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Abductive Plan Recognition By Extending Bayesian Logic Programs

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Abductive Plan Recognition By Extending Bayesian Logic Programs

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  1. Abductive Plan Recognition By Extending Bayesian Logic Programs Sindhu V. Raghavan & Raymond J. Mooney The University of Texas at Austin

  2. Plan Recognition • Predict an agent’s top-level plans based on the observed actions • Abductive reasoning involving inference of cause from effect • Applications • Story Understanding • Strategic Planning • Intelligent User Interfaces

  3. Plan Recognition in Intelligent User Interfaces $ cd test-dir $ cp test1.txt my-dir $ rm test1.txt What task is the user performing? move-file Which files and directories are involved? test1.txt and test-dir Data is relational in nature - several files and directories and several relations between them

  4. Related Work • First-order logic based approaches[Kautz and Allen, 1986; Ng and Mooney, 1992] • Knowledge base of plans and actions • Default reasoning or logical abduction to predict the best plan based on the observed actions • Unable to handle uncertainty in data or estimate likelihood of alternative plans • Probabilistic graphical models[Charniak and Goldman, 1989; Huber et al., 1994; Pynadath and Wellman, 2000; Bui, 2003; Blaylock and Allen, 2005] • Encode the domain knowledge using Bayesian networks, abstract hidden Markov models, or statistical n-gram models • Unable to handle relational/structured data • Statistical Relational Learning based approaches • Markov Logic Networks for plan recognition[Kate and Mooney, 2009; Singla and Mooney, 2011]

  5. Our Approach • Extend Bayesian Logic Programs (BLPs) [Kersting and De Raedt, 2001] for plan recognition • BLPs integrate first-order logic and Bayesian networks • Why BLPs? • Efficient grounding mechanism that includes only those variables that are relevant to the query • Easy to extend by incorporating any type oflogical inferenceto construct networks • Well suited for capturing causal relations in data

  6. Outline • Motivation • Background • Logical Abduction • Bayesian Logic Programs (BLPs) • Extending BLPs for Plan Recognition • Experiments • Conclusions

  7. Logical Abduction • Abduction • Process of finding the best explanation for a set of observations • Given • Background knowledge, B, in the form of a set of (Horn) clauses in first-order logic • Observations, O, in the form of atomic facts in first-order logic • Find • A hypothesis, H, a set of assumptions(atomic facts) that logically entail the observations given the theory: B  H  O • Best explanation is the one with the fewest assumptions

  8. Bayesian Logic Programs (BLPs) [Kersting and De Raedt, 2001] • Set of Bayesian clauses a|a1,a2,....,an • Definite clauses that are universally quantified • Range-restricted, i.e variables{head} variables{body} • Associated conditional probability table (CPT) • P(head|body) • Bayesian predicates a, a1, a2, …, an have finite domains • Combining rule like noisy-or for mapping multiple CPTs into a single CPT.

  9. Inference in BLPs[Kersting and De Raedt, 2001] • Logical inference • Given a BLP and a query, SLD resolution is used to construct proofs for the query • Bayesian network construction • Each ground atom is a random variable • Edges are added from ground atoms in the body to the ground atom in head • CPTs specified by the conditional probability distribution for the corresponding clause • P(X) = P(Xi | Pa(Xi)) • Probabilistic inference • Marginal probability given evidence • Most Probable Explanation (MPE) given evidence

  10. BLPs for Plan Recognition • SLD resolution is deductive inference, used for predicting observations from top-level plans • Plan recognition is abductive in nature and involves predicting the top-level plan from observations BLPs cannot be used as is for plan recognition

  11. Extending BLPs for Plan Recognition BLPs Logical Abduction + = BALPs BALPs – Bayesian Abductive Logic Programs

  12. Logical Abduction in BALPs • Given • A set of observation literals O = {O1, O2,….On} and a knowledge base KB • Compute a set abductive proofs of O using Stickel’s abduction algorithm[Stickel 1988] • Backchain on each Oi until it is proved or assumed • A literal is said to be proved if it unifies with a fact or the head of some rule in KB, otherwise it is said to be assumed • Construct a Bayesian network using the resulting set of proofs as in BLPs.

