## Expressiveness of a spatial logic for trees (2005)

Venue: | In LICS 2005 |

Citations: | 13 - 2 self |

### BibTeX

@INPROCEEDINGS{Boneva05expressivenessof,

author = {Iovka Boneva and Jean-marc Talbot and Sophie Tison},

title = {Expressiveness of a spatial logic for trees},

booktitle = {In LICS 2005},

year = {2005},

pages = {280--289},

publisher = {Press}

}

### OpenURL

### Abstract

In this paper we investigate the quantifier-free fragment of the TQL logic proposed by Cardelli and Ghelli. The TQL logic, inspired from the ambient logic, is the core of a query language for semistructured data represented as unranked and unordered trees. The fragment we consider here, named STL, contains as main features spatial composition and location as well as a fixed point construct. We prove that satisfiability for STL is undecidable. We show also that STL is strictly more expressive than the Presburger monadic second-order logic (PMSO) of Seidl, Schwentick and Muscholl when interpreted over unranked and unordered edge-labelled trees. We define a class of tree automata whose transitions are conditioned by arithmetical constraints; we show then how to compute from a closed STL formula a tree automaton accepting precisely the models of the formula. Finally, still using our tree automata framework, we exhibit some syntactic restrictions over STL formulae that allow us to capture precisely the logics MSO and PMSO. 1

### Citations

705 | Separation logic: a logic for shared mutable data structures
- Reynolds
(Show Context)
Citation Context ...ver STL formulae that allow us to capture precisely the logics MSO and PMSO. 1 Introduction Spatial logics allow one to express properties about structures such as trees [8], graphs [6, 14] and heaps =-=[19]-=-. The main ingredient of spatial logics is an operator called composition (or separation). This operator permits compositional reasoning over concurrent and mobile processes [10, 4] or over imperative... |

268 | Local reasoning about programs that alter data structures
- O’Hearn, Reynolds, et al.
- 2001
(Show Context)
Citation Context ...operator called composition (or separation). This operator permits compositional reasoning over concurrent and mobile processes [10, 4] or over imperative programs with shared mutable data structures =-=[18]-=-. The TQL logic [7, 8] is the core of a query language for semistructured data represented as unranked and unordered trees. The TQL logic is based on the static fragment of the ambient logic: it conta... |

162 | Anytime, anywhere. Modal logics for mobile ambients - Cardelli, Gordon - 2000 |

136 | A spatial logic for concurrency (part I
- Caires, Cardelli
- 2001
(Show Context)
Citation Context ...phs [6, 14] and heaps [19]. The main ingredient of spatial logics is an operator called composition (or separation). This operator permits compositional reasoning over concurrent and mobile processes =-=[10, 4]-=- or over imperative programs with shared mutable data structures [18]. The TQL logic [7, 8] is the core of a query language for semistructured data represented as unranked and unordered trees. The TQL... |

62 | A Query Language Based on the Ambient Logic
- Cardelli, Ghelli
- 2004
(Show Context)
Citation Context ...osition (or separation). This operator permits compositional reasoning over concurrent and mobile processes [10, 4] or over imperative programs with shared mutable data structures [18]. The TQL logic =-=[7, 8]-=- is the core of a query language for semistructured data represented as unranked and unordered trees. The TQL logic is based on the static fragment of the ambient logic: it contains spatial primitives... |

61 | A spatial logic for querying graphs
- Cardelli, Gardner, et al.
(Show Context)
Citation Context ...tic restrictions over STL formulae that allow us to capture precisely the logics MSO and PMSO. 1 Introduction Spatial logics allow one to express properties about structures such as trees [8], graphs =-=[6, 14]-=- and heaps [19]. The main ingredient of spatial logics is an operator called composition (or separation). This operator permits compositional reasoning over concurrent and mobile processes [10, 4] or ... |

52 | Deciding Validity in a Spatial Logic for Trees
- Calcagno, Cardelli, et al.
- 2005
(Show Context)
Citation Context ...n over names is crucial for this result. However, decidable fragments of the logic TQL could be useful in several domains: for constructing type systems for semistructured data as the one proposed in =-=[5]-=- or for testing query correctness, query containment and defining constraints, as suggested in [8]. The logic TL introduced by Dal Zilio et al. in [12, 13] is also inspired by the ambient logic. This ... |

51 | D.: Rudiments of µ-Calculus
- Arnold, Niwiński
- 2001
(Show Context)
Citation Context ...ch that T = �φ�. System of fixed point equations We present here an alternative representation of the semantics of a closed STL formula as the solution of a system of fixed point equations. Following =-=[1]-=-, a system of fixed point equations κ Σ over the variables ξ1,...,ξn is a sequence ξ1 = κ f1(ξ1,...,ξn),...,ξn = fn(ξ1,...,ξn) where the fi denote functionals that are intended to be interpreted as mo... |

