Regular n-ary Queries in Trees and Variable Independence

Transcription

1 Regular n-ary Queries in Trees and Variable Independence Emmanuel Filiot Sophie Tison Laboratoire d Informatique Fondamentale de Lille (LIFL) INRIA Lille Nord-Europe, Mostrare Project IFIP TCS, 2008 E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

2 Motivations n-ary queries φ(x 1,...,x n ) in trees t: select n-tuples of nodes fundamental to XML processing tasks the set of answers can grow exponentially in t (O( t n ) in the worst case) the answers share common information a compact representation is needed E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

3 Aggregated Answers (Meuss, Schulz, Bry, ICDT 01) join-free term over {, } and constants [x n]; Example (aggregated answers) XML document DVDs Aggregated Answer DVD 0 7 DVD0 DVD lang prices prices sub E $... fr ende sub language subtitles E $... ende 8 9 en fr ench Query: (DVD, Lang, Subtitles) E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

4 Advantages of this representation efficient model-checking: τ Ans(φ, t)? efficient enumeration: τ 1,...,τ i,... advanced query answering: answer searching, statistics answer browsing cascade style query-answering view computing E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

5 Objective How compact are aggregated answers? How to measure compactness? E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

6 Objective How compact are aggregated answers? How to measure compactness? compactness is related to variable independence variable independence in the context of infinite constraint databases (Chomicki et. al., PODS 96) (Libkin, TOCL 03) only between blocks of variables, and fixed database we propose a more general notion: dependency forests MSO in ordered finite binary trees E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

7 Objective How compact are aggregated answers? How to measure compactness? compactness is related to variable independence variable independence in the context of infinite constraint databases (Chomicki et. al., PODS 96) (Libkin, TOCL 03) Question only between blocks of variables, and fixed database we propose a more general notion: dependency forests MSO in ordered finite binary trees Given a dependency forest, and a query, are the variables independent w.r.t. this forest? E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

8 Outline 1 variable independence w.r.t. a partition 2 variable independence w.r.t. a dependency forest E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

9 Trees and MSO finite ordered binary trees t are viewed as structures over the signature S 1 (x,y),s 2 (x,y),lab a (x), a Σ, with domain nodes(t) first-order variables x, y, z,... denote nodes second-order variables X, Y, Z,... denote set of nodes existential quantifiers x, X, Boolean operators,,, and membership x X; an MSO formula φ(x 1,...,x n ) defines an n-ary query: Ans(φ,t) = {(u 1,...,u n ) nodes(t) n t = φ(u 1,...,u n )} E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

10 Trees and MSO finite ordered binary trees t are viewed as structures over the signature S 1 (x,y),s 2 (x,y),lab a (x), a Σ, with domain nodes(t) first-order variables x, y, z,... denote nodes second-order variables X, Y, Z,... denote set of nodes existential quantifiers x, X, Boolean operators,,, and membership x X; an MSO formula φ(x 1,...,x n ) defines an n-ary query: Ans(φ,t) = {(u 1,...,u n ) nodes(t) n t = φ(u 1,...,u n )} Example Select all the DVDs: φ(x) = lab DV D (x) E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

11 Trees and MSO finite ordered binary trees t are viewed as structures over the signature S 1 (x,y),s 2 (x,y),lab a (x), a Σ, with domain nodes(t) first-order variables x, y, z,... denote nodes second-order variables X, Y, Z,... denote set of nodes existential quantifiers x, X, Boolean operators,,, and membership x X; an MSO formula φ(x 1,...,x n ) defines an n-ary query: Ans(φ,t) = {(u 1,...,u n ) nodes(t) n t = φ(u 1,...,u n )} Example Select all the triples (dvd,lang,sub): φ(x,y,z) = lab DV D (x) lab lang (y) lab sub (z) desc(x,y) desc(x,z) E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

12 Variable Independence w.r.t. a Partition input: a formula φ(x 1,...,x n ), a partition P = {B 1,...,B k } of {x 1,...,x n } output: is φ equivalent to a formula of the form: N φ 1 i (B 1) φ k i (B k) i=1 where freevar(φ j i ) = B j. We say that φ conforms to P. Theorem Variable independence w.r.t. a partition is decidable, and a decomposition is computable. E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

