Abductive Plan Recognition By Extending Bayesian Logic Programs

Sindhu V. Raghavan & Raymond J. Mooney

The University of Texas at Austin

Plan RecognitionPredict an agent’s top-level plans based on the

observed actions

Abductive reasoning involving inference of cause from effect

ApplicationsStory UnderstandingStrategic Planning Intelligent User Interfaces

$ cd test-dir$ cp test1.txt my-dir$ rm test1.txt

$ cd test-dir$ cp test1.txt my-dir$ rm test1.txtWhat task is the user performing?move-file

Which files and directories are involved?test1.txt and test-dir

Plan Recognition in Intelligent User Interfaces

Data is relational in nature - several files and directories and several relations between them

Related WorkFirst-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]

Our Approach

Extend Bayesian Logic Programs (BLPs) [Kersting and

De Raedt, 2001] for plan recognitionBLPs integrate first-order logic and Bayesian

networks

Why BLPs?Efficient grounding mechanism that includes only those

variables that are relevant to the queryEasy to extend by incorporating any type of logical

inference to construct networksWell suited for capturing causal relations in data

Outline

MotivationBackground

Logical AbductionBayesian Logic Programs (BLPs)

Extending BLPs for Plan RecognitionExperimentsConclusions

Logical AbductionAbduction

Process of finding the best explanation for a set of observations

GivenBackground knowledge, B, in the form of a set of (Horn) clauses in

first-order logicObservations, O, in the form of atomic facts in first-order logic

FindA hypothesis, H, a set of assumptions (atomic facts) that logically

entail the observations given the theory:

B H OBest explanation is the one with the fewest assumptions

Bayesian Logic Programs (BLPs) [Kersting and De Raedt, 2001]

Set of Bayesian clauses a|a1,a2,....,an

Definite clauses that are universally quantifiedRange-restricted, i.e variables{head} variables{body}Associated conditional probability table (CPT)

o P(head|body)

Bayesian predicates a, a1, a2, …, an have finite domainsCombining rule like noisy-or for mapping multiple CPTs

into a single CPT.

Inference in BLPs[Kersting and De Raedt, 2001]

Logical inferenceGiven a BLP and a query, SLD resolution is used to construct

proofs for the query

Bayesian network constructionEach ground atom is a random variableEdges are added from ground atoms in the body to the ground

atom in headCPTs specified by the conditional probability distribution for the

corresponding clauseP(X) = P(Xi | Pa(Xi))

Probabilistic inferenceMarginal probability given evidenceMost Probable Explanation (MPE) given evidence

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

Extending BLPs for Plan Recognition

BLPsBLPs Logical Abduction

Logical Abduction

BALPsBALPs

BALPs – Bayesian Abductive Logic Programs

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.

Example – Intelligent User InterfacesTop-level plan predicates

copy-file, move-file, remove-file

Action predicatescp, 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 actionscp(test1.txt, mydir) rm(test1.txt)

Abductive Inference

copy-file(test1.txt,mydir)

cp(test1.txt,mydir)

cp(Filename,Destdir) | copy-file(Filename,Destdir)

Assumed literalAssumed literal

Abductive Inference

cp(test1.txt,mydir)

move-file(test1.txt,mydir)

cp(Filename,Destdir) | move-file(Filename,Destdir)

Abductive Inference

cp(test1.txt,mydir)

rm(Filename) | move-file(Filename,Destdir)

rm(test1.txt)

Match existing assumptionMatch existing assumption

Abductive Inference

cp(test1.txt,mydir)

rm(Filename) | remove-file(Filename)

rm(test1.txt)

remove-file(test1)

Structure of Bayesian network

cp(test1.txt,mydir)

rm(test1.txt)

remove-file(test1)

Probabilistic Inference

Specifying probabilistic parametersNoisy-and

o Specify the CPT for combining the evidence from conjuncts in the body of the clause

Noisy-oro Specify the CPT for combining the evidence from

disjunctive contributions from different ground clauses with the same head

o Models “explaining away”Noisy-and and noisy-or models reduce the number of

parameters learned from data

cp(test1.txt,mydir)

rm(test1.txt)

remove-file(test1)

Noisy-or Noisy-or

Most Probable Explanation (MPE)For multiple plans, compute MPE, the most likely

combination of truth values to all unknown literals given this evidence

Marginal ProbabilityFor single top level plan prediction, compute marginal

probability for 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

cp(test1.txt,mydir)

rm(test1.txt)

remove-file(test1)

Noisy-or Noisy-or

cp(test1.txt,mydir)

rm(test1.txt)

remove-file(test1)

Noisy-or Noisy-or

Evidence

cp(test1.txt,mydir)

rm(test1.txt)

remove-file(test1)

