Queries, Modalities, Relations, Trees, XPath Lecture VII Core XPath and beyond

Transcription

1 Queries, Modalities, Relations, Trees, XPath Lecture VII Core XPath and beyond Tadeusz Litak Department of Computer Science University of Leicester July 2010: draft Tadeusz Litak Lecture VII: CXPath and beyond (1/46)

2 Basic Axioms I: Idempotent Semirings ISAx1 (A B) C A (B C) ISAx2 A B B A ISAx3 A A A ISAx4 A/(B/C) (A/B)/C ISAx5{ /A A A/ A A/(B C) A/B A/C ISAx6{ (A B)/C A/C B/C ISAx7 A Distributive lattices, Kleene algebras, Tarski s relation algebras: they all have idempotent semiring reducts. Idempotency is the axiom ISAx3. abbreviates [ ] Tadeusz Litak Lecture VII: CXPath and beyond (2/46)

3 Basic Axioms II: Predicate Axioms PrAx1 A [ B ] /B PrAx2 A [φ ψ] A [φ] A [ψ] PrAx3 (A/B) [φ] A/B [φ] PrAx4 [ ] In Tarski s relation algebras and XPath 2.0, predicates can be defined away Note that PrAx3 would not be valid if we allowed unrestricted positional predicates Tadeusz Litak Lecture VII: CXPath and beyond (3/46)

4 Basic Axioms III: Node Axioms NdAx1 φ ( φ ψ) ( φ ψ) NdAx2 A B A B NdAx3 A/B A [ B ] NdAx4 [φ] φ Note how little was needed to ensure booleanity! (by Huntington s result from the 1930 s) And NdAx2 NdAx4 just mimick PrAx2 PrAx4 (redundancy: price to pay for two-sorted signature) Tadeusz Litak Lecture VII: CXPath and beyond (4/46)

5 Axioms in one-sorted signature Recall all the two-sorted axioms for predicates and expressions: PrAx1 A [ B ] /B PrAx2 A [φ ψ] A [φ] A [ψ] PrAx3 (A/B) [φ] A/B [φ] PrAx4 [ ] NdAx1 φ ( φ ψ) ( φ ψ) NdAx2 A B A B NdAx3 A/B A [ B ] NdAx4 [φ] φ Tadeusz Litak Lecture VII: CXPath and beyond (5/46)

6 Axioms in one-sorted signature Here is a one-sorted axiomatization for over idempotent semi-ring axioms found by Hollenberg: A/A A/A A (A/B)/A ( (A/B)/A)/ B (A B) A/ B A B ( A B) We need to add one more axiom for tests:?p?p Tadeusz Litak Lecture VII: CXPath and beyond (5/46)

7 Now, you may have the feeling that there was nothing XPath-specific yet Tadeusz Litak Lecture VII: CXPath and beyond (6/46)

8 Now, you may have the feeling that there was nothing XPath-specific yet But in fact there is a fragment for which it is all there is: Tadeusz Litak Lecture VII: CXPath and beyond (6/46)

9 Theorem Now, you may have the feeling that there was nothing XPath-specific yet But in fact there is a fragment for which it is all there is: Core XPath( ), the child-axis-only fragment! The axioms presented so far are complete for all valid equivalences of Core XPath( ). Tadeusz Litak Lecture VII: CXPath and beyond (6/46)

10 Theorem Now, you may have the feeling that there was nothing XPath-specific yet But in fact there is a fragment for which it is all there is: Core XPath( ), the child-axis-only fragment! The axioms presented so far are complete for all valid equivalences of Core XPath( ). In order to find more interesting equivalences, we have to move to other fragments Tadeusz Litak Lecture VII: CXPath and beyond (6/46)

11 Axioms for Linear Axes The non-transitive case: LinAx1 s [ φ] [ s [φ] ] /s for s {,, } This forces functionality of the corresponding axis Tadeusz Litak Lecture VII: CXPath and beyond (7/46)

12 Axioms for Transitive Axes One for node expressions, one for path expressions: TransAx1 s + [φ] s + [φ s + [φ] ] TransAx2 s + s + s + /s + The first one is called the Löb axiom and forces well-foundedness Don t get modal logicians started on it people wrote books about this formula In particular, all the consequences of TransAx2 for node expressions can be already derived from TransAx1 I can neither prove nor disprove that for path expressions TransAx2 is (ir-)redundant Tadeusz Litak Lecture VII: CXPath and beyond (8/46)

13 Finally, Axes which Are Both Transitive and Linear LinAx2 [ s + [φ] ] /s + s + [φ] s + [φ] /s + s + [ s + [φ] ] for s {,, } together with transitivity axioms This forces the corresponding axis is a linear order Tadeusz Litak Lecture VII: CXPath and beyond (9/46)

