Probabilistic Theorem Proving. Vibhav Gogate (Joint work with Pedro Domingos)

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1 Probabilistic Theorem Proving Vibhav Gogate (Joint work with Pedro Domingos)

2 Progress to Date First-order variable elimination [Poole, 2003; de Salvo Braz, 2007; etc.] Lifted belief propagation [Singla & Domingos, 2008; etc.] Hot topic in UAI, IJCAI, etc.

3 Limitations FOVE does not scale Lifted BP cannot handle determinism, etc. Neither exploits logical structure Unclear relation to logical inference

4 Probabilistic Theorem Proving addresses all of these limitations. Probabilistic Theorem Proving is a single inference procedure with graphical model inference and theorem proving as special cases.

5 Probabilistic Theorem Proving Given Probabilistic knowledge base K Query formula Q Output P(Q K)

6 Propositional Logic Atoms: Symbols representing propositions Logical connectives:, Λ, V, etc. Knowledge base: Set of formulas Every KB can be converted to CNF CNF: Conjunction of clauses Clause: Disjunction of literals Literal: Atom or its negation Entailment: Does KB entail query?

7 Theorem Proving TP(KB, Query) KB Q KB U { Query} return SAT(CNF(KB Q ))

8 Satisfiability (DPLL) SAT(CNF) if CNF is empty return True if CNF contains empty clause return False choose an atom A return SAT(CNF(A)) V SAT(CNF( A))

9 First-Order Logic Atom: Predicate(Variables,Constants) E.g.: Ground atom: All arguments are constants Quantifiers: Friends ( Anna, x), This talk: Finite, Herbrand interpretations

10 First-Order Theorem Proving Propositionalization 1. Form all possible ground atoms 2. Apply propositional theorem prover Lifted Inference: Resolution Resolve pairs of clauses until empty clause derived x Bob Friends ( Anna, Bob) Unify literals by substitution, e.g.: unifies Friends ( Anna, x) and Friends ( Anna, x) Happy( x) Friends ( Anna, Bob) Happy( Bob)

11 Probabilistic Knowledge Bases PKB = Set of formulas and their probabilities + Consistency + Maximum entropy = Set of formulas and their weights (a.k.a. Markov logic network) = Set of formulas and their potentials i (1 if formula true, if formula false) 1 P( world ) Z i n i i ( world)

12 Weighted Model Counting ModelCount(CNF) = # worlds that satisfy CNF Assign a weight to each literal Weight(world) = П weights(true literals) Weighted model counting: Given CNF C and literal weights W Output Σ weights(worlds that satisfy C) PTP is reducible to lifted WMC

13 Inference Problems

14 Propositional Case All conditional probabilities are ratios of partition functions: P( Query PKB) worlds ( world ) Z( PKB) All partition functions can be computed by weighted model counting 1 Query Z( PKB {( Query,0)}) Z( PKB) i i ( world )

15 Conversion to CNF + Weights WCNF(PKB) for all (F i, Φ i ) є PKB s.t. Φ i > 0 do PKB PKB U {(F i A i, 0)} \ {(F i, Φ i )} CNF CNF(PKB) for all A i literals do W Ai Φ i for all other literals L do w L 1 return (CNF, weights)

16 Probabilistic Theorem Proving PTP(PKB, Query) PKB Q PKB U {(Query,0)} return WMC(WCNF(PKB Q )) / WMC(WCNF(PKB))

17 Probabilistic Theorem Proving PTP(PKB, Query) PKB Q PKB U {(Query,0)} return WMC(WCNF(PKB Q )) / WMC(WCNF(PKB)) Compare: TP(KB, Query) KB Q KB U { Query} return SAT(CNF(KB Q ))

18 Weighted Model Counting WMC(CNF, weights) if all clauses in CNF are satisfied return Base Case ( w w ) A A( CNF) A A if CNF has empty unsatisfied clause return 0

19 Weighted Model Counting WMC(CNF, weights) if all clauses in CNF are satisfied return ( w w A A( CNF) A A if CNF has empty unsatisfied clause return 0 if CNF can be partitioned into CNFs C 1,, C k sharing no atoms Decomp. k return i 1WMC( C i, weights ) Step )

