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Practical Partition-Based Theorem Proving for Large Knowledge Bases

8/14/03. Bill MacCartney, Stanford KSL. 2. Motivation. Goal: to enable automated reasoners to exploit the implicit structure of large knowledge basesReasoners in big KBs face combinatorial explosionMaking headway often requires KB-specific manual tuningBut, large commonsense KBs contain structure

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Practical Partition-Based Theorem Proving for Large Knowledge Bases

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    1. Practical Partition-Based Theorem Proving for Large Knowledge Bases Bill MacCartney (Stanford KSL) Sheila A. McIlraith (Stanford KSL) Eyal Amir (UC Berkeley) Tomas Uribe (SRI) with thanks to Mark Stickel (SRI)

    2. 8/14/03 Bill MacCartney, Stanford KSL 2 Motivation Goal: to enable automated reasoners to exploit the implicit structure of large knowledge bases Reasoners in big KBs face combinatorial explosion Making headway often requires KB-specific manual tuning But, large commonsense KBs contain structure Loosely-coupled clusters of domain knowledge Partitioning aims to speed reasoning by: Decomposing graph structure of KB into a tree of partitions Propagating results between partitions using message-passing Thereby, focusing proof search and ignoring the irrelevant

    3. 8/14/03 Bill MacCartney, Stanford KSL 3 Outline Background: partition-based reasoning Algorithms for automatic partitioning of large KBs The MP algorithm for reasoning with partitions Experimental evaluation of MP Partition-derived ordering (PDO) Automatic alternative to hand-crafted symbol orderings MP with focused support (MFS) Enhancing vanilla MP with a smart within-partition strategy Combinations of strategies Can outperform set-of-support by 10x or more

    4. 8/14/03 Bill MacCartney, Stanford KSL 4 The espresso machine theory

    5. 8/14/03 Bill MacCartney, Stanford KSL 5 Outline Background: partition-based reasoning Algorithms for automatic partitioning of large KBs The MP algorithm for reasoning with partitions Experimental evaluation of MP Partition-derived ordering (PDO) Automatic alternative to hand-crafted symbol orderings MP with focused support (MFS) Enhancing vanilla MP with a smart within-partition strategy Combinations of strategies Can outperform set-of-support by 10x or more

    6. 8/14/03 Bill MacCartney, Stanford KSL 6 Automatic partitioning

    7. 8/14/03 Bill MacCartney, Stanford KSL 7 Automatic partitioning

    8. 8/14/03 Bill MacCartney, Stanford KSL 8 Automatic partitioning

    9. 8/14/03 Bill MacCartney, Stanford KSL 9 Outline Background: partition-based reasoning Algorithms for automatic partitioning of large KBs The MP algorithm for reasoning with partitions Experimental evaluation of MP Partition-derived ordering (PDO) Automatic alternative to hand-crafted symbol orderings MP with focused support (MFS) Enhancing vanilla MP with a smart within-partition strategy Combinations of strategies Can outperform set-of-support by 10x or more

    10. 8/14/03 Bill MacCartney, Stanford KSL 10 Reasoning with MP MP Algorithm [Amir & McIlraith 2000]

    11. 8/14/03 Bill MacCartney, Stanford KSL 11 MP in action A simple propositional theory

    12. 8/14/03 Bill MacCartney, Stanford KSL 12 Reasoning is performed locally in each partition Relevant results propagate toward goal partition Globally sound & complete provided each local reasoner is sound & complete for Li-consequence finding [Amir & McIlraith 2000] Performance is worst-case exponential within partitions, but linear in tree structure Characteristics of MP

    13. 8/14/03 Bill MacCartney, Stanford KSL 13 Outline Background: partition-based reasoning Algorithms for automatic partitioning of large KBs The MP algorithm for reasoning with partitions Experimental evaluation of MP Partition-derived ordering (PDO) Automatic alternative to hand-crafted symbol orderings MP with focused support (MFS) Enhancing vanilla MP with a smart within-partition strategy Combinations of strategies Can outperform set-of-support by 10x or more

