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Logical and Probabilistic Reasoning to Support
Information Analysis in Uncertain Domains
Marco Valtorta, John Byrnes, and Michael Huhns
valtorta@sc.edu
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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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
LogicalReasonerFacts & Rules
Causality, Conventions, Hypotheses, and Norms
BayesianNetwork
Generator
Proof Trees
BN FragmentsMatcher &Composer
BN
BNs
BayesianNetwork
Reasoner
BN Situation
Conclusionsand Advice
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
LogicalReasonerFacts & Rules
Causality, Conventions, Hypotheses, and Norms
BayesianNetwork
Generator
Proof Trees
BN FragmentsMatcher &Composer
BN
BNs
BayesianNetwork
Reasoner
BN Situation
Conclusionsand Advice
EvidenceVarious Sources
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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
LogicalReasonerFacts & Rules
Causality, Conventions, Hypotheses, and Norms
BayesianNetwork
Generator
Proof Trees
BN FragmentsMatcher &Composer
BN
BNs
BayesianNetwork
Reasoner
BN Situation
Conclusionsand Advice
EvidenceVarious Sources
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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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