Islands of tractability in ontology-based data access

Transcription

1 Islands of tractability in ontology-based data access Michael Zakharyaschev Department of Computer Science and Information Systems, Birkbeck, University of London supported by EPSRC grants ExODA EP/H05099X and itract EP/M012670

2 Data access in industry (from Norwegian Petroleum Directorate s FactPages) show me the wellbores completed before 2008 where Statoil as a drilling operator sampled less than 10 meters of cores 5 days later: SELECT DISTINCT cores.wlbname, cores.lenghtm, wellbore.wlbdrillingoperator, wellbore.wlbcompletionyear FROM ( (SELECT wlbname, wlbnpdidwellbore, (wlbtotalcorelength * ) AS lenghtm FROM wellbore core WHERE wlbcoreintervaluom = [ft ] ) UNION (SELECT wlbname, wlbnpdidwellbore, wlbtotalcorelength AS lenghtm FROM wellbore core WHERE wlbcoreintervaluom = [m ] ) ) as cores, ( (SELECT wlbnpdidwellbore, wlbdrillingoperator, wlbcompletionyear FROM wellbore development all UNION (SELECT wlbnpdidwellbore, wlbdrillingoperator, wlbcompletionyear FROM wellbore exploration all ) UNION 2,000 tables (SELECT wlbnpdidwellbore, wlbdrillingoperator, wlbcompletionyear FROM wellbore shallow all ) ) as wellbore WHERE wellbore.wlbnpdidwellbore = cores.wlbnpdidwellbore... In STATOIL: 1,000 TB of relational data different schemas 30 70% of time on data gathering UCL

3 Ontology-based data access (OBDA) (the Romans 2007) SELECT DISTINCT?unit?well WHERE { query [] npdv:stratumforwellbore?wellboreuri ; npdv:inlithostratigraphicunit [ npdv:name?unit ].?wellboreuri npdv:name?well.?core a npdv:wellborecore ; npdv:coreforwellbore?wellboreuri. } ProductionWellbore Wellbore coreforwellbore WellboreCore [] rdf:type rr:triplesmap; rr:logicaltable "select * from wellbore core"; rr:subjectmap [ a rr:termmap; rr:template "&npd-v2;wellbore/{wlbnpdidwellbore}/";]; rr:propertyobjectmap [ rr:property npdv:coreintervalbottom; rr:column "wlbcoreintervalbottom" ];... mappings stratumforwellbore WellboreStratum ontology CREATE TABLE wellbore core ( wlbname varchar(60) NOT NULL, wlbcorenumber int(11) NOT NULL, wlbcoreintervaltop decimal(13,6),... ) A B C D data sources Ontology gives a high-level conceptual view of the data provides a convenient & natural vocabulary for user queries enriches incomplete data with background knowledge UCL

4 OBDA via FO-rewriting query q + rewriting q + unfolding ontology T npdv:moveablefacility npdv:facility... mapping npdv:moveablefacility (URI( &npdv;facility/{},t7)) :- facility moveable(t1,...,t6,t7,t8,...,t10)... canonical model derived triples virtual ABox A + + triples database n-ary relations for all A and a, T, A = q( a) I A = q ( a) reduction to DB query evaluation UCL

5 OWL 2 QL profile of OWL 2 (W3C 2012) Roles ϱ(x, y) ::= P (x, y) P (y, x) R ::= P P Basic concepts τ (x) ::= A(x) y ϱ(x, y) B ::= A R TBoxes x ( τ (x) τ (x) ) B B x, y ( ϱ(x, y) ϱ (x, y) ) R R x ϱ(x, x) x ( τ (x) τ (x) ) x, y ( ϱ(x, y) ϱ (x, y) ) x ( ϱ(x, x) ) R is reflexive B B R R R is irreflexive Sugar x ( τ (x) y (ϱ 1 (x, y) ϱ k (x, y) τ (y)) ) B R.B ABoxes {A(a), P (a, b),...} (expressible via additional role inclusions) based on the DL-Lite family designed by the Romans ( 2005) and extended by Artale, Calvanese, Kontchakov & Z (2007 9) UCL

