partition-based logical reasoning

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?


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


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


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 ?


• 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


• 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

Characteristics of MP

Minimizesbetween-partition

deduction

Supports parallel processing

Different reasonersin different partitions

Focuseswithin-partition

deduction


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


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


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?


“Vanilla” MP vs. common strategies

Performance of MP and other common strategies(relative to no strategy)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

None Ordering Set-of-support MP (alg 5) MP (alg 6)

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“Smart” MP vs. set-of-support

Performance of MP and MP + SoS(relative to no strategy)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

SoS MP5 MP6 MP5 + SoS MP6 + SoS

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Partition-derived ordering (PDO)

Performance of partition-derived ordering(relative to no strategy)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

SoS PDO5 PDO6 PDO5 + SoS PDO6 + SoS

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q1

q2

q3

q4

q5

q6

q7


MP and PDO vs. SoS

Performance of MP + SoS and PDO + SoS(relative to SoS)

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

SoS MP5 + SoS MP6 + SoS PDO5 + SoS PDO6 + SoS

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q1

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q5

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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


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


Webwww.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.

References


The End


Example

(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

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) coffee tea

Example: partitioning

hot-drink tea

coffeesteam

on-boiler

ok-boiler

water

on-pump

ok-pumpman-fill

Step 1: construct symbol graph



hot-drink tea

coffee

steam

on-boiler

ok-boiler

water

on-pump

ok-pumpman-fill

steam

water

hot-drink tea

coffee

steam on-boiler

ok-boiler

water on-pump

ok-pumpman-fill

water

steam

Step 2: graph decomposition

steam

water



steam

water

hot-drink tea

coffee

steam on-boiler

ok-boiler

water on-pump

ok-pumpman-fill

water

steam

(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

steam

water

Step 3: generate partition graph


Example: add query to partition graph




steam

water

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

(13) ok-boiler(14) on-boiler

(15) hot-drink





steam

water

(12) ok-pump

(13) ok-boiler(14) on-boiler

(15) hot-drink

Example of MP

(16) on-pump water(17) man-fill water(18) water

water

steam

(19) ok-boiler on-boiler steam(20) steam

(21) steam tea hot-drink(22) steam hot-drink(23) hot-drink

Using set-of-support, SNARK took 28 steps to prove this.Using partitioning, SNARK took just 11 steps.

(24)


Automatic partitioning


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?

partition-based logical reasoning

Documents

partition sizes

libill maccartney

stanford kslreasoning

stanford ksloverviewalgorithms

based reasoning

folefficient reasoning

efficient global reasoning

connected partitions