Concurrent Reasoning with Inference Graphs

1. Concurrent Reasoning with Inference Graphs Department of Computer Science and Engineering University at Buffalo, The State University of New York Buffalo, New York, USA <drschleg,shapiro>@buffalo.edu Daniel R. Schlegel and Stuart C. Shapiro D. R. Schlegel and S. C. Shapiro

2. Problem Statement • Rise of multi-core computers • Lack of concurrent natural deduction systems • Application to natural language understanding for terrorist plot detection. A Motivation D. R. Schlegel

3. What are Inference Graphs? • Graphs for natural deduction • Four types of inference: • Forward • Backward • Bi-directional • Focused • Retain derived formulas for later re-use. • Propagate disbelief. • Built upon Propositional Graphs. • Take advantage of multi-core computers • Concurrency and scheduling • Near-linear speedup. D. R. Schlegel

4. Propositional Graphs • Directed acyclic graph • Every well-formed expression is a node • Individual constants • Functional terms • Atomic formulas • Non-atomic formulas (“rules”) • Each node has an identifier, either • Symbol, or • wfti[!] • No two nodes with same identifier. D. R. Schlegel and S. C. Shapiro

5. Propositional Graphs a wft1! c and-ant cq and-ant b If a, and b are true, then c is true. D. R. Schlegel and S. C. Shapiro

6. Inference Graphs • Extend Propositional Graphs • Add channels for information flow (messages): • i-channelsreport truth of an antecedent to a rule node. • u-channelsreport truth of a consequent from a rule node. • Channels contain valves. • Hold messages back, or allow them through. u-channel i-channel a wft1! c and-ant cq and-ant b D. R. Schlegel and S. C. Shapiro

7. Messages • Five kinds • I-INFER – “I’ve been inferred” • U-INFER – “You’ve been inferred” • BACKWARD-INFER – “Open valves so I might be inferred” • CANCEL-INFER – “Stop trying to infer me (close valves)” • UNASSERT – “I’m no longer believed” D. R. Schlegel and S. C. Shapiro

8. Priorities • Messages have priorities. • UNASSERT is top priority • CANCEL-INFER is next • I-INFER/U-INFER are higher priority closer to a result • BACKWARD-INFER is lowest D. R. Schlegel and S. C. Shapiro

9. Rule Node Inference • Message arrives at node. Assume we have a KB with a ^ b -> c, and b. Then a is asserted with forward inference. i-infer a : true a! wft1! c and-ant cq and-ant b! A message is sent from a to wft1 D. R. Schlegel and S. C. Shapiro

10. Rule Node Inference 2. Message is translated to Rule Use Information 1 Positive Antecedent, a 0 Negative Antecedents 2 Total Antecedents a : true a! wft1! c and-ant cq and-ant b! Rule Use Information is stored in rule nodes to be combined later with others that arrive. D. R. Schlegel and S. C. Shapiro

11. Rule Node Inference 3. Combine RUIs with any existing ones 2 Positive Antecedents, a,b 0 Negative Antecedents 2 Total Antecedents 1 Positive Antecedent, b 0 Negative Antecedents 2 Total Antecedents 1 Positive Antecedent, a 0 Negative Antecedents 2 Total Antecedents + = a! wft1! c and-ant cq and-ant b! Combine the RUI for a with the one which already exists in wft1 for b. D. R. Schlegel and S. C. Shapiro

12. Rule Node Inference 4. Determine if the rule can fire. 2 Positive Antecedents, a,b 0 Negative Antecedents 2 Total Antecedents a! wft1! c and-ant cq and-ant b! We have two positive antecedents, and we need two. The rule can fire. D. R. Schlegel and S. C. Shapiro

13. Rule Node Inference 5. Send out new messages. u-infer c : true a! wft1! c and-ant cq and-ant b! c will receive the message, and assert itself. D. R. Schlegel

