scholarly journals The Computational Complexity of Some Problems of Linear Algebra

1999 ◽  
Vol 58 (3) ◽  
pp. 572-596 ◽  
Author(s):  
Jonathan F Buss ◽  
Gudmund S Frandsen ◽  
Jeffrey O Shallit
Keyword(s):  
Computational Complexity ◽  
Linear Algebra

scholarly journals What Can You Do with a Rock? Affordance Extraction via Word Embeddings

2017 ◽  
Author(s):  
Nancy Fulda ◽  
Daniel Ricks ◽  
Ben Murdoch ◽  
David Wingate
Keyword(s):  
Reinforcement Learning ◽  
Computational Complexity ◽  
Linear Algebra ◽  
Autonomous Agents ◽  
Common Knowledge ◽  
Search Space ◽  
Word Embeddings ◽  
Knowledge Database ◽  
Learning Agent ◽  
Action Spaces

Autonomous agents must often detect affordances: the set of behaviors enabled by a situation. Affordance extraction is particularly helpful in domains with large action spaces, allowing the agent to prune its search space by avoiding futile behaviors. This paper presents a method for affordance extraction via word embeddings trained on a tagged Wikipedia corpus. The resulting word vectors are treated as a common knowledge database which can be queried using linear algebra. We apply this method to a reinforcement learning agent in a text-only environment and show that affordance-based action selection improves performance in most cases. Our method increases the computational complexity of each learning step but significantly reduces the total number of steps needed. In addition, the agent's action selections begin to resemble those a human would choose.

The computational complexity of some problems of linear algebra

1997 ◽  
pp. 451-462 ◽  
Author(s):  
Jonathan F. Buss ◽  
Gudmund S. Frandsen ◽  
Jeffrey O. Shallit
Keyword(s):  
Computational Complexity ◽  
Linear Algebra

scholarly journals The Computational Complexity of Some Problems of Linear Algebra

1996 ◽  
Vol 3 (33) ◽  
Author(s):  
Jonathan F. Buss ◽  
Gudmund Skovbjerg Frandsen ◽  
Jeffery O. Shallit
Keyword(s):  
Computational Complexity ◽  
Commutative Ring ◽  
Linear Algebra ◽  
Polynomial Time ◽  
Matrix Rank ◽  
Maximum Rank ◽  
The Matrix ◽  
Np Complete

We consider the computational complexity of some problems dealing with matrix rank.<br /> Let E, S be subsets of a commutative ring R.<br />Let x1, x2, ..., xt be variables. Given a matrix M = M(x1, x2, ..., xt)<br />with entries chosen from E union {x1, x2, ..., xt}, we want to determine<br />maxrankS(M) = max rank M(a1, a2, ... , at)<br />and<br />minrankS(M) = min rank M(a1, a2, ..., at). <br />There are also variants of these problems that specify more about the<br />structure of M, or instead of asking for the minimum or maximum rank, <br />ask if there is some substitution of the variables that makes the matrix<br /> invertible or noninvertible.<br />Depending on E, S, and on which variant is studied, the complexity<br />of these problems can range from polynomial-time solvable to random<br />polynomial-time solvable to NP-complete to PSPACE-solvable to<br />unsolvable.

scholarly journals A novel architecture with scalable security having expandable computational complexity for stream ciphers

2017 ◽  
Vol 30 (4) ◽  
pp. 459-475
Author(s):  
Prathap Siddavaatam ◽  
Reza Sedaghat
Keyword(s):  
Computational Complexity ◽  
Nonlinear System ◽  
Linear Algebra ◽  
Stream Cipher ◽  
Algebraic Degree ◽  
Stream Ciphers ◽  
Polynomial Equations ◽  
Hardware Complexity ◽  
Multivariate Polynomial ◽  
Algebraic Cryptanalysis

Stream cipher designs are difficult to implement since they are prone to weaknesses based on usage, with properties being similar to one-time pad besides keystream is subjected to very strict requirements. Contemporary stream cipher designs are highly vulnerable to algebraic cryptanalysis based on linear algebra, in which the inputs and outputs are formulated as multivariate polynomial equations. Solving a nonlinear system of multivariate equations will reduce the complexity, which in turn yields the targeted secret information. Recently, Addition Modulo has been suggested over logic XOR as a mixing operator to guard against such attacks. However, it has been observed that the complexity of Modulo Addition can be drastically decreased with the appropriate formulation of polynomial equations and probabilistic conditions. A new design for Addition Modulo is proposed. The framework for the new design is characterized by user-defined expandable security for stronger encryption and does not impose changes in existing layout for any stream cipher such as SNOW 2.0, SOSEMANUK, CryptMT, Grain Family, etc. The structure of the proposed design is highly scalable, which boosts the algebraic degree and thwarts the probabilistic conditions by maintaining the original hardware complexity without changing the integrity of the Addition Modulo.

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