14 Single Axis Completeness Result Theorem Base axioms are complete for Core XPath( ) Base axioms with LinAx1 are complete for other intransitive single axis fragments Base axioms with TransAx1 and TransAx2 are complete for Core XPath( + ) Base axioms with TransAx1, TransAx2 and LinAx2 are complete for other transitive single axis fragments Tadeusz Litak Lecture VII: CXPath and beyond (10/46)

15 A Few Words About Proofs First, rewrite node expressions to simple node expressions: sinode ::= p a [sinode] sinode sinode sinode Tadeusz Litak Lecture VII: CXPath and beyond (11/46)

16 A Few Words About Proofs First, rewrite node expressions to simple node expressions: sinode ::= p a [sinode] sinode sinode sinode They are isomorphic variants of modal formulas Tadeusz Litak Lecture VII: CXPath and beyond (11/46)

17 A Few Words About Proofs First, rewrite node expressions to simple node expressions: sinode ::= p a [sinode] sinode sinode sinode They are isomorphic variants of modal formulas Using normal form theorems for modal logic, we provide a completeness proof for node expressions Tadeusz Litak Lecture VII: CXPath and beyond (11/46)

18 A Few Words About Proofs cntd. Then we rewrite all path expressions as sums of sum-free expressions of the form (all β i are normal forms of S = [β 1 ] /a [β 2 ] /... /a [β l ], the same nesting degree in case of transitive axes strictly decreasing degree for intransitive axes) In case of linear axes, we can even guarantee that every formula is witnessed further down the chain Tadeusz Litak Lecture VII: CXPath and beyond (12/46)

19 A Few Words About Proofs cntd. Then we rewrite all path expressions as sums of sum-free expressions of the form (all β i are normal forms of S = [β 1 ] /a [β 2 ] /... /a [β l ], the same nesting degree in case of transitive axes strictly decreasing degree for intransitive axes) In case of linear axes, we can even guarantee that every formula is witnessed further down the chain We prove that for every two such expressions either Tadeusz Litak Lecture VII: CXPath and beyond (12/46)

20 A Few Words About Proofs cntd. Then we rewrite all path expressions as sums of sum-free expressions of the form (all β i are normal forms of S = [β 1 ] /a [β 2 ] /... /a [β l ], the same nesting degree in case of transitive axes strictly decreasing degree for intransitive axes) In case of linear axes, we can even guarantee that every formula is witnessed further down the chain We prove that for every two such expressions either one is a subsequence of the other provably contained or Tadeusz Litak Lecture VII: CXPath and beyond (12/46)

21 A Few Words About Proofs cntd. Then we rewrite all path expressions as sums of sum-free expressions of the form (all β i are normal forms of S = [β 1 ] /a [β 2 ] /... /a [β l ], the same nesting degree in case of transitive axes strictly decreasing degree for intransitive axes) In case of linear axes, we can even guarantee that every formula is witnessed further down the chain We prove that for every two such expressions either one is a subsequence of the other provably contained or there is a countermodel for containment Tadeusz Litak Lecture VII: CXPath and beyond (12/46)

22 Aside: the issue of labels There is a fact about XML trees we did not take into account (unless we opt to render attribute-value pairs as additional labels) Tadeusz Litak Lecture VII: CXPath and beyond (13/46)

23 Aside: the issue of labels There is a fact about XML trees we did not take into account (unless we opt to render attribute-value pairs as additional labels) The labels are disjoint! Tadeusz Litak Lecture VII: CXPath and beyond (13/46)

24 Aside: the issue of labels There is a fact about XML trees we did not take into account (unless we opt to render attribute-value pairs as additional labels) The labels are disjoint! However, this is easy to fix: add node axiom p q for distinct p and q This axiom itself is not substitution-invariant, this is why we do not like it Tadeusz Litak Lecture VII: CXPath and beyond (13/46)

25 Aside: the issue of labels There is a fact about XML trees we did not take into account (unless we opt to render attribute-value pairs as additional labels) The labels are disjoint! However, this is easy to fix: add node axiom p q for distinct p and q This axiom itself is not substitution-invariant, this is why we do not like it But as our proofs used only Birkhoff s rules they are quite flexible and adding this axiom does not hurt Tadeusz Litak Lecture VII: CXPath and beyond (13/46)

26 Starting from the Other End Instead of beginning with single axes and then trying to combine two or more Tadeusz Litak Lecture VII: CXPath and beyond (14/46)

27 Starting from the Other End Instead of beginning with single axes and then trying to combine two or more LET S GO FOR THE WHOLE CORE XPATH! Tadeusz Litak Lecture VII: CXPath and beyond (14/46)

28 Axiom For Axes Dependencies TreeAx1{ s+ /s s s + s/s + s s + TreeAx2 s [φ] /s [ s [φ] ] ( for s distinct than ) TreeAx3 [φ] / ( + + ) [ [φ] ] TreeAx4{ + + [ ] + + [ ] TreeAx1 says: s + is a transitive closure of s TreeAx2 says non-child axes are functional and describes their converse TreeAx3 forces is the converse of (non-functional) with TreeAx4, it also describes how horizontal and vertical axes interplay Tadeusz Litak Lecture VII: CXPath and beyond (15/46)