13 Towards a Characterization of Variable Independence For all i {1,...,k}, consider: Swap i (x,y) = B 1... B i 1 B i+1... B n φ(b 1,...,B i 1, x,b i+1,...,b n ) φ(b 1,...,B i 1, y,b i+1,...,x Bn ) E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

14 Towards a Characterization of Variable Independence For all i {1,...,k}, consider: Swap i (x,y) = B 1... B i 1 B i+1... B n φ(b 1,...,B i 1, x,b i+1,...,b n ) φ(b 1,...,B i 1, y,b i+1,...,x Bn ) intuition: if t = Swap i (u,v) then u and v are not distinguished by φ. E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

15 Towards a Characterization of Variable Independence For all i {1,...,k}, consider: Swap i (x,y) = B 1... B i 1 B i+1... B n φ(b 1,...,B i 1, x,b i+1,...,b n ) φ(b 1,...,B i 1, y,b i+1,...,x Bn ) intuition: if t = Swap i (u,v) then u and v are not distinguished by φ. Every formula Swap i (x,y) defines on a tree t an equivalence relation Ans(Swap i,t) between tuples of size B i. E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

16 Variable Independence Reduces to Query Boundedness Theorem φ conforms to {B 1,...,B k } iff for all i = 1,...,k, Swap i is of bounded index, i.e.: b i 0, t, nodes(t) B i / Ans(Swapi,t) b i The decomposition has the following form: φ(x 1,...,x n ) j φ j cl j 1 (B 1) cl j k (B k) E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

17 On Deciding Boundedness Theorem Given a formula φ(x), we can decide if there is a bound b 0 such that: t, Ans(φ,t) b Moreover, some bound b is computable. 1 translate φ(x) into a canonical {0,1}-labeling transducer T φ 2 decide whether the number of images of any tree by T φ is bounded by some constant [Seidl, habilitation thesis]. Extension to n-ary queries φ(x 1,...,x n ) is bounded iff for each i, the following is bounded: proj i (x) = x 1... x i 1 x i+1... x n φ(x 1,...,x i 1,x,x i+1,...,x n ) E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

18 Bounded Index Property Corollary Bounded index property is decidable for every Swap i (x,y), moreover, an index is computable. 1 define a total order x n y on n-tuples of nodes (by an MSO formula); 2 decide boundedness of Min(x), which selects the minimal representatives of the relation defined by Swap i (x,y). E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

19 Orthographic Dimension Definition (Grumbach, Rigaux, Segoufin, ICDT 99) Let φ(x 1,...,x n ) be a formula, and P the set of partitions P such that φ conforms to P. The orthographic dimension d(φ) is defined by: d(φ) = min {B1,...,B k } P max i B i how to compute it? try every partition of {x 1,...,x n }. improvement: consider only 2-partitions, thanks to the following theorem, adapted from (Cosmadakis, Kuper, Libkin, 01): E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

20 Orthographic Dimension Definition (Grumbach, Rigaux, Segoufin, ICDT 99) Let φ(x 1,...,x n ) be a formula, and P the set of partitions P such that φ conforms to P. The orthographic dimension d(φ) is defined by: d(φ) = min {B1,...,B k } P max i B i how to compute it? try every partition of {x 1,...,x n }. improvement: consider only 2-partitions, thanks to the following theorem, adapted from (Cosmadakis, Kuper, Libkin, 01): Theorem If φ conforms to P 1 and P 2, then φ conforms to P 1 P 2. E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

21 Relation to answer set representation We can compute a formula equivalent to φ(x 1,...,x n ) which corresponds to the orthographic dimension, i.e. a formula N φ 1 i (B 1 ) φ k i (B k ) where d(φ) = max i B i i=1 E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

