Concurrent Reasoning with Inference Graphs
Daniel R. Schlegel
Stuart C. Shapiro
Department of Computer Science and Engineering
Problem Summary
• Rise of multi-core computers, BUT:
• Lack of concurrent natural deduction systems.
Inference Capabilities
• Forward, backward, bi-directional, and focused
inference.
• Retains all derived formulas for later re-use.
• Propagates disbelief.
Only concurrent inference system with these
capabilities.
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.
Example:
Propositional graph for the assertions that if a, b,
and c are true, then d is true, and if d or e are true,
then f is true.
Inference Graphs
Concurrency and Scheduling
• Extend Propositional Graphs
Example:
• Adds channels for information flow:
• i-channels report truth of an
antecedent to a rule node.
• u-channels report truth of
a consequent from a rule node.
Channels represented by dashed lines
• Channels contain valves.
are i-channels and are drawn from
• Hold messages back, or allow
antecedents to rule nodes. Channels
them through.
represented by dotted lines are u channels and are drawn from rule nodes
• Channels relay messages
to consequents.
• I-INFER (“I’ve been inferred”)
• U-INFER (“You’ve been inferred”)
• BACKWARD-INFER (“Open valves so messages that might infer me can arrive”)
• CANCEL-INFER (“Stop inferring me (close valves)”)
• UNASSERT (“I’m no longer believed”)
• Different message types have different relative priorities (important for scheduling).
The area between two valves is called an
inference segment.
Rule Node Inference
1.
2.
3.
4.
5.
Message arrives at node.
Message translated to a RUI, containing
positive and negative instances of antecedents
contained in the message.
New RUI combined with existing ones.
Output is a set of new RUIs which are used to
decide of the rule can fire.
When a rule fires, new messages are sent out.
Example: We assume backward inference has been
initiated, opening all the valves in the graph. First, in (a),
messages about the truth of a, b, and c flow through ichannels to wft1. Since wft1 is and-entailment, each of
its antecedents must be true for it to fire. Since they are,
in (b) the message that d is true flows through wft1’s uchannel. d becomes asserted and reports its new status
through its i-channel (c). In (d), wft2 receives this
information, and since it is an or-entailment rule and
requires only a single antecedent to be true for it to fire,
it reports to its consequents that they are now true, and
cancels inference in e. Finally, in (e), f is asserted, and
inference is complete.
When a message passes through a valve:
• A task is created with the same priority as the
message, and is the application of the
inference segment’s function to the
message.
• The task is added to a queue which puts
higher priority tasks towards its head.
A task only operates within a task segment.
1. tasks for relaying newly derived
information using segments to the right are
executed before those to the left, and
2. once a node is known to be true or false,
all tasks attempting to derive it (left of it in
the graph) are canceled, as long as their
results are not needed elsewhere.
There is minimal shared state between tasks,
allowing many tasks to operate concurrently.
Evaluation
Concurrency:
• Near linear performance improvement with the
number of processors
• Performance resilient 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.
See GKR paper (below) for more details.
References
• Daniel R. Schlegel and Stuart C. Shapiro, Concurrent Reasoning with
Inference Graphs. In Proceedings of the Third International IJCAI Workshop
on Graph Structures for Knowledge Representation and Reasoning (GKR
2013), 2013, in press.
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.