6 Example Staff ontology T x ( ProjectManager(x) y (isassistedby(x, y) PA(y)) ) x ( y managesproject(x, y) ProjectManager(x) ) x ( ProjectManager(x) Staff(x) ) x ( PA(x) Secretary(x) ) User query q: find the staff assisted by secretaries q(x) = y (Staff(x) isassistedby(x, y) Secretary(y))) PE-rewriting of ontology-mediated query (T, q) q (x) = y [ Staff(x) isassistedby(x, y) (Secretary(y) PA(y)) ] ProjectManager(x) z managesproject(x, z) UCL

7 Why are OWL 2 QL OMQs FO-rewritable? Canonical model (chase) C T,A of a given consistent (T, A) homomorphically embeddable into every model of (T, A) T, A = q C T,A = q for any CQ q Example: T = {A R. R.B, B S.B} A = {A(a)} C T,A a A R R B S B S B all Horn DLs have canonical models but OMQ ({ R.A A}, A(x)) is not FO-rewritable (recursive datalog needed) Bounded depth derivation property: there is a function f such that T, A = q C N T,A = q with CN T,A constructed in N = f( T, q ) steps FO-rewritability f is polynomial for OWL 2 QL UCL

8 What is the price of OBDA? reduction to DB query evaluation could be too expensive OBDA would not be viable 1 what is the size of rewritings? depending on the type of OMQs depending on the type of rewritings new research (succinctness) problem 2 what is the combined complexity of OMQ answering? depending on the type of OMQs well-known problem in DB theory it may turn out that reduction to DB query evaluation is not most optimal way of OMQ answering UCL

9 Tree-witness rewriting of OMQ Q = (T, q) q q t 2 h C τ 2(a 2 ) T q t 1 h C τ 1 (a 1 ) T C T,A q tw ( x) = Θ independent set of tree witnesses ( y S( z) S( z) q\q Θ t Θ tw t ) Θ is independent if q t q t =, for any distinct t, t Θ UCL

10 The number of tree witnesses B q(x 1, x 2, x 3 ) B a C T,{A(a)} x 1 x 2 x 3 A exponentially-many tree witnesses huge tw-rewriting however, it can be simplified to a polynomial-size PE-rewriting: q(x 1, x 2, x 3 ) z [ A(z) n ( i=1 (xi = z) y (R(y, x i ) R(y, z)) )] can we always do this? UCL

11 Circuit complexity P/poly: the class of problems decidable by polynomial-size circuit families P P/poly if NP P/poly then P NP almost all Boolean functions with n inputs require circuits of size Θ(2 n /n) (Shannon 1949) are there complex Boolean functions f n in NP? (known lower bound: 5n o(n)) nobody knows, but... UCL

12 Monotone circuit complexity (Razborov, Raz, et al. 1985) Boolean variables e ij give graph G = (V, E): V = {1,..., n}, E = { {i, j} e ij = 1 } CLIQUE n,k ( e) = 1 iff G contains a k-clique (e.g., for k n 1/4 ) monotone circuits: exp (2 ε k ) monotone formulas: exp formulas with : superpoly unless NP P/poly MATCHING n ( e) = 1 iff the bipartite graph e with n vertices in each part has a perfect matching (subset of edges containing every node once) monotone formulas: formulas with : exp poly UCL

13 Tree-witness rewriting as a Boolean function OMQ Q = (T, q) a hypergraph H Q = (V, E) where vertices V = atoms of q hyperedges E = tree witnesses q t monotone Boolean hypergraph function for Q (or hypergraph H Q ) f Q = E E independent ( v V \V E p v e E p e (some tweaks required in case of exponentially-many tree witnesses) Boolean formula ϕ for f Q FO-rewriting of size O( ϕ Q ) monotone Boolean formula ϕ for f Q PE-rewriting monotone Boolean circuit ϕ for f Q NDL-rewriting (nonrecursive datalog) tool for obtaining upper succinctness and complexity bounds ) using classical circuit complexity UCL

14 Tool for lower bounds For any OMQ Q = (T, q) and assignment α : predicates(q) {0, 1}, A α = {A(a) α(a) = 1} {P (a, a) α(p ) = 1} ABox with a single individual a Primitive evaluation function: g Q (α) = 1 T, A α = q( a) FO-rewriting q of Q Boolean formula for g Q of size O( q ) PE-rewriting q of Q monotone Boolean formula for g Q NDL-rewriting q of Q monotone Boolean circuit for g Q (proof by quantifier elimination) tool for obtaining lower succinctness bounds using classical circuit complexity UCL