29 Theorem The axioms presented so far are complete for Core XPath node expressions Tadeusz Litak Lecture VII: CXPath and beyond (16/46)

30 Theorem The axioms presented so far are complete for Core XPath node expressions Proof. By reduction to simple node expressions and derivation of all axioms of modal logic of finite trees by Blackburn, Meyer-Viol, de Rijke Tadeusz Litak Lecture VII: CXPath and beyond (16/46)

31 (boolean axioms) s [ ] s [φ ψ] s [φ] s [ψ] φ s [ s [φ] ] s [ φ] s [φ] ( for s distinct than ) s [φ] s [ s + [φ] ] s + [φ] s [φ] s + [φ] s + [ φ s [φ] ] s [ ] s + [ s [ ] ] TransAx1 for + and + [ [φ] ] [φ] [φ] [ ] [ ] Tadeusz Litak Lecture VII: CXPath and beyond (17/46)

32 A Nasty Trick Tadeusz Litak Lecture VII: CXPath and beyond (18/46)

33 A Nasty Trick We can use this to provide an axiomatization for path expressions... Tadeusz Litak Lecture VII: CXPath and beyond (18/46)

34 A Nasty Trick We can use this to provide an axiomatization for path expressions of a sort a non-orthodox one! Tadeusz Litak Lecture VII: CXPath and beyond (18/46)

35 A Nasty Trick We can use this to provide an axiomatization for path expressions of a sort a non-orthodox one! Add the separability rule: (Sep) IF A [p] B [p] for p not occurring in A, B THEN A B. Tadeusz Litak Lecture VII: CXPath and beyond (18/46)

36 A Nasty Trick We can use this to provide an axiomatization for path expressions of a sort a non-orthodox one! Add the separability rule: (Sep) IF A [p] B [p] for p not occurring in A, B THEN A B. Except for spoiling the whole equational story, it does not sit too well with the labelling axiom... Tadeusz Litak Lecture VII: CXPath and beyond (18/46)

37 The Nasty Trick Does Its Job... but it s perfect for obtaining complexity results for query equivalence problem by using reductions to corresponding modal logics Tadeusz Litak Lecture VII: CXPath and beyond (19/46)

38 Complexity Theorem Theorem Query equivalence of Core XPath( +, + ), Core XPath( + ), Core XPath(s) (for s {,, }) is conp-complete. Query equivalence of Core XPath( +,, +,, +, ) is PSPACE-complete. Thus, the PSPACE upper bound applies to all its sublanguages. Query equivalence of Core XPath( ) and Core XPath( + ) is PSPACE-complete. Thus, all extensions of this fragment are PSPACE-hard. Query equivalence of Core XPath(, + ) is EXPTIME-complete. Thus, all extensions of this fragment are EXPTIME-hard. Tadeusz Litak Lecture VII: CXPath and beyond (20/46)

39 Proofs... by reductions to complexity results for modal logics like K, K4, Alt.1 and fragments of tense/temporal logic on linear and branching orders. The most interesting one is for the second clause somewhat tricky embedding into a logic of Sistla and Clarke. Tadeusz Litak Lecture VII: CXPath and beyond (21/46)

40 What we have seen so far... We have seen: equational axiomatizations for path equivalences of all eight single axis fragments of Core XPath Tadeusz Litak Lecture VII: CXPath and beyond (22/46)

41 What we have seen so far... We have seen: equational axiomatizations for path equivalences of all eight single axis fragments of Core XPath equational axiomatizations for node equivalences of full Core XPath 1.0 Tadeusz Litak Lecture VII: CXPath and beyond (22/46)

42 What we have seen so far... We have seen: equational axiomatizations for path equivalences of all eight single axis fragments of Core XPath equational axiomatizations for node equivalences of full Core XPath 1.0 non-orthodox axiomatization for path equivalences of full Core XPath 1.0 Tadeusz Litak Lecture VII: CXPath and beyond (22/46)

43 What we have seen so far... We have seen: equational axiomatizations for path equivalences of all eight single axis fragments of Core XPath equational axiomatizations for node equivalences of full Core XPath 1.0 non-orthodox axiomatization for path equivalences of full Core XPath 1.0 computational complexity results for path equivalences in most meaningful sublanguages of Core XPath 1.0 Tadeusz Litak Lecture VII: CXPath and beyond (22/46)

44 What we have not seen so far... Definability and expressivity results (for finite sibling-ordered trees... ) Results for fragments of XPath stronger than CoreXPath 1.0 From now on, I am going to use Balder Ten Cate s M4M 2007 slides Tadeusz Litak Lecture VII: CXPath and beyond (23/46)

