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URDF Query-Time Reasoning in Uncertain RDF Knowledge Bases

URDF Query-Time Reasoning in Uncertain RDF Knowledge Bases . Ndapandula Nakashole Mauro Sozio Fabian Suchanek Martin Theobald. Information Extraction. YAGO/DBpedia et al. bornOn(Jeff, 09/22/42). gradFrom(Jeff, Columbia). hasAdvisor(Jeff, Arthur). hasAdvisor(Surajit, Jeff).

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URDF Query-Time Reasoning in Uncertain RDF Knowledge Bases

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  1. URDFQuery-Time Reasoning in Uncertain RDF Knowledge Bases Ndapandula Nakashole Mauro Sozio Fabian Suchanek Martin Theobald

  2. Information Extraction YAGO/DBpedia et al. bornOn(Jeff, 09/22/42) gradFrom(Jeff, Columbia) hasAdvisor(Jeff, Arthur) hasAdvisor(Surajit, Jeff) knownFor(Jeff, Theory) >120 M facts for YAGO2 (mostly from Wikipedia infoboxes) type(Jeff, Author)[0.9] author(Jeff, Drag_Book)[0.8] New fact candidates author(Jeff,Cind_Book)[0.6] worksAt(Jeff, Bell_Labs)[0.7] type(Jeff, CEO)[0.4] 100’s M additional facts from Wikipedia text

  3. Outline • Motivation & Problem Setting • URDF running example: people graduating from universities • Efficient MAP Inference • MaxSAT solving with soft & hard constraints • Grounding • Deductive grounding of soft rules (SLD resolution) • Iterative grounding of hard rules (closure) • MaxSAT Algorithm • MaxSAT algorithm in 3 steps • Experiments & Future Work Query-Time Reasoning in Uncertain RDF Knowledge Bases

  4. URDF: Uncertain RDF Data Model • Extensional Layer (information extraction & integration) • High-confidence facts:existing knowledge base (“ground truth”) • New fact candidates: extracted facts with confidence values • Integration of different knowledge sources: Ontology merging or explicit Linked Data(owl:sameAs, owl:equivProp.) Large “Uncertain Database” of RDFfacts • Intensional Layer (query-time inference) • Soft rules:deductive grounding & lineage (Datalog/SLD resolution) • Hard rules:consistency constraints (more general FOL rules) • Propositional&probabilisticconsistency reasoning Query-Time Reasoning in Uncertain RDF Knowledge Bases

  5. Soft Rules vs. Hard Rules (Soft) Deduction Rules vs. (Hard) Consistency Constraints • People may live inmore than one place livesIn(x,y)  marriedTo(x,z)  livesIn(z,y) livesIn(x,y)  hasChild(x,z)  livesIn(z,y) • People are not born indifferent places/on different dates bornIn(x,y)  bornIn(x,z)  y=z • People are not married to more than one person (at the same time, in most countries?) marriedTo(x,y,t1)  marriedTo(x,z,t2)  y≠z  disjoint(t1,t2) [0.8] [0.5] Query-Time Reasoning in Uncertain RDF Knowledge Bases

  6. Soft Rules vs. Hard Rules Rule-based (deductive) reasoning: Datalog, RDF/S, OWL2-RL, etc. (Soft) Deduction Rules vs. (Hard) Consistency Constraints • People may live inmore than one place livesIn(x,y)  marriedTo(x,z)  livesIn(z,y) livesIn(x,y)  hasChild(x,z)  livesIn(z,y) • People are not born indifferent places/on different dates bornIn(x,y)  bornIn(x,z)  y=z • People are not married to more than one person (at the same time, in most countries?) marriedTo(x,y,t1)  marriedTo(x,z,t2)  y≠z  disjoint(t1,t2) [0.8] [0.5] FOL constraints (in particular mutex): Datalog with constraints, X-tuples in Prob. DB’s owl:FunctionalProperty, etc. Query-Time Reasoning in Uncertain RDF Knowledge Bases

  7. URDF Running Example KB:RDF Base Facts First-OrderRules hasAdvisor(x,y)  worksAt(y,z) graduatedFrom(x,z) [0.4] graduatedFrom(x,y)  graduatedFrom(x,z)  y=z Computer Scientist type[1.0] type[1.0] type[1.0] hasAdvisor[0.7] hasAdvisor[0.8] Jeff Surajit David graduatedFrom[0.9] graduatedFrom[?] graduatedFrom[0.6] graduatedFrom[?] graduatedFrom[?] graduatedFrom[?] graduatedFrom[0.7] Stanford Princeton • Derived Facts • gradFrom(Surajit,Stanford) • gradFrom(David,Stanford) worksAt[0.9] type[1.0] type[1.0] University Query-Time Reasoning in Uncertain RDF Knowledge Bases