Noisy-or Noisy-or

Evidence

Query variables

cp(test1.txt,mydir)

rm(test1.txt)

remove-file(test1)

Noisy-or Noisy-or

Evidence

Query variablesTRUE FALSEFALSE

cp(test1.txt,mydir)

rm(test1.txt)

remove-file(test1)

Noisy-or Noisy-or

Evidence

Query variablesTRUE FALSEFALSE

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 observableEvidence is present only for observed actions and top-level

plans; sub-goals, noisy-or, and noisy-and nodes are not observed

Simplify learning problemLearn noisy-or parameters onlyUsed logical-and instead of noisy-and to combine evidence

from conjuncts in the body of a clause

Experimental Evaluation

Monroe (Strategic planning)

Linux (Intelligent user interfaces)

Story Understanding (Story understanding)

Monroe and Linux [Blaylock and Allen, 2005]

TaskMonroe 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

No. examples

Avg. observations/ example

Total top-level plan predicates

Total observed action predicates

Monroe 1000 10.19 10 30

Linux 457 6.1 19 43

Monroe and Linux

MethodologyManually encoded the knowledge base Learned noisy-or parameters using EMComputed marginal probability for plan instances

Systems comparedBALPsMLN-HCAM [Singla and Mooney, 2011]

o 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 metricConvergence score - measures the fraction of examples for

which the plan predicate was predicted correctly

Results on Monroe

94.2 *

BALPs MLN-HCAM Blaylock & Allen

* - Differences are statistically significant wrt BALPs

Results on Linux

BALPs MLN-HCAM Blaylock & Allen

36.1 *

* - Differences are statistically significant wrt BALPs

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 observationsPerform plan recognition after observing the first 25%,

50%, 75%, and 100% of the observationsAccuracy – Assign partial credit for the predicting plan

predicate and a subset of the arguments correctly

Results on Monroe

34Percent observations seen

Results on Linux

Percent observations seen

Story Understanding [Charniak and Goldman, 1991; Ng and Mooney, 1992]

TaskRecognize character’s top level plans based on actions

described in narrative textMultiple top-level plans in each example

Data25 examples in development set and 25 examples in test

set12.6 observations per example8 top-level plan predicates

Story UnderstandingMethodology

Knowledge base was created for ACCEL [Ng and Mooney, 1992]

Parameters set manuallyo Insufficient number of examples in the development set to learn

parameters

Computed MPE to get the best set of plans

o Best performing MLN model

ACCEL-Simplicity [Ng and Mooney, 1992]

ACCEL-Coherence [Ng and Mooney, 1992]

o Specific for Story Understanding

Results on Story Understanding

38* - Differences are statistically significant wrt BALPs

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

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

Questions

Backup

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

First-order LogicTerms

Constants – individual entities like anna, bobVariables – 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 literalsHorn clause has at most one positive literalDefinite clause has exactly one positive literal

First-order Logic

QuantifiersUniversal quantification - true for all objects in the domain

Existential quantification - true for some objects in the domain

Logical InferenceForward Chaining– For every implication pq, if p is true,

then q is concluded to be trueBackward Chaining – For a query literal q, if an implication

pq is present and p is true, then q is concluded to be true, otherwise backward chaining tries to prove p

∀( )

∃( )

Forward chaining

For every implication pq, if p is true, then q is concluded to be true

Results in addition of a new fact to KBEfficient, but incompleteInference can explode and forward chaining may

never terminateAddition of new facts might result in rules being satisfied

It is data-driven, not goal-drivenMight result in irrelevant conclusions

Backward chaining

For a query literal q, if an implication pq is present and p is true, then q is concluded to be true, otherwise backward chaining tries to prove p

Efficient, but not completeMay never terminate, might get stuck in infinite

loopExponentialGoal-driven

Herbrand Model Semantics

Herbrand universeAll constants in the domain

Herbrand baseAll ground atoms atoms over Herbrand universe

Herbrand interpretationA set of ground atoms from Herbrand base that are true

Herbrand modelHerbrand interpretation that satisfies all clauses in the

knowledge base

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

Parameter sharing in SRL

father(john)father(john)

parent(john)parent(john)

father(mary)father(mary)

parent(mary)parent(mary)

father(alice)father(alice)

parent(alice)parent(alice)

dummydummy

θ1θ1 θ2θ2 θ3θ3

Parameter sharing in SRL

father(X) parent(X)

father(john)father(john)

parent(john)parent(john)

father(mary)father(mary)

parent(mary)parent(mary)

father(alice)father(alice)

parent(alice)parent(alice)

dummydummy

θθ θθ θθ

Noisy-and Model

Several causes ci have to occur simultaneously if event e has to occur

ci fails to trigger e with probability pi

inh accounts for some unknown cause due to which e has failed to trigger

P(e) = (I – inh) Πi(1-pi)^(1-ci)

