Path Querying on Graph Databases

Transcription

1 Path Querying on Graph Databases Jelle Hellings Hasselt University and transnational University of Limburg 1/38

2 Overview Graph Databases Motivation Walk Logic Relations with FO and MSO Relations with CTL and Hybrid CTL Conjunctive Regular Path Queries Open Problems and Conclusion 2/38

3 Overview Graph Databases Motivation Walk Logic Relations with FO and MSO Relations with CTL and Hybrid CTL Conjunctive Regular Path Queries Open Problems and Conclusion 3/38

4 Graphs Pieces of data (nodes) Relations between the pieces of data (edges) Example (Social networks) workswith Jan Jelle worksat UHasselt worksat 4/38

5 Applications XML and RDF, Social networks, Transportation networks, The World Wide Web,... 5/38

6 Graph Database: Google Maps Nodes: points of interest, addresses,... Edges: road network Queries: Example (Distance based query) university close to <my address> (answer: Universiteit Hasselt; 5.2 km) Example (Route-planning query) From: <my address>, to: <university> (answer: options for university; followed by route) 6/38

7 Challenges Engineering: big data Storage, distributed processing, hardware failures,... Conceptual: semantics and consistency Structured data (facebook) versus structured? data (the web) Conceptual: data querying Local/navigational based versus graph-wide path based No widely used general purpose languages Current practice: application specific languages 7/38

8 Challenges Engineering: big data Storage, distributed processing, hardware failures,... Conceptual: semantics and consistency Structured data (facebook) versus structured? data (the web) Conceptual: data querying Local/navigational based versus graph-wide path based No widely used general purpose languages Current practice: application specific languages Our focus Path-based graph querying 7/38

9 Overview Graph Databases Motivation Walk Logic Relations with FO and MSO Relations with CTL and Hybrid CTL Conjunctive Regular Path Queries Open Problems and Conclusion 8/38

10 Motivation Expressing graph-queries Properties of paths, walks,... Route planning We want to travel from our office to a cafetaria and from this cafetaria get back to the office using a different route 9/38

11 Graph querying Graphs as traditional relations worksat(person, company) Already deep knowledge of these systems First-order based query languages: Q(n) := m workswith(n, m) worksat(m, UHasselt) Largely restricted to local reasoning no paths, no or only limited reachability,... 10/38

12 Higher-order logics Monadic second-order logic Extend first-order logic with quantification over sets Strong theoretical background Sets with only nodes versus sets with nodes and edges Some graph problems are naturally expressible with sets: Graph coloring, bipartite graph,... S T ( x (x S x T ) (x S = x T ) (x T = x S) y edge(x, y) = ((x S y T ) (y S x T ))) Paths non-straightforward: y is reachable from x S [(x S) u v (u S edge(u, v) = v S) = y S] 11/38

13 Conjunctive Regular Path Queries Idea Query nodes based on labelling of paths between nodes Express labelling by a regular expression Example Q(a, b) := aπb, (αβ + γδ) + (π) α n 1 n 2 β n 3 γ n 4 δ n 5 γ n 6 α n 7 β n 8 δ 12/38

14 Conjunctive Regular Path Queries Idea Query nodes based on labelling of paths between nodes Express labelling by a regular expression Example Q(a, b) := aπb, (αβ + γδ) + (π) α n 1 n 2 β n 3 γ n 4 δ n 5 γ n 6 α n 7 β n 8 δ 12/38

15 Conjunctive Regular Path Queries Idea Query nodes based on labelling of paths between nodes Express labelling by a regular expression Example Q(a, b) := aπb, (αβ + γδ) + (π) α n 1 n 2 β n 3 γ n 4 δ n 5 γ n 6 α n 7 β n 8 δ 12/38

16 Extended Conjunctive Regular Path Queries Idea Comparing labelling of paths Regular expressions over n-tuples Use special symbol to specify end-of-path Example Q(a, b) := aπ 1 b, aπ 2 b, ([ α β ] + [ α ])(π 1, π 2 ) α n 1 n 2 α n 3 α n 4 β n 6 β n 7 β n 8 α 13/38

17 Computation tree logic Usage: verification of formal models Describe behaviour by a transition system (graph) Write propositions that should hold Example shutdown buy start idle work crash giveup 14/38

18 Computation tree logic Usage: verification of formal models Describe behaviour by a transition system (graph) Write propositions that should hold Example shutdown buy start idle work crash giveup Machine never crashes: A G crash 14/38

19 Computation tree logic Usage: verification of formal models Describe behaviour by a transition system (graph) Write propositions that should hold Example shutdown buy start idle work crash giveup Machine can work without crashing: E G crash 14/38

20 Hybrid CTL Idea CTL has only implicit paths and nodes Add ability to name nodes in our formulae Example a b We can get from a to b in two different ways: E v1 E F( v2 E X F(b v1 E( v 2 U v 3 ))) 15/38