  8. Basic Types of Inference • Maximum-A-Posteriori (MAP) Inference • Find the most likely assignment to query variables y under a given evidence x. • Compute: argmaxy P( y | x)(NP-hard for propositional formulas, e.g., MaxSAT over CNFs) • Marginal/Success Probabilities • Probability that query y is true in a random world under a given evidence x. • Compute: ∑y P( y | x)(#P-hard for propositional formulas) Query-Time Reasoning in Uncertain RDF Knowledge Bases

  9. General Route: Grounding & MaxSAT Solving Query graduatedFrom(x, y) • 1) Grounding • Consider only facts (and rules)which are relevant for answering the query • 2) Propositional formula in CNF, consisting of • Grounded hard & soft rules • Uncertain base facts • 3) Propositional Reasoning • Find truth assignment to facts such that the total weight of the satisfied clauses is maximized  MAP inference: compute “most likely”possible world CNF (graduatedFrom(Surajit, Stanford) graduatedFrom(Surajit, Princeton)) (graduatedFrom(David, Stanford) graduatedFrom(David, Princeton))  (hasAdvisor(Surajit, Jeff) worksAt(Jeff, Stanford) graduatedFrom(Surajit, Stanford))  (hasAdvisor(David, Jeff) worksAt(Jeff, Stanford) graduatedFrom(David, Stanford)) worksAt(Jeff, Stanford) hasAdvisor(Surajit, Jeff) hasAdvisor(David, Jeff) graduatedFrom(Surajit, Princeton) graduatedFrom(Surajit, Stanford)  graduatedFrom(David, Princeton) 1000 1000 0.4 0.4 0.9 0.8 0.7 0.6 0.7 0.9 Query-Time Reasoning in Uncertain RDF Knowledge Bases

  10. Why are high weights for hard rules not enough? CNF (graduatedFrom(Surajit, Stanford) graduatedFrom(Surajit, Princeton)) graduatedFrom(Surajit, Princeton) graduatedFrom(Surajit, Stanford) • Consider the following CNF (for A,B > 0, A >> B) • The optimal solution has weight A+B • The next-best solution has weight A+0 • Hence the ratio of the optimal over the approximate solution is A+B / A • In general, any (1+) approximation algorithm, with > 0, may set graduatedFrom(Surajit, Princeton)to true, as A+B /A 1 for A  . A 0 B Query-Time Reasoning in Uncertain RDF Knowledge Bases

  11. URDF: MaxSAT Solving with Soft & Hard Rules Special case:Horn-clauses as soft rules & mutex-constraintsas hard rules • Find:argmaxy P( y | x) • Resolves to a variant of MaxSAT for propositional formulas 0.4 0.4 0.9 0.8 0.7 0.6 0.7 0.9 { graduatedFrom(Surajit, Stanford), graduatedFrom(Surajit, Princeton) } { graduatedFrom(David, Stanford), graduatedFrom(David, Princeton) } S: Mutex-const. MaxSAT Alg. Compute W0 = ∑clauses C w(C) P(C is satisfied); For each hard constraint S { For each fact f in St { Compute Wf+t = ∑clauses C w(C) P(C is sat. | f = true); } Compute WS-t= ∑clauses C w(C) P(C is sat. | St= false); Choose truth assignment to f in St that maximizes Wf+t , WS-t ; Remove satisfied clauses C; t++; } (hasAdvisor(Surajit, Jeff) worksAt(Jeff, Stanford) graduatedFrom(Surajit, Stanford))  (hasAdvisor(David, Jeff) worksAt(Jeff, Stanford) graduatedFrom(David, Stanford)) worksAt(Jeff, Stanford) hasAdvisor(Surajit, Jeff) hasAdvisor(David, Jeff) graduatedFrom(Surajit, Princeton) graduatedFrom(Surajit, Stanford)  graduatedFrom(David, Princeton) C: Weighted Horn clauses (CNF) • Runtime: O(|S||C|) • Approximation guarantee of 1/2 Query-Time Reasoning in Uncertain RDF Knowledge Bases

  12. Deductive Grounding Algorithm (SLD Resolution/Datalog) First-Order Rules hasAdvisor(x,y)  worksAt(y,z) graduatedFrom(x,z) [0.4] graduatedFrom(x,y)  graduatedFrom(x,z)  y=z Query graduatedFrom(Surajit, y) • graduatedFrom • (Surajit, Princeton) • graduatedFrom • (Surajit, Stanford)  /\ Base Facts • graduatedFrom(Surajit, Princeton) [0.7] • graduatedFrom(Surajit, Stanford) [0.6] • graduatedFrom(David, Princeton) [0.9] • hasAdvisor(Surajit, Jeff) [0.8] hasAdvisor(David, Jeff) [0.7] worksAt(Jeff, Stanford) [0.9] • type(Princeton, University) [1.0] type(Stanford, University) [1.0] • type(Jeff, Computer_Scientist) [1.0] • type(Surajit, Computer_Scientist) [1.0] • type(David, Computer_Scientist) [1.0] • hasAdvisor • (Surajit,Jeff) • worksAt • (Jeff,Stanford) Grounded Rules • hasAdvisor(Surajit, Jeff)  • worksAt(Jeff, Stanford)  • gradFrom(Surajit, Stanford) • gradFrom(Surajit, Stanford) • gradFrom(Surajit, Princeton) Query-Time Reasoning in Uncertain RDF Knowledge Bases