14. Cycles • Graphs may contain cycles. • No rule node will infer on the same message more than once. • RUIs with no new information are ignored. • Already open valves can’t be opened again. wft1! cq ant a b ant cq wft2! D. R. Schlegel and S. C. Shapiro

15. Concurrency and Scheduling • Inference Segment: the area between two valves. • When messages reach a valve: • A task is created with the same priority as the message. • Task: application of the segment’s function to the message. • Task is added to a prioritized queue. • Tasks have minimal shared state, easing concurrency. D. R. Schlegel and S. C. Shapiro

16. Concurrency and Scheduling • A task only operates within a single segment. • tasks for relaying newly derived information using segments “later” in the derivation are executed before “earlier” ones, and • once a node is known to be true or false, all tasks attempting to derive it are canceled, as long as their results are not needed elsewhere. D. R. Schlegel and S. C. Shapiro

17. Example cq Backchain on cq. Assume every node requires a single one of its incoming nodes to be true for it to be true(simplified for easy viewing). Two processors will be used. D. R. Schlegel and S. C. Shapiro

18. Example cq Backward Inference (Open valve) Inferred Inferring Cancelled D. R. Schlegel and S. C. Shapiro

19. Example cq Backward Inference (Open valve) Inferred Inferring Cancelled D. R. Schlegel and S. C. Shapiro

20. Example cq Backward Inference (Open valve) Inferred Inferring Cancelled D. R. Schlegel and S. C. Shapiro

21. Example cq Backward Inference (Open valve) Inferred Inferring Cancelled D. R. Schlegel and S. C. Shapiro

22. Example cq Backward Inference (Open valve) Inferred Inferring Cancelled D. R. Schlegel and S. C. Shapiro

23. Example cq Backward Inference (Open valve) Inferred Inferring Cancelled D. R. Schlegel and S. C. Shapiro

24. Example cq Backward Inference (Open valve) Inferred Inferring Cancelled D. R. Schlegel and S. C. Shapiro

25. Example cq Backward Inference (Open valve) Inferred Inferring Cancelled D. R. Schlegel and S. C. Shapiro

26. Example cq Backward Inference (Open valve) Inferred Inferring Cancelled D. R. Schlegel and S. C. Shapiro

27. Example cq Backward Inference (Open valve) Inferred Inferring Cancelled D. R. Schlegel and S. C. Shapiro

28. Example cq Backward Inference (Open valve) Inferred Inferring Cancelled D. R. Schlegel and S. C. Shapiro

29. Example cq Backward Inference (Open valve) Inferred Inferring Cancelled D. R. Schlegel and S. C. Shapiro

30. Example cq Backward Inference (Open valve) Inferred Inferring Cancelled D. R. Schlegel and S. C. Shapiro

31. Example cq Backward Inference (Open valve) Inferred Inferring Cancelled D. R. Schlegel and S. C. Shapiro

32. Example cq Backward Inference (Open valve) Inferred Inferring Cancelled D. R. Schlegel and S. C. Shapiro

33. Example cq Backward Inference (Open valve) Inferred Inferring Cancelled D. R. Schlegel and S. C. Shapiro

34. Example cq Backward Inference (Open valve) Inferred Inferring Cancelled D. R. Schlegel and S. C. Shapiro

35. Example cq Backward Inference (Open valve) Inferred Inferring Cancelled D. R. Schlegel and S. C. Shapiro

36. Evaluation • Concurrency: • Near linear performance improvement with the number of processors • Performance robustto graph depth and branching factor changes. • Scheduling Heuristics: • Backward-inference with or-entailment shows 10x improvement over LIFO queues, and 20-40x over FIFO queues. D. R. Schlegel and S. C. Shapiro

37. Acknowledgements This work has been supported by a Multidisciplinary University Research Initiative (MURI) grant (Number W911NF-09- 1-0392) for Unified Research on Network-based Hard/Soft Information Fusion, issued by the US Army Research Office (ARO) under the program management of Dr. John Lavery. D. R. Schlegel