Concurrent Reasoning with Inference Graphs - PowerPoint Presentation

Slide1

Concurrent Reasoning with Inference Graphs

Daniel R. Schlegel and Stuart C. Shapiro

Department of Computer Science and Engineering

University at Buffalo, The State University of New York

Buffalo, New York, USA<drschleg,shapiro>@buffalo.edu

D. R. Schlegel and S. C. Shapiro

1Slide2

Problem Statement

Rise of multi-core computersLack of concurrent natural deduction systemsApplication to natural language understanding

for terrorist plot detection.D. R. Schlegel2

A MotivationSlide3

What are Inference Graphs?

Graphs for natural deductionFour types of inference:ForwardBackwardBi-directional

FocusedRetain derived formulas for later re-use.Propagate disbelief.Built upon Propositional Graphs.Take advantage of multi-core computersConcurrency and schedulingNear-linear speedup.

D. R. Schlegel

3Slide4

Propositional Graphs

Directed acyclic graphEvery well-formed expression is a nodeIndividual constants

Functional termsAtomic formulasNon-atomic formulas (“rules”)Each node has an identifier, eitherSymbol, orwfti

[!]No two nodes with same identifier.D. R. Schlegel and S. C. Shapiro

4Slide5

Propositional Graphs

D. R. Schlegel and S. C. Shapiro5

If a, and b are true, then c is true.

a

c

and-ant

cq

wft1!

b

and-antSlide6

Inference Graphs

Extend Propositional GraphsAdd channels for information flow

(messages):i-channels report truth of an antecedent to a rule node.u-channels

report truth of a consequent from a rule node.Channels contain

valves.Hold messages back, or allow them through.D. R. Schlegel and S. C. Shapiro

6

i

-channel

u

-channel

a

c

and-ant

cq

wft1!

b

and-antSlide7

Messages

Five kindsI-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

7Slide8

Priorities

Messages have priorities.UNASSERT is top priorityCANCEL-INFER is nextI-INFER/U-INFER are higher priority closer to a result

BACKWARD-INFER is lowestD. R. Schlegel and S. C. Shapiro8Slide9

Rule Node Inference

Message arrives at node.

D. R. Schlegel and S. C. Shapiro9

a!

c

Assume we have a KB with a ^ b -> c, and b. Then a is asserted with forward inference.

A message is sent from a to wft1

and-ant

cq

a : true

wft1!

i

-infer

b

!

and-antSlide10

Rule Node Inference

2. Message is translated to

Rule Use InformationD. R. Schlegel and S. C. Shapiro

10

1 Positive Antecedent, a0 Negative Antecedents2 Total Antecedents

a : true

Rule Use Information is stored in rule nodes to be combined later with others that arrive.

a!

c

and-ant

cq

wft1!

b

!

and-antSlide11

Rule Node Inference

3

. Combine RUIs with any existing onesD. R. Schlegel and S. C. Shapiro

11

1 Positive Antecedent, b0 Negative Antecedents2 Total AntecedentsCombine the RUI for a with the one which already exists in wft1 for b.

1 Positive Antecedent,

a

0 Negative Antecedents

2

Total Antecedents

+

2

Positive Antecedents,

a,b

0 Negative Antecedents

2

Total Antecedents

=

a!

c

and-ant

cq

wft1!

b

!

and-antSlide12

Rule Node Inference

4. Determine if the rule can fire.

D. R. Schlegel and S. C. Shapiro12

We have two positive antecedents, and we need two. The rule can fire.

2 Positive Antecedents, a,b0 Negative Antecedents2

Total Antecedents

a!

c

and-ant

cq

wft1!

b

!

and-antSlide13

Rule Node Inference

5. Send out new messages.

D. R. Schlegel13

c will receive the message, and assert itself.

c : true

u-infer

a!

c

and-ant

cq

wft1!

b

!

and-antSlide14

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.D. R. Schlegel and S. C. Shapiro14

a

b

ant

cq

wft2!

wft1!

ant

cqSlide15

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

15Slide16

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, andonce 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. Shapiro16Slide17

Example

D. R. Schlegel and S. C. Shapiro17

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.Slide18

Example

D. R. Schlegel and S. C. Shapiro18

Backward Inference

(Open valve)

Inferring

Inferred

Cancelled

cqSlide19

Example

D. R. Schlegel and S. C. Shapiro19

Backward Inference

(Open valve)

Inferring

Inferred

Cancelled

cqSlide20

Example

D. R. Schlegel and S. C. Shapiro20

Backward Inference

(Open valve)

Inferring

Inferred

Cancelled

cqSlide21

Example

D. R. Schlegel and S. C. Shapiro21

Backward Inference

(Open valve)

Inferring

Inferred

Cancelled

cqSlide22

Example

D. R. Schlegel and S. C. Shapiro22

Backward Inference

(Open valve)

Inferring

Inferred

Cancelled

cqSlide23

Example

D. R. Schlegel and S. C. Shapiro23

Backward Inference

(Open valve)

Inferring

Inferred

Cancelled

cqSlide24

Example

D. R. Schlegel and S. C. Shapiro24

Backward Inference

(Open valve)

Inferring

Inferred

Cancelled

cqSlide25

Example

D. R. Schlegel and S. C. Shapiro25

Backward Inference

(Open valve)

Inferring

Inferred

Cancelled

cqSlide26

Example

D. R. Schlegel and S. C. Shapiro26

Backward Inference

(Open valve)

Inferring

Inferred

Cancelled

cqSlide27

Example

D. R. Schlegel and S. C. Shapiro27

Backward Inference

(Open valve)

Inferring

Inferred

Cancelled

cqSlide28

Example

D. R. Schlegel and S. C. Shapiro28

Backward Inference

(Open valve)

Inferring

Inferred

Cancelled

cqSlide29

Example

D. R. Schlegel and S. C. Shapiro29

Backward Inference

(Open valve)

Inferring

Inferred

Cancelled

cqSlide30

Example

D. R. Schlegel and S. C. Shapiro30

Backward Inference

(Open valve)

Inferring

Inferred

Cancelled

cqSlide31

Example

D. R. Schlegel and S. C. Shapiro31

Backward Inference

(Open valve)

Inferring

Inferred

Cancelled

cqSlide32

Example

D. R. Schlegel and S. C. Shapiro32

Backward Inference

(Open valve)

Inferring

Inferred

Cancelled

cqSlide33

Example

D. R. Schlegel and S. C. Shapiro

33

Backward Inference

(Open valve)

Inferring

Inferred

Cancelled

cqSlide34

Example

D. R. Schlegel and S. C. Shapiro

34

Backward Inference

(Open valve)

Inferring

Inferred

Cancelled

cqSlide35

Example

D. R. Schlegel and S. C. Shapiro35

Backward Inference

(Open valve)

Inferring

Inferred

Cancelled

cqSlide36

Evaluation

Concurrency:Near linear performance improvement with the number of processorsPerformance

robust to 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

36Slide37

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. Schlegel37