  13. Dependency Graph of a Query • SLD grounding always starts from a query literal and first pursues over the soft deduction rules. • Grounding is also iterated over the hard rules in a top-down fashion by using the literals in each hard rule as new subqueries. • Cycles (due to recursive rules) are detected and resolved via a form of tabling known from Datalog. • Grounding terminates when a closure is reached, i.e., when no new facts can be grounded from the rules and all subgoals are either resolved or form the root of a cycle. Query-Time Reasoning in Uncertain RDF Knowledge Bases

  14. Weighted MaxSAT Algorithm General idea Compute a potential function Wt that iterates over all hard rules St and set the fact f  Stthat maximizes Wt(or none of them) to true;set all other facts in St to false. • At iteration 0, we have • At any intermediate iteration t, we compare • At the final iteration t_max, all facts are assigned either true or false. • Wt_max is equal to the total weight of all clauses that are satisfied. Query-Time Reasoning in Uncertain RDF Knowledge Bases

  15. Step 1 { gradFrom(Surajit, Stanford), gradFrom(Surajit, Princeton) } { gradFrom(David, Stanford), gradFrom(David, Princeton) } S: Mutex-const. • Weights w(fi) and probabilities pi (hasAdvisor(Surajit, Jeff) worksAt(Jeff, Stanford) gradFrom(Surajit, Stanford))0.4  (hasAdvisor(David, Jeff) worksAt(Jeff, Stanford) gradFrom(David, Stanford))0.4 worksAt(Jeff, Stanford) 0.9 hasAdvisor(Surajit, Jeff) 0.8 hasAdvisor(David, Jeff) 0.7 gradFrom(Surajit, Princeton) 0.6 gradFrom(Surajit, Stanford)0.7  gradFrom(David, Princeton) 0.9 C: Weighted Horn clauses (CNF) Query-Time Reasoning in Uncertain RDF Knowledge Bases

  16. Step 2 { gradFrom(Surajit, Stanford), gradFrom(Surajit, Princeton) } { gradFrom(David, Stanford), gradFrom(David, Princeton) } S: Mutex-const. • Weights w(fi) and probabilities pi (hasAdvisor(Surajit, Jeff) worksAt(Jeff, Stanford) gradFrom(Surajit, Stanford))0.4  (hasAdvisor(David, Jeff) worksAt(Jeff, Stanford) gradFrom(David, Stanford))0.4 worksAt(Jeff, Stanford) 0.9 hasAdvisor(Surajit, Jeff) 0.8 hasAdvisor(David, Jeff) 0.7 gradFrom(Surajit, Princeton) 0.6 gradFrom(Surajit, Stanford)0.7  gradFrom(David, Princeton) 0.9 C: Weighted Horn clauses (CNF) Query-Time Reasoning in Uncertain RDF Knowledge Bases

  17. Step 2 { gradFrom(Surajit, Stanford), gradFrom(Surajit, Princeton) } { gradFrom(David, Stanford), gradFrom(David, Princeton) } S: Mutex-const. • Weights w(fi) and probabilities pi (hasAdvisor(Surajit, Jeff) worksAt(Jeff, Stanford) gradFrom(Surajit, Stanford))0.4  (hasAdvisor(David, Jeff) worksAt(Jeff, Stanford) gradFrom(David, Stanford))0.4 worksAt(Jeff, Stanford) 0.9 hasAdvisor(Surajit, Jeff) 0.8 hasAdvisor(David, Jeff) 0.7 gradFrom(Surajit, Princeton) 0.6 gradFrom(Surajit, Stanford)0.7  gradFrom(David, Princeton) 0.9 C1: hasAdvisor(Surajit, Jeff) worksAt(Jeff, Stanford) gradFrom(Surajit, Stanford) P(C1) = 1 – (1-(1-1))(1-(1-1))(1-1) = 1 single partition, negated: 1 - pi C: Weighted Horn clauses (CNF) single partition, negated: 1 - pi single partition, positive: pi Query-Time Reasoning in Uncertain RDF Knowledge Bases

