Safety Properties for Querying XML Streams

Transcription

1 Safety Properties for Querying XML Streams Olivier Gauwin University of Mons joint work with: J. Niehren, S. Tison, M. Benedikt and G. Puppis Workshop on Timed and Infinite Systems Warwick March 30th, 2010 Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

2 XML A Format for Semi-Structured Data XML Document <bib> <book> <author> Hopcroft </author> <author> Ullman </author> <title> Introduction... </title> </book> <book>. author. </book> Hopcroft <book>. </book> </bib> Corresponding Tree bib book book book author title Ullman Introduction... Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

3 XML A Format for Semi-Structured Data XML Document <bib> Corresponding Tree bib </bib> Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

4 XML A Format for Semi-Structured Data <bib> <book> XML Document Corresponding Tree bib </book> <book> book book book </book> <book> </book> </bib> Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

5 XML A Format for Semi-Structured Data XML Document <bib> <book> <author> Hopcroft </author> <author> Ullman </author> <title> Introduction... </title> </book> <book>. author. </book> Hopcroft <book>. </book> </bib> Corresponding Tree bib book book book author title Ullman Introduction... Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

6 XML A Format for Semi-Structured Data XML Document <bib> <book> <author> Hopcroft </author> <author> Ullman </author> <title> Introduction... </title> </book> <book>. author. </book> Hopcroft <book>. </book> </bib> Corresponding Tree bib book book book author title Ullman Introduction... finite labeled ordered unranked trees Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

7 Streaming process data on-the-fly objective: low memory consumption (buffering) use cases: huge data (larger than main memory) natural stream sources: network sockets sensors subscribed feeds etc. Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

8 XML Streams <bib> <book> <author> Hopcroft </author> <author> Ullman </author> <title> Introduction... </title> </book> <book> book bib book. </book> </bib> author Hopcroft author Ullman title Introduction Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

9 XML Streams <bib> bib Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

10 XML Streams <bib> <book> bib book Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

11 XML Streams <bib> <book> <author> bib book author Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

12 XML Streams <bib> <book> <author> Hopcroft bib book author Hopcroft Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

13 XML Streams <bib> <book> <author> Hopcroft </author> bib book author Hopcroft Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

14 XML Streams <bib> <book> <author> Hopcroft </author> <author> bib book author author Hopcroft Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

15 XML Streams <bib> <book> <author> Hopcroft </author> <author> Ullman bib book author author Hopcroft Ullman Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

16 XML Streams <bib> <book> <author> Hopcroft </author> <author> Ullman </author> bib book author author Hopcroft Ullman Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

17 XML Streams <bib> <book> <author> Hopcroft </author> <author> Ullman </author> <title> book bib author author title Hopcroft Ullman Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

18 XML Streams <bib> <book> <author> Hopcroft </author> <author> Ullman </author> <title> Introduction... book bib author author title Hopcroft Ullman Introduction... Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

19 XML Streams <bib> <book> <author> Hopcroft </author> <author> Ullman </author> <title> Introduction... </title> book bib author author title Hopcroft Ullman Introduction... Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

20 XML Streams <bib> <book> <author> Hopcroft </author> <author> Ullman </author> <title> Introduction... </title> </book> book bib author author title Hopcroft Ullman Introduction... Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

21 XML Streams <bib> <book> <author> Hopcroft </author> <author> Ullman </author> <title> Introduction... </title> </book> <book> book bib book. </book> author author title... Hopcroft Ullman Introduction... Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

22 XML Streams <bib> <book> <author> Hopcroft </author> <author> Ullman </author> <title> Introduction... </title> </book> <book> book bib book. </book> </bib> author Hopcroft author Ullman title Introduction Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

23 Data exchange html body bib table book book tr tr author author title... td td td... Hopcroft Ullman Hopcroft Ullman Introduction... b Introduction... conform to schema 1 conform to schema 2 requires transformations usually based on selection of tuples of nodes via queries Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

24 Queries Queries A query Q selects a set of nodes in every tree t: Q(t) nodes(t) For instance, for Q = select the first author of each book bib book book author author title author title Hopcroft Ullman Introduction... Knuth The art of CP Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

25 Objectives process queries defined by: XPath expressions (W3C) tree automata in an earliest manner: select nodes as soon as possible reject nodes as soon as possible Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