45 Expressive power Possible yardsticks for expressive power on trees: First-order logic (FO), (cf. Codd completeness of SQL/RA) Monadic second-order logic (MSO)... e.g., in between FO and MSO lies FO(TC) Tadeusz Litak Lecture VII: CXPath and beyond (24/46)

46 Expressive power Possible yardsticks for expressive power on trees: First-order logic (FO), (cf. Codd completeness of SQL/RA) Monadic second-order logic (MSO)... e.g., in between FO and MSO lies FO(TC) What kind of queries do we want to characterize Tadeusz Litak Lecture VII: CXPath and beyond (24/46)

47 Expressive power Possible yardsticks for expressive power on trees: First-order logic (FO), (cf. Codd completeness of SQL/RA) Monadic second-order logic (MSO)... e.g., in between FO and MSO lies FO(TC) What kind of queries do we want to characterize Binary relations definable by path expressions? Node sets definable by node expressions? Properties of trees definable by node expressions evaluated at the root? Tadeusz Litak Lecture VII: CXPath and beyond (24/46)

48 Expressive power Possible yardsticks for expressive power on trees: First-order logic (FO), (cf. Codd completeness of SQL/RA) Monadic second-order logic (MSO)... e.g., in between FO and MSO lies FO(TC) What kind of queries do we want to characterize Binary relations definable by path expressions? Node sets definable by node expressions? Properties of trees definable by node expressions evaluated at the root? Possible types of characterizations: Tadeusz Litak Lecture VII: CXPath and beyond (24/46)

49 Expressive power Possible yardsticks for expressive power on trees: First-order logic (FO), (cf. Codd completeness of SQL/RA) Monadic second-order logic (MSO)... e.g., in between FO and MSO lies FO(TC) What kind of queries do we want to characterize Binary relations definable by path expressions? Node sets definable by node expressions? Properties of trees definable by node expressions evaluated at the root? Possible types of characterizations: Syntactic (e.g. L is equivalent to the two variable... ) versus semantic (e.g., bisimulation invariant fragment... ) Tadeusz Litak Lecture VII: CXPath and beyond (24/46)

50 Expressive power Possible yardsticks for expressive power on trees: First-order logic (FO), (cf. Codd completeness of SQL/RA) Monadic second-order logic (MSO)... e.g., in between FO and MSO lies FO(TC) What kind of queries do we want to characterize Binary relations definable by path expressions? Node sets definable by node expressions? Properties of trees definable by node expressions evaluated at the root? Possible types of characterizations: Syntactic (e.g. L is equivalent to the two variable... ) versus semantic (e.g., bisimulation invariant fragment... ) Decidable characterizations? Tadeusz Litak Lecture VII: CXPath and beyond (24/46)

51 Expressive power (ct d) Descendant-only fragment CoreXPath( ) node expressions have the same expressive power as MSO formulas ϕ(x) for which (i) truth of ϕ(x) at a node depends only on the subtree (ii) ϕ(x) does not distinguish children from descendants, i.e., the following operation preserves truth/falsity at the root: Tadeusz Litak Lecture VII: CXPath and beyond (25/46)

52 Expressive power (ct d) Descendant-only fragment CoreXPath( ) node expressions have the same expressive power as MSO formulas ϕ(x) for which (i) truth of ϕ(x) at a node depends only on the subtree (ii) ϕ(x) does not distinguish children from descendants, i.e., the following operation preserves truth/falsity at the root: Easy proof from De Jongh-Sambin fixed point theorem for GL and Janin-Walukiewicz theorem for µ-calculus, see M4M proceedings paper. Tadeusz Litak Lecture VII: CXPath and beyond (25/46)

53 Expressive power (ct d) Descendant-only fragment CoreXPath( ) node expressions have the same expressive power as MSO formulas ϕ(x) for which (i) truth of ϕ(x) at a node depends only on the subtree (ii) ϕ(x) does not distinguish children from descendants, i.e., the following operation preserves truth/falsity at the root: Easy proof from De Jongh-Sambin fixed point theorem for GL and Janin-Walukiewicz theorem for µ-calculus, see M4M proceedings paper. Moreover, the proof is effective: it yields a decision procedure. Tadeusz Litak Lecture VII: CXPath and beyond (25/46)

54 A Lost Exercise Exercise 1 Prove that finite sibling-ordered trees are bisimiliar iff they are ismorphic 2 Does this result hold for arbitrary trees? Tadeusz Litak Lecture VII: CXPath and beyond (26/46)

55 Expressive power (ct d) What about the full Core XPath language? Tadeusz Litak Lecture VII: CXPath and beyond (27/46)

56 Expressive power (ct d) What about the full Core XPath language? No decidable characterization in terms of MSO is known. All we have is: Tadeusz Litak Lecture VII: CXPath and beyond (27/46)