  18. Step 2 { gradFrom(Surajit, Stanford), gradFrom(Surajit, Princeton) } { gradFrom(David, Stanford), gradFrom(David, Princeton) } S: Mutex-const. • Weights w(fi) and probabilities pi P(C1 is satisfied) = 1-(1-(1-1))(1-(1-1))(1-1) = 1 P(C2is satisfied) = 1-(1-(1-1))(1-(1-1))(1-0) = 0 ... (hasAdvisor(Surajit, Jeff) worksAt(Jeff, Stanford) gradFrom(Surajit, Stanford))0.4  (hasAdvisor(David, Jeff) worksAt(Jeff, Stanford) gradFrom(David, Stanford))0.4 worksAt(Jeff, Stanford) 0.9 hasAdvisor(Surajit, Jeff) 0.8 hasAdvisor(David, Jeff) 0.7 gradFrom(Surajit, Princeton) 0.6 gradFrom(Surajit, Stanford)0.7  gradFrom(David, Princeton) 0.9 C: Weighted Horn clauses (CNF) • W0 = 0.4 + 0.9 + 0.8 + 0.7 + 0.6 + 0.7 + 0.9 = 5.0 Query-Time Reasoning in Uncertain RDF Knowledge Bases

  19. Step 3 { gradFrom(Surajit, Stanford), gradFrom(Surajit, Princeton) } { gradFrom(David, Stanford), gradFrom(David, Princeton) } S: Mutex-const. • Weights w(fi), probabilities pi, truth values P(C1 is satisfied | f1=true) = 1-(1-(1-1))(1-(1-1))(1-1) = 1 P(C1is satisfied | f2=true) = 1-(1-(1-1))(1-(1-1))(1-0) = 0 ... (hasAdvisor(Surajit, Jeff) worksAt(Jeff, Stanford) gradFrom(Surajit, Stanford))0.4  (hasAdvisor(David, Jeff) worksAt(Jeff, Stanford) gradFrom(David, Stanford))0.4 worksAt(Jeff, Stanford) 0.9 hasAdvisor(Surajit, Jeff) 0.8 hasAdvisor(David, Jeff) 0.7 gradFrom(Surajit, Princeton) 0.6 gradFrom(Surajit, Stanford)0.7  gradFrom(David, Princeton) 0.9 C: Weighted Horn clauses (CNF) • W1 = 0.4 + 0.4 + 0.9 + 0.8 + 0.7 + 0.7 + 0.9 = 4.8 • W2 = 0.4 + 0.9 + 0.8 + 0.7 + 0.7 + 0.9 = 4.4 Query-Time Reasoning in Uncertain RDF Knowledge Bases

  20. Experiments – Setup • YAGO Knowledge Base • 2 Mio entities,20 Mio facts • Soft Rules • 16 soft rules (hand-crafted deduction rules with weights) • Hard Rules • 5 predicates with functional properties (bornIn, diedIn, bornOnDate, diedOnDate, marriedTo) • Queries • 10 conjunctive SPARQL queries • Markov Logic as Competitor (based on MCMC) • MAP inference: Alchemy employs a form of MaxWalkSAT • MC-SAT: Iterative MaxSAT & Gibbs sampling Query-Time Reasoning in Uncertain RDF Knowledge Bases

  21. YAGO Knowledge Base: URDF vs. Markov Logic • First run: ground each query against the rules • (SLD grounding + MaxSAT solving) & report sum of runtimes • Asymptotic runtime checks: synthetic soft rule expansions URDF: SLD grounding & MaxSat solving • URDF vs. Markov Logic • (MAP inference & MC-SAT) • |C| - # ground literals in soft rules • |S| - # ground literals in hard rules Query-Time Reasoning in Uncertain RDF Knowledge Bases

  22. Recursive Rules & LUBM Benchmark • 42 inductively learned (partly recursive) rules over 20 Mio facts in YAGO • URDF grounding with different maximum SLD levels • URDF (SLD grounding + MaxSAT) vs. Jena (only grounding) over the LUBM benchmark • SF-1: 103,397 triplets • SF-5: 646,128 triplets • SF-10: 1,316,993 triplets Query-Time Reasoning in Uncertain RDF Knowledge Bases

  23. Current & Future Topics... • Temporal consistency reasoning • Soft/hard ruleswith temporal predicates • Soft deduction rules: deduce confidence distribution of derived facts • Learning soft rules & consistency constraints • Explore how Inductive Logic Programming can be applied to large, uncertain & incomplete knowledge bases • More solving/sampling • Linear-time constrained&weightedMaxSATsolver • Improved Gibbs sampling with soft & hard rules • Scale-out • Distributed grounding via message passing • Updates/versioning for (linked) RDF data • Non-monotonicanswers for rules with negation! Query-Time Reasoning in Uncertain RDF Knowledge Bases

  24. Online Demo! urdf.mpi-inf.mpg.de Query-Time Reasoning in Uncertain RDF Knowledge Bases

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