D. R. Schlegel and S. C. Shapiro 1

Concurrent Reasoning with Inference GraphsDaniel R. Schlegel and Stuart C. Shapiro

Department of Computer Science and EngineeringUniversity at Buffalo, The State University of New York

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

D. R. Schlegel 2

Problem Statement

• Rise of multi-core computers• Lack of concurrent natural deduction systems

• Application to natural language understanding for terrorist plot detection.

A Motivation

D. R. Schlegel 3

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.

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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.

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Propositional Graphs

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

a cand-ant cqwft1!

b and-ant

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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.

i-channel u-channel

a cand-ant cqwft1!

b 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”

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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

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Rule Node Inference

1. Message arrives at node.

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 : truewft1!

i-infer

b! and-ant

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Rule Node Inference

2. Message is translated to Rule Use Information

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! cand-ant cqwft1!

b! and-ant

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Rule Node Inference

3. Combine RUIs with any existing ones

1 Positive Antecedent, b0 Negative Antecedents2 Total Antecedents

Combine the RUI for a with the one which already exists in wft1 for b.

1 Positive Antecedent, a0 Negative Antecedents2 Total Antecedents

+2 Positive Antecedents, a,b0 Negative Antecedents2 Total Antecedents

=

a! cand-ant cqwft1!

b! and-ant

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Rule Node Inference

4. Determine if the rule can fire.

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

2 Positive Antecedents, a,b0 Negative Antecedents2 Total Antecedents

a! cand-ant cqwft1!

b! and-ant

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Rule Node Inference

5. Send out new messages.

c will receive the message, and assert itself.

c : trueu-infer

a! cand-ant cqwft1!

b! and-ant

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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.

a bant

cq

wft2!

wft1!ant

cq

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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.

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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.

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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.

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Example

Backward Inference(Open valve) Inferring Inferred Cancelled

cq

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Example

Backward Inference(Open valve) Inferring Inferred Cancelled

cq

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Example

Backward Inference(Open valve) Inferring Inferred Cancelled

cq

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Example

Backward Inference(Open valve) Inferring Inferred Cancelled

cq

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Example

Backward Inference(Open valve) Inferring Inferred Cancelled

cq

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Example

Backward Inference(Open valve) Inferring Inferred Cancelled

cq

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Example

Backward Inference(Open valve) Inferring Inferred Cancelled

cq

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Example

Backward Inference(Open valve) Inferring Inferred Cancelled

cq

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Example

Backward Inference(Open valve) Inferring Inferred Cancelled

cq

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Example

Backward Inference(Open valve) Inferring Inferred Cancelled

cq

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Example

Backward Inference(Open valve) Inferring Inferred Cancelled

cq

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Example

Backward Inference(Open valve) Inferring Inferred Cancelled

cq

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Example

Backward Inference(Open valve) Inferring Inferred Cancelled

cq

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Example

Backward Inference(Open valve) Inferring Inferred Cancelled

cq

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Example

Backward Inference(Open valve) Inferring Inferred Cancelled

cq

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Example

Backward Inference(Open valve) Inferring Inferred Cancelled

cq

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Example

Backward Inference(Open valve) Inferring Inferred Cancelled

cq

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Example

Backward Inference(Open valve) Inferring Inferred Cancelled

cq

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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.

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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.