57 Expressive power (ct d) What about the full Core XPath language? No decidable characterization in terms of MSO is known. All we have is: Syntactic characterization of Core XPath (Marx-De Rijke 05) Core XPath node expressions have the same expressive power as formulas φ(x) in the two-variable fragment of FO[R, R, R, R ]. There is a similar characterization for path expressions. Tadeusz Litak Lecture VII: CXPath and beyond (27/46)

58 Brief recap In summary... Core XPath is reasonably expressive yet computationally attractive. Tadeusz Litak Lecture VII: CXPath and beyond (28/46)

59 Brief recap In summary... Core XPath is reasonably expressive yet computationally attractive. In the remainder of this talk, we consider two extensions: Tadeusz Litak Lecture VII: CXPath and beyond (28/46)

60 Brief recap In summary... Core XPath is reasonably expressive yet computationally attractive. In the remainder of this talk, we consider two extensions: 1 Regular XPath: the extension of Core XPath with full transitive closure. Tadeusz Litak Lecture VII: CXPath and beyond (28/46)

61 Brief recap In summary... Core XPath is reasonably expressive yet computationally attractive. In the remainder of this talk, we consider two extensions: 1 Regular XPath: the extension of Core XPath with full transitive closure. 2 Core XPath 2.0: the navigational core of XPath 2.0, featuring path intersection and complementation and more. Tadeusz Litak Lecture VII: CXPath and beyond (28/46)

62 Motivation for adding transitive closure Several people have proposed extending XPath with transitive closure, for various reasons. Tadeusz Litak Lecture VII: CXPath and beyond (29/46)

63 Motivation for adding transitive closure Several people have proposed extending XPath with transitive closure, for various reasons. Practical reasons: some applications require the use of transitive closure. Tadeusz Litak Lecture VII: CXPath and beyond (29/46)

64 Motivation for adding transitive closure Several people have proposed extending XPath with transitive closure, for various reasons. Practical reasons: some applications require the use of transitive closure. Core XPath extended with transitive closure has full first-order expressive power, is rich enough to express DTDs, and admits view based query rewriting with recursive views. Tadeusz Litak Lecture VII: CXPath and beyond (29/46)

65 Motivation for adding transitive closure Several people have proposed extending XPath with transitive closure, for various reasons. Practical reasons: some applications require the use of transitive closure. Core XPath extended with transitive closure has full first-order expressive power, is rich enough to express DTDs, and admits view based query rewriting with recursive views. From the perspective of PDL, extending XPath with transitive closure seems a very natural thing to do. Tadeusz Litak Lecture VII: CXPath and beyond (29/46)

66 Motivation for adding transitive closure Several people have proposed extending XPath with transitive closure, for various reasons. Practical reasons: some applications require the use of transitive closure. Core XPath extended with transitive closure has full first-order expressive power, is rich enough to express DTDs, and admits view based query rewriting with recursive views. From the perspective of PDL, extending XPath with transitive closure seems a very natural thing to do. The extension of Core XPath with transitive closure is called Regular XPath. Tadeusz Litak Lecture VII: CXPath and beyond (29/46)

67 Syntax of Regular XPath Regular XPath has two types of expressions: Tadeusz Litak Lecture VII: CXPath and beyond (30/46)

68 Syntax of Regular XPath Regular XPath has two types of expressions: path expressions α ::=. α/β α β α α[φ] Tadeusz Litak Lecture VII: CXPath and beyond (30/46)

69 Syntax of Regular XPath Regular XPath has two types of expressions: path expressions α ::=. α/β α β α α[φ] node expressions φ ::= p φ φ ψ α Tadeusz Litak Lecture VII: CXPath and beyond (30/46)

70 An example Go to the next book that has at least two authors. In Regular XPath: Tadeusz Litak Lecture VII: CXPath and beyond (31/46)

71 An example Go to the next book that has at least two authors. In Regular XPath: ( [ twoauthorbook]) / [twoauthorbook] where twoauthorbook stands for book [author]/ + [author]. Tadeusz Litak Lecture VII: CXPath and beyond (31/46)

72 Another example The following can be expressed in Regular XPath: The tree has an even number of nodes To see this, note that Tadeusz Litak Lecture VII: CXPath and beyond (32/46)

73 Another example The following can be expressed in Regular XPath: The tree has an even number of nodes To see this, note that Let (α while φ) be shorthand for (.[φ]/α). Tadeusz Litak Lecture VII: CXPath and beyond (32/46)

74 Another example The following can be expressed in Regular XPath: The tree has an even number of nodes To see this, note that Let (α while φ) be shorthand for (.[φ]/α). Let root be short for. Let leaf be short for. Let first be short for. Let last be short for. Tadeusz Litak Lecture VII: CXPath and beyond (32/46)

75 Another example The following can be expressed in Regular XPath: The tree has an even number of nodes To see this, note that Let (α while φ) be shorthand for (.[φ]/α). Let root be short for. Let leaf be short for. Let first be short for. Let last be short for. Let suc be shorthand for [first].[leaf ]/ ( while last ) / (the successor in depth first left-to-right ordering). Tadeusz Litak Lecture VII: CXPath and beyond (32/46)

