URDF Query-Time Reasoning in Uncertain RDF Knowledge Bases Ndapandula Nakashole Mauro Sozio Fabian Suchanek Martin Theobald

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

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 3

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 RDF facts  Intensional Layer (query-time inference)  Soft rules: deductive grounding & lineage (Datalog/SLD resolution)  Hard rules: consistency constraints (more general FOL rules)  Propositional & probabilistic consistency reasoning Query-Time Reasoning in Uncertain RDF Knowledge Bases 4

Soft Rules vs. Hard Rules (Soft) Deduction Rules vs. (Hard) Consistency Constraints PPeople may live in more than one place livesIn(x,y)  marriedTo(x,z)  livesIn(z,y) livesIn(x,y)  hasChild(x,z)  livesIn(z,y) PPeople are not born in different places/on different dates bornIn(x,y)  bornIn(x,z)  y=z PPeople are not married to more than one person (at the same time, in most countries?) marriedTo(x,y,t 1 )  marriedTo(x,z,t 2 )  y≠z  disjoint(t 1,t 2 ) [0.8] [0.5] Query-Time Reasoning in Uncertain RDF Knowledge Bases 5

Soft Rules vs. Hard Rules (Soft) Deduction Rules vs. (Hard) Consistency Constraints  People may live in more 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 in different 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,t 1 )  marriedTo(x,z,t 2 )  y≠z  disjoint(t 1,t 2 ) [0.8] [0.5] Query-Time Reasoning in Uncertain RDF Knowledge Bases 6 Rule-based (deductive) reasoning: Datalog, RDF/S, OWL2-RL, etc. FOL constraints (in particular mutex): Datalog with constraints, X-tuples in Prob. DB’s owl:FunctionalProperty, etc. FOL constraints (in particular mutex): Datalog with constraints, X-tuples in Prob. DB’s owl:FunctionalProperty, etc.

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

Basic Types of Inference  Maximum-A-Posteriori (MAP) Inference  Find the most likely assignment to query variables y under a given evidence x.  Compute: arg max y 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 8

9 General Route: Grounding & MaxSAT Solving Query graduatedFrom(x, y) Query graduatedFrom(x, y) 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) 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) ) 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

Why are high weights for hard rules not enough?  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  . Query-Time Reasoning in Uncertain RDF Knowledge Bases 10 CNF (graduatedFrom(Surajit, Stanford)  graduatedFrom(Surajit, Princeton))  graduatedFrom(Surajit, Princeton)  graduatedFrom(Surajit, Stanford) CNF (graduatedFrom(Surajit, Stanford)  graduatedFrom(Surajit, Princeton))  graduatedFrom(Surajit, Princeton)  graduatedFrom(Surajit, Stanford) A0BA0B

 Find: arg max y P( y | x)  Resolves to a variant of MaxSAT for propositional formulas URDF: MaxSAT Solving with Soft & Hard Rules Query-Time Reasoning in Uncertain RDF Knowledge Bases { graduatedFrom(Surajit, Stanford), graduatedFrom(Surajit, Princeton) } { graduatedFrom(David, Stanford), graduatedFrom(David, Princeton) } { 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) (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) S: Mutex-const. Special case: Horn-clauses as soft rules & mutex-constraints as hard rules C: Weighted Horn clauses (CNF) Compute W 0 = ∑ clauses C w(C) P(C is satisfied); For each hard constraint S { For each fact f in S t { Compute W f+ t = ∑ clauses C w(C) P(C is sat. | f = true); } Compute W S- t = ∑ clauses C w(C) P(C is sat. | S t = false); Choose truth assignment to f in S t that maximizes W f+ t, W S- t ; Remove satisfied clauses C; t++; } Compute W 0 = ∑ clauses C w(C) P(C is satisfied); For each hard constraint S { For each fact f in S t { Compute W f+ t = ∑ clauses C w(C) P(C is sat. | f = true); } Compute W S- t = ∑ clauses C w(C) P(C is sat. | S t = false); Choose truth assignment to f in S t that maximizes W f+ t, W S- t ; Remove satisfied clauses C; t++; } Runtime: O(|S||C|) Approximation guarantee of 1/2 Runtime: O(|S||C|) Approximation guarantee of 1/2 11 MaxSAT Alg.

