stratification Criteria and Rewriting Techniques for Checking Chase Termination Constraints Rewriting Technique A New Rewriting Algorithm Example Base (Σ) = r i 1 : N b ( x ) → ∃ y E bf 1 ( x , y ) r 1 : N ( x ) → ∃ y E ( x , y ) Σ = r 2 : E ( x , y ) → N ( y ) r i 2 : E bb ( x , y ) → N b ( y ) Derived (Σ) = r 1 < r 2 r ii 2 : E bf 1 ( x , y ) → N f 1 ( y ) r 2 < r 1 r ii 1 : N f 1 ( x ) → ∃ y E f 1 f 2 ( x , y ) cyc = false r iii 2 : E f 1 f 2 ( x , y ) → N f 2 ( y ) r iii 1 : N f 2 ( x ) → ∃ y E f 2 f 3 ( x , y ) S. Greco, F. Spezzano and I. Trubitsyna Stratification Criteria and Rewriting Techniques for Checking

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