20 Weighted Model Counting WMC(CNF, weights) if all clauses in CNF are satisfied return ( w w A A( CNF) A A if CNF has empty unsatisfied clause return 0 if CNF can be partitioned into CNFs C 1,, C k sharing no atoms i k return 1WMC( C i, weights ) choose an atom A Splitting return w WMC( CNF A, weights ) Step A WMC( CNF A, weights ) w A )

21 First-Order Case PTP schema remains the same Conversion of PKB to hard CNF and weights: New atom in F i A i is now Predicate i (variables in F i, constants in F i ) Lift each step of WMC

22 Lifted Weighted Model Counting LWMC(CNF, substs, weights) if all clauses in CNF are satisfied Base return ( w w ) na ( substs) A A( CNF) A A Case if CNF has empty unsatisfied clause return 0

23 Lifted Weighted Model Counting LWMC(CNF, substs, weights) if all clauses in CNF are satisfied return if CNF has empty unsatisfied clause return 0 if there exists a lifted decomposition of CNF return A ( w w ) n A( CNF) A A k i 1 [ ( CNFi, 1 A ( substs) LWMC, weights)] m i Decomp. Step

24 Lifted Weighted Model Counting LWMC(CNF, substs, weights) if all clauses in CNF are satisfied return if CNF has empty unsatisfied clause return 0 if there exists a lifted decomposition of CNF return choose an atom A return l i A ( w w ) n A( CNF) A A k i 1 [ ( CNFi, 1 A ( substs) LWMC, weights)] Splitting Step ti fi 1 niwaw ALWMC( CNF j, weights) m i

25 Lifting the Decomposition and Splitting Steps: Details We will assume Normal forms Step 1: Convert to Normal form if not in one

26 Lifting the Decomposition Step Propositional Decomposition: Independent Lifted Decomposition: Independent and Identical (symmetry) How to find independent and identical decompositions by just looking at the logical structure? POWER RULE (Jha-Gogate-Meliou-Suciu, 2010)

27 POWER RULE Decomposer: Given a predicate R, a logical variable x at position i in R is a decomposer if by grounding x to any two constants A and B in its domain yields a decomposable WCNF having two components such that Both components are identical subject to renaming of predicates and constants. R(x) v S(x); S(y) v T(y) --- x is a decomposer R(x) v S(x); S(x) v T(y) --- No decomposer

28 How to use the decomposer? Substitute x with a constant in the domain Return [LWMC (x\a)] n Where n is the number of constants in the domain of x

29 Lifting the Conditioning or Splitting Step Singleton Rule If an atom R is singleton, namely if the predicate symbol contains only one logical variable then splitting on it creates exactly n+1 distinct WCNFs. Moreover, each of these WCNFs appear exactly n choose k times where 0<=k<=n. Examples: R(x) v T(y) R(x) v S(x); S(x) v T(y)

30 How to use the Singleton Rule Split the WCNF into n+1 WCNFs In the i-th WCNF For each clause in which R appears Replace the domain size of all variables which are shared with R by n-i if the literal of R is positive and by i if the literal of R is negative Keep other domain sizes the same Multiply the result by n choose i

31 Extensions Unit propagation, etc. Caching / Memoization Knowledge-based model construction

32 Formal properties PTP can be exponentially more efficient than FOVE (first-order variable elimination) PTP has the same worst-case time and space complexity as FOVE

33 Approximate Inference WMC(CNF, weights) if all clauses in CNF are satisfied return A A( CNF) ( wa w A) if CNF has empty unsatisfied clause return 0 if CNF can be partitioned into CNFs C 1,, C k sharing no atoms k return i 1WMC( C i, weights ) Splitting choose an atom A Step w return A WMC( CNF A, weights ) Q( A CNF, weights ) with probability Q( A CNF, weights ), etc.

34 Link Prediction

35 Coreference (Cora)

36 Conclusion We need inference for first-order models Probabilistic theorem proving is the most powerful approach to date PTP is reducible to lifted weighted model counting A single algorithm solves a lot of problems Empirically very efficient Available in the Alchemy system (alchemy.cs.washington.edu)

Lifted Inference: Exact Search Based Algorithms

Lifted Inference: Exact Search Based Algorithms Vibhav Gogate The University of Texas at Dallas Overview Background and Notation Probabilistic Knowledge Bases Exact Inference in Propositional Models First-order