XPath with transitive closure

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1 XPath with transitive closure Logic and Databases Feb XPath with transitive closure Logic and Databases Feb Navigating XML trees XPath with transitive closure Newton Institute: Logic and Databases Maarten Marx Feb 2006 Navigation is at the core of XML processing. Navigation is about paths between nodes. The two main XML query languages XSLT and XQuery have delegated the navigation part to XPath. This talk is about binary relations between nodes in finite, unbounded, sibling ordered, node labelled trees. XPath with transitive closure Logic and Databases Feb XPath XPath 2.0 is explicitly designed to be first order complete. XQuery is strong enough to define whatever more you want. So why would we consider stronger versions of XPath? XPath with transitive closure Logic and Databases Feb XPath with transitive closure: Why? It s natural Regular Path Queries [ABS 2000] Core XPath is a handicapped version of Propositional Dynamic Logic (PDL): only Kleene star on the basic steps. It s needed Both Saxon and Xalan provide implementations of the EXSLT closure function. Nentwich et al 2002 describe several use-cases. (They have an example of a university curriculum, with rules like A course cannot be a pre-requisite of itself.) It s clean and algebraic Simple extension of regular languages with a few extras. Easier to optimize than user-defined functions. It s fun Well, let s see... XPath with transitive closure Logic and Databases Feb XPath with transitive closure Logic and Databases Feb Regular Path Queries, a.k.a. Caterpillars Regular Path Queries are the navigation language of [ABS2000, Data on the Web]. Caterpillars originate with [Brüggemann-Klein, Wood 2000]. On XML documents, they are the same. Design idea: a path in a tree is like a string of child, parent, left, right steps and tests on node labels. So use regular expressions over these and the four endtests root leaf first last. XPath with transitive closure Logic and Databases Feb The depth first traversal from the root to the last node: as a string: child child right parent right child... as a regular expression: XPath with transitive closure Logic and Databases Feb

2 XPath with transitive closure Logic and Databases Feb XPath with transitive closure Logic and Databases Feb XPath with transitive closure Logic and Databases Feb XPath with transitive closure Logic and Databases Feb Regular Path Queries, a.k.a. Caterpillars: Syntax The atomic symbols of the language are axis ::= self child parent right left. test ::= p root leaf first last. The expressions are generated by the grammar: R ::= axis test R R R/R R +. We can take either or + as primitive. XPath with transitive closure Logic and Databases Feb Regular Path Queries, a.k.a. Caterpillars: Semantics Every expression denotes a path, a binary relation: [p] = {(x, x) x is labelled by p} [leaf ] = {(x, x) x is a leaf} [child] = {(x, y) y is a child of x} etc. [R S ] = union of [R] and [S ]. [R/S ] = [R] [S ] (composition of binary relations) [R + ] = transitive closure of [R]. XPath with transitive closure Logic and Databases Feb Caterpillars: results Positive The expressive power of the language is characterized by tree walk automata. Query evaluation is tractable. Very expressible as a tree language (e.g., DTD s). Theorem [Bojanczyk Colcombet 2004] Caterpillars cannot define all regular tree languages. Every expression defines a monadic second order definable binary relation. XPath with transitive closure Logic and Databases Feb Caterpillars as a navigation language Caterpillars are too weak for a navigation language: Simple structural tests on nodes are not expressible: {(x, x) the parent of x is labelled p} {(x, x) all nodes in the subtree of x are labelled p}. To summarize: F O caterpillars MSO. XPath with transitive closure Logic and Databases Feb