26 Objectives process queries defined by: XPath expressions (W3C) tree automata in an earliest manner: select nodes as soon as possible reject nodes as soon as possible Consider Q = select authors of The Art of CP bib book book author author title author title Hopcroft Ullman Introduction... Knuth The art of CP Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

27 Objectives process queries defined by: XPath expressions (W3C) tree automata in an earliest manner: select nodes as soon as possible reject nodes as soon as possible Consider Q = select authors of The Art of CP bib book book author author title author title Hopcroft Ullman Introduction... Knuth The art of CP schemas may help Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

28 In this talk 1 we focus on the simpler case of filtering A filter F expresses some properties of trees specification φ in verification example: validation wrt a schema: DTD, XML Schema, Relax NG XPath filters evaluation Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

29 In this talk 1 we focus on the simpler case of filtering A filter F expresses some properties of trees specification φ in verification example: validation wrt a schema: DTD, XML Schema, Relax NG XPath filters evaluation 2 we only deal with filters defined by tree automata schemas are tree automata XPath tree automata: in PTIME for most streamable XPath expressions Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

30 In this talk 1 we focus on the simpler case of filtering A filter F expresses some properties of trees specification φ in verification example: validation wrt a schema: DTD, XML Schema, Relax NG XPath filters evaluation 2 we only deal with filters defined by tree automata schemas are tree automata XPath tree automata: in PTIME for most streamable XPath expressions The task becomes: Given a tree automaton A, and a tree t accessed in streaming, detect at the earliest position, whether t L(A). Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

31 In this talk 1 we focus on the simpler case of filtering A filter F expresses some properties of trees specification φ in verification example: validation wrt a schema: DTD, XML Schema, Relax NG XPath filters evaluation 2 we only deal with filters defined by tree automata schemas are tree automata XPath tree automata: in PTIME for most streamable XPath expressions The task becomes: Given a tree automaton A, and a tree t accessed in streaming, detect at the earliest position, whether t L(A). online verification ([Kupferman and Vardi (2001)]: earliest rejection) Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

32 Hardness Task: Given a tree automaton A, and a tree t accessed in streaming, detect at the earliest position, whether t L(A). t e harder than universality of A consider the starting event Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

33 Hardness Task: Given a tree automaton A, and a tree t accessed in streaming, detect at the earliest position, whether t L(A). t e t harder than universality of A consider the starting event Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

34 Earliest Validation vs Earliest Rejection in this talk, we consider earliest validation (t L(A)?) do all continuations succeed? but not earliest rejection (t / L(A)?) do all continuations fail? Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

35 Earliest Validation vs Earliest Rejection in this talk, we consider earliest validation (t L(A)?) do all continuations succeed? but not earliest rejection (t / L(A)?) do all continuations fail? these problems are different! earliest validation = are all accessible configurations accepting? earliest rejection = is there an accessible accepting configurations? Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

36 Outline 1 Visibly Pushdown Automata (VPAs) as tree automata Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

37 Outline 1 Visibly Pushdown Automata (VPAs) as tree automata 2 Algorithm 1 for earliest validation propagation of safe states [Gauwin et al. (2009b)] Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

38 Outline 1 Visibly Pushdown Automata (VPAs) as tree automata 2 Algorithm 1 for earliest validation propagation of safe states [Gauwin et al. (2009b)] 3 Algorithm 2 for earliest validation reachability of configurations inspired from pushdown systems [Bouajjani et al. (1997)] Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

39 1 Introduction 2 Visibly Pushdown Automata 3 Algorithm 1: Propagation of Safe States 4 Algorithm 2: Reachability of Configurations 5 Conclusion Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

40 Visibly Pushdown Automata recent model [Alur and Madhusudan (2004), Alur and Madhusudan (2009)] a pushdown automaton over Σ Σ visible means that 1 action (push/pop) is performed by each letter: rules using a Σ only push rules using a Σ only pop Visibly Pushdown Automaton (VPA) A VPA is a tuple (Q,I,F,Γ,,Σ Σ) with I,F Q and is a set of rules with the form: q 1,a γ,q 2 written as q 1 a:γ q 2 q 1,a,γ q 2 written as q 1 a:γ q 2 with q 1,q 2 Q, a Σ, a Σ, γ Γ. The semantics is defined as for usual pushdown automata (ends on final states, not on empty stack). Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