21 Hybrid CTL Idea CTL has only implicit paths and nodes Add ability to name nodes in our formulae Example v 1 a b We can get from a to b in two different ways: E v1 E F( v2 E X F(b v1 E( v 2 U v 3 ))) 15/38

22 Hybrid CTL Idea CTL has only implicit paths and nodes Add ability to name nodes in our formulae Example v 2 v 1 a b We can get from a to b in two different ways: E v1 E F( v2 E X F(b v1 E( v 2 U v 3 ))) 15/38

23 Hybrid CTL Idea CTL has only implicit paths and nodes Add ability to name nodes in our formulae Example v 2 v 1 v 3 a b We can get from a to b in two different ways: E v1 E F( v2 E X F(b v1 E( v 2 U v 3 ))) 15/38

24 Hybrid CTL Idea CTL has only implicit paths and nodes Add ability to name nodes in our formulae Example v 2 v 1 v 3 a b v 2 v 2 We can get from a to b in two different ways: E v1 E F( v2 E X F(b v1 E( v 2 U v 3 ))) 15/38

25 Overview Graph Databases Motivation Walk Logic Relations with FO and MSO Relations with CTL and Hybrid CTL Conjunctive Regular Path Queries Open Problems and Conclusion 16/38

26 Walk Logic Idea: extend first-order logic Add walks Add positions on walks Necessary operators to compare positions Route planning We want to travel from our office to a cafetaria (R) and from this cafetaria get back to the office using a different route (S) R S t 1 R t 2 R u 1 S u 2 S u 3 S (office(t 1 ) t 1 < t 2 cafeteria(t 2 ) u 1 < u 3 < u 2 u 1 t 2 u 2 t 1 t 3 R (t 1 < t 3 < t 2 = t 3 u 3 )) 17/38

27 Definitions Definition (Directed node-labeled graph) A directed node-labeled graph is a triple G = (N, E, l): N is a finite set of nodes E N N is the set of edges l : N 2 AP is a node-label function 18/38

28 Walk Logic Walk variables Position variables per walk variable Atomic Formulae a(t) Node referred to by position variable t has labelling a t 1 t 2 Position variables t 1, t 2 refer to the same node t 1 < t 2 Position variable t 1 comes before t 2 in walk W Position variables t 1 and t 2 must be of the same sort ϕ, ψ are formulae ϕ, ϕ ψ W ϕ t W ϕ Negation and disjunction Quantification over walks Quantification over positions on walks 19/38

29 Semantics: walks? Definition Infinite walk A finite or infinite sequence v 1... of nodes such that (v i, v i+1 ) E for each 1 i v 1... Walk A nonempty finite sequence v 1... v n of nodes such that (v i, v i+1 ) E for each 1 i n Trail A walk without edge repetition Path A walk without node repetition 20/38

30 Semantics: walks? Definition Infinite walk A finite or infinite sequence v 1... of nodes such that (v i, v i+1 ) E for each 1 i v 1... Walk A nonempty finite sequence v 1... v n of nodes such that (v i, v i+1 ) E for each 1 i n Trail A walk without edge repetition Path A walk without node repetition CTL and Hybrid CTL : primarily infinite walks CRPQs: primarily walks 20/38

31 Semantics: expressive power Hierarchy of expressive power Infinite walk A walk W is finite: t W u W t < u Walk A walk is a trail (informally): t W u W (t u t +1 u +1 ) = t = u Trail A trail W is a path (informally): t W u W t u = t = u Path Logic Trail Logic Walk Logic Infinite Walk Logic 21/38

32 Walk-based Graph Properties Example (Hamiltonian Path (in Path Logic)) P Q t Q u P (t u) Example (Eulerian Trail (in Trail Logic)) T Q t Q u T (t u) (t +1 u +1 ) Example (Strongly Connected) P Q t P u Q R v R w R (v < w t v u w) 22/38

33 Properties on undirected graphs Theorem Weakly Connected is not expressible on directed graphs Proof. n 1 n 2 n 3 n 4 n 5 n 6 All walks contain at most 2 nodes: reduce to first-order logic Direction matters! On undirected graphs: Weakly Connected same way as strongly connected Planar Graph using Kuratowski s Theorem 23/38

34 Overview Graph Databases Motivation Walk Logic Relations with FO and MSO Relations with CTL and Hybrid CTL Conjunctive Regular Path Queries Open Problems and Conclusion 24/38

35 MSO(nodes, edges) and paths Observations Path: sequence of connected edges No node repetition: nodes and positions coincide Node a before node b on path P if and only if Node b is reachable from a using the edges in P Theorem Path Logic MSO(nodes, edges) 25/38