76 Another example The following can be expressed in Regular XPath: The tree has an even number of nodes To see this, note that Let (α while φ) be shorthand for (.[φ]/α). Let root be short for. Let leaf be short for. Let first be short for. Let last be short for. Let suc be shorthand for [first].[leaf ]/ ( while last ) / (the successor in depth first left-to-right ordering). Then (suc/suc) [leaf ]/ ( while last ) [root] is true at the root iff the number of nodes is even. Tadeusz Litak Lecture VII: CXPath and beyond (32/46)

77 One more example Consider game trees: leafs are labeled by Anne-wins or Bob-wins inner nodes are labeled by Anne s-move or Bob s-move Tadeusz Litak Lecture VII: CXPath and beyond (33/46)

78 One more example Consider game trees: leafs are labeled by Anne-wins or Bob-wins inner nodes are labeled by Anne s-move or Bob s-move Puzzle: Show that Anne has a winning strategy is expressible. Tadeusz Litak Lecture VII: CXPath and beyond (33/46)

79 Expressive power of Regular XPath Tadeusz Litak Lecture VII: CXPath and beyond (34/46)

80 Expressive power of Regular XPath What is the expressive power of Regular XPath? We know that FO Regular XPath FO(TC) (The first inclusion follows from results by Marx 2004). Tadeusz Litak Lecture VII: CXPath and beyond (34/46)

81 Expressive power of Regular XPath What is the expressive power of Regular XPath? We know that FO Regular XPath FO(TC) (The first inclusion follows from results by Marx 2004). A natural conjecture: Regular XPath FO(TC) (after all, Regular XPath has a transitive closure operator!) Tadeusz Litak Lecture VII: CXPath and beyond (34/46)

82 Expressive power of Regular XPath What is the expressive power of Regular XPath? We know that FO Regular XPath FO(TC) (The first inclusion follows from results by Marx 2004). A natural conjecture: Regular XPath FO(TC) (after all, Regular XPath has a transitive closure operator!) We managed to prove this only if we extend Regular XPath with a within operator W : T, n = W φ iff T n, n = φ (cf. temporal logics with forgettable past) Tadeusz Litak Lecture VII: CXPath and beyond (34/46)

83 Expressive power of Regular XPath (ct d) Tadeusz Litak Lecture VII: CXPath and beyond (35/46)

84 Expressive power of Regular XPath (ct d) FO(TC) is the extension of first-order logic with a transitive closure operator for binary relations. Tadeusz Litak Lecture VII: CXPath and beyond (35/46)

85 Expressive power of Regular XPath (ct d) FO(TC) is the extension of first-order logic with a transitive closure operator for binary relations. Theorem: (Ten Cate and Segoufin, PODS 2008, JACM 2010) Regular XPath(W) path expressions define the same binary relations as FO(TC) formulas with two free variables. Similarly for node expressions. Tadeusz Litak Lecture VII: CXPath and beyond (35/46)

86 Expressive power of Regular XPath (ct d) FO(TC) is the extension of first-order logic with a transitive closure operator for binary relations. Theorem: (Ten Cate and Segoufin, PODS 2008, JACM 2010) Regular XPath(W) path expressions define the same binary relations as FO(TC) formulas with two free variables. Similarly for node expressions. Corollary: Regular XPath(W) is closed under path intersection and complementation. Tadeusz Litak Lecture VII: CXPath and beyond (35/46)

87 Axiomatizations and complexity No axiomatizations are known yet for Regular XPath and Regular XPath(W). Tadeusz Litak Lecture VII: CXPath and beyond (36/46)

88 Axiomatizations and complexity No axiomatizations are known yet for Regular XPath and Regular XPath(W). As for complexity, Tadeusz Litak Lecture VII: CXPath and beyond (36/46)

89 Axiomatizations and complexity No axiomatizations are known yet for Regular XPath and Regular XPath(W). As for complexity, Query evaluation can still be performed in PTime even for Regular XPath(W). Query containment is still ExpTime-complete for Regular XPath but it is 2ExpTime-complete for Regular XPath(W) Tadeusz Litak Lecture VII: CXPath and beyond (36/46)

90 Core XPath 2.0 Tadeusz Litak Lecture VII: CXPath and beyond (37/46)

91 XPath 2.0 extends XPath 1.0 with many features, including the following new navigational operations: Intersection and complementation of path expressions. α intersect β and α except β Tadeusz Litak Lecture VII: CXPath and beyond (38/46)

92 XPath 2.0 extends XPath 1.0 with many features, including the following new navigational operations: Intersection and complementation of path expressions. α intersect β and α except β Example: [p] except [q]/ [p] Tadeusz Litak Lecture VII: CXPath and beyond (38/46)

