View-based query determinacy and rewritings over graph databases - PowerPoint PPT Presentation

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Example 2 [Afrati] V 3 = � a 3 � V 4 = � a 4 � Q 5 = � a 5 � R( x , y ) = ∃ z , V 4 ( x , z ) ∧ ( ∀ z ′ V 3 ( z ′ , z ) ⇒ V 4 ( z ′ , y )) Q ⇒ R � z ′ y x z 16 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Example 2 [Afrati] V 3 = � a 3 � V 4 = � a 4 � Q 5 = � a 5 � R( x , y ) = ∃ z , V 4 ( x , z ) ∧ ( ∀ z ′ V 3 ( z ′ , z ) ⇒ V 4 ( z ′ , y )) Q ⇒ R � R ⇒ Q z ′ y x z 16 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Example 2 [Afrati] V 3 = � a 3 � V 4 = � a 4 � Q 5 = � a 5 � R( x , y ) = ∃ z , V 4 ( x , z ) ∧ ( ∀ z ′ V 3 ( z ′ , z ) ⇒ V 4 ( z ′ , y )) Q ⇒ R � R ⇒ Q y z ′ y x z x 16 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Example 2 [Afrati] V 3 = � a 3 � V 4 = � a 4 � Q 5 = � a 5 � R( x , y ) = ∃ z , V 4 ( x , z ) ∧ ( ∀ z ′ V 3 ( z ′ , z ) ⇒ V 4 ( z ′ , y )) Q ⇒ R � R ⇒ Q y z ′ y x z x z 16 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Example 2 [Afrati] V 3 = � a 3 � V 4 = � a 4 � Q 5 = � a 5 � R( x , y ) = ∃ z , V 4 ( x , z ) ∧ ( ∀ z ′ V 3 ( z ′ , z ) ⇒ V 4 ( z ′ , y )) Q ⇒ R � R ⇒ Q y z ′ y z ′ x z x z 16 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Example 2 [Afrati] V 3 = � a 3 � V 4 = � a 4 � Q 5 = � a 5 � R( x , y ) = ∃ z , V 4 ( x , z ) ∧ ( ∀ z ′ V 3 ( z ′ , z ) ⇒ V 4 ( z ′ , y )) Q ⇒ R � R ⇒ Q y z ′ y z ′ x z x z 16 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Example 2 [Afrati] V 3 = � a 3 � V 4 = � a 4 � Q 5 = � a 5 � R( x , y ) = ∃ z , V 4 ( x , z ) ∧ ( ∀ z ′ V 3 ( z ′ , z ) ⇒ V 4 ( z ′ , y )) Q ⇒ R � R ⇒ Q y z ′ y z ′ x z x z 16 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Example 2 [Afrati] V 3 = � a 3 � V 4 = � a 4 � Q 5 = � a 5 � R( x , y ) = ∃ z , V 4 ( x , z ) ∧ ( ∀ z ′ V 3 ( z ′ , z ) ⇒ V 4 ( z ′ , y )) Q ⇒ R � R ⇒ Q � y z ′ y z ′ x z x z 16 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Example 2 [Afrati] V 3 = � a 3 � V 4 = � a 4 � Q 5 = � a 5 � R( x , y ) = ∃ z , V 4 ( x , z ) ∧ ( ∀ z ′ V 3 ( z ′ , z ) ⇒ V 4 ( z ′ , y )) R ′ ( x , y ) = ∃ z , t , V 4 ( x , z ) ∧ V 3 ( x , t ) ∧ ∀ z ′ V 3 ( z ′ , z ) ⇒ V 4 ( z ′ , y ) R ′ is also a rewriting : ∀ D , R( V (D)) = R ′ ( V (D)) R and R ′ behave differently outside of view images : ∃ E , R(E) � = R ′ (E) 16 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Certain answers [Abiteboul, Duschka] cert exact � Q , V (E) = Q(D) (exact views) D | V (D)=E cert sound � Q , V (E) = Q(D) (sound views) D | V (D) ⊇ E 17 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Certain answers [Abiteboul, Duschka] cert exact � Q , V (E) = Q(D) (exact views) D | V (D)=E cert sound � Q , V (E) = Q(D) (sound views) D | V (D) ⊇ E Remarks If V determines Q, then cert exact is a rewriting. Q , V cert Q , V ( · ) has coNP-complete data complexity. 17 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Certain answers [Abiteboul, Duschka] cert exact � Q , V (E) = Q(D) (exact views) D | V (D)=E cert sound � Q , V (E) = Q(D) (sound views) D | V (D) ⊇ E Remarks If V determines Q, then cert exact is a rewriting. Q , V cert Q , V ( · ) has coNP-complete data complexity. Theorem [F., Segoufin, Sirangelo] If an RPQ view V determines an RPQ query Q, then there exist a rewriting of Q using V with NP data complexity, and one with coNP data complexity. → cert exact is likely not optimal! Q , V 17 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Positive and negative results [Afrati]: Determinacy is decidable for single path queries and views. Rewritings are in FO. [Calvanese, De Giacomo, Lenzerini, Vardi]: Monotone determinacy is decidable for RPQ views and queries. [Nash, Segoufin, Vianu]: Determinacy is undecidable for UCQ views and queries. [F., Segoufin, Sirangelo]: Determinacy is undecidable for CRPQ views and RPQ queries. Open questions Decidability is still open for two key query languages: Conjunctive queries and views. (WIP - [Marcinkowski et al.]) Regular path queries and views. 