    14. 8/14/03 Bill MacCartney, Stanford KSL 14 Experimental Evaluation of MP Do real world KBs exhibit inherent structure? Do they have good tree decompositions (partition graphs)? Can partition-based reasoning outperform other strategies? Experimental testbed Theorem prover: SNARK KB: Cyc A subset on spatial relationships, ~750 axioms, ~150 symbols Were working on adding SUMO, others Queries from outside source Number of resolution steps used as chief performance metric Normalized to number of steps required using no strategy

    15. 8/14/03 Bill MacCartney, Stanford KSL 15 Comparison to conventional strategies Restriction strategies focus proof search Disallow some resolution steps to speed search Completeness issues are critical Set-of-support restriction Place the negated query into a designated set of support Allow only resolutions involving a clause from the set of support Add newly-derived clauses to set of support Ordering restriction Define a global ordering among predicates Resolve on predicates in order from greatest to least (SNARK provides a default ordering, which is arbitrary)

    16. 8/14/03 Bill MacCartney, Stanford KSL 16 Experimental results: vanilla MP

    17. 8/14/03 Bill MacCartney, Stanford KSL 17 Outline Background: partition-based reasoning Algorithms for automatic partitioning of large KBs The MP algorithm for reasoning with partitions Experimental evaluation of MP Partition-derived ordering (PDO) Automatic alternative to hand-crafted symbol orderings MP with focused support (MFS) Enhancing vanilla MP with a smart within-partition strategy Combinations of strategies Can outperform set-of-support by 10x or more

    18. 8/14/03 Bill MacCartney, Stanford KSL 18 Motivation for PDO

    19. 8/14/03 Bill MacCartney, Stanford KSL 19 How PDO works Generate a partition-derived ordering Direct edges of partition graph toward goal partition Perform topological sort on partitions Beginning with partitions furthest from goal, progressively append symbols from each partition to ordering

    20. 8/14/03 Bill MacCartney, Stanford KSL 20 Experimental results: PDO

    21. 8/14/03 Bill MacCartney, Stanford KSL 21 Outline Background: partition-based reasoning Algorithms for automatic partitioning of large KBs The MP algorithm for reasoning with partitions Experimental evaluation of MP Partition-derived ordering (PDO) Automatic alternative to hand-crafted symbol orderings MP with focused support (MFS) Enhancing vanilla MP with a smart within-partition strategy Combinations of strategies Can outperform set-of-support by 10x or more

    22. 8/14/03 Bill MacCartney, Stanford KSL 22 MP with focused support (MFS)

    23. 8/14/03 Bill MacCartney, Stanford KSL 23 Experimental results: MFS

    24. 8/14/03 Bill MacCartney, Stanford KSL 24 Outline Background: partition-based reasoning Algorithms for automatic partitioning of large KBs The MP algorithm for reasoning with partitions Experimental evaluation of MP Partition-derived ordering (PDO) Automatic alternative to hand-crafted symbol orderings MP with focused support (MFS) Enhancing vanilla MP with a smart within-partition strategy Combinations of strategies Can outperform set-of-support by 10x or more

    25. 8/14/03 Bill MacCartney, Stanford KSL 25 Strategy combinations

    26. 8/14/03 Bill MacCartney, Stanford KSL 26 Experimental results: strategy combos

    27. 8/14/03 Bill MacCartney, Stanford KSL 27 Experimental results: strategy combos

    28. 8/14/03 Bill MacCartney, Stanford KSL 28 Conclusions and Future Work

    29. 8/14/03 Bill MacCartney, Stanford KSL 29 References

    30. 8/14/03 Bill MacCartney, Stanford KSL 30 Thanks!

    31. 8/14/03 Bill MacCartney, Stanford KSL 31 Results: automatic partitioning Partition graph is largely independent of query But edges may need to be redirected Were experimenting with multiple algorithms

    32. 8/14/03 Bill MacCartney, Stanford KSL 32 Queries

    33. 8/14/03 Bill MacCartney, Stanford KSL 33 Automatic partitioning

    34. 8/14/03 Bill MacCartney, Stanford KSL 34 MP in action

    35. 8/14/03 Bill MacCartney, Stanford KSL 35 MP in action

    36. 8/14/03 Bill MacCartney, Stanford KSL 36 Ongoing research

    37. 8/14/03 Bill MacCartney, Stanford KSL 37 Recap: automatic partitioning Begin with a KB in PL or FOL

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