## First order paths in ordered trees (2005)

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Venue: | In ICDT’05 |

Citations: | 37 - 5 self |

### BibTeX

@INPROCEEDINGS{Marx05firstorder,

author = {Maarten Marx},

title = {First order paths in ordered trees},

booktitle = {In ICDT’05},

year = {2005},

pages = {114--128},

publisher = {Springer}

}

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### Abstract

Abstract. We give two sufficient conditions on XPath like languages for having first order expressivity, meaning that every first order definable set of paths in an ordered node-labeled tree is definable in that XPath language. They are phrased in terms of expansions of navigational (sometimes called “Core”) XPath. Adding either complementation, or the more elegant conditional paths is sufficient. A conditional path is an axis relation of the form (one step axis::n[F]) +, denoting the transitive closure of the relation expressed by one step axis::n[F]. As neither is expressible in navigational XPath we also give characterizations in terms of first order logic of the answer sets and the sets of paths navigational XPath can define. The first in terms of a suitable two variable fragment, the second in terms of unions of conjunctive queries. 1

### Citations

252 | Efficient algorithms for processing XPath queries
- Gottlob, Koch, et al.
- 2005
(Show Context)
Citation Context ... | fexpr and fexpr. Here ntst is a node test consisting of a tag name or the wild card ∗. The steps correspond to the four basic steps in an ordered tree. The semantics is as with standard Core XPath =-=[5]-=-, with (·) + interpreted as the transitive closure. Although the choice of the syntax can be motivated by its close relation to temporal logic with since and until, it may still seem rather ad hoc. Mo... |

240 |
Tense logic and the theory of linear order
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- 1968
(Show Context)
Citation Context ...riables, even in the signature expanded with the child relation. Whence there are sets of nodes which are not definable in Core XPath, by the previous theorem. The following example goes back to Kamp =-=[10]-=-: the set of all nodes x such that ∃y(x descendant y ∧ A(y) ∧ ∀z((x descendant z ∧ z descendant y) → B(z))). (2) Note that this set is expressible in Conditional XPath. By the node wff 〈(child?B) ∗ /c... |

131 |
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(Show Context)
Citation Context ...ode wffs of Core XPath with negation have been characterized in Theorem 1. We can combine these results using a relaxation of positive existential first order logic, reminiscent of the Carin language =-=[7]-=-. 4 [1] does not consider the horizontal axis relations, but their proof is easily adjusted. 8sT heorem 2 ������������������ ����� ����� � Lemma 2 Lemma�4 ����� ��� ����������� �� Lemma 1 Lemma�6 Lemm... |

105 |
Why is modal logic so robustly decidable
- Vardi
- 1997
(Show Context)
Citation Context ... the node wffs and only prove the second equivalence in the theorem. By the standard translation well known from modal logic each node wff translates into a one free variable F O tree 2 formula (cf., =-=[17]-=- which takes care to use only two variables). The translation is just the definition from Table 2 written in first order logic. This takes care of the easy direction. For the other direction, let φ(x)... |

89 | Structural properties of XPath fragments
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(Show Context)
Citation Context ...ence for the great expressive power of the construction (step::ntst[fexpr]) + , and obtain an expressive completeness result for location paths. Extensive motivation for such a result can be found in =-=[1]-=-. In the above definition it was not needed to close the location path expressions under composition (the ′ / ′ ) and union (the ′ | ′ ). This is because we dealt with filter expressions only. When de... |

77 | Definability with bounded number of bound variables
- Immerman, Kozen
- 1989
(Show Context)
Citation Context ...at most 3 free variables is equivalent to a F O tree formula in the same signature which uses at most 3 free and bound variables. This theorem can be shown using Ehrenfeucht–Fraïssé pebble games from =-=[9]-=-. The first part of Theorem 2 can now be derived as follows. Proof. Let L be an expansion as in the Theorem. Then L can express every binary relation expressible in Tarski’s relation algebras. Tarski’... |

65 | Conjunctive queries over trees
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(Show Context)
Citation Context ...e finish the section with a characterization of Core XPath’s path wffs. For positive Core XPath (without negation but with disjunction of node wffs), such a characterization is provided in [1] and in =-=[6]-=-. It is exactly positive existential first order logic (of course in the signature expanded with the child and right-sibling relation). 4 The node wffs of Core XPath with negation have been characteri... |

