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Concurrent Inference Graphs. Daniel R. Schlegel. Department of Computer Science and Engineering. Problem Summary. Subsumption Inference. L A – A Logic of Arbitrary and Indefinite Objects.

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Concurrent Inference Graphs

Daniel R. Schlegel

Department of Computer Science and Engineering

Problem Summary

Subsumption Inference

LA – A Logic of Arbitrary and Indefinite Objects

Inference graphs2 in their current form only support propositional logic. We expand it to support LA – A Logic of Arbitrary and Indefinite Objects.3

Note: Much of this is work in progress, advice and criticism are very welcome!

  • CSNePS will support at least 2 types of subsumption.4

  • Structural Subsumption: by their formal definitions, C1 is more general than C2.

    • Example: Since the arbitrary domesticated dog is friendly, the arbitrary white domesticated dog is friendly.

  • Recorded Subsumption: C1 is above C2 in a subsumption data structure.

  • A logic designed by Stuart C. Shapiro for:

    • KRR Systems

    • NL Understanding / Generation

    • Commonsense Reasoning

  • Uses arbitrary/indefinite terms, not universally/existentially quantified variables.

    • Structure sharing between terms.

    • Makes term subsumption possible.

Inference Components

  • Draws influences from:

  • RETE Networks

    • Alpha Networks

    • Beta Networks

    • Terminal Node

    • Token

  • Truth Maintenance Systems

    • LATMS Constraints

    • LTMS Node

  • Active Connection Graphs

    • Report

    • Filter

    • Switch

    • P-Tree

    • S-Index

    • p-node

  • Some of these components are, in many ways, equivalent:

  • ACG Report = RETE Token

    • “Message”

  • Beta Network = P-Tree

    • “Binary Conjunct Tree”

  • “Chain” of Alpha Network = ACG Filter

    • “Verifier”

  • Terminal Node = LATMS Constraint = ACG Rule Node

    • “Rule Node”

  • LTMS Node = p-node

    • “Propositional Node”

Example (Structure Sharing):

  • Example: If the arbitrary Animal is alive, then according to this hierarchy, so are any Dogs, Huskies, or Cats – including the arbitrary ones.

  • Example: The property of having two different colored eyes (which a Husky has) would not be inherited by Dog.

The arbitrary domesticated dog is both loyal and friendly. Notice that only one arbitrary domesticated dog has been created in the graph.

Enhanced Channels

  • Channels appear not only within a rule, but also from rule consequents to unifiablerule antecedents, and from consequents to unifiablequestions.

    • When formulas are added to the graph, the are unified with all others using a kind of substitution tree1.

  • Channels (and inference segments) are extended to contain:

    • Verifiers – Verify substitution is applicable.

    • Switches – Changes variable context.

    • Valves – Prevent or allow substitutions to pass through.

  • The type hierarchy used in recorded subsumption works in concert with unification, to limit unifiers.

  • What (if any) types of deduced subsumption are possible is still under consideration. Belief revision is a complicating factor.

From MGU factorization


Hoder, K., & Voronkov, A. Comparing unification algorithms in first-order theorem proving. In KI 2009: Advances in Artificial Intelligence (pp. 435-443). Springer Berlin Heidelberg, 2009.

Schlegel, D. R. & Shapiro, S. C.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.

Shapiro, S. C.A Logic of Arbitrary and Indefinite Objects. In D. Dubois, C. Welty, & M. Williams, Principles of Knowledge Representation and Reasoning: Proceedings of the Ninth International Conference (KR2004), AAAI Press, Menlo Park, CA, 2004, 565-575.

Woods, W. A. Understanding subsumption and taxonomy. In Sowa, J., ed., Principles of Semantic Networks. Los Altos, CA: Morgan Kaufmann. 45–94, 1991.


This graph contains the proposition that every arbitrary entity is friends with their arbitrary child. It also contains the wh-question “Who is Dave friends with?” Two i-channels are created from wft1 to wft3, since the friends relation is symmetric. Note that arb1, arb2, qvar1, and Dave are all entities, but that data has been omitted from the graph for readability.

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.