18 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Main contributions Theorem [ICDT 2014] If an RPQ view V determines an RPQ query Q in a monotone way, then there exists a Datalog rewriting of Q using V . → Cor: there exists a rewriting with PTime data complexity. → Cor: given V and Q, it is decidable whether there exists a Datalog rewriting of Q using V . 19 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Main contributions Theorem [ICDT 2014] If an RPQ view V determines an RPQ query Q in a monotone way, then there exists a Datalog rewriting of Q using V . → Cor: there exists a rewriting with PTime data complexity. → Cor: given V and Q, it is decidable whether there exists a Datalog rewriting of Q using V . Theorem [ICDT 2015, Best student paper] The asymptotic determinacy problem is decidable for UPQ views and SPQ queries over graphs, and results in first-order rewritings. → Cor: first-order is almost complete for rewritings of SPQ queries using UPQ views. 19 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Monotone Determinacy 20 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Example 2 revisited V 3 = � a 3 � V 4 = � a 4 � Q = � a 5 � V ։ Q but f is not monotone. V (D) ⊆ V (D ′ ) but Q(D ′ ) = ∅ Q(D) = { ( x 0 , x 5 ) } x 1 x 5 x 2 x 0 x 1 x 2 x 3 x 4 x 5 x 0 x 3 x 4 D ′ D 21 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Monotone determinacy V ։ Q and f is monotone. Equivalently: V (D) ⊆ V (D ′ ) ⇒ Q(D) ⊆ Q(D ′ ) ∀ D , D ′ Motivations Many interesting languages are monotone: conjunctive queries, regular path queries, Datalog, etc. [Nash, Segoufin, Vianu]: for CQ views and queries, monotone determinacy ⇒ CQ rewriting. Key observation If V determines Q in a monotone way, then cert sound is a rewriting. Q , V → Idea: look for approximations of cert sound Q , V . 22 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Constraint Satisfaction Problems as Queries Fix a template structure T with source and target nodes (E , u , v ) ∈ CSP( T ) iff there exists h such that h E − → T h ( u ) is a source node h ( v ) is a target node CSP( T ) selects all u , v such that (E , u , v ) ∈ CSP( T ) Theorem [Calvanese, De Giacomo, Lenzerini, Vardi] There exists a structure T Q , V such that, for all D, ( u , v ) ∈ cert sound Q , V ( V (D)) ⇔ ( V (D) , u , v ) / ∈ CSP( T Q , V ) 23 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy A CSP approximation: the ( ℓ, k )-two-player-game Arena : (E , u , v ) and T with source and target nodes. u v E T 24 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy A CSP approximation: the ( ℓ, k )-two-player-game Player 1 starts by selecting k nodes on E. u v E T 24 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy A CSP approximation: the ( ℓ, k )-two-player-game Player 2 provides a homomorphism from those nodes to T. u v E T 24 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy A CSP approximation: the ( ℓ, k )-two-player-game Player 1 selects k nodes, keeping at most ℓ previous nodes. u v E T 24 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy A CSP approximation: the ( ℓ, k )-two-player-game Player 2 extends previous homomorphism to new nodes. u v E T 24 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy A CSP approximation: the ( ℓ, k )-two-player-game If Player 1 selects u ... u v E T 24 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy A CSP approximation: the ( ℓ, k )-two-player-game Then Player 2 must send it to a source node. u v E T 24 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy A CSP approximation: the ( ℓ, k )-two-player-game Similarly, if Player 1 selects v ... u v E T 24 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy A CSP approximation: the ( ℓ, k )-two-player-game Then Player 2 must send it to a target node. u v E T 24 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy A CSP approximation: the ( ℓ, k )-two-player-game If Player 2 has no legal move... u v E T 24 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy A CSP approximation: the ( ℓ, k )-two-player-game Then Player 1 wins. u WIN LOSS v E T 24 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy A CSP approximation: the ( ℓ, k )-two-player-game Player 2 wins if she can play forever. u v E T 24 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Approximations The game on T Q , V over-approximates CSP( T Q , V ). → If ( E , u , v ) ∈ CSP( T Q , V ), then P2 has a winning strategy. The game on T Q , V under-approximates cert sound Q , V . → If P1 has a winning strategy, then ( u , v ) ∈ cert sound Q , V ( E ). Theorem [Feder, Vardi] There exists a Datalog ℓ, k program Q ℓ, k such that ( u , v ) ∈ Q ℓ, k (E) if and only if Player 1 has a winning strategy for the game played on (E , u , v ) and T Q , V . 25 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy The case of simple paths ∃ ℓ, k such that Q ℓ, k is exact on views of simple paths. ( u , v ) ∈ Q ℓ, k ( V ( π )) if and only if ( u , v ) ∈ cert sound Q , V ( V ( π )). • This is the technical part of the proof! Lifting paths to arbitrary databases Assume that some candidate query R is: a rewriting on views of simple paths, an under-approximation of a rewriting, closed under homomorphism. Then it is a rewriting on views of arbitrary databases. 26 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy V ։ Q and f ր 27 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy V ։ Q and f ր . . . Certain answers are a (coNP-hard) rewriting. 27 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy V ։ Q and f ր . . . Certain answers are a (coNP-hard) rewriting. . . . Certain answers ≡ ¬ CSP [Calvanese et al.] 27 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy V ։ Q and f ր . . . Certain answers are a (coNP-hard) rewriting. . . . Certain answers ≡ ¬ CSP [Calvanese et al.] . . . ¬ CSP ∼ Q ℓ, k in Datalog [Feder, Vardi] 27 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy V ։ Q and f ր . . . Certain answers are a (coNP-hard) rewriting. . . . Certain answers ≡ ¬ CSP [Calvanese et al.] . . . ¬ CSP ∼ Q ℓ, k in Datalog [Feder, Vardi] . . . Q ℓ, k is exact on simple paths, for some ℓ, k 27 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy V ։ Q and f ր . . . Certain answers are a (coNP-hard) rewriting. . . . Certain answers ≡ ¬ CSP [Calvanese et al.] . . . ¬ CSP ∼ Q ℓ, k in Datalog [Feder, Vardi] . . . Q ℓ, k is exact on simple paths, for some ℓ, k . . . Paths can be lifted to arbitrary databases. 27 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy V ։ Q and f ր . . . Certain answers are a (coNP-hard) rewriting. . . . Certain answers ≡ ¬ CSP [Calvanese et al.] . . . ¬ CSP ∼ Q ℓ, k in Datalog [Feder, Vardi] . . . Q ℓ, k is exact on simple paths, for some ℓ, k . . . Paths can be lifted to arbitrary databases. . . . Q ℓ, k is a (PTime) rewriting. 27 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Asymptotic Determinacy 28 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Restricted setting Graph databases → Graphs Regular path queries → Single path queries Notation: � a k � → � k � Regular path views “ → ” Union of single path queries Notation: � a i , a j � → � i , j � 29 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Restricted setting Graph databases → Graphs Regular path queries → Single path queries Notation: � a k � → � k � Regular path views “ → ” Union of single path queries Notation: � a i , a j � → � i , j � Example 3 V 2 = � 2 � V 1 , 2 = � 1 , 2 � V 2 , 3 = � 2 , 3 � Q 3 = � 3 � Q 5 = � 5 � V � ։ Q 3 V ։ Q 5 29 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Key observation 1 If V ։ Q k then ∃ c , Q c ∈ V . 