41 VPAs as Tree Automata a run of a VPA on a tree consists in: when opening node n: update the current state, and assign γ Γ to n when closing node n: update the current state according to γ a β b β a α a b b a 0 α β γ β b γ A: VPA on T Σ with Σ = {a,b} and Q = {0,1,2,3,4,5} and Γ = {α,β,γ} 5 Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

42 VPAs as Tree Automata a run of a VPA on a tree consists in: when opening node n: update the current state, and assign γ Γ to n when closing node n: update the current state according to γ a β b β a α a b b a 0 α β γ β b γ A: VPA on T Σ with Σ = {a,b} and Q = {0,1,2,3,4,5} and Γ = {α,β,γ} (0, ) 5 a 0 a b b Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

43 VPAs as Tree Automata a run of a VPA on a tree consists in: when opening node n: update the current state, and assign γ Γ to n when closing node n: update the current state according to γ a β b β a α a b b a 0 α β γ β b γ A: VPA on T Σ with Σ = {a,b} and Q = {0,1,2,3,4,5} and Γ = {α,β,γ} (0, ) a (1,α) 5 a 0 1 α a b b Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

44 VPAs as Tree Automata a run of a VPA on a tree consists in: when opening node n: update the current state, and assign γ Γ to n when closing node n: update the current state according to γ a β b β a α a b b a 0 α β γ β b γ A: VPA on T Σ with Σ = {a,b} and Q = {0,1,2,3,4,5} and Γ = {α,β,γ} 1 (0, ) a (1,α) a (1,α.β) 5 β a 0 1 α a b b Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

45 VPAs as Tree Automata a run of a VPA on a tree consists in: when opening node n: update the current state, and assign γ Γ to n when closing node n: update the current state according to γ a β b β a α a b b a 0 α β γ β b γ A: VPA on T Σ with Σ = {a,b} and Q = {0,1,2,3,4,5} and Γ = {α,β,γ} 1 (0, ) a (1,α) a (1,α.β) b (4, α.β.β) 5 4 β a β b 0 1 α a b Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

46 VPAs as Tree Automata a run of a VPA on a tree consists in: when opening node n: update the current state, and assign γ Γ to n when closing node n: update the current state according to γ a β b β a α a b b a 0 α β γ β b γ A: VPA on T Σ with Σ = {a,b} and Q = {0,1,2,3,4,5} and Γ = {α,β,γ} 1 (0, ) a (1,α) a (1,α.β) b (4, α.β.β) b (3,α.β) 5 4 β a β b α a b Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

47 VPAs as Tree Automata a run of a VPA on a tree consists in: when opening node n: update the current state, and assign γ Γ to n when closing node n: update the current state according to γ a β b β a α a b b a 0 α β γ β b γ A: VPA on T Σ with Σ = {a,b} and Q = {0,1,2,3,4,5} and Γ = {α,β,γ} 1 (0, ) a (1,α) a (1,α.β) b (4, α.β.β) b (3,α.β) a (2,α) 5 4 β a β b α a b Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

48 VPAs as Tree Automata a run of a VPA on a tree consists in: when opening node n: update the current state, and assign γ Γ to n when closing node n: update the current state according to γ a β b β a α a b b a 0 α β γ β b γ A: VPA on T Σ with Σ = {a,b} and Q = {0,1,2,3,4,5} and Γ = {α,β,γ} 1 (0, ) a (1,α) a (1,α.β) b (4, α.β.β) b (3,α.β) a (2,α) b (4,α.γ) 5 4 β a β b α a 2 4 γ b Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

49 VPAs as Tree Automata a run of a VPA on a tree consists in: when opening node n: update the current state, and assign γ Γ to n when closing node n: update the current state according to γ a β b β a α a b b a 0 α β γ β b γ A: VPA on T Σ with Σ = {a,b} and Q = {0,1,2,3,4,5} and Γ = {α,β,γ} 1 (0, ) a (1,α) a (1,α.β) b (4, α.β.β) b (3,α.β) a (2,α) b (4,α.γ) b (3,α) 5 4 β a β b α a 2 4 Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34 γ b 3