36 Set-based Graph Properties Theorem Bipartite graph is not expressible on directed graphs Lemma (Dénes Kőnig) A graph is bipartite iff it does not contain an odd cycle Proof. n 2 n 3 m 2 m 3 m 4 n 1 m 1 m 6 m 5 All walks contain at most 3 nodes: reduce to first-order logic MSO(nodes) can express bipartite graph Is Walk Logic strictly subsumed by MSO? 26/38

37 Eulerian Trail Theorem MSO(nodes, edges) cannot express Eulerian Trail Lemma (well known result) MSO cannot distinguish sets with i from sets with j elements Proof. For MSO: existence of Eulerian Trail in the graph a n v 2. a 1 v 1 a n b m b m. =.. b 1 a 1 b 1 Reduces to sets A and B having the equal number of elements 27/38

38 Relations with FO and MSO We have FO Path Logic MSO(nodes, edges) Trail Logic, Walk Logic, and Infinite Walk Logic are incomparable with MSO(nodes) and MSO(nodes, edges) Lemma (Courcelle and Engelfriet) MSO(nodes) cannot express Hamiltonian Path Path Logic and MSO(nodes) are incomparable Path Logic Trail Logic 1 Walk Logic Infinite Walk Logic 1 The proof for Trail Logic Walk Logic is omitted but is similar to the proof of Path Logic Trail Logic 28/38

39 Overview Graph Databases Motivation Walk Logic Relations with FO and MSO Relations with CTL and Hybrid CTL Conjunctive Regular Path Queries Open Problems and Conclusion 29/38

40 Review: Hybrid CTL Definition Let a be an atomic proposition, x a node variable, ϕ 1 and ϕ 2 node formulas, ψ 1 and ψ 2 path formulas Node formulas ϕ ::= a x ϕ 1 ϕ 1 ϕ 2 x ϕ x ϕ 1 E ψ Path formulas ψ ::= ϕ ψ 1 ψ 1 ψ 1 X ψ 1 ψ 1 U ψ 2 30/38

41 Translating Hybrid CTL to Walk Logic Idea Node formulas Properties of single node: translate to properties of single position Hybrid extensions Named nodes: translate to named position variables Path formulas Properties on a single path with forward navigation: translate to walk variable; keep track of current position using position variables and < 31/38

42 Translating Hybrid CTL to Walk Logic Idea Node formulas Properties of single node: translate to properties of single position Hybrid extensions Named nodes: translate to named position variables Path formulas Properties on a single path with forward navigation: translate to walk variable; keep track of current position using position variables and < CTL Hybrid CTL Infinite Walk Logic 31/38

43 Hybrid CTL Infinite Walk Logic? Theorem Hybrid CTL Infinite Walk Logic Proof. CTL Infinite Walk Logic as CTL is invariant under bisimulation Hybrid CTL Infinite Walk Logic as Hybrid CTL is invariant under generated submodels 32/38

44 Hybrid CTL Infinite Walk Logic? Theorem Hybrid CTL Infinite Walk Logic Proof. CTL Infinite Walk Logic as CTL is invariant under bisimulation Hybrid CTL Infinite Walk Logic as Hybrid CTL is invariant under generated submodels CTL Hybrid CTL Infinite Walk Logic 32/38

45 Hybrid CTL Infinite Walk Logic? Theorem Hybrid CTL Infinite Walk Logic Proof. CTL Infinite Walk Logic as CTL is invariant under bisimulation Hybrid CTL Infinite Walk Logic as Hybrid CTL is invariant under generated submodels CTL Hybrid CTL Infinite Walk Logic Walk Logic Infinite Walk Logic 32/38

46 Overview Graph Databases Motivation Walk Logic Relations with FO and MSO Relations with CTL and Hybrid CTL Conjunctive Regular Path Queries Open Problems and Conclusion 33/38

47 CRPQs versus Walk Logics Different languages! Focus on path labelling versus focus on path-structure of graphs All CRPQs are incomparable with all Walk Logics. 34/38

48 CRPQs versus Walk Logics Different languages! Focus on path labelling versus focus on path-structure of graphs All CRPQs are incomparable with all Walk Logics. Similar semantically questions Similar proof techniques 34/38

49 Some results Hamiltonian path cannot be expressed Eulerian trail cannot be expressed 35/38

50 Some results Hamiltonian path cannot be expressed Eulerian trail cannot be expressed Paths versus Walks CRPQ with paths can express queries not expressible in the strongest language with Walks! 35/38

51 Overview Graph Databases Motivation Walk Logic Relations with FO and MSO Relations with CTL and Hybrid CTL Conjunctive Regular Path Queries Open Problems and Conclusion 36/38

52 Open Problems Walk Logic versus Infinite Walk Logic: Infinite walks are the standard in verification logics Can we express the verification logics in Walk Logic? Also interesting: finite CTL versus infinite CTL Complexity bounds on model checking for WL: WL model checking is decidable Current approach has horrible complexity 37/38

53 Conclusion General walk-based reasoning on graphs Relates to practical graph languages Framework for studying expressivity 38/38