22 Relation to answer set representation We can compute a formula equivalent to φ(x 1,...,x n ) which corresponds to the orthographic dimension, i.e. a formula N φ 1 i (B 1 ) φ k i (B k ) where d(φ) = max i B i i=1 The aggregated answer on a tree t is of size O(f( φ ). t d(φ) ). compute the answer sets A j i of each φj i represent the answer set by a union (whose size only depends on φ, which is fixed) of cartesian products A 1 i Ak i. E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

23 Outline 1 variable independence w.r.t. a partition 2 variable independence w.r.t. a dependency forest E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

24 Dependency Forest F over a set V = {x 1,...,x n } of variables B 1 B 6 B 2 B 5 B 7 where {B1,...,B7} partitions V B 3 B 4 E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

25 Conformance to a Dependency Forest F {B 1,...,B 7 } partitions V = {x 1,...,x n } B 1 B 6 φ 2 (B 1,B 2 ) φ 3 (B 1,B 5 ) φ 1 (B 6,B 7 ) B 2 B 5 B 7 φ 4 (B 2,B 3 ) φ 5 (B 2,B 4 ) B 3 B 4 µ : edges(f) MSO formulas µ(f) = φ 1 (B 6,B 7 ) φ 2 (B 1,B 3 ) φ 3 (B 1,B 5 ) φ 4 (B 2,B 3 ) φ 5 (B 2,B 5 ) E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

26 Conformance to a Dependency Forest F {B 1,...,B 7 } partitions V = {x 1,...,x n } B 1 B 6 φ 2 (B 1,B 2 ) φ 3 (B 1,B 5 ) φ 1 (B 6,B 7 ) B 2 B 5 B 7 φ 4 (B 2,B 3 ) φ 5 (B 2,B 4 ) B 3 B 4 Definition A formula φ(x 1,...,x n ) conforms to F if there is a finite sequence µ 1,...,µ k such that φ i µ i(f). E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

27 Example x The query φ(x,y,z) : t (DV D,Lang,Sub) conforms to y z E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

28 Relation to Answer Set Representation The set of answers can be represented by an aggregated answer of size O(f( φ ). t 2b ), where b = max i y i. B 1 B 6 B 1 B 6 B 2 B 5 B 7 B 2 B 7 B 5 B 3 B 4 B 3 B 4 E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

29 Main Result Theorem It is decidable whether a formula φ(x 1,...,x n ) conforms to a dependency forest F. 1 decide it for forests of the form B 1 (B 2,B 3 ) signature Σ {0,1} B 1 E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

30 Main Result Theorem It is decidable whether a formula φ(x 1,...,x n ) conforms to a dependency forest F. 1 decide it for forests of the form B 1 (B 2,B 3 ) signature Σ {0,1} B 1 2 inductively: if F = {T 1,...,T k } decompose φ w.r.t. the partition {var(t1 ),..., var(t k )} get a disjunction of the form i α1 i (var(t 1)) α k i (var(t k)) decompose each α j i w.r.t. T j E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

31 Main Result Theorem It is decidable whether a formula φ(x 1,...,x n ) conforms to a dependency forest F. 1 decide it for forests of the form B 1 (B 2,B 3 ) signature Σ {0,1} B 1 2 inductively: if F = B 1 (B 2 (F 1 ),F 2 ): B 1 i φi 1 (B 1, B 2 var(f 1 )) φ i 2 (B 1,var(F 2 )) B 2 var(f 1 )var(f 2 ) B 2 B 1 B 1 F 1 F 2 E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

32 Are tuples of variables really needed in F? Proposition There is an MSO formula φ which does not conform to any dependency forest whose nodes are labeled by single variables. take φ(x, y, z) defined by lca(x, y) = lca(x, z) find counter-examples for every forest F over {x,y,z} E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

33 Perspectives extend the structures beyond trees need an MSO-definable total order decidability of boundedness = variable independence is decidable for unranked trees E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22

34 Perspectives extend the structures beyond trees need an MSO-definable total order decidability of boundedness = variable independence is decidable for unranked trees tree pattern queries n-ary tree patterns with desc, child, next sibling, label tests fragments for which the aggregated answers have size O(poly( φ ). t ) E.Filiot S.Tison (LIFL&INRIA) Variable Independence IFIP TCS, / 22