Deductive Grounding Algorithm (SLD Resolution/Datalog)   /\ graduatedFrom (Surajit, Princeton) graduatedFrom (Surajit, Princeton) hasAdvisor (Surajit,Jeff ) hasAdvisor (Surajit,Jeff ) worksAt (Jeff,Stanford ) worksAt (Jeff,Stanford ) graduatedFrom (Surajit, Stanford) graduatedFrom (Surajit, Stanford) Query graduatedFrom(Surajit, y) Query graduatedFrom(Surajit, y) First-Order Rules hasAdvisor(x,y)  worksAt(y,z)  graduatedFrom(x,z) [0.4] graduatedFrom(x,y)  graduatedFrom(x,z)  y=z First-Order Rules hasAdvisor(x,y)  worksAt(y,z)  graduatedFrom(x,z) [0.4] graduatedFrom(x,y)  graduatedFrom(x,z)  y=z 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] 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] Query-Time Reasoning in Uncertain RDF Knowledge Bases 12 Grounded Rules hasAdvisor(Surajit, Jeff)  worksAt(Jeff, Stanford)  gradFrom(Surajit, Stanford) gradFrom(Surajit, Stanford)  gradFrom(Surajit, Princeton) Grounded Rules hasAdvisor(Surajit, Jeff)  worksAt(Jeff, Stanford)  gradFrom(Surajit, Stanford) gradFrom(Surajit, Stanford)  gradFrom(Surajit, Princeton)

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 13

Weighted MaxSAT Algorithm General idea Compute a potential function W t that iterates over all hard rules S t and set the fact f  S t that maximizes W t (or none of them) to true; set all other facts in S t to false. Query-Time Reasoning in Uncertain RDF Knowledge Bases 14  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.  W t_max is equal to the total weight of all clauses that are satisfied.

Step 1  Weights w(f i ) and probabilities p i Query-Time Reasoning in Uncertain RDF Knowledge Bases 15 { gradFrom(Surajit, Stanford), gradFrom(Surajit, Princeton) } { gradFrom(David, Stanford), gradFrom(David, Princeton) } { gradFrom(Surajit, Stanford), gradFrom(Surajit, Princeton) } { gradFrom(David, Stanford), gradFrom(David, Princeton) } (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 (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 S: Mutex-const. 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) } { gradFrom(Surajit, Stanford), gradFrom(Surajit, Princeton) } { gradFrom(David, Stanford), gradFrom(David, Princeton) } S: Mutex-const. C: Weighted Horn clauses (CNF)  Weights w(f i ) and probabilities p i (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 (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

(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 (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  Weights w(f i ) and probabilities p i Query-Time Reasoning in Uncertain RDF Knowledge Bases 17 Step 2 { gradFrom(Surajit, Stanford), gradFrom(Surajit, Princeton) } { gradFrom(David, Stanford), gradFrom(David, Princeton) } { gradFrom(Surajit, Stanford), gradFrom(Surajit, Princeton) } { gradFrom(David, Stanford), gradFrom(David, Princeton) } S: Mutex-const. C: Weighted Horn clauses (CNF) C 1 : hasAdvisor(Surajit, Jeff)  worksAt(Jeff, Stanford)  gradFrom(Surajit, Stanford) P(C 1 ) = 1 – (1-(1-1))(1-(1-1))(1-1) = 1 C1:C1: hasAdvisor(Surajit, Jeff)  worksAt(Jeff, Stanford)  gradFrom(Surajit, Stanford) P(C 1 ) = 1 – (1-(1-1))(1-(1-1))(1-1) = 1 single partition, negated: 1 - p i single partition, positive: p i

Query-Time Reasoning in Uncertain RDF Knowledge Bases 18 Step 2 { gradFrom(Surajit, Stanford), gradFrom(Surajit, Princeton) } { gradFrom(David, Stanford), gradFrom(David, Princeton) } { gradFrom(Surajit, Stanford), gradFrom(Surajit, Princeton) } { gradFrom(David, Stanford), gradFrom(David, Princeton) } S: Mutex-const. C: Weighted Horn clauses (CNF)  Weights w(f i ) and probabilities p i P(C 1 is satisfied) = 1-(1-(1-1))(1-(1-1))(1-1) = 1 P(C 2 is satisfied) = 1-(1-(1-1))(1-(1-1))(1-0) = 0...  W 0 = = 5.0 (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 (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

(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 (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 Query-Time Reasoning in Uncertain RDF Knowledge Bases 19 Step 3 { gradFrom(Surajit, Stanford), gradFrom(Surajit, Princeton) } { gradFrom(David, Stanford), gradFrom(David, Princeton) } { gradFrom(Surajit, Stanford), gradFrom(Surajit, Princeton) } { gradFrom(David, Stanford), gradFrom(David, Princeton) } S: Mutex-const. C: Weighted Horn clauses (CNF)  W 1 = = 4.8  W 2 = = 4.4

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 20

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

Recursive Rules & LUBM Benchmark  42 inductively learned (partly recursive) rules over 20 Mio facts in YAGO  URDF grounding with different maximum SLD levels Query-Time Reasoning in Uncertain RDF Knowledge Bases 22  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

Current & Future Topics...  Temporal consistency reasoning  Soft/hard rules with 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 & weighted MaxSAT solver  Improved Gibbs sampling with soft & hard rules  Scale-out  Distributed grounding via message passing  Updates/versioning for (linked) RDF data  Non-monotonic answers for rules with negation! Query-Time Reasoning in Uncertain RDF Knowledge Bases 23

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