3 XPath with transitive closure Logic and Databases Feb Caterpillars loose their way. Try to program {(x, x) some child of x is labelled p} as a caterpillar expression. child/p/parent Now try its inverse {(x, x) the parent of x is labelled p}. parent/p/child only works on unary trees. We want parent/p/child self XPath with transitive closure Logic and Databases Feb Tree walk automata (TWA s) TWA s are a sequential alternative to (parallel) tree automata. A TWA is just a finite state automaton on the language axis test. axis ::= self child parent right left. test ::= p root leaf first last. The input of a TWA is a tree and two nodes. A formal proof of this inexpressibility result can be given using tree walk automata. XPath with transitive closure Logic and Databases Feb TWA s, transitions and acceptance The transition relation between state node pairs is defined from the transition function δ: t, q test t, q q δ( q, test ), and T = test(t) t, q axis t, q q δ( q, axis ), and T = t(axis)t. A TWA accepts a path from t to t in tree T if there exists an initial state q i and a final state q f such that t, q i t, q f. XPath with transitive closure Logic and Databases Feb Adding expressive power to caterpillars The literature shows three ways for expressing structural tests: 1. add explicit memory-devices: pebbles, variables (Pebble TWA s); 2. add a second sort of test expressions, with their own operations (e.g. XPath filters); 3. add new operations on binary relations. Theorem [Brüggeman-Klein, Wood 2000] TWA s and caterpillar expressions can define exactly the same paths. XPath with transitive closure Logic and Databases Feb XPath with transitive closure Logic and Databases Feb Caterpillars with variable binders I Engelfriet Hoogeboom 2006 add pebbles to Tree Walk Automata. Pebbles can be placed on nodes in the tree, inspected, and picked up. Using pebbles one can remember nodes, and for instance, come back. Equivalently one can add variables and a binder to the caterpillar exppresions. XPath with transitive closure Logic and Databases Feb Caterpillars with variable binders II Idea: add a syntactic equivalent of pebbles to caterpillars: add variables to the language; allow testing of a variable; add a binder x (R). Example: In XPath 2.0: x (parent/p/child/x) I have a p parent. for $x in. return $x/parent::p/child::*[. is $x] XPath with transitive closure Logic and Databases Feb Caterpillars with variable binders: results The language is characterized by pebble TWA s (Gorris, M. 2005). Pebble TWA s are weaker than MSO (Bojańczyk, Samuelides, Schwentick, Segoufin 2006). Thus caterpillars with variable binders are weaker than MSO. Every FO relation with quantifier depth k is definable by a deterministic k-pebble TWA (Engelfriet, Hoogeboom 1999). From the proof it follows that there is a linear translation from FO to Caterpillars with variable binders. Hence, the combined complexity of query evaluation of caterpillars with binders is PSPACE complete.

4 XPath with transitive closure Logic and Databases Feb XPath with transitive closure Logic and Databases Feb Adding structural tests I Idea Add formulas in one free variable as tests. Bloem Engelfriet 1997 added all MSO definable tests φ(x) to caterpillars. Then every MSO definable binary relation becomes definable. One can implement this using e.g. the modal µ-calculus, or using Gottlob and Koch s unary Monadic Datalog for trees. XPath with transitive closure Logic and Databases Feb Adding structural tests II Both Propositional Dynamic Logic (PDL) and XPath add the same structural tests: domain(r) and all booleans. domain(r) = {x y : xry}. I have a p parent becomes expressible: PDL parent p? XPath self :: [parent :: p]. There is also an alternative way to achieve this: through the counterdomain operator. XPath with transitive closure Logic and Databases Feb Counterdomain and safety for bisimulation van Benthem considered the following puzzle: Consider a bisimulation B which preserves atomic labels P and bisimulates the relations R 1,... R n. Then B also bisimulates other relations: for instance, 1. R i /R j ; 2. R i ; 3. R i R j ; Question What are the first order definable binary relations which are safe for bisimulations? Answer Exactly those definable from R 1,... R n and atomic tests on P and using only composition, union, and counterdomain. XPath with transitive closure Logic and Databases Feb Counterdomain For R a binary relation, R = {(x, x) not y : xry}. Examples: 1. (parent/p) defines I am not in the domain of parent/p. 2. (parent/p) defines I have a p parent. 3. (child + / p) defines all my descendants are p XPath with transitive closure Logic and Databases Feb Caterpillars with counterdomain Adding counterdomain to caterpillars gives an alternative definition of PDL over ordered trees, because ( R φ)? = (R/φ?) ( φ)? = (φ?) (φ ψ)? = (φ)?/(ψ)? With counterdomain we can also give a very short definition of Core XPath. XPath with transitive closure Logic and Databases Feb Aside: Clean algebraic version of Core XPath The atomic symbols of the language are axis ::= child parent right left child + parent + right + left +. test ::= p. The expressions are generated by the grammar: R ::= axis test R R R/R R. self is definable as (child child). XPath with transitive closure Logic and Databases Feb is definable as self.