50 VPAs as Tree Automata a run of a VPA on a tree consists in: when opening node n: update the current state, and assign γ Γ to n when closing node n: update the current state according to γ a β b β a α a b b a 0 α β γ β b γ A: VPA on T Σ with Σ = {a,b} and Q = {0,1,2,3,4,5} and Γ = {α,β,γ} 1 (0, ) a (1,α) a (1,α.β) b (4, α.β.β) b (3,α.β) a (2,α) b (4,α.γ) b (3,α) a (5, ) 5 4 β a β b α a 2 4 Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34 5 γ b 3

51 VPAs: related models From now on, we will consider VPAs as tree automata. Other equivalent models (in expressiveness): nested word automata [Alur (2007)]: reformulation of VPAs pushdown forest automata [Neumann and Seidl (1998)]: VPAs on forests (i.e. hedges) streaming tree automata [Gauwin et al. (2008)]: VPAs on trees Translations in [Gauwin (2009)] (and also from DTDs, stepwise tree automata...) Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

52 VPA: determinism determinism of VPAs corresponds to determinism along the XML stream in the sequel we only consider filters defined by deterministic VPAs (dvpas) Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

53 1 Introduction 2 Visibly Pushdown Automata 3 Algorithm 1: Propagation of Safe States 4 Algorithm 2: Reachability of Configurations 5 Conclusion Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

54 E A : a VPA for earliest validation A E A dvpa VPA defines the filter F defines the filter F detects validity Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

55 E A : a VPA for earliest validation A E A dvpa VPA defines the filter F defines the filter F detects validity states of E A are of the form (q,s) where: q QA S QA are safe states for validation, i.e., Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

56 E A : a VPA for earliest validation A E A dvpa VPA defines the filter F defines the filter F detects validity states of E A are of the form (q,s) where: q QA S QA are safe states for validation, i.e., if r(e) = (q,s) where r is a run of E A on t then t L(A) is sure at e iff q S Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

57 E A : a VPA for earliest validation A E A dvpa VPA defines the filter F defines the filter F detects validity states of E A are of the form (q,s) where: q QA S QA are safe states for validation, i.e., if r(e) = (q,s) where r is a run of E A on t then t L(A) is sure at e iff q S remark: for words, all states q are already safe or unsafe... Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

58 E A : a VPA for earliest validation A E A dvpa VPA defines the filter F defines the filter F detects validity states of E A are of the form (q,s) where: q QA S QA are safe states for validation, i.e., if r(e) = (q,s) where r is a run of E A on t then t L(A) is sure at e iff q S remark: for words, all states q are already safe or unsafe......but not for VPAs: it depends on the context (i.e. the stack) Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

59 Earliest Validation with a VPA Safe States via Propagation Rules a a b b a b a Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

60 Earliest Validation with a VPA Safe States via Propagation Rules a S 0 a b b a b a (R1) safe(,root) = F A Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

61 Earliest Validation with a VPA Safe States via Propagation Rules S 1 a S 0 q γ l close A l,γ (q ) a b h q = acc A h (q) b a b a (R1) safe(,root) = F A (R2) safe(,n) = S n nodes(t) l = lab t (n) γ = r A (n) safe(,n) = {q h. close A l,γ acc A h (q) S} Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

62 Earliest Validation with a VPA Safe States via Propagation Rules S 1 a S 0 a q l S 1 b S 1 γ h close A l,γ (q ) q = acc A h (q) b a b a (R1) safe(,root) = F A (R2) (R3) safe(,n) = S n nodes(t) l = lab t (n) γ = r A (n) safe(,n) = {q h. close A l,γ acc A h (q) S} safe(,n) = S n nodes(t) n.i nodes(t) safe(,n.i) = S Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

63 Earliest Validation with a VPA Safe States via Propagation Rules S 1 a S 0 S 2 a S 1 b S 1 b a b a (R1) safe(,root) = F A (R2) (R3) safe(,n) = S n nodes(t) l = lab t (n) γ = r A (n) safe(,n) = {q h. close A l,γ acc A h (q) S} safe(,n) = S n nodes(t) n.i nodes(t) safe(,n.i) = S Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

64 Earliest Validation with a VPA Safe States via Propagation Rules S 1 a S 0 S 2 a S 1 b S 1 b S 2 a b a (R1) safe(,root) = F A (R2) (R3) safe(,n) = S n nodes(t) l = lab t (n) γ = r A (n) safe(,n) = {q h. close A l,γ acc A h (q) S} safe(,n) = S n nodes(t) n.i nodes(t) safe(,n.i) = S Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

