logical and probabilistic reasoning to support information analysis in uncertain domains marco...

Logical and Probabilistic Reasoning to Support

Information Analysis in Uncertain Domains

Marco Valtorta, John Byrnes, and Michael Huhns

[email protected]

September 6, 2007

Acknowledgments: This work was funded in part by the Disruptive Technology Office Collaboration and Analyst System Effectiveness (CASE) Program, contract FA8750-06-C-0194 issued by Air Force Research Laboratory (AFRL). The views and conclusions are those of the authors, not of the US Government or its agencies.

The contributions of Scott Langevin, Laura Zavala, Jingsong Wang, Jingshan Huang, and Dylan Kane (who prepared several of the slides) are appreciated.

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USC/HNC BALER Project

Overview

The BALER ProjectMotivation and AimsArchitecture

Conversion of Natural Deduction to Bayesian Network FragmentsNatural DeductionConverter Program

Use of BN Fragments Derived From Proofs Examples (throughout) Proof of Correctness Conclusions

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Bayesian and Logical Reasoning: BALER

BALER makes it possible for analysts to confront problems of credibility, relevance, contradictory evidence, and pervasive uncertainty, using

A unique combination of the power of logical and probabilistic reasoning

Numerical analysis of competing hypotheses

Automated linking of relevant evidence Automated propagation of uncertainty

values: good arguments from uncertain data still add strength to a conclusion

Robust reasoning over contradictory information allows analysts to exploit maximal amounts of information

Analysts can enter their own knowledge directly, allowing the system to learn from its users

Probabilities quantify belief in hypotheses to support optimal decision making according to the principle of maximum expected utility

Democracies are stable

Palestine is a

democracy

Palestine is stable

Palestine is an anarchy

Palestinian Parliament

to be dissolved

ContradictoryReports

Information Extraction

Error?

Text Document

Source Reliability?

Logical Inference

HowCertain?

Uncertain rule

How Relevant?

Causal Link

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BALER Architectural Concept

Some knowledge is best formalized in logic. This kind of knowledge includesClass-subclass statements, such as “dogs are mammals”Part-whole statements, such as “intake valves are parts of

cylinders”Definitional statements, such as “triangles have three sides”Temporal statements, such as “3:00 p.m. occurs before 4:00

p.m.”Spatial statements, such as “London is located in the UK”

Other knowledge is naturally probabilistic in nature. Examples are“Terrorist cell X planned the bombing”“Suspect Y met with cell leader Z in Syria last March”

BALER reasons both logically and probabilistically, permitting each piece of knowledge to be represented in the most appropriate way

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Overview of the BALER Architecture

BALER first generates a logical proof tree to focus the reasoning, then augments it with probabilities, and finally uses Bayesian reasoning to handle uncertainty, credibility, and relevance. The resultant Bayesian network structure is smaller, and thus the computation is tractable

LogicalReasonerFacts & Rules

Causality, Conventions, Hypotheses, and Norms

BayesianNetwork

Generator

Proof Trees

BN FragmentsMatcher &Composer

BN

BNs

BayesianNetwork

Reasoner

BN Situation

Conclusionsand Advice

EvidenceVarious Source

s

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BALER Logical Reasoner

Consumes knowledge, which could be provided by: Standard Upper Merged Ontology (SUMO) Magellan ontology IKRIS formalizations http://

nrrc.mitre.org/NRRC/ikris.htm Databases of interest

Provides proofs of the type consumed by the BN Constructor. Initial investigations focus on natural deduction proofs

Features the ability to search in classical, intuitionistic or minimal logic

Features the ability to present high-level outline of proof

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Bayesian Network Generator

Consumes a proof Consumes partial conditional

probability information Generates a Bayesian

network Features the ability to

estimate missing probabilities through maximum entropy, and possibly other techniques

We describe a prototype implementation and show its correctness in this talk



BayesianNetwork

Generator

Proof Trees


BN

BNs

BayesianNetwork

Reasoner

BN Situation


EvidenceVarious Sources

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BN Fragment Matcher and Composer

Retrieves BN fragments based on data (evidence) and instantiates the attributes of the nodes

Joins a set of BN fragments Stores instantiated and

composed fragments in a repository



BayesianNetwork

Generator

Proof Trees


BN

BNs

BayesianNetwork

Reasoner

BN Situation



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Bayesian Network Reasoner

Provides updates to a composed Bayesian network given hard, virtual, or soft evidence

Provides value of information computation

Provides analysis of sensitivity to parameters and evidence



BayesianNetwork

Generator

Proof Trees


BN

BNs

BayesianNetwork

Reasoner

BN Situation



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Natural Deduction

Abstract system of first-order logic Designed to mimic the natural reasoning process, as

follows:Make assumptions (“A” is true)

The set of assumptions being relied on at a given step is called the context

Use inference rules to draw conclusionsDischarge assumptions as they become no longer

necessary We use a sound and complete system of rules for classical

first-order logic; variations for intuitionistic and minimal logics require only small modifications

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Input Syntax for a Proof Step

<proofStep id="2"><rule>if I</rule><discharge>

<formula>(A)</formula></discharge><premises>

<formula contextID="2">(and (A) (B))</formula>

</premises><conclusion>

<formula>(if (A) (and (A) (B)))</formula></conclusion>

</proofStep>

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XMLSchema for Proof Language

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Creating Nodes by Formula Create a node for every formula Parse formula into operator and subformulas Create nodes for each subformula and make them

parents of the current node Assign the node a truth table based on the operator Recursively repeat the process for the subformulas until

they cannot be decomposed anymore

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Creating Nodes: Contexts

Contexts are provided by the logical reasoner

Their contents are stored as premises

Make each premise a parent of the context

Set the truth table such that the context is only true if all of its subformulas are trueFor a context of two

formulas, this looks like the table for AND

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Example: Brown Liquids

We want to show B

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Brown Liquids Proof and Corresponding BN

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B Logically Follows from the Axioms

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Representing Expert Judgment

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Composing a Fragment Derived from a Logical Theorywith a Fragment Representing Expert Judgment

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Using the Composed Model

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Adding One More Fragment

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Soft Evidence

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Extension of the language meaning that the formula holds for all terms other than a and b

Extending the Proof Converter toFirst-Order Logic

)()( xSxRx

)()( aSaR

baSR ,)()(

)()( bSbR

)(bS)(bR

Explicitly list instantiations occurring in proof

treated like infinite , like . Occurrences from the proof are explicitly represented; a single node represents “all others”

Correctness: show that defined distribution Pr satisfies Pr( A=True ) = Pr*( {M | M ╞ A} )

for Pr* over the class of term models If ├ A then Pr(A=T | =T ) = 1

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Conclusions

Our approach enables first-order logic sentences to be combined with Bayesian networks

Our results are based on the assumption that logic proofs and Bayesian reasoning can be handled separately and serially, and that the Bayesian network nodes can attach only to proof nodes without parents

Our converter successfully generates Bayesian networks for any first-order natural deduction proof (that uses the Reeves-Clarke inference rules)

We emphasize that our approach can handle formulas beyond Horn clauses

Additional work underway:Applying more real world examples and probabilistic

knowledge bases

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