93 XPath 2.0 extends XPath 1.0 with many features, including the following new navigational operations: Intersection and complementation of path expressions. α intersect β and α except β Example: [p] except [q]/ [p] Variables and for loops for $x in α return β α[. is $x] and Tadeusz Litak Lecture VII: CXPath and beyond (38/46)

94 XPath 2.0 extends XPath 1.0 with many features, including the following new navigational operations: Intersection and complementation of path expressions. α intersect β and α except β Example: [p] except [q]/ [p] Variables and for loops for $x in α return β α[. is $x] and Example: for $x in. return [p [q]/ [. is $x] ] Tadeusz Litak Lecture VII: CXPath and beyond (38/46)

95 XPath 2.0 extends XPath 1.0 with many features, including the following new navigational operations: Intersection and complementation of path expressions. α intersect β and α except β Example: [p] except [q]/ [p] Variables and for loops for $x in α return β α[. is $x] and Example: for $x in. return [p [q]/ [. is $x] ] Core XPath 2.0 is the extension of Core XPath with these features. Tadeusz Litak Lecture VII: CXPath and beyond (38/46)

96 The path intersection and complementation turn Core XPath 2.0 into a version of Tarski s relation algebra (interpreted on finite ordered trees). Tadeusz Litak Lecture VII: CXPath and beyond (39/46)

97 The path intersection and complementation turn Core XPath 2.0 into a version of Tarski s relation algebra (interpreted on finite ordered trees). The variables and for-loops make it possible to give a linear translation from first-order logic to Core XPath 2.0: TR(φ(x, y)) = for $x in., $y in return $y[tr (φ)] TR (x = y) = [. is $x. is $y] TR (R xy) = [. is $x [. is $y] ] TR (R xy) = [. is $x [. is $y] ] TR (R xy) = [. is $x [. is $y] ] TR (R xy) = [. is $x [. is $y] ] TR (φ ψ) = TR (φ) TR (ψ) TR ( φ) = TR (φ) TR ( x.φ) = for $x in return TR (φ) where is shorthand for / (the universal relation) Tadeusz Litak Lecture VII: CXPath and beyond (39/46)

98 Expressivity and complexity Core XPath 2.0 has the same expressive power as first-order logic, both with and without variables (in the case with variables there is a linear translation). Tadeusz Litak Lecture VII: CXPath and beyond (40/46)

99 Expressivity and complexity Core XPath 2.0 has the same expressive power as first-order logic, both with and without variables (in the case with variables there is a linear translation). The complexity of the query equivalence problem is non-elementary, both with and without variables. Tadeusz Litak Lecture VII: CXPath and beyond (40/46)

100 Expressivity and complexity Core XPath 2.0 has the same expressive power as first-order logic, both with and without variables (in the case with variables there is a linear translation). The complexity of the query equivalence problem is non-elementary, both with and without variables. (even adding only path intersection to Core XPath makes it 2ExpTime-complete.) Tadeusz Litak Lecture VII: CXPath and beyond (40/46)

101 Expressivity and complexity Core XPath 2.0 has the same expressive power as first-order logic, both with and without variables (in the case with variables there is a linear translation). The complexity of the query equivalence problem is non-elementary, both with and without variables. (even adding only path intersection to Core XPath makes it 2ExpTime-complete.) Is there anything interesting left to say about XPath 2.0? Tadeusz Litak Lecture VII: CXPath and beyond (40/46)

102 Expressivity and complexity Core XPath 2.0 has the same expressive power as first-order logic, both with and without variables (in the case with variables there is a linear translation). The complexity of the query equivalence problem is non-elementary, both with and without variables. (even adding only path intersection to Core XPath makes it 2ExpTime-complete.) Is there anything interesting left to say about XPath 2.0? Sure! For example, axiomatization. Tadeusz Litak Lecture VII: CXPath and beyond (40/46)

103 Expressivity and complexity Core XPath 2.0 has the same expressive power as first-order logic, both with and without variables (in the case with variables there is a linear translation). The complexity of the query equivalence problem is non-elementary, both with and without variables. (even adding only path intersection to Core XPath makes it 2ExpTime-complete.) Is there anything interesting left to say about XPath 2.0? Sure! For example, axiomatization. We have two complete axiomatizations of path equivalence in Core XPath 2.0: one with and one without variables. Tadeusz Litak Lecture VII: CXPath and beyond (40/46)

104 The case without variables Recall that, without variables, Core XPath 2.0 is essentially a version of Relation Algebra interpreted on finite sibling ordered trees. Tadeusz Litak Lecture VII: CXPath and beyond (41/46)

105 The case without variables Recall that, without variables, Core XPath 2.0 is essentially a version of Relation Algebra interpreted on finite sibling ordered trees. One apparent problem: Relation Algebra has no node tests. However, these can easily be translated away: Pred1. α[φ ψ] α[φ][ψ] Pred2. α[φ ψ] α[φ] α[ψ] Pred3. α[ φ] α α[φ] Pred4. α[ β ] α/((β/ ).) Tadeusz Litak Lecture VII: CXPath and beyond (41/46)