55 | XPath with conditional axis relations
- Marx
- 2004
(Show Context)
Citation Context ...l point of view, Conditional XPath is not harder than Core XPath: the query evaluation problem is still solvable in time O(|Q| · |D|), with |Q|, |D|, the sizes of the query and the data, respectively =-=[12]-=-. Its easy syntax, visual attractiveness, and low complexity combined with its expressive completeness make Conditional XPath an excellent candidate for succeeding XPath 1.0. Acknowledgments Maarten M... |

52 | Conditional XPath, the first order complete XPath dialect
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- 2004
(Show Context)
Citation Context ...ic of the answer sets and the sets of paths navigational XPath can define. The first in terms of a suitable two variable fragment, the second in terms of unions of conjunctive queries. 1 Introduction =-=[11]-=- showed how a simple addition to Core XPath led to expressive completeness: every first order definable set of nodes in an XML tree is definable as the answer set of an expression //[fexpr] in which t... |

44 |
A Formalization of Set Theory without Variables, volume 41. AMS Colloquium publications
- Tarski, Givant
- 1987
(Show Context)
Citation Context ...F O 2 3, first order logic in a signature with at most binary relations symbols in which every formula contains at most three free and bound (possibly reused) variables and at most two free variables =-=[16]-=-. The desired result now follows from Theorem 3. Remark 4. It is tempting to think that the three variable property for ordered trees is derivable from results about trees and CTL like languages, perh... |

43 | Two semantics for XPath
- WADLER
- 2000
(Show Context)
Citation Context ...written as [R] M . As promised, path wffs denote sets of pairs, and node wffs sets of nodes. Table 2 contains the definition of [ · ] M . The equivalence with the W3C syntax and semantics (cf., e.g., =-=[5,18]-=-) should be clear. Let us spell out the semantics of the conditional axis relation, as it does not occur in standard navigational XPath. The path wff (child?A) + denotes all pairs (n, n ′ ) for which ... |

22 |
First-Order Logic with Two Variables and Unary Temporal Logic
- Wilke
- 2002
(Show Context)
Citation Context ...on follows more or less directly from the argument used to show a similar statement for linear orders —characterizing temporal logic with only unary temporal connectives— by Etessami, Vardi and Wilke =-=[4]-=-. The proof shows that a similar statement holds for the version of Core XPath of Gottlob, Koch and Pichler [5] which does not have the right and left sibling axis but just their transitive closures. ... |

16 |
A Descriptive Approach to Language Theoretic Complexity
- Rogers
- 1998
(Show Context)
Citation Context ...computational linguistics is the class of finite ordered trees. In the beginning of the field of model theoretic syntax Monadic Second Order Logic was invariably used to reason about these structures =-=[15]-=-. Later, formalisms based on modal logic were proposed as alternatives. Arguments for the alternatives were both based on computational complexity (which is lower both for model checking and theorem p... |

14 | A proof system for finite trees
- Blackburn, Meyer-Viol, et al.
- 1996
(Show Context)
Citation Context ...lower both for model checking and theorem proving) and on “naturalness” of expressing properties (in this case of grammars). In fact, both Core and Conditional XPath have their roots in the nineties: =-=[2]-=- and [14] define isomorphic variants of the filter expressions of Core and Conditional XPath, respectively. From a theoretical point of view, Conditional XPath is not harder than Core XPath: the query... |

8 |
Rijke. Semantic characterizations of XPath
- Marx, de
- 2004
(Show Context)
Citation Context ...h temporal logic. These connections, together with further related work and fairly complete proofs are in the full version of the paper, which is available from the author. For related work, see also =-=[11,13]-=-. We note that the expressive completeness result for Conditional XPath’s answer sets (shown in [11]) follows from the result presented here, but not conversely. The results about Core XPath have been... |

7 |
Propostional tense logic for trees
- Palm
- 1999
(Show Context)
Citation Context ...th for model checking and theorem proving) and on “naturalness” of expressing properties (in this case of grammars). In fact, both Core and Conditional XPath have their roots in the nineties: [2] and =-=[14]-=- define isomorphic variants of the filter expressions of Core and Conditional XPath, respectively. From a theoretical point of view, Conditional XPath is not harder than Core XPath: the query evaluati... |

1 | The k-variable property is stronger than H-dimension k
- Hodkinson, Simon
- 1997
(Show Context)
Citation Context ...ed trees is derivable from results about trees and CTL like languages, perhaps in case of finite trees. This is partly due to inconsistent terminology, partly because all notions are closely related. =-=[8]-=- give a clear picture of these notions and their relations and especially the relations that do not exist. In particular they show that the three variable property is strictly stronger than the proper... |