Departmental Papers (ESE)

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Two examples of the fundamental construction motivating this paper: the dual cubing---or model space---of an abstract "system of implications", P. Here, both instances of P encode different implication relations among three binary queries a,b,c and their complements. P is depicted as a Hasse diagram (left). Starting with the full Hamming cube (faces of all dimensions included), remove any vertices contradicting an implication in P, and all incident faces. The resulting complex, cube(P), may serve as a model for *any* space where the relations in P are satisfied (center). The middle hyper-planes of the full Hamming cube restrict to separating hyper-planes of the resulting cubical complex, whose complementary components satisfy the same relations as those in P (right). The comparison of the top row with the bottom illustrates the dimensional reduction resulting from introducing more implications into P.

Abstract

We propose a variant of iterated belief revision designed for settings with limited computational resources, such as mobile autonomous robots.

The proposed memory architecture---called the universal memory architecture (UMA)---maintains an epistemic state in the form of a system of default rules similar to those studied by Pearl and by Goldszmidt and Pearl (systems Z and Z+). A duality between the category of UMA representations and the category of the corresponding model spaces, extending the Sageev-Roller duality between discrete poc sets and discrete median algebras provides a two-way dictionary from inference to geometry, leading to immense savings in computation, at a cost in the quality of representation that can be quantified in terms of topological invariants. Moreover, the same framework naturally enables comparisons between different model spaces, making it possible to analyze the deficiencies of one model space in comparison to others.

This paper develops the formalism underlying UMA, analyzes the complexity of maintenance and inference operations in UMA, and presents some learning guarantees for different UMA-based learners. Finally, we present simulation results to illustrate the viability of the approach, and close with a discussion of the strengths, weaknesses, and potential development of UMA-based learners.

Sponsor Acknowledgements

This research was developed in part with funding from Air Force Research Lab (AFRL) grant FA8650-15-D-1845 (subcontract 669737-1), and in part with funding from the Defense Advanced Research Projects Agency (DARPA) and the Air Force Research Lab (AFRL) under agreement number FA8650-18-2-7840.

The views, opinions and/or findings expressed are those of the authors and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government.

Document Type

Working Paper

Subject Area

Kodlab, GRASP

Date of this Version

12-18-2018

Bib Tex

@article{Guralnik-Koditschek-UMA_IBR,
AUTHOR = {Guralnik, D.P. and Koditschek, D.E.},
TITLE = {Iterated Belief Revision Under Resource Constraints: Logic as Geometry},
YEAR = {2018},
JOURNAL = {preprint},
}

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Date Posted:19 December 2018