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Multi/infinite dimensional neural networks, multi/infinite dimensional logic theory.

Garimella Rama Murthy¹

¹International Institute of Information Technology (IIIT), Gachibowli, Hyderabad-500019, AP, India. rammurthy@iiit.net

International Journal of Neural Systems

|July 14, 2005

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A mathematical model for multi-dimensional neural networks using fully symmetric tensors is presented. This model relates network states to energy function minima, enabling the realization of logic functions and circuits.

Area of Science:

* Computational Neuroscience
* Mathematical Logic
* Tensor Theory

Background:

* Neural networks are complex systems with interconnected nodes.
* Logic gates perform Boolean operations, forming the basis of computation.
* Representing these systems mathematically is crucial for understanding their behavior.

Purpose of the Study:

* To develop a mathematical model for multi-dimensional neural networks.
* To establish a convergence theorem for these networks.
* To explore the relationship between energy functions and logic operations.

Main Methods:

* Development of a mathematical model for arbitrary multi-dimensional neural networks.
* Representation of network connectivity using fully symmetric tensors.

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* Definition of energy functions relating input/output states to tensor structures.

Main Results:

* A convergence theorem for multi-dimensional neural networks represented by fully symmetric tensors is proven.
* Input and output states are linked via an energy function, with minima corresponding to logic gate outputs.
* Logic circuits are modeled using block symmetric tensors and associated energy functions.

Conclusions:

* The developed model provides a framework for analyzing multi-dimensional neural networks.
* The energy function approach unifies neural network dynamics and logic gate operations.
* Infinite dimensional logic theory can be explored using higher-order tensors.