106 The case without variables Recall that, without variables, Core XPath 2.0 is essentially a version of Relation Algebra interpreted on finite sibling ordered trees. One apparent problem: Relation Algebra has no node tests. However, these can easily be translated away: Pred1. α[φ ψ] α[φ][ψ] Pred2. α[φ ψ] α[φ] α[ψ] Pred3. α[ φ] α α[φ] Pred4. α[ β ] α/((β/ ).) Besides these axioms, our axiomatization for variable free Core XPath 2.0 contains two groups of axioms: General axioms of Boolean Algebra and Relation Algebra Axioms describing (first-order) properties of trees. Tadeusz Litak Lecture VII: CXPath and beyond (41/46)

107 Axioms of Boolean algebra BA1. α (β γ) (α β) γ BA2. α (β γ) (α β) γ BA3. α β β α BA4. α β β α BA5. α (β γ) (α β) (α γ) BA6. α (β γ) (α β) (α γ) BA7. α (α β) α BA8. α (α β) α BA9. α ( α) BA10. α ( α) BA11. α ( β) α β Tadeusz Litak Lecture VII: CXPath and beyond (42/46)

108 The axioms of Relation algebra RA1. α/(β/γ) (α/β)/γ RA2. α/. α RA3. (α β)/γ α/γ β/γ RA4. (α β) α β RA5. (α/β) β /α RA6. (α ) α RA7. (α/( (α /β)) except β To completely axiomatize relation algebra, normally, one needs to add also Venema s Rule: If X is a relation variable not occurring in α and X (((.)/X/ ) ( /X/(.))) α then α. Fortunately, this rule turns out to be derivable in our case. Tadeusz Litak Lecture VII: CXPath and beyond (43/46)

109 The axioms for finite sibling ordered trees Tr1. + / + + Tr Tr3a. + ( / + ) Tr3b. + ( + / + ) Tr4..[ ].[ + [ ]] Tr5. + / + +.[ ].[ ]/ + Tr6. + / + + Tr Tr8a. + ( / + ) Tr8b. + ( + / + ) Tr9..[ ].[ + [ ] ] Tr ( / ). Tr ( / + / ) ( / + / ) Ind. [ α ] [ α (α/ <<) ] Tadeusz Litak Lecture VII: CXPath and beyond (44/46)

110 Rounding up Three languages: Tadeusz Litak Lecture VII: CXPath and beyond (45/46)

111 Rounding up Three languages: Core XPath: the navigational core of XPath 1.0 Expressivity: FO 2 Query evaluation: PTime Query equivalence: ExpTime-complete Tadeusz Litak Lecture VII: CXPath and beyond (45/46)

112 Rounding up Three languages: Core XPath: the navigational core of XPath 1.0 Expressivity: FO 2 Query evaluation: PTime Query equivalence: ExpTime-complete Regular XPath(W): the extension with and W. Expressivity: same as FO(TC) Query evaluation: PTime Query equivalence: 2ExpTime-complete. Tadeusz Litak Lecture VII: CXPath and beyond (45/46)

113 Rounding up Three languages: Core XPath: the navigational core of XPath 1.0 Expressivity: FO 2 Query evaluation: PTime Query equivalence: ExpTime-complete Regular XPath(W): the extension with and W. Expressivity: same as FO(TC) Query evaluation: PTime Query equivalence: 2ExpTime-complete. Core XPath 2.0: the navigational core of XPath 2.0 Expressivity: same as FO. Query evaluation: PSpace-complete Query equivalence: non-elementary hard. Tadeusz Litak Lecture VII: CXPath and beyond (45/46)

114 Some references Tadeusz Thank Litak you! Lecture VII: CXPath and beyond (46/46) Expressive power: Marx and De Rijke. Semantic characterizations of navigational XPath. SIGMOD Record 34(2), 2005 Ten Cate, Fontaine and Litak. Some modal aspects of XPath. M4M 07. Journal version for a special issue of JANCL 2010 in preparation Ten Cate and Segoufin. XPath, transitive closure logic, and nested tree walking automata. Journal of the ACM, Extended abstract appeared in PODS Axiomatization: Ten Cate, Fontaine and Litak. Some modal aspects of XPath. M4M 07. Journal version for a special issue of JANCL 2010 in preparation Ten Cate, Litak and Marx. Complete axiomatizations of XPath fragments. JAL Extended abstract presented at LiD Ten Cate and Marx. Axiomatizing the logical core of XPath 2.0. ICDT 07. Complexity: Gottlob, Koch and Pichler. Efficient algorithms for processing XPath queries. TODS 30(2), 2005 Ten Cate and Lutz. Query containment in very expressive XPath dialects. PODS 07.