30 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Key observation 1 If V ։ Q k then ∃ c , Q c ∈ V . By contraposition, if ∀ c , Q c / ∈ V , then V � ։ Q k . Proof by example, with V 2 , 4 � ։ Q 4 . x 0 x 1 x 2 x 3 x 4 30 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Key observation 1 If V ։ Q k then ∃ c , Q c ∈ V . By contraposition, if ∀ c , Q c / ∈ V , then V � ։ Q k . Proof by example, with V 2 , 4 � ։ Q 4 . V 2 , 4 x 0 x 1 x 2 x 3 x 4 30 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Key observation 1 If V ։ Q k then ∃ c , Q c ∈ V . By contraposition, if ∀ c , Q c / ∈ V , then V � ։ Q k . Proof by example, with V 2 , 4 � ։ Q 4 . x 0 x 1 x 2 x 3 x 4 30 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Key observation 1 If V ։ Q k then ∃ c , Q c ∈ V . By contraposition, if ∀ c , Q c / ∈ V , then V � ։ Q k . Proof by example, with V 2 , 4 � ։ Q 4 . x 0 x 1 x 2 x 3 x 4 V 2 , 4 30 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Key observation 1 If V ։ Q k then ∃ c , Q c ∈ V . By contraposition, if ∀ c , Q c / ∈ V , then V � ։ Q k . Proof by example, with V 2 , 4 � ։ Q 4 . x 0 x 1 x 2 x 3 x 4 30 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Key observation 1 If V ։ Q k then ∃ c , Q c ∈ V . By contraposition, if ∀ c , Q c / ∈ V , then V � ։ Q k . Proof by example, with V 2 , 4 � ։ Q 4 . x 0 x 1 x 2 x 3 x 4 30 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Key observation 1 If V ։ Q k then ∃ c , Q c ∈ V . By contraposition, if ∀ c , Q c / ∈ V , then V � ։ Q k . Proof by example, with V 2 , 4 � ։ Q 4 . x 0 x 1 x 2 x 3 x 4 Key observation 2 If V ։ Q k then V ։ Q k + c . 30 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Observations 1 & 2 → possible behaviors for a view V : 31 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Observations 1 & 2 → possible behaviors for a view V : V ։ ? Q 1 Q 2 Q 3 Q 4 Q 5 Q c . . . 31 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Observations 1 & 2 → possible behaviors for a view V : V ։ ? Q 1 Q 2 Q 3 Q 4 Q 5 Q c . . . Q 1+ c Q 2+ c Q 3+ c Q 4+ c Q 5+ c Q 2 c . . . 31 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Observations 1 & 2 → possible behaviors for a view V : V ։ ? Q 1 Q 2 Q 3 Q 4 Q 5 Q c . . . Q 1+ c Q 2+ c Q 3+ c Q 4+ c Q 5+ c Q 2 c . . . Q 1+2 c Q 2+2 c Q 3+2 c Q 4+2 c Q 5+2 c Q 3 c . . . 31 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Observations 1 & 2 → possible behaviors for a view V : V ։ ? Q 1 Q 2 Q 3 Q 4 Q 5 Q c . . . Q 1+ c Q 2+ c Q 3+ c Q 4+ c Q 5+ c Q 2 c . . . Q 1+2 c Q 2+2 c Q 3+2 c Q 4+2 c Q 5+2 c Q 3 c . . . Q 1+3 c Q 2+3 c Q 3+3 c Q 4+3 c Q 5+3 c Q 4 c . . . 31 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Observations 1 & 2 → possible behaviors for a view V : V ։ ? Q 1 Q 2 Q 3 Q 4 Q 5 Q c . . . Q 1+ c Q 2+ c Q 3+ c Q 4+ c Q 5+ c Q 2 c . . . Q 1+2 c Q 2+2 c Q 3+2 c Q 4+2 c Q 5+2 c Q 3 c . . . Q 1+3 c Q 2+3 c Q 3+3 c Q 4+3 c Q 5+3 c Q 4 c . . . . . . . . . . . . . . . . . . . . . Q 1+ α c Q 2+ α c Q 3+ α c Q 4+ α c Q 5+ α c Q c + α c . . . . . . . . . . . . . . . . . . . . . 31 / 38

Introduction Determinacy and Query Rewriting Monotone Determinacy Asymptotic Determinacy Observations 1 & 2 → possible behaviors for a view V : V ։ ? Q 1 Q 2 Q 3 Q 4 Q 5 Q c . . . Q 1+ c Q 2+ c Q 3+ c Q 4+ c Q 5+ c Q 2 c . . . Q 1+2 c Q 2+2 c Q 3+2 c Q 4+2 c Q 5+2 c Q 3 c . . . Q 1+3 c Q 2+3 c Q 3+3 c Q 4+3 c Q 5+3 c Q 4 c . . . . . . . . . . . . . . . . . . . . . Q 1+ α c Q 2+ α c Q 3+ α c Q 4+ α c Q 5+ α c Q c + α c . . . . . . . . . . . . . . . . . . . . . Asymptotic determinacy problem Given V , produce full picture at some line α past stabilization. 31 / 38