65 Earliest Validation with a VPA Safe States via Propagation Rules S 1 a S 0 S 2 a S 1 b S 1 S 3 b S 2 a b a (R1) safe(,root) = F A (R2) (R3) safe(,n) = S n nodes(t) l = lab t (n) γ = r A (n) safe(,n) = {q h. close A l,γ acc A h (q) S} safe(,n) = S n nodes(t) n.i nodes(t) safe(,n.i) = S Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

66 Earliest Validation with a VPA Safe States via Propagation Rules S 1 a S 0 S 2 a S 1 S 4 b S 1 S 3 b S 2 a b a (R1) safe(,root) = F A (R2) (R3) safe(,n) = S n nodes(t) l = lab t (n) γ = r A (n) safe(,n) = {q h. close A l,γ acc A h (q) S} safe(,n) = S n nodes(t) n.i nodes(t) safe(,n.i) = S Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

67 Earliest Validation with a VPA Safe States via Propagation Rules S 1 a S 0 S 2 a S 1 S 4 b S 1 S 3 b S 2 a S 4 b S 4 a S 4 (R1) safe(,root) = F A (R2) (R3) safe(,n) = S n nodes(t) l = lab t (n) γ = r A (n) safe(,n) = {q h. close A l,γ acc A h (q) S} safe(,n) = S n nodes(t) n.i nodes(t) safe(,n.i) = S Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

68 Earliest Validation with a VPA Safe States via Propagation Rules S 1 a S 0 S 2 a S 1 S 4 b S 1 S 3 b S 2 S 5 a S 4 S 6 b S 4 S 7 a S 4 (R1) safe(,root) = F A (R2) (R3) safe(,n) = S n nodes(t) l = lab t (n) γ = r A (n) safe(,n) = {q h. close A l,γ acc A h (q) S} safe(,n) = S n nodes(t) n.i nodes(t) safe(,n.i) = S Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

69 Earliest Validation with a VPA Safe States via Propagation Rules S 1 a S 0 S 2 a S 1 S 4 b S 1 S 3 b S 2 S 5 a S 4 S 6 b S 4 S 7 a S 4 (R1) safe(,root) = F A (R2) (R3) safe(,n) = S n nodes(t) l = lab t (n) γ = r A (n) safe(,n) = {q h. close A l,γ acc A h (q) S} safe(,n) = S n nodes(t) n.i nodes(t) safe(,n.i) = S we can do this with a VPA! Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

70 Computing Runs of E A Problem: E A is in O(2 A ) Efficient implementation A is deterministic E A is deterministic we don t build E A, but its runs on the fly to do so, we compute the safe states in 2 steps precomputation of an accessibility relation (in PTIME) at reception of an event, use this relation to compute the safe states (in PTIME) drawback: computation of safe states cannot be stopped/resumed anywhere Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

71 1 Introduction 2 Visibly Pushdown Automata 3 Algorithm 1: Propagation of Safe States 4 Algorithm 2: Reachability of Configurations 5 Conclusion Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

72 Configurations of a VPA 0 1 α a 5 a β b β a α 0 a b b a α β γ β b 2 γ β a β b γ b 3 configurations of the VPA A: (q,w) with q Q A and w Γ A (0,) a (1,α) a (1,α.β) b (4,α.β.β)... b (4,α.γ) b (3,α) a (5,) Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

73 Configurations of a VPA 0 1 α a 5 a β b β a α 0 a b b a α β γ β b 2 γ β a β b γ b 3 configurations of the VPA A: (q,w) with q Q A and w Γ A (0,) a (1,α) a (1,α.β) b (4,α.β.β)... b (4,α.γ) b (3,α) a (5,) when running on trees: (q,w) with q Q A and w (Γ A Σ) (0,) a (1,(α,a)) a (1,(α,a).(β,a))... b (3,(α,a)) a (5,) Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

74 Regular Sets of Configurations VPA A: : λ a : λ c : λ a : β c : β b : a : 0 q b 1 : β c : λ b : c : β : λ : A set of configurations C of A is regular if there exists an NFA N such that L conf (N) = C. For instance, for C = {(0,),(q b,)} {(1,(β,a).(λ,b) n )} n 0 : NFA N: 0 s 1 (β, a) q b (λ,b) Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

