PartitionBased Logical Reasoning

1
Partition-BasedLogical 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

2
Motivation

• With large KBs, general-purpose reasonerssuffer
  from combinatorial explosion.
• Can we focus reasoning by decomposing the KB into
  anetwork 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?

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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

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Automatic partitioning

• Begin with a KB in PL or FOL

• 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

Efficient reasoning depends on keepingpartition
sizes and link sizes small
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Reasoning with partitions an example

• A simple propositional theory

Query Q ? U ? V ? Z ?
(21) W
(16) ?R ? T
(22) ?W ? Y ? Z
(17) S ? T
(18) T
(23) ?W ? Z
(18) T
(19) ?U ? ?V ? W
(24) Z
(20) ?V ? W
(25) ?
(21) W
Using partitioning, this query took just 10
resolution steps. Using set-of-support, the same
query can take 28 steps.
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Reasoning with partitions using MP

• MP Algorithm
• Amir McIlraith 2000

• Start with a tree-structured partition graph

• Identify goal partition

• Direct edges toward goal
• (fixing outbound link language Li for each
  partition)

• Concurrently, in each partition
• Generate consequences in Li

• Pass messages in Li toward goal

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Characteristics of MP

• 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
• Performance is worst-caseexponential within
  partitions, but linear in tree structure

Minimizesbetween-partitiondeduction
Focuseswithin-partitiondeduction
Supports parallel processing
Different reasoners in different partitions
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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
• Were working on adding SUMO, Geo-Logica, RCC-8
• Queries
• Cyc queries provided by Vinay Chaudhri

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Results automatic partitioning

• Partition graph is largely independent of query
• But edges may need to be redirected
• Were experimenting with multiple algorithms

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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-supportto
  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?

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Vanilla MP vs. common strategies
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Smart MP vs. set-of-support
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Partition-derived ordering (PDO)
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MP and PDO vs. SoS
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Ongoing research

• Testing on more KBs
• Partition-derived symbol orderings
• Can we beat hand-crafted symbol orderings?
• Within-partition restriction strategies
• Focus reasoning on Li-consequence finding
• Completeness results
• When is partitioning set-of-support complete?
• Distributed implementations
• Demonstrate integration of heterogeneous reasoners

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Conclusions

• Partitioning can speed up reasoning
• Makes large KBs tractable by exploiting implicit
  structure
• Reasoning becomes significantly more focused and
  efficient
• Smarter local strategies should do even better
• Partition-derived ordering is surprisingly
  effective
• Especially when combined with set-of-support
• Automatic alternative to hand-crafted orderings
• Partitioning supports heterogeneouslocal
  reasoners
• Efficient special-purpose reasoners can be
  cleanly integrated
• MP ensures global soundness completeness

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References

• Web
• www.ksl.stanford.edu/projects/RKF/Partitioning/
• 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.

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The End

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Example
The espresso machine theory
(1) ok-pump ? on-pump ? water (2) man-fill ?
water (3) man-fill ? ?on-pump (4) ?man-fill ?
on-pump (5) water ? ok-boiler ? on-boiler ?
steam (6) ?water ? ?steam (7) ?on-boiler ?
?steam (8) ?ok-boiler ? ?steam (9) steam ? coffee
? hot-drink (10) steam ? tea ? hot-drink (11) coff
ee ? tea
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Example partitioning
Step 1 construct symbol graph
(1) ?ok-pump ? ?on-pump ? water (2) ?man-fill ?
water (3) ?man-fill ? ?on-pump (4) man-fill ?
on-pump (5) ?water ? ?ok-boiler ? ?on-boiler ?
steam (6) water ? ?steam (7) on-boiler ?
?steam (8) ok-boiler ? ?steam (9) ?steam ?
?coffee ? hot-drink (10) ?steam ? ?tea ?
hot-drink (11) coffee ? tea
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Example partitioning
Step 2 graph decomposition
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Example partitioning
Step 3 generate partition graph
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Example add query to partition graph
Query If the pump is OK and the boiler is OK and
the boiler is on, do we get a hot drink?
(12) ok-pump
(1) ?ok-pump ? ?on-pump ? water (2) ?man-fill ?
water (3) ?man-fill ? ?on-pump (4) man-fill ?
on-pump
water
(13) ok-boiler
(5) ?water ? ?ok-boiler ? ?on-boiler ?
steam (6) water ? ?steam (7) on-boiler ?
?steam (8) ok-boiler ? ?steam
(14) on-boiler
steam
(9) ?steam ? ?coffee ? hot-drink (10) ?steam ?
?tea ? hot-drink (11) coffee ? tea
(15) ?hot-drink
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Example of MP
Using set-of-support, SNARK took 28 steps to
prove this. Using partitioning, SNARK took just
11 steps.
(16) ?on-pump ? water
(17) man-fill ? water
(18) water
(19) ?ok-boiler ? ?on-boiler ? steam
(20) steam
(21) ?steam ? tea ? hot-drink
(22) ?steam ? hot-drink
(23) hot-drink
(24) ?
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Automatic partitioning
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Queries
hd-q1 If the pump is OK and the boiler is OK and
the boiler is on, do we get a hot
drink? cyc-p5 If A and B are inside C, can C be
inside A? cyc-p7 If A and B are part of C and C
is at D, where is A? cyc-p1 Suppose that A is
touching B and B is inside C and C is at D. Is A
at D? cyc-v5 A has parts B, C, and D. B has
parts E, and F. Is F near A? cyc-p3 If C is
between A and B, and both A and B are inside D,
and D is at E, is C at E? cyc-p4 If C is between
A and B, and both A and B are at D, is C also at
D?