Partition-Based Logical Reasoning Bill MacCartney (KSL), Sheila A. McIlraith (KSL), Eyal Amir (FRG/Berkeley), Tomas Uribe (SRI) Richard Fikes and John McCarthy Knowledge Systems Lab | Formal Reasoning Group Stanford University with thanks to Mark Stickel and Vinay Chaudhri of SRI
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Partition-Based Logical Reasoning Bill MacCartney (KSL), Sheila A. McIlraith (KSL), Eyal Amir (FRG/Berkeley), Tomas Uribe (SRI) Richard Fikes and John.
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Partition-BasedLogical Reasoning
Bill MacCartney (KSL), Sheila A. McIlraith (KSL),Eyal Amir (FRG/Berkeley), Tomas Uribe (SRI)
Richard Fikes and John McCarthyKnowledge Systems Lab | Formal Reasoning Group
Stanford University
with thanks to Mark Stickel and Vinay Chaudhri of SRI
11/14/02Bill MacCartney, Stanford KSL
Motivation
• With large KBs, general-purpose reasonerssuffer from combinatorial explosion. Can we focus reasoning by decomposing the KB into a
network of minimally-connected partitions?
• Special-purpose reasoners can be highly efficient in specific domains, but how to integrate them? Given a network of (possibly heterogeneous) knowledge
systems, how can we achieve efficient global reasoning?
• Can we exploit implicit structure of knowledge to make reasoning more focused & efficient?
11/14/02Bill MacCartney, Stanford KSL
Overview
• Algorithms and theoretical results Automatic partitioning of large KBs Reasoning with partitions using message passing (MP)
• Experimental testing Empirical validation of the effectiveness of partitioning Even better when combined with good local strategies
• Surprising, productive results Partitioning can induce near-optimal symbol orderings MP can integrate special-purpose reasoners Many new research questions
11/14/02Bill MacCartney, Stanford KSL
Automatic partitioning
• Begin with a KB in PL or FOL
Efficient reasoning depends on keeping
partition sizes and link sizes small
• Construct symbol graph Edges join symbols which appear together in an
axiom
• Apply tree decomposition algorithm “Alg 5”: a variant of min-fill “Alg 6”: a divide-and-conquer tree-width algorithm• Partition axioms correspondingly Each partition has its own vocabulary “Link languages” are defined by shared vocabulary
11/14/02Bill MacCartney, Stanford KSL
Reasoning with partitions: an example
A simple propositional theory
Theory {Q R S T U V W X Y Z}Partition 1 {Q R S T} Partition 2 {T U V W} Partition 3 {W X Y Z}{T} {W}
Partition 1 {Q R S T} Partition 2 {T U V W} Partition 3 {W X Y Z}{T} {W}
(1) Q R T(2) S T(3) S R
(4) S R
(5) T U V W(6) T W
(7) U W
(8) V W
(9) W X Z(10) X Y(11) W Y Z(15) Z
(12) Q (13) U
(14) V(16) R T(17) S T(18) T
(18) T
(19) U V W(20) V W(21) W
(21) W
(22) W Y Z(23) W Z(24) Z
(25)
Using partitioning, this query took just 10 resolution steps.
Using set-of-support, the same query can take 28 steps.
Query: Q U V Z ?
11/14/02Bill MacCartney, Stanford KSL
• Start with a tree-structured partition graph
Reasoning with partitions using MP
MP Algorithm[Amir & McIlraith 2000]
Pass messages in Li toward goal
• Identify goal partition
• Direct edges toward goal(fixing outbound link language Li for each partition)
• Concurrently, in each partition: Generate consequences in Li
11/14/02Bill MacCartney, Stanford KSL
• Reasoning is performed locally in each partition
• Globally sound & complete… provided each local reasoner is sound & complete for Li-consequence finding
• Performance is worst-caseexponential within partitions, but linear in tree structure
Characteristics of MP
Minimizesbetween-partition
deduction
Supports parallel processing
Different reasonersin different partitions
Focuseswithin-partition
deduction
11/14/02Bill MacCartney, Stanford KSL
Experimental Testing
• Do “real world” KBs exhibit inherent structure? Can we generate partitionings in which both partition sizes and
link language sizes are small? Can partition-based reasoning outperform other strategies?
• Experimental testbed Theorem prover: SNARK
– Thanks to Mark Stickel and SRI KB: Cyc
– A subset on spatial relationships, ~750 axioms, ~150 symbols– We’re working on adding SUMO, Geo-Logica, RCC-8
Queries– Cyc queries provided by Vinay Chaudhri
11/14/02Bill MacCartney, Stanford KSL
Results: automatic partitioning
• Partition graph is largely independent of query But edges may need to be redirected
• We’re experimenting with multiple algorithms
Alg 5 Alg 6
Number of partitions 124 40
Max symbols/partition 16 19
Max symbols/link 14 17
Max axioms/partition 80 95
Max partitions/axiom 25 28
Axioms in multiple partitions 152 152
11/14/02Bill MacCartney, Stanford KSL
Testing MP
• “Vanilla” MP vs. common restriction strategies Use MP with no local strategy Compare to no strategy, ordered resolution, set-of-support
• “Smart” MP vs. set-of-support In SNARK testbed, we use MP + set-of-support
to approximate MP with smart local strategy Within-partition restriction strategies should do better
• Partition-derived symbol ordering Use partitioning to induce symbol ordering Compare partition-derived ordering with set-of-support What if we combine them?
11/14/02Bill MacCartney, Stanford KSL
“Vanilla” MP vs. common strategies
Performance of MP and other common strategies(relative to no strategy)
Papers• Amir, E. and McIlraith, S., “Partition-Based Logical Reasoning for
First-Order and Propositional Theories,” Artificial Intelligence journal, accepted for publication.
• McIlraith, S. and Amir, E., “Theorem Proving with Structured Theories,” 17th International Joint Conference on Artificial Intelligence (IJCAI-01), 2001.
• Amir, E., “Efficient Approximation for Triangulation of Minimum Treewidth,” 17th Conference on Uncertainty in Artificial Intelligence (UAI ’01), 2001.
• Amir, E. and McIlraith, S., “Solving Satisfiability using Decomposition and the Most Constrained Subproblem.” Proceedings of SAT 2001, 2001.
• Amir, E. and McIlraith, S., “Partition-Based Logical Reasoning,” 7th International Conference on Principles of Knowledge Representation and Reasoning (KR ’2000), 2000.