5 XPath with transitive closure Logic and Databases Feb We just saw counterdomain. Adding new operations XPath 2.0 even adds intersection and complementation of path expressions to the language. Then counterdomain becomes definable: R self R/all. Of particular interest is intersection with the identity. XPath with transitive closure Logic and Databases Feb Intersection with identity I have a p parent is expressed by parent/p/child self. This operation resembles equality in XPath: [. = parent :: p/child :: ]. [$R = $S] is true if there exists nodes r $R, s $S which have the same atomic value. We call this operation loop: R loop R self. XPath with transitive closure Logic and Databases Feb Looping Caterpillars: results Looping Caterpillars is caterpillars with the extra loop operation. (Gorris, M 2005) saw it as the synthesis between Regular Path expressions and XPath 2.0: every FO definable relation is expressible it contains the caterpillar expressions XPath with transitive closure Logic and Databases Feb Looping Caterpillars: open problems Are all PDL tests definable? More general, is the language closed under counterdomain? Query evaluation is tractable. Expressive power is characterized by a restricted version of pebble TWA s. So it is still a strict subset of MSO. XPath with transitive closure Logic and Databases Feb XPath with transitive closure Logic and Databases Feb Caterpillars with counterdomain: Regular XPath ten Cate (2005) tried to characterize Caterpillars with counterdomain in terms of a natural logical language. One can also think of this as PDL on trees (also known as Regular XPath). He conjectures that the following relation is not expressible in Caterpillars with counterdomain: {(x, x) x has an even number of descendants}. XPath with transitive closure Logic and Databases Feb I have an even number of descendants {(x, x) x has an even number of descendants} is expressible using loop. Let next denote the successor relation in the document order. This is already caterpillar definable. Then ((next/next) /(last/parent) ) loop does the job. Note that without loop one could shoot back too much. XPath with transitive closure Logic and Databases Feb Caterpillars with counterdomain and loop Ten Cate added loop to Regular XPath and calls it RegularXPath =. It has the following syntax: axis ::= child parent right left. test ::= p. R ::= axis test R R R/R R + R loop R. An alternative is to add the test loop(r) to PDL on trees. n = loop(r) (n, n) = R. Then R loop loop(r)? and loop(r) R loop.

6 XPath with transitive closure Logic and Databases Feb Caterpillars with counterdomain and loop: results Theorem [ten Cate 2005] The following are equivalent: 1. a binary relation is definable in caterpillars with counterdomain and loop; 2. a binary relation is definable as a formula φ(x, y) in first order logic expanded with the Kleene star. M = φ (a, b) if and only if there are a = c 1,..., c n = b such that M = φ(c i, c i+1 ). One can only apply the Kleene star to formulas with exactly two free variables. XPath with transitive closure Logic and Databases Feb Caterpillars with counterdomain and loop: open problems 1. Does loop contribute to the expressive power? 2. Is the language strictly less expressive than MSO? 3. What is a good automata model for this language? 4. What is the complexity of the containment problem for Regular XPath extended with loop? (Without loop it is known to be EXPTIME complete). 5. Find efficient algorithms for query evaluation. (Without loop, a linear algorithm exists). XPath with transitive closure Logic and Databases Feb Conclusion Caterpillars with counterdomain and loop form a nice synthesis between a number of approaches to XML reasoning: Caterpillar expressions, regular path queries, DTD s First order logic, XPath 2.0 Propositional Dynamic Logic Its expressive completeness shows that it is a natural choice of language.

Semantic Characterizations of XPath

Semantic Characterizations of XPath Maarten Marx Informatics Institute, University of Amsterdam, The Netherlands CWI, April, 2004 1 Overview Navigational XPath is a language to specify sets and paths in