2 Search Results for "Cohn, Henry"

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Current and Future Challenges in Knowledge Representation and Reasoning (Dagstuhl Perspectives Workshop 22282)

Authors: James P. Delgrande, Birte Glimm, Thomas Meyer, Miroslaw Truszczynski, and Frank Wolter

Published in: Dagstuhl Manifestos, Volume 10, Issue 1 (2024)

Abstract

Knowledge Representation and Reasoning is a central, longstanding, and active area of Artificial Intelligence. Over the years it has evolved significantly; more recently it has been challenged and complemented by research in areas such as machine learning and reasoning under uncertainty. In July 2022,sser a Dagstuhl Perspectives workshop was held on Knowledge Representation and Reasoning. The goal of the workshop was to describe the state of the art in the field, including its relation with other areas, its shortcomings and strengths, together with recommendations for future progress. We developed this manifesto based on the presentations, panels, working groups, and discussions that took place at the Dagstuhl Workshop. It is a declaration of our views on Knowledge Representation: its origins, goals, milestones, and current foci; its relation to other disciplines, especially to Artificial Intelligence; and on its challenges, along with key priorities for the next decade.

Cite as

James P. Delgrande, Birte Glimm, Thomas Meyer, Miroslaw Truszczynski, and Frank Wolter. Current and Future Challenges in Knowledge Representation and Reasoning (Dagstuhl Perspectives Workshop 22282). In Dagstuhl Manifestos, Volume 10, Issue 1, pp. 1-61, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@Article{delgrande_et_al:DagMan.10.1.1,
  author =	{Delgrande, James P. and Glimm, Birte and Meyer, Thomas and Truszczynski, Miroslaw and Wolter, Frank},
  title =	{{Current and Future Challenges in Knowledge Representation and Reasoning (Dagstuhl Perspectives Workshop 22282)}},
  pages =	{1--61},
  journal =	{Dagstuhl Manifestos},
  ISSN =	{2193-2433},
  year =	{2024},
  volume =	{10},
  number =	{1},
  editor =	{Delgrande, James P. and Glimm, Birte and Meyer, Thomas and Truszczynski, Miroslaw and Wolter, Frank},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagMan.10.1.1},
  URN =		{urn:nbn:de:0030-drops-201403},
  doi =		{10.4230/DagMan.10.1.1},
  annote =	{Keywords: Knowledge representation and reasoning, Applications of logics, Declarative representations, Formal logic}
}

Document

DOI: 10.4230/LIPIcs.ITCS.2023.19

Matrix Multiplication via Matrix Groups

Authors: Jonah Blasiak, Henry Cohn, Joshua A. Grochow, Kevin Pratt, and Chris Umans

Published in: LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Abstract

In 2003, Cohn and Umans proposed a group-theoretic approach to bounding the exponent of matrix multiplication. Previous work within this approach ruled out certain families of groups as a route to obtaining ω = 2, while other families of groups remain potentially viable. In this paper we turn our attention to matrix groups, whose usefulness within this framework was relatively unexplored. We first show that groups of Lie type cannot prove ω = 2 within the group-theoretic approach. This is based on a representation-theoretic argument that identifies the second-smallest dimension of an irreducible representation of a group as a key parameter that determines its viability in this framework. Our proof builds on Gowers' result concerning product-free sets in quasirandom groups. We then give another barrier that rules out certain natural matrix group constructions that make use of subgroups that are far from being self-normalizing. Our barrier results leave open several natural paths to obtain ω = 2 via matrix groups. To explore these routes we propose working in the continuous setting of Lie groups, in which we develop an analogous theory. Obtaining the analogue of ω = 2 in this potentially easier setting is a key challenge that represents an intermediate goal short of actually proving ω = 2. We give two constructions in the continuous setting, each of which evades one of our two barriers.

Cite as

Jonah Blasiak, Henry Cohn, Joshua A. Grochow, Kevin Pratt, and Chris Umans. Matrix Multiplication via Matrix Groups. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 19:1-19:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{blasiak_et_al:LIPIcs.ITCS.2023.19,
  author =	{Blasiak, Jonah and Cohn, Henry and Grochow, Joshua A. and Pratt, Kevin and Umans, Chris},
  title =	{{Matrix Multiplication via Matrix Groups}},
  booktitle =	{14th Innovations in Theoretical Computer Science Conference (ITCS 2023)},
  pages =	{19:1--19:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-263-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{251},
  editor =	{Tauman Kalai, Yael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.19},
  URN =		{urn:nbn:de:0030-drops-175226},
  doi =		{10.4230/LIPIcs.ITCS.2023.19},
  annote =	{Keywords: Fast matrix multiplication, representation theory, matrix groups}
}

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