  13. Example – Intelligent User Interfaces • Top-level plan predicates • copy-file, move-file, remove-file • Action predicates • cp, rm • Knowledge Base (KB) • cp(Filename,Destdir) | copy-file(Filename,Destdir) • cp(Filename,Destdir) | move-file(Filename,Destdir) • rm(Filename) | move-file(Filename,Destdir) • rm(Filename) | remove-file(Filename) • Observed actions • cp(test1.txt, mydir) • rm(test1.txt)

  14. Abductive Inference Assumed literal copy-file(test1.txt,mydir) cp(test1.txt,mydir) cp(Filename,Destdir) | copy-file(Filename,Destdir)

  15. Abductive Inference Assumed literal copy-file(test1.txt,mydir) move-file(test1.txt,mydir) cp(test1.txt,mydir) cp(Filename,Destdir) | move-file(Filename,Destdir)

  16. Abductive Inference Match existing assumption copy-file(test1.txt,mydir) move-file(test1.txt,mydir) rm(test1.txt) cp(test1.txt,mydir) • rm(Filename) | move-file(Filename,Destdir)

  17. Abductive Inference Assumed literal copy-file(test1.txt,mydir) move-file(test1.txt,mydir) remove-file(test1) rm(test1.txt) cp(test1.txt,mydir) • rm(Filename) | remove-file(Filename)

  18. Structure of Bayesian network copy-file(test1.txt,mydir) move-file(test1.txt,mydir) remove-file(test1) rm(test1.txt) cp(test1.txt,mydir)

  19. Probabilistic Inference • Specifying probabilistic parameters • Noisy-and • Specify the CPT for combining the evidence from conjuncts in the body of the clause • Noisy-or • Specify the CPT for combining the evidence from disjunctive contributions from different ground clauses with the same head • Models “explaining away” • Noisy-and and noisy-or models reduce the number of parameters learned from data

  20. Probabilistic Inference copy-file(test1.txt,mydir) move-file(test1.txt,mydir) remove-file(test1) Noisy-or Noisy-or rm(test1.txt) cp(test1.txt,mydir)

  21. Probabilistic Inference • Most Probable Explanation (MPE) • For multiple plans, compute MPE, the most likely combination of truth values to all unknown literals given this evidence • Marginal Probability • For single top level plan prediction, compute marginal probabilityfor all instances of plan predicate and pick the instance with maximum probability • When exact inference is intractable, SampleSearch[Gogate and Dechter, 2007], an approximate inference algorithm for graphical models with deterministic constraints is used

  22. Probabilistic Inference copy-file(test1.txt,mydir) move-file(test1.txt,mydir) remove-file(test1) Noisy-or Noisy-or rm(test1.txt) cp(test1.txt,mydir)

  23. Probabilistic Inference copy-file(test1.txt,mydir) move-file(test1.txt,mydir) remove-file(test1) Noisy-or Noisy-or rm(test1.txt) cp(test1.txt,mydir) Evidence

  24. Probabilistic Inference Query variables copy-file(test1.txt,mydir) move-file(test1.txt,mydir) remove-file(test1) Noisy-or Noisy-or rm(test1.txt) cp(test1.txt,mydir) Evidence

  25. Probabilistic Inference MPE Query variables FALSE FALSE TRUE copy-file(test1.txt,mydir) move-file(test1.txt,mydir) remove-file(test1) Noisy-or Noisy-or rm(test1.txt) cp(test1.txt,mydir) Evidence

  26. Probabilistic Inference MPE Query variables FALSE FALSE TRUE move-file(test1.txt,mydir) copy-file(test1.txt,mydir) remove-file(test1) Noisy-or Noisy-or rm(test1.txt) cp(test1.txt,mydir) Evidence

  27. Parameter Learning • Learn noisy-or/noisy-and parameters using the EM algorithm adapted for BLPs [Kersting and De Raedt, 2008] • Partial observability • In plan recognition domain, data is partially observable • Evidence is present only for observed actions and top-level plans; sub-goals, noisy-or, and noisy-and nodes are not observed • Simplify learning problem • Learn noisy-or parameters only • Used logical-and instead of noisy-and to combine evidence from conjuncts in the body of a clause

  28. Experimental Evaluation • Monroe (Strategic planning) • Linux (Intelligent user interfaces) • Story Understanding (Story understanding)

  29. Monroe and Linux [Blaylock and Allen, 2005] • Task • Monroe involves recognizing top level plans in an emergency response domain (artificially generated using HTN planner) • Linux involves recognizing top-level plans based on linux commands • Single correct plan in each example • Data

  30. Monroe and Linux • Methodology • Manually encoded the knowledge base • Learned noisy-or parameters using EM • Computed marginal probability for plan instances • Systems compared • BALPs • MLN-HCAM [Singla and Mooney, 2011] • MLN-PC and MLN-HC do not run on Monroe and Linux due to scaling issues • Blaylock and Allen’s system [Blaylock and Allen, 2005] • Performance metric • Convergence score - measures the fraction of examples for which the plan predicate was predicted correctly