15 Case study: OMQs with ontologies of depth 1 no axioms such as A P, P R depth 1 depth 2 b a b a A A Q = (T, q) with T of depth 1 hypergraph H Q is of degree 2 each vertex belongs to 2 hyperedges hypergraph H of degree 2 OMQ Q H with T of depth 1 and H = H QH What can hypergraph functions of degree 2 compute? UCL

16 Hypergraph programs (HGPs) An HGP is a hypergraph H = (V, E) with every vertex labelled by 0, 1, p i or p i computes f: f( α) = 1 there is an independent E E covering all zeros (contains all vertices whose labels evaluate to 0 under α) monotone if no p i among the labels Any monotone HGP based on H computes a sub-function of f H HGPs based on hypergraphs of degree 2 are polynomially equivalent to nondeterministic branching programs (NBPs) e f p 3 p s t HGP 2 = NBP = NL/poly p 1 p 6 d functions computable by NLogSpace TMs with polynomial advice functions (non-uniform analogue of NLOGSPACE) UCL

17 Rewritings for OMQs with ontologies of depth 1 HGP 2 = NL/poly P/poly (for monotone functions) polynomial-size NDL-rewritings there is a monotone f computable by a polynomial-size NBP, but Ω(log n) any monotone Boolean formula computing f is of size n OMQ with superpolynomial PE-rewritings only all OMQs have polynomial FO-rewritings NC 1 = NL/poly all OMQs with CQs of bounded treewidth have polynomial PE-rewritings all tree-shaped OMQs have polynomial-size Π 4 -rewritings ( ) (SPARQL queries under OWL 2 QL entailment regime) UCL

18 poly NDL, but no poly PE polyndl, FO iff NL/poly but no poly NC 1 PE Succinctness landscape for OMQ rewritings Number of leaves Treewidth arb btw... tw 2 trees l... 2 poly PE, NDL, & FO no poly PE or NDL poly FO iff NP/poly NC 1 poly NDL, but no poly PE poly FO iff LOGCFL/poly NC 1 poly NDL, but no poly PE poly FO iff NL/poly NC 1... TBox depth d no poly FO unless NP/poly NC 1 arb Bienvenu, Kikot, Kontchakov, Podolskii, Z UCL

19 Combined complexity landscape Number of leaves Treewidth arb btw... tw 2 trees l... 2 NP-complete LOGCFL-complete NL-complete TBox depth d NP-complete LOGCFL-c arb CQ evaluation over databases is NP-complete L NL LOGCFL NC 2 P NP bounded treewidth CQ evaluation is LOGCFL-complete (logspace reducible to a CFL) Gottlob et al UCL

20 query + rewriting + unfolding ontology npdv:moveablefacility npdv:facility... compile mapping npdv:moveablefacility... + SQO (URI( &npdv;facility/{},t7)) :- facility moveable(t1,...,t6,t7,t8,...,t10) canonical model derived triples virtual ABox + + triples database n-ary relations ABox constraints integrity constraints Rodriguez-Muro, Calvanese, Kontchakov, Rezk, Xiao, Z UCL

21 Ontop in practice T-mappings compile (big parts of) OWL 2 QL ontologies into mappings (domain and range constraints, concept and role hierarchies) can be optimised offline few tree witnesses in real-world OBDA polynomial-size rewritings database constraints and SQO significantly simplify T-mappings efficient SQL queries over the data some important conceptual modelling constructs are missing in OWL 2 QL A B C R.A B owl:sameas? islands of OMQ rewritability & succinctness for expressive languages UCL

22 itract: Islands of Tractability in Ontology-Based Data Access EPSRC UK project: (i) establish a novel, OMQ-centric approach to OBDA aiming to identify islands of tractable OMQs in rich ontology and query languages (ii) develop uniformly efficient OMQ answering techniques for the identified islands (iii) implement and test these techniques in practice, using state-of-the-art OBDA systems Team: London: MZ (PI), S Kikot (RA), R Kontchakov (co-i), I Razgon (co-i) Liverpool: F Wolter (PI), F Papacchini (RA), A Hernich, B Konev Project partners: University of Bozen-Bolzano (Diego Calvanese) University of Bremen (Carsten Lutz) University of Oslo (Arild Waaler) IBM Watson, New York (Mariano Rodriguez-Muro) UCL