Logical and Probabilistic Reasoning to Support Information Analysis in Uncertain Domains

1. 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.

2. Overview • The BALER Project • Motivation and Aims • Architecture • Conversion of Natural Deduction to Bayesian Network Fragments • Natural Deduction • Converter Program • Use of BN Fragments Derived From Proofs • Examples (throughout) • Proof of Correctness • Conclusions USC/HNC BALER Project

3. 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 Uncertain rule Democracies are stable Palestine is a democracy Logical Inference Contradictory Reports Palestine is stable Palestine is an anarchy HowCertain? Causal Link Source Reliability? Palestinian Parliament to be dissolved InformationExtraction Error? How Relevant? Text Document USC/HNC BALER Project

4. BALER Architectural Concept • Some knowledge is best formalized in logic. This kind of knowledge includes • Class-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 USC/HNC BALER Project

5. Logical Reasoner Facts & Rules Proof Trees Causality, Conventions, Hypotheses, and Norms Bayesian NetworkGenerator BN BNs Matcher &Composer BN Fragments BN Situation Various Sources Bayesian NetworkReasoner Evidence Conclusionsand Advice 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 USC/HNC BALER Project

6. 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 USC/HNC BALER Project

7. Facts & Rules Logical Reasoner Proof Trees Causality, Conventions, Hypotheses, and Norms Bayesian NetworkGenerator BN BNs BN Fragments Matcher &Composer BN Situation Various Sources Bayesian NetworkReasoner Evidence Conclusionsand Advice 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 USC/HNC BALER Project

8. Facts & Rules Logical Reasoner Proof Trees Causality, Conventions, Hypotheses, and Norms Bayesian NetworkGenerator BN BNs BN Fragments Matcher &Composer BN Situation Various Sources Bayesian NetworkReasoner Evidence Conclusionsand Advice 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 USC/HNC BALER Project

9. Facts & Rules Logical Reasoner Proof Trees Causality, Conventions, Hypotheses, and Norms Bayesian NetworkGenerator BN BNs BN Fragments Matcher &Composer BN Situation Various Sources Bayesian NetworkReasoner Evidence Conclusionsand Advice 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 USC/HNC BALER Project

10. 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 conclusions • Discharge 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 USC/HNC BALER Project

11. 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> USC/HNC BALER Project

12. XMLSchema for Proof Language USC/HNC BALER Project

13. 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 USC/HNC BALER Project

14. 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 true • For a context of two formulas, this looks like the table for AND USC/HNC BALER Project

15. Example: Brown Liquids • We want to show B USC/HNC BALER Project

16. Brown Liquids Proof and Corresponding BN USC/HNC BALER Project

17. B Logically Follows from the Axioms USC/HNC BALER Project

18. Representing Expert Judgment USC/HNC BALER Project

19. Composing a Fragment Derived from a Logical Theorywith a Fragment Representing Expert Judgment USC/HNC BALER Project

20. Using the Composed Model USC/HNC BALER Project

21. Adding One More Fragment USC/HNC BALER Project

22. Soft Evidence USC/HNC BALER Project

23. Extending the Proof Converter toFirst-Order Logic Extension of the language meaning that the formula holds for all terms other than a and b 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 USC/HNC BALER Project

24. 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 USC/HNC BALER Project