36 | The decidability of model checking mobile ambients, in
- Charatonik, Talbot
- 2010
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Citation Context ...nd the other one φ ′ . The least fixed point operator considers formulae as mappings over sets of trees. An important question about logics is the decidability of satisfiability. It has been shown in =-=[11]-=- that satisfiability is undecidable for a fragment of the ambient logic contained in TQL. The use of quantification over names is crucial for this result. However, decidable fragments of the logic TQL... |

32 | Numerical document queries
- Seidl, Schwentick, et al.
- 2003
(Show Context)
Citation Context ... not be defined in MSO; the formula µξ.(a[ξ] | b[ξ] | ξ ∨ 0) expresses that any node has as many outgoing edges labelled with a as outgoing edges labelled with b. Recently, Seidl et al. introduced in =-=[20]-=- the Presburger monadic second order (PMSO) as an extension of MSO with some counting constraints allowing one to express relationship between the number of children of a given node satisfying some pr... |

25 | Tql: a query language for semistructured data based on the ambient logic
- Cardelli, Ghelli
(Show Context)
Citation Context ... some syntactic restrictions over STL formulae that allow us to capture precisely the logics MSO and PMSO. 1 Introduction Spatial logics allow one to express properties about structures such as trees =-=[8]-=-, graphs [6, 14] and heaps [19]. The main ingredient of spatial logics is an operator called composition (or separation). This operator permits compositional reasoning over concurrent and mobile proce... |

18 | D.: XML schema, tree logic and sheaves automata
- Zilio, Lugiez
- 2003
(Show Context)
Citation Context ...ms for semistructured data as the one proposed in [5] or for testing query correctness, query containment and defining constraints, as suggested in [8]. The logic TL introduced by Dal Zilio et al. in =-=[12, 13]-=- is also inspired by the ambient logic. This logic is quantifierfree and equipped with a restricted form of recursion but it allows an additional spatial primitive (composition adjunct). However, this... |

16 | A logic you can count on
- Zilio, Lugiez, et al.
- 2004
(Show Context)
Citation Context ...ms for semistructured data as the one proposed in [5] or for testing query correctness, query containment and defining constraints, as suggested in [8]. The logic TL introduced by Dal Zilio et al. in =-=[12, 13]-=- is also inspired by the ambient logic. This logic is quantifierfree and equipped with a restricted form of recursion but it allows an additional spatial primitive (composition adjunct). However, this... |

15 | Expressiveness and complexity of graph logic
- Dawar, Gardner, et al.
(Show Context)
Citation Context ...tic restrictions over STL formulae that allow us to capture precisely the logics MSO and PMSO. 1 Introduction Spatial logics allow one to express properties about structures such as trees [8], graphs =-=[6, 14]-=- and heaps [19]. The main ingredient of spatial logics is an operator called composition (or separation). This operator permits compositional reasoning over concurrent and mobile processes [10, 4] or ... |

12 |
Counting and equality constraints for multitree automata
- Lugiez
- 2003
(Show Context)
Citation Context ...adapted to edge-labelled trees and for which Presburger formulae are replaced by more general arithmetical constraints. Our automata are also close to sheaves automata [13, 12] and multitree automata =-=[16]-=-. We also relate these automata to the logics PMSO and MSO. In the following, for any sets S, T we denote T S the set of mappings from S into T . Moreover, we freely identify elements from N S with mu... |

10 | Automata and logics for unranked and unordered trees
- Boneva, Talbot
- 2005
(Show Context)
Citation Context ...ar sets and solutions of Presburger formulae, the result easily follows from the equivalence of expressiveness of Presburger automata and PMSO logic sentences as established in [20]. ✷ Proposition 5 (=-=[3]-=-) A set of trees is accepted by some starfree constrained automaton iff it is MSO definable. 4 Satisfiability of STL The satisfiability problem is, given a closed STL formula φ, decide whether �φ� is ... |

6 | On complexity of model-checking for the TQL logic - Boneva, Talbot - 2004 |

3 |
Alternating twoway AC-tree automata. Research Report LSV-02-11, LSV. Available at http://www.lsv.ens-cachan.fr/Publis/RAPPORTS_LSV
- Goubault-Larrecq, Verma
- 2002
(Show Context)
Citation Context ...idable for STL. Sketch of proof The proof is done using a reduction of emptiness of two-counter machines inspired from the proof of undecidability of emptiness for alternating twoway AC-tree automata =-=[15]-=-. A two-counter machine [17] M = 〈Q, qi,qf , ∆〉 is a finite labelled transition system where Q is the set of states, qi is the initial state, qf is the final state and ∆ is a set of transitions of the... |

1 |
Recursive insolvability of Post’s problem of ”tag” ard other topics in the theory of turing machines
- Minsky
- 1961
(Show Context)
Citation Context ...oof The proof is done using a reduction of emptiness of two-counter machines inspired from the proof of undecidability of emptiness for alternating twoway AC-tree automata [15]. A two-counter machine =-=[17]-=- M = 〈Q, qi,qf , ∆〉 is a finite labelled transition system where Q is the set of states, qi is the initial state, qf is the final state and ∆ is a set of transitions of the form (q, r, q ′ ), where q,... |