75 Configurations Reachability Given a VPA A, pre A (C) = {c c C. c A c } Regularity Preservation pre A preserves regularity, i.e.: C regular pre (C) regular Computation of pre (C) An NFA recognizing pre (C) can be computed in PTIME from an NFA recognizing C. This also holds for pushdown automata [Bouajjani et al. (1997)]. Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

76 Earliest Validation, rephrased t can be safely validated at event e (wrt. A) Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

77 Earliest Validation, rephrased t can be safely validated at event e (wrt. A) A accepts all possible continuations from current config. c Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

78 Earliest Validation, rephrased t can be safely validated at event e (wrt. A) A accepts all possible continuations from current config. c from c, there is no continuation leading to a failing config. Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

79 Earliest Validation, rephrased t can be safely validated at event e (wrt. A) A accepts all possible continuations from current config. c from c, there is no continuation leading to a failing config. c / pre A (C fail) with C fail = {(q,) q / F A } Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

80 Computing pre (C fail ) Let N fail be an NFA such that L(N fail ) = C fail. From N fail, we build N fail such that L(N fail ) = pre (C fail ). Saturation process, according to 2 inference rules: a : γ (γ,a) q q s A N fail N fail a : γ q q s A (γ,a) N fail N fail infered rule Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

81 Computing N fail from N fail Saturation algorithm 0 s 1 : λ a : λ c : λ a : β c : β b : 0 q b 1 : β c : λ b : c : β a : a : γ (γ,a) q q s A Nfail N fail : λ : q b N fail a : γ q q s A Nfail (γ,a) N fail Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

82 Computing N fail from N fail Saturation algorithm (λ,a) (λ,c) (β,a) (β,c) (,b) 0 (λ,c) (, ) s 1 (, ) : λ a : λ c : λ a : β c : β b : 0 q b 1 : β c : λ b : c : β a : a : γ (γ,a) q q s A Nfail : λ : q b (,b) (,b) (,c) (,a) N fail a : γ q q s A Nfail (γ,a) N fail Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

83 Computing N fail from N fail Saturation algorithm (λ,a) (λ,c) (β,a) (β,c) (,b) 0 (λ,c) (, ) s 1 (, ) : λ a : λ c : λ a : β c : β b : 0 q b 1 : β c : λ b : c : β a : a : γ (γ,a) q q s A Nfail : λ : (,b) (,c) N fail q b (,b) (,a) a : γ q q s A Nfail (γ,a) N fail Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

84 Computing N fail from N fail Saturation algorithm (λ,a) (λ,c) (β,a) (β,c) (,b) (λ,a) (λ,c) 0 q b (λ,c) (, ) (,b) (,c) (,b) (,c) s (β,a) (β,c) (,b) 1 (,a) (,b) (, ) : λ a : λ c : λ a : β c : β b : 0 q b 1 : β c : λ b : c : β a : a : γ (γ,a) q q s A Nfail N fail a : γ q q s A Nfail (γ,a) N fail : λ : Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

85 Computing N fail from N fail Saturation algorithm (λ,a) (λ,c) (β,a) (β,c) (,b) (λ,a) (λ,c) 0 q b (λ,c) (, ) (,b) (,c) (,b) (,c) s (β,a) (β,c) (,b) 1 (,a) (,b) (, ) : λ a : λ c : λ a : β c : β b : 0 q b 1 : β c : λ b : c : β a : a : γ (γ,a) q q s A Nfail N fail a : γ q q s A Nfail (γ,a) N fail : λ : Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

86 Computing N fail from N fail Saturation algorithm (λ,a) (λ,c) (β,a) (β,c) (,b) (λ,a) (λ,c) 0 q b (λ,c) (, ) (,b) (,c) (,b) (,c) s (β,a) (β,c) (,b) (λ, ) 1 (,a) (,b) (,c) (, ) : λ a : λ c : λ a : β c : β b : 0 q b 1 : β c : λ b : c : β a : a : γ (γ,a) q q s A Nfail N fail a : γ q q s A Nfail (γ,a) N fail : λ : Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