Noisy-or Model

Several causes ci cause event e has to occur

ci independently triggers e with probability pi

leak accounts for some unknown cause due to which e has triggered

P(e) = 1 – [(I – inh) Πi (1-pi)^(1-ci)]

Models explaining away If there are several causes of an event, and if there is

evidence for one of the causes, then the probability that the other causes have caused the event goes down

Noisy-and And Noisy-or Models

alarm(james)

burglary(james)

neighborhood(james)

lives(james,yorkshire) tornado(yorkshire)

Noisy-and And Noisy-or Models

55alarm(james)

burglary(james)

neighborhood(james)

dummy1 dummy2

Noisy/logical-and

Noisy/logical-andNoisy/logical-and

Noisy-or

Logical Inference in BLPs SLD Resolution

lives(james,yorkshire).lives(stefan,freiburg).neighborhood(james).tornado(yorkshire).

burglary(X) | neighborhood(X).alarm(X) | burglary(X).alarm(X) | lives(X,Y), tornado(Y).

Queryalarm(james)

alarm(james)

Example from Ngo and Haddawy, 1997

Queryalarm(james)

alarm(james)

burglary(james)

Queryalarm(james)

alarm(james)

burglary(james)

neighborhood(james)

Queryalarm(james)

alarm(james)

burglary(james)

neighborhood(james)

Queryalarm(james)

alarm(james)

burglary(james)

neighborhood(james)

Queryalarm(james)

alarm(james)

burglary(james)

neighborhood(james)

Queryalarm(james)

alarm(james)

burglary(james)

neighborhood(james)

lives(james,Y) tornado(Y)

Queryalarm(james)

alarm(james)

burglary(james)

neighborhood(james)

lives(james,yorkshire)

Queryalarm(james)

alarm(james)

burglary(james)

neighborhood(james)

tornado(yorkshire)

Queryalarm(james)

alarm(james)

burglary(james)

neighborhood(james)

tornado(yorkshire)

Bayesian Network Construction

alarm(james)

burglary(james)

neighborhood(james)

Each ground atom becomes a node (random variable) in the Bayesian network

Edges are added from ground atoms in the body of a clause to the ground atom in the head

Specify probabilistic parameters using the CPTs associated with Bayesian clauses

alarm(james)

burglary(james)

neighborhood(james)

alarm(james)

burglary(james)

neighborhood(james)

alarm(james)

burglary(james)

neighborhood(james)

Use combining rule to combine multiple CPTs into a single CPT

lives(stefan,freiburg)✖

copy-file(Test1,txt,Mydir)

cp(Test1.txt,Mydir)

move-file(Test1,txt,Mydir)

rm(Test1.txt)

remove-file(Test1)

4 parameters

copy-file(Test1,txt,Mydir)

cp(Test1.txt,Mydir)

move-file(Test1,txt,Mydir)

rm(Test1.txt)

remove-file(Test1)

2 parameters 2 parameters

θ1 θ2 θ3θ4

Noisy models require parameters linear in the number of parents

Learning in BLPs[Kersting and De Raedt, 2008]

Parameter learningExpectation Maximization Gradient-ascent based learningBoth approaches optimize likelihood

Structure learningHill climbing search through the space of possible

structures Initial structure obtained from CLAUDIEN [De Raedt and

Dehaspe, 1997]

Learns from only positive examples

Probabilistic Inference and Learning

Probabilistic inferenceMarginal probability given evidenceMost Probable Explanation (MPE) given evidence

Learning [Kersting and De Raedt, 2008]

Parameterso Expectation Maximizationo Gradient-ascent based learning

Structureo Hill climbing search through the space of possible

structures

Expectation Maximization for BLPs/BALPs

• Perform logical inference to construct a ground Bayesian network for each example

• Let r denote rule, X denote a node, and Pa(X) denote parents of X

• E Step

• The inner sum is over all groundings of rule r

• M Step

* From SRL tutorial at ECML 07 74

Decomposable Combining Rules

Express different influences using separate nodes

These nodes can be combined using a deterministic function

Combining Rules

76alarm(james)

burglary(james)

neighborhood(james)

dummy1 dummy2

Logical-and

Logical-andLogical-and

Noisy-or

Decomposable Combining Rules

77alarm(james)

burglary(james)

neighborhood(james)

dummy1 dummy2

Logical-and

Logical-andLogical-and

Noisy-or Noisy-or

dummy1-new dummy2 -new

Logical-or

BLPs vs. PLPs

Differences in representation In BLPs, Bayesian atoms take finite set of values, but in

PLPs, each atom is logical in nature and it can take true or false

Instead of having neighborhood(x) = bad, in PLPs, we have neighborhood(x,bad)

To compute probability of a query alarm(james), PLPs have to construct one proof tree for all possible values for all predicates