  31. Results on Monroe Convergence Score 94.2 * MLN-HCAM Blaylock & Allen BALPs * - Differences are statistically significant wrt BALPs

  32. Results on Linux Convergence Score 36.1 * MLN-HCAM Blaylock & Allen BALPs * - Differences are statistically significant wrt BALPs

  33. Experiments with partial observability • Limitations of convergence score • Does not account for predicting the plan arguments correctly • Requires all the observations to be seen before plans can be predicted • Early plan recognition with partial set of observations • Perform plan recognition after observing the first25%, 50%, 75%, and 100% of the observations • Accuracy – Assign partial credit for the predicting plan predicate and a subset of the arguments correctly • Systems compared • BALPs • MLN-HCAM [Singla and Mooney, 2011]

  34. Results on Monroe Accuracy Percent observations seen

  35. Results on Linux Accuracy Percent observations seen

  36. Story Understanding [Charniak and Goldman, 1991; Ng and Mooney, 1992] • Task • Recognize character’s top level plans based on actions described in narrative text • Multiple top-level plans in each example • Data • 25 examples in development set and 25 examples in test set • 12.6 observations per example • 8 top-level plan predicates

  37. Story Understanding • Methodology • Knowledge base was created for ACCEL [Ng and Mooney, 1992] • Parameters set manually • Insufficient number of examples in the development set to learn parameters • Computed MPE to get the best set of plans • Systems compared • BALPs • MLN-HCAM[Singla and Mooney, 2011] • Best performing MLN model • ACCEL-Simplicity [Ng and Mooney, 1992] • ACCEL-Coherence [Ng and Mooney, 1992] • Specific for Story Understanding

  38. Results on Story Understanding * * * - Differences are statistically significant wrt BALPs

  39. Conclusion • BALPS – Extension of BLPs for plan recognition by employing logical abduction to construct Bayesian networks • Automatic learning of model parameters using EM • Empirical results on all benchmark datasets demonstrate advantages over existing methods

  40. Future Work • Learn abductive knowledge base automatically from data • Compare BALPs with other probabilistic logics like ProbLog [De Raedt et. al, 2007], PRISM [Sato, 1995] and Poole’s Horn Abduction [Poole, 1993] on plan recognition

  41. Questions

  42. Backup

  43. Completeness in First-order Logic • Completeness - If a sentence is entailed by a KB, then it is possible to find the proof that entails it • Entailment in first-order logic is semidecidable, i.e it is not possible to know if a sentence is entailed by a KB or not • Resolution is complete in first-order logic • If a set of sentences is unsatisfiable, then it is possible to find a contradiction

  44. First-order Logic • Terms • Constants – individual entities like anna, bob • Variables – placeholders for objects like X, Y • Predicates • Relations over entities like worksFor, capitalOf • Literal – predicate or its negation applied to terms • Atom – Positive literal like worksFor(X,Y) • Ground literal – literal with no variables like worksFor(anna,bob) • Clause – disjunction of literals • Horn clause has at most one positive literal • Definite clause has exactly one positive literal

  45. First-order Logic • Quantifiers • Universal quantification - true for all objects in the domain • Existential quantification - true for some objects in the domain • Logical Inference • Forward Chaining– For every implication pq, if p is true, then q is concluded to be true • Backward Chaining – For a query literal q, if an implication pq is present and p is true, then q is concluded to be true, otherwise backward chaining tries to prove p

  46. Forward chaining • For every implication pq, if p is true, then q is concluded to be true • Results in addition of a new fact to KB • Efficient, but incomplete • Inference can explode and forward chaining may never terminate • Addition of new facts might result in rules being satisfied • It is data-driven, not goal-driven • Might result in irrelevant conclusions

  47. Backward chaining • For a query literal q, if an implication pq is present and p is true, then q is concluded to be true, otherwise backward chaining tries to prove p • Efficient, but not complete • May never terminate, might get stuck in infinite loop • Exponential • Goal-driven

  48. Herbrand Model Semantics • Herbrand universe • All constants in the domain • Herbrand base • All ground atoms atoms over Herbrand universe • Herbrand interpretation • A set of ground atoms from Herbrand base that are true • Herbrand model • Herbrand interpretation that satisfies all clauses in the knowledge base

  49. Advantages of SRL models over vanilla probabilistic models • Compactly represent domain knowledge in first-order logic • Employ logical inference to construct ground networks • Enables parameter sharing

  50. Parameter sharing in SRL father(john) father(mary) father(alice) θ1 θ2 θ3 parent(john) parent(mary) parent(alice) dummy