Noncommutative Bilinear Algorithms for $3 \times 3$ Matrix Multiplication

Rodney W. Johnson; Aileen M. McLoughlin

doi:10.1137/0215043

Noncommutative Bilinear Algorithms for $3 \times 3$ Matrix Multiplication

SIAM Journal on Computing ◽

10.1137/0215043 ◽

1986 ◽

Vol 15 (2) ◽

pp. 595-603 ◽

Cited By ~ 18

Author(s):

Rodney W. Johnson ◽

Aileen M. McLoughlin

Keyword(s):

Matrix Multiplication ◽

Bilinear Algorithms

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On the inequivalence of bilinear algorithms for matrix multiplication

Information Processing Letters ◽

10.1016/j.ipl.2013.05.011 ◽

2013 ◽

Vol 113 (17) ◽

pp. 640-645 ◽

Cited By ~ 8

Author(s):

Jinsoo Oh ◽

Jin Kim ◽

Byung-Ro Moon

Keyword(s):

Matrix Multiplication ◽

Bilinear Algorithms

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Stable and Supported Semantics in Continuous Vector Spaces

Proceedings of the Seventeenth International Conference on Principles of Knowledge Representation and Reasoning ◽

10.24963/kr.2020/7 ◽

2020 ◽

Author(s):

Yaniv Aspis ◽

Krysia Broda ◽

Alessandra Russo ◽

Jorge Lobo

Keyword(s):

Logic Program ◽

Matrix Multiplication ◽

Vector Spaces ◽

Continuous Space ◽

Continuous Vector ◽

Novel Approach ◽

Gradient Based ◽

Normal Logic ◽

Parameter Values ◽

Normal Logic Program

We introduce a novel approach for the computation of stable and supported models of normal logic programs in continuous vector spaces by a gradient-based search method. Specifically, the application of the immediate consequence operator of a program reduct can be computed in a vector space. To do this, Herbrand interpretations of a propositional program are embedded as 0-1 vectors in $\mathbb{R}^N$ and program reducts are represented as matrices in $\mathbb{R}^{N \times N}$. Using these representations we prove that the underlying semantics of a normal logic program is captured through matrix multiplication and a differentiable operation. As supported and stable models of a normal logic program can now be seen as fixed points in a continuous space, non-monotonic deduction can be performed using an optimisation process such as Newton's method. We report the results of several experiments using synthetically generated programs that demonstrate the feasibility of the approach and highlight how different parameter values can affect the behaviour of the system.

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