Inference is cumbersome

BLPs subsume PLPs

BLPs vs. Poole's Horn Abduction

Differences in representationFor example, if P(x) and R(x) are two competing

hypothesis, then either P(x) could be true or R(x) could be true

Prior probabilities of P(x) and R(x) should sum to 1 Restrictions of these kind are not there in BLPsPLPs and hence BLPs are more flexible and have a richer

representation

BLPs vs. PRMs

BLPs subsume PRMsPRMs use entity-relationship models to represent

knowledge and they use KBMC-like construction to construct a ground Bayesian networkEach attribute becomes a random variable in the ground

network and relations over the entities are logical constraints In BLP, each attribute becomes a Bayesian atom and

relations become logical atoms Aggregator functions can be transformed into combining

BLPs vs. RBNs

BLPs subsume RBNs In RBNs, each node in BN is a predicate and

probability formulae are used to specify probabilitiesCombining rules can be used to represent these

probability formulae in BLPs.

BALPs vs. BLPs

Like BLPs, BALPs use logic programs as templates for constructing Bayesian networks

Unlike BLPs, BALPs uses logical abduction instead of deduction to construct the network

Monroe [Blaylock and Allen, 2005]

TaskRecognize top level plans in an emergency response

domain (artificially generated using HTN planner)Plans include set-up-shelter, clear-road-wreck, provide-

medical-attentionSingle correct plan in each exampleDomain consists of several entities and sub-goalsTest the ability to scale to large domains

DataContains 1000 examples

Monroe

MethodologyKnowledge base constructed based on the domain knowledge

encoded in plannerLearn noisy-or parameters using EMCompute marginal probability for instances of top level plans

and pick the one with the highest marginal probabilitySystems compared

o BALPso MLN-HCAM [Singla and Mooney, 2011]

o Blaylock and Allen’s system [Blaylock and Allen, 2005]

Convergence score - measures the fraction of examples for which the plan schema was predicted correctly

Learning Results - Monroe

MW MW-Start Rand-Start

Conv Score 98.4 98.4 98.4

Acc-100 79.16 79.16 79.86

Acc-75 46.06 44.63 44.73

Acc-50 20.67 20.26 19.7

Acc-25 7.2 7.33 10.46

Linux [Blaylock and Allen, 2005]

TaskRecognize top level plans based on Linux commandsHuman users asked to perform tasks in Linux and commands

were recordedTop-level plans include find-file-by-ext, remove-file-by-ext,

copy-file, move-fileSingle correct plan in each exampleTests the ability to handle noise in data

o Users indicate success even when they have not achieved the task correctly

o Some top-level plans like find-file-by-ext and file-file-by-name have identical actions

DataContains 457 examples

MethodologyKnowledge base constructed based on the knowledge of Linux

commandsLearn noisy-or parameters using EMCompute marginal probability for instances of top level plans

and pick the one with the highest marginal probabilitySystems compared

o Blaylock and Allen’s system [Blaylock and Allen, 2005]

Convergence score - measures the fraction of examples for which the plan schema was predicted correctly o

Learning Results - Linux

Partial Observability

MW MW-Start Rand-Start

Conv Score 39.82 46.6 41.57

Story Understanding [Charniak and Goldman, 1991; Ng and Mooney, 1992]

TaskRecognize character’s top level plans based on actions

described in narrative textLogical representation of actions literals providedTop-level plans include shopping, robbing, restaurant

dining, partying Multiple top-level plans in each exampleTests the ability to predict multiple plans

Data25 development examples25 test examples

Story UnderstandingMethodology

Knowledge base constructed for ACCEL by Ng and Mooney [1992]

Insufficient number of examples to learn parameterso Noisy-or parameters set to 0.9o Noisy-and parameters set to 0.9o Priors tuned on development set

Compute MPE to get the best set of plansSystems compared

o ACCEL-Simplicity [Ng and Mooney, 1992]

o ACCEL-Coherence [Ng and Mooney, 1992]

– Specific for Story Understanding

Results on Story Understanding

91* - Differences are statistically significant wrt BALPs

Other Applications of BALPs

Medical diagnosisTextual entailmentComputational biology

Inferring gene relations based on the output of micro-array experiments

Any application that requires abductive reasoning

ACCEL[Ng and Mooney, 1992]

First-order logic based system for plan recognition

Simplicity metric selects explanations that have the least number of assumptions

Coherence metric selects explanations that connect maximum number of observationsMeasures explanatory coherenceSpecific to text interpretation

System by Blaylock and Allen[2005]

Statistical n-gram models to predict plans based on observed actions

Performs plan recognition in two phasesPredicts the plan schema firstPredicts arguments based on the predicted schema

Abductive Plan Recognition By Extending Bayesian Logic Programs

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