87 Example Run and Analysis (β,a) (β,c) (,b) (λ,a) (λ,c) 0 q b (λ,a) (λ,c) (λ,c) (,b) (,c) (, ) (,b) (,c) s N fail (β,a) (β,c) (,b) (λ, ) 1 (,a) (,b) (,c) (, ) : λ a : λ c : λ a : β c : β b : 0 q b 1 c b : β c : λ b a b b : c : β a : : λ : a a Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

88 Example Run and Analysis (β,a) (β,c) (,b) (λ,a) (λ,c) 0 q b (λ,a) (λ,c) (λ,c) (,b) (,c) (, ) (,b) (,c) s N fail (β,a) (β,c) (,b) (λ, ) 1 (,a) (,b) (,c) (, ) : λ a : λ c : λ a : β c : β b : 0 q b 1 : β c : λ 0 λ 0 c 0 0 λ 0 b qb λ 0 b 1 λ 0 a 1 b : c : β λ b qb a : : λ : β 0 a q b λ 0 a 0 Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

89 Remarks synthesis of bad continuations Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

90 Remarks synthesis of bad continuations flexibility: verification can be performed/stopped at any time point (garbage collector) Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

91 Beyond dvpas earliest validation earliest rejection dvpas PTIME PTIME VPAs EXPTIME-complete PTIME dpdas PTIME PTIME PDAs undecidable PTIME Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

92 1 Introduction 2 Visibly Pushdown Automata 3 Algorithm 1: Propagation of Safe States 4 Algorithm 2: Reachability of Configurations 5 Conclusion Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

93 Perspectives experimental comparison between both algorithms work in progress (EvoXs [Gauwin et al. (2009a)]) worst-case complexity is similar find efficient data structures/representations non-deterministic VPAs efficient techniques for earliest validation theoretical perspectives which class is tractable when allowing EXPTIME precomputation? which restrictions on logics (MSO, XPath, etc.) ensure tractability? Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

94 Safety Properties for Querying XML Streams Olivier Gauwin University of Mons joint work with: J. Niehren, S. Tison, M. Benedikt and G. Puppis Workshop on Timed and Infinite Systems Warwick March 30th, 2010 Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

95 References Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

96 [Alur (2007)] Rajeev Alur. Marrying words and trees. In 26th ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, pages ACM-Press, [Alur and Madhusudan (2004)] Rajeev Alur and P. Madhusudan. Visibly pushdown languages. In 36th ACM Symposium on Theory of Computing, pages ACM-Press, URL [Alur and Madhusudan (2009)] Rajeev Alur and P. Madhusudan. Adding nesting structure to words. Journal of the ACM, 56(3):1 43, URL [Bouajjani et al. (1997)] Ahmed Bouajjani, Javier Esparza, and Oded Maler. Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

97 Reachability analysis of pushdown automata: Application to model-checking. In 8th International Conference on Concurrency Theory, volume 1243 of Lecture Notes in Computer Science, pages Springer, [Gauwin (2009)] Olivier Gauwin. Streaming Tree Automata and XPath. PhD thesis, Université Lille 1, URL [Gauwin et al. (2008)] Olivier Gauwin, Joachim Niehren, and Yves Roos. Streaming tree automata. Information Processing Letters, 109(1):13 17, December URL [Gauwin et al. (2009a)] Olivier Gauwin, Christophe Munilla, Joachim Niehren, and Benoît Papegay. EvoXs project, 2009a. Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34

98 [Gauwin et al. (2009b)] Olivier Gauwin, Joachim Niehren, and Sophie Tison. Earliest query answering for deterministic nested word automata. In 17th International Symposium on Fundamentals of Computer Theory, volume 5699 of Lecture Notes in Computer Science, pages Springer Verlag, 2009b. URL [Kupferman and Vardi (2001)] Orna Kupferman and Moshe Y. Vardi. Model checking of safety properties. Formal Methods in System Design, 19(3): , [Neumann and Seidl (1998)] Andreas Neumann and Helmut Seidl. Locating matches of tree patterns in forests. In 18th Conference on Foundations of Software Technology and Theoretical Computer Science, volume 1530 of Lecture Notes in Computer Science, pages Springer Verlag, Olivier Gauwin (UMons) Safety Properties for Querying XML Streams March 30th, / 34