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
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Problem Statement
• Rise of multi-core computers
• Lack of concurrent natural deduction systems
A Motivation
• Application to natural language understanding
for terrorist plot detection.
D. R. Schlegel
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What are Inference Graphs?
• Graphs for natural deduction
– Four types of inference:
•
•
•
•
Forward
Backward
Bi-directional
Focused
– Retain derived formulas for later re-use.
– Propagate disbelief.
– Built upon Propositional Graphs.
• Take advantage of multi-core computers
– Concurrency and scheduling
– Near-linear speedup.
D. R. Schlegel
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Propositional Graphs
•
•
Directed acyclic graph
Every well-formed expression is a node
– Individual constants
– Functional terms
– Atomic formulas
– Non-atomic formulas (“rules”)
• Each node has an identifier, either
–
Symbol, or
– wfti[!]
• No two nodes with same identifier.
D. R. Schlegel and S. C. Shapiro
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Propositional Graphs
a
b
and-ant
wft1!
cq
c
and-ant
If a, and b are true, then c is true.
D. R. Schlegel and S. C. Shapiro
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Inference Graphs
• Extend Propositional Graphs
• Add 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.
u-channel
i-channel
a
b
and-ant
wft1!
cq
c
and-ant
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Messages
• Five kinds
–
–
–
–
–
I-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
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Priorities
• Messages have priorities.
–
–
–
–
UNASSERT is top priority
CANCEL-INFER is next
I-INFER/U-INFER are higher priority closer to a result
BACKWARD-INFER is lowest
D. R. Schlegel and S. C. Shapiro
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Rule Node Inference
1. Message arrives at node.
Assume we have a KB with a ^ b -> c, and b.
Then a is asserted with forward inference.
i-infer
a!
b!
a : true
and-ant
wft1!
cq
c
and-ant
A message is sent from a to wft1
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Rule Node Inference
2. Message is translated to Rule Use Information
a : true
a!
b!
and-ant
1 Positive Antecedent, a
0 Negative Antecedents
2 Total Antecedents
wft1!
cq
c
and-ant
Rule Use Information is stored in rule nodes
to be combined later with others that arrive.
D. R. Schlegel and S. C. Shapiro
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Rule Node Inference
3. Combine RUIs with any existing ones
1 Positive Antecedent, b
0 Negative Antecedents
2 Total Antecedents
a!
b!
1 Positive Antecedent, a
+ 0 Negative Antecedents
2 Total Antecedents
and-ant
wft1!
=
2 Positive Antecedents, a,b
0 Negative Antecedents
2 Total Antecedents
cq
c
and-ant
Combine the RUI for a with the one which
already exists in wft1 for b.
D. R. Schlegel and S. C. Shapiro
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Rule Node Inference
4. Determine if the rule can fire.
2 Positive Antecedents, a,b
0 Negative Antecedents
2 Total Antecedents
a!
b!
and-ant
wft1!
cq
c
and-ant
We have two positive antecedents, and we
need two. The rule can fire.
D. R. Schlegel and S. C. Shapiro
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Rule Node Inference
5. Send out new messages.
u-infer
c : true
a!
b!
and-ant
wft1!
cq
c
and-ant
c will receive the message, and assert itself.
D. R. Schlegel
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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.
wft1!
cq
ant
a
b
ant
cq
wft2!
D. R. Schlegel and S. C. Shapiro
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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
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Concurrency and Scheduling
• A task only operates within a single segment.
1.
tasks for relaying newly derived
information using segments “later” in the
derivation are executed before “earlier”
ones, and
2.
once 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. Shapiro
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Example
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.
D. R. Schlegel and S. C. Shapiro
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Example
cq
Backward Inference
(Open valve)
Inferring
D. R. Schlegel and S. C. Shapiro
Inferred
Cancelled
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Example
cq
Backward Inference
(Open valve)
Inferring
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Inferred
Cancelled
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Example
cq
Backward Inference
(Open valve)
Inferring
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Inferred
Cancelled
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Example
cq
Backward Inference
(Open valve)
Inferring
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Inferred
Cancelled
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Example
cq
Backward Inference
(Open valve)
Inferring
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Inferred
Cancelled
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Example
cq
Backward Inference
(Open valve)
Inferring
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Inferred
Cancelled
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Example
cq
Backward Inference
(Open valve)
Inferring
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Inferred
Cancelled
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Example
cq
Backward Inference
(Open valve)
Inferring
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Inferred
Cancelled
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Example
cq
Backward Inference
(Open valve)
Inferring
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Inferred
Cancelled
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Example
cq
Backward Inference
(Open valve)
Inferring
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Inferred
Cancelled
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Example
cq
Backward Inference
(Open valve)
Inferring
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Inferred
Cancelled
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Example
cq
Backward Inference
(Open valve)
Inferring
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Inferred
Cancelled
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Example
cq
Backward Inference
(Open valve)
Inferring
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Inferred
Cancelled
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Example
cq
Backward Inference
(Open valve)
Inferring
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Inferred
Cancelled
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Example
cq
Backward Inference
(Open valve)
Inferring
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Inferred
Cancelled
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Example
cq
Backward Inference
(Open valve)
Inferring
D. R. Schlegel and S. C. Shapiro
Inferred
Cancelled
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Example
cq
Backward Inference
(Open valve)
Inferring
D. R. Schlegel and S. C. Shapiro
Inferred
Cancelled
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Example
cq
Backward Inference
(Open valve)
Inferring
D. R. Schlegel and S. C. Shapiro
Inferred
Cancelled
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Evaluation
• Concurrency:
– Near linear performance improvement with the
number